Transformation Groups: Proceedings of the Conference in the University of Newcastle Upon Tyne, August 1976 9780521215091, 0521215099

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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Editor: PROFESSOR G. C. SHEPHARD, University of East Anglia Already published in this series 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

General cohomology theory and K-theory, PETER HILTON. Numerical ranges of operators on normed spaces and of elements of normed algebras, F. F. BONSALL andJ. DUNCAN. Convex polytopes and the upper bound conjecture, P. McMULLEN and G. C. SHEPHARD. Algebraic topology: A student's guide, J. F. ADAMS. Commutative algebra,]. T. KNIGHT. Finite groups of automorphisms, NORMAN BIGGS. Introduction to combinatory logic, J. R. HINDLEY, B. LERCHER and J.P. SELDIN. Integration and harmonic analysis on compact groups, R. E. EDWARDS. Elliptic functions and elliptic curves, PATRICK DUVAL. Numerical rang~s II, F. F. BONSALL andJ. DUNCAN. New developments in topology, G. SEGAL (ed.). Symposium on complex analysis Canterbury, 1973,]. CLUNIE and W. K. HAYMAN (eds). Combinatorics, Proceedings of the British combinatorial conference 1973, T. P. McDONOUGH and V. C. MAVRON (eds). Analytic theory of abelian varieties, H. P. F. SWINNERTON-DYER. An introduction to topological groups, P. J. HIGGINS. Topics in finite groups, TERENCE M. GAGEN. Differentiable germs and catastrophies, THEODOR BROCKER and L. LANDER. A geometric approach to homology theory, S. BUONCRISTIANO, C. P. ROURKE and B.J. SANDERSON. Graph theory, coding theory and block designs, P. J. CAMERON and J. H. VAN LINT. Sheaf theory, B. R. TENNISON. Automatic continuity of linear operators, ALLAN M. SINCLAIR. Presentations of groups, D. L. JOHNSON. Parallelisms of complete .designs, PETERJ. CAMERON. The topology of Stiefel manifolds, I. M. JAMES. Lie groups and compact groups,]. F. PRICE.

London Mathematical Society Lecture Note Series.

Transformation Groups Proceedings of the Conference in the University of Newcastle upon Tyne, August 1976

Edited by CZES KOSN IOWSKI

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON NEW YORK

MELBOURNE

26

Published by the Syndics of the Cambridge University Press The Pitt Building, Trumpington Street, Cambridge CB2 1RP Bentley House, 200 Euston Road, London NW1 2DB 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia ©Cambridge University Press 1977 ISBN 0 521 21509 9 First published 1977 Printed in Great Britain at the University Printing House, Cambridge (Harry Myers, University Printer)

CONTENTS

PART ONE Herbert Abels.

Generators and relations for groups of homeomorphisms

Nguiffo B. Boyom.

3

Affine embeddings of real Lie groups

.

Robert Delver.

Equivariant differential operators of a Lie group

Allan L. Edmonds.

V. Giambalvo.

21

40

Equivariant regular neighbourhoods

51

Characteristic numbers and equivariant

...

spin cobordism

Equivariant K-theory and cyclic

Stefan Jackowski.

76

subgroups ~p manifolds with low dimensional

Czes Kosniowski.

92

fixed point set Hsu-Tung Ku and Mei-Chin Ku.

Gaps in the relative degree 121

of symmetry Arunas Liulevicius. Gerhard X. Ritter.

Characters do not lie Actions of Z/2n on

Gerhard X. Ritter and Bradd E. Clark.

s3

139 147

Periodic homeomorphisms

on non-compact 3 manifolds Reinhard Schultz.

70

154

Equivariant function spaces and equivariant stable homotopy theory

169

Haruo Suzuki.

A property of a characteristic class of an orbit foliation

• • • 190

Orbit structure for Lie group actions on

Per Tomter.

higher cohomology projective spaces Steven H. Weintraub.

• •• 204

On the existence of group actions on

certain manifolds

•.. 228

PART TWO (SUMMARIES AND SURVEYS) H. Abels.

237

Proper transformation groups

R. D. Anderson.

Problems on group actions on Q manifolds

L. Auslander, B. Kolb and R. Tolimieri.

249

A non-abelian view

of abelian varieties M. P. Heble.

259

Non compact Lie groups of transformation and invariant operator measures on homogeneous spaces in Hilbert space

Soren Illman.

... 267

Approximation of simplicial G-maps by equivariantly non degenerate maps

Katsuo Kawakubo.

M. Kreck.

Equivariant Riemann-Roch type theorems and related topics

284

Knots and diffeomorphisms

295

Peter L3ffler.

Gordon Lukesh.

Some remarks on free differentiable involutions on homotopy spheres •••

299

Compact transitive isometry spaces

301

W. J. R. Mitchell.

A problem of Bredon concerning

homology manifolds

vi

279

•.• 305

PREFACE In August 1976 a conference on Transformation Groups was held at the University of Newcastle upon Tyne with participants from the following countries:

Canada, Eire, Finland, France, Japan,

Norway, Poland, United Kingdom, United States of America, West Germany. Manuscripts were received from all the speakers at the conference.

Other papers were also submitted, four of which were

accepted for these proceedings.

The articles have been divided into

two parts (with summaries and surveys appearing in the second part).

ACKNOWLEDGEMENTS I would like to to thank very much indeed the following: The London Mathematical Society·and the University of Newcastle upon Tyne for financial assistance towards the conference. The secretaries of the School of Mathematics, University of Newcastle upon Tyne, for typing the papers. typing was by Pauline Harley and Janet McKay.

The majority of the Additional typing

was by Joyce Edger and Carol Reynolds The referees of papers - their assistance and advice was invaluable.

CZES KOSNIOWSKI NEWCASTLE UPON TYNE SEPTEMBER 1976 vii

PART ONE

GENERATORS AND RELATIONS FOR GROUPS OF HOMEOMORPHISMS HERBERT ABELS

The aim of the present paper is to unify and generalize the proofs of results of Behr, Gerstenhaber and Macbeath concerning the theme of the title.

I.

RESULTS 1.1

NOTATIONS.

Let the group G act on the topological

space X, i.e. suppose a homeomorphism of G into the group of homeomorphisms of X is given.

By gx we denote the image of the

point x E X under the homeomorphism corresponding to g E G. M c G, A c X let MA

= {gx;

For

gEM, x E A}.

For any two subsets A, B of X define G(A, B) := { g E G; gA

(1.1.1)

n B "f. ¢}.

Obviously (1.1.2)

G(B,A)

= [G(A,B)]- 1

(1.1.3)

G(A,gB)

= g·G(A,B)

(1.1.4)

G(gA,B)

= G(A,B)·g- 1

Let F be a non empty subset of X (think of F as a "fundamental set"). (1.1.5)

Define

E := G(F,F)

(think of E "Erzeugendenmenge"). homomorphism.

Let H be a group, q:H-+ G be a

Suppose a section s:E _, H is given, i.e. a map s:E-+ H 3

such that qos

= id\E.

In applications H will always be a group with generators E and certain relations, which hold in G, q:H - G will be the homomorphism induced by the inclusion E- G, the section s is the obvious one.

The problem is:

Under which conditions is q an

isomorphism?

We express the relations that hold in H by

multiplicative properties of the section s. 1. 2

(1.2.1)

H'{POTHESES

(Multiplicative hypothesis)

If

gl n g 2F n F f

Both sides of this equality are defined because E and g1-1 g2 E G(F,F)

= E- 1

¢ we have

by 1.1.2

= E.

Further hypotheses are:

(1.2.2)

GF =X

(1.2.3)

s(E) generates H

(1.2.4)

F is connected

(1.2.5)

X is connected and simply connected. 1. 3

RESULTS

THEOREM 1.

q:H- G is an isomorphism if (1.2.1) through 0

(1.2.5) hold and GF THEOREM 2.

= X(e.g.

if F is open).

q:H- G is an isomorphism if (1.2.1) through

(1.2.5) hold, F is closed in X and {gF; g E G} is a locally finite cover of X. Macbeath [6] proved Theorem 1 for open F, Theorem 2 for groups of isometries.

The case of f.1n1•t e E 1n · Th eorem 2 was proved

by Behr [1], with a bigger set of relations. 4

Cf also [5].

Swan

= n 0 (X) = 0 n 1 (X) 1 0.

[11] considers the case F open, n (F) 0

detailed description of ker(q) if

and gives a

We actually prove a common generalization of Theorems 1 A result of Soul~ [10] is an easy

and 2 (Theorem 4.5). application (see 4.6).

For our next result we need to following definition.

The

cover {gF; g E G} of X is called G-numerable if there is a partition of unity {p ; g E G} with supp(p ) c gF such that p (gx) g g g for every x E X, g E G.

= p e (x)

For example, if X is normal, F is open

and {gF; g E G} is a locally finite cover of X, or if X is normal and F is a neighbourhood of a closed subset ofF' and {gF'; g E G} is a locally finite cover of X, then {gF; g E G} is a G-numerable cover of X. THEOREM 3.

q:H

~

G is an isomorphism if (1.2.1) and (1.2.3)

hold, {gF; g E G} is a G-numerable cover of X and n 0 (F) n 1 (X)

=n

0

(X)

=

= 0. 1.4

IDEA OF PROOF.

surjective (see Section II).

It is easy to prove that q is If in Theorems 1 and 2 we drop the

assumption that X be simply connected, we obtain a covering space

Y~X (i.e. a locally trivial sheaf) with the following properties:

= q(h)p(y).

(1)

H acts on Y and pis an H-map, i.e. p(hy)

(2)

ker q acts as a group of covering transformations of p.

The

action is free and transitive on the fibres of p. (3)

There is a section for p over F.

Hence if any such covering space of X is trivial, q:H

~

G is an

isomorphism, in particular if X is simply connected,

5

The main difficulty of the proof is to define a topology on

Y. The proof of Theorem 3 makes use of the nerve of the covering and its geometric realization. instead of covering spaces.

It makes use of fundamental groups

There is a similar generalization as

above (see Theorem 5.4). I thank H. Behr for helpful conversations.

This paper

actually grew out of a talk in a seminar of Behr's.

I also thank

the referee for useful hints to the literature.

II

SURJECTIVITY OF q Notations as in 1.1.

The following result is well known

[9, no.9]. 2.1.

THEOREM.

and X is connected. PROOF.

Suppose GF =X, EF is a neighbourhood ofF

~

E generates G.

Let G0 be the subgroup of G generated by E.

set {X.l = g,G F; g.G E G/G} is a cover of X, since GF =X. l 0 l 0 0 Xi's are disjoint:

g1G0 F

n g2G0 F f

The The

¢ implies that G0 g2-1 g1G0

contains an element of E c G0 , so g2 -1 g1 E G, hence g G = g G • 1 0 2 0 Since EF is a neighbourhood of F, each X. = g.G F = g.G (EF) is l

a neighbourhood of itself, i.e. open.

l

0

them: # G/G

6

0

0

So the X. = g.G F form an l

open disjoint cover of X.

l

l

0

If X is connected, there is only one of

= 1, so G = G • 0

III

THE SET Y

For the whole Section III we use the notations of 1.1 and assume only the multiplicative hypothesis (1.2.1): g1F

n g2F n F f

If

¢ we have

(3. 1 )

This implies for g1

= g2 = e

(3. 2)

the neutral element s(e)

For 91 g 2

=e

= e.

we obtain s(g -1) = (s(g) )-1

(3, 3)

for g E E

'

which we sometimes denote by s(g)- 1 . We have an action of H on X defined by (3. 4)

hx := q(h)x,

' h E H,

X

E X.

Define } Z:= { (x,h)EXxH;h -1 xEF.

(3. 5)

The relation on Z if and only if is an equivalence relation by our multiplicative hypotheis (3.1). We define

(3. 7)

Y=

z/'-'.

The main point of the proof will be to endow Y with a suitable topology.

We need some preparations.

equivalence class of (x,h) E Z by [x,h]. on Y defined by h1 [x,h] p([x,h]) (3.8)

=x

= [h1 x,h1

is an H-map. t(x)

h].

We have an action of H

The projection p:Y ~X,

We have a section t:F

= [x,e]

for

We denote the

~

Y, namely

x E F.

7

Our definitions imply

= Y.

(3. 9)

H t (F)

(3.10)

H(t(x),t(F)) = s(G(x,F)) for x E F,

hence H(t(F), t(F)) = s(E).

(3. 11)

The homomorphism q:H

~

G is to be analysed.

Set K = ker q.

The next lemma shows that K is a good candidate for the group of covering transformations of p. 3.12

LEMMA.

K acts freely on Y and simply transitively on

the non-empty fibres p PROOF.

-1

(x) of p.

We have to show first that K = {k E K; ky = y} y

contains only the neutral element.

By (3.9) it suffices to prove

that claim for y E t(F), say y = [x,e]. have k E s(E)

n K.

But

s(E)

nK=

If ky = [kx,k] = [x,e], we

{ e}, since qj s(E): s(E) ..... E

is bijective with inverse mapping g ..... s(g). K acts on the fibres of p.

It remains to prove that K acts

transitively on the non empty fibres p- 1p(y) of p, y t Y. we may assume, y = [x,e].

Suppose z E p

-1

p(y).

Again,

By (3.9) there is

an element hE Hand a point [x 1 ,e] E t(F) such that z = h [x 1 ,e] = Since p(y) = p(z) we have x = hx 1 = q(h)x 1 , so q(h) E E.

[hx 1 ,g].

For h1 = s(q(h)) E s(E) we have h1 [x 1,e] = [x,e] = y. h[x 1,e] = h·h1 - 1 and k = h·h 1 - 1 E K.

Hence z =

Note that k is the unique element of K such that k~= z. In particular:

hs(q(h))- 1 is the same element of K for every h E H

such that z = [x,h]. mapping of

8

So the proof actually yields the inverse

KX F (3.13)

(k,x)

namely

(h s(q(h))-1, x)

... kt(x) = [x,k] [x,h].

The next lemma makes explicit the properties of the topology of Y we want,

3.14 LEMMA,

Suppose Y is endowed with a topology such that

(a)

Every h E H acts as a homeomorphism on Y,

(b)

p:Y ... X is a sheaf,

(•1, e • a 1 oca 1 homeomorp h.1sm, 1,e, · every

pointy E Y has an open neighbourhood U such that p!U: U ... p(U) is a homeomorphism and p(U) is open in

x1 ).

Then p:Y ... X is a covering, i.e. a locally trivial sheaf, K acts as a group of covering transformations of p, transitively on the non empty fibres of p.

PROOF.

Then p- 1 (p(U))

Suppose U as in (b).

=

U

kU is

kEK Endow K with the discrete

the disjoint union of the open sets kU, topology.

In the commutative diagram

KXu

~ p

K X p(U)

~

p(U)

-1

(p(U))

/

the upper diagonal maps are homeomorphisms, hence so is the horizontal map, yielding the local triviality of the sheaf p:Y ... X. We need a more technical version of 3.14. r:A ~ Y a section _(for p) if par to have a topology yet.

= idA'

We call a map

Note that Y is not supposed

If r:A ... Y is a section, so is hr:hA ... Y

9

defined by hr ( hx)

(3.15)

= hr ( x) •

As usual, two maps defined in neighbourhoods of the same point x E X are said to have the same germ at x E X, if and only if they coincide in some neighbourhood of x. LEMMA.

3.16

Suppose we are given for every point x EX

an open neiqhbourhood UXof x ahd a section r X :E X ~ Y with the following properties: r X IU X

(a) (b)

n F =t!UX n F

= hx1 ,

If x1 E F, x2 E F, x2

hE s(E), then rx 2 and hrx 1 have

the same germ at x2 • (c)

There is an h E H such that hx Let x1 E F, x E U x1 and-rx and hr have the same -germ at x 2 • -x 2

= x2

E F

1

Then Y has a unique topology satisfying 3.14 and such that the sections r

X

are continuous.

In particular t:F

~

Y is a continuous

section. PROOF.

The proof is given in the language of presheaves.

One could give it also by defining neighbourhood bases of the points of Y.

Let U be an open subset of X.

sections r:U

~

Define R(U) to be the set of

Y with the following property:

is an h E H and an x1 E F such that hx 1

have the x1 The R(U) obviously form a presheaf, satisfying the

same germ at x.

two Serre conditions. Furthermore, (i)

If r E R(U)

(ii)

r

10

X

=x

For any x E U there

E R(U ) X

then for

hr E R(hU) x E F

and r and hr

(iii)

r (x) = t(x)

(iv)

For any point y E Y there is an open neighbourhood U of

X

for

x E F

p(y) = x and an r E R(U), such that r(x) = y. The set {x E U; ·r1 (x) = r 2 (x)} of points where two sections

(v)

r 1 , r 2 of R(U) coincide, is open. (i) is immediate from the definition of R(U); (ii) it follows from hypothesis (c); (iii) from (a); (iv) from (i), (iii) and 3.9. (v) follows from hypothesis (b):

Note that h1 x1

=x =

Finall

h2 x2, h.rx. (x) ].

= r(x) fori= 1,2, or more explicitly h r .

X.

(x) = h.• r

r.].].

h.t(x.) = [h.x.,h.] = [x,h 2.] = r(x) implies h1 ]. ]. ]. ]. ].

J.

-1

X.

(x.) ].

].

=

].

·h2 E s(E) by 3.6,

-1 -1 so hypothesis ( b) can be applied for h = (h1 ·h2 ) E s(E) by 3.3.

Now let R be the sheaf associated to the presheaf R. the evaluation map R

~

Y is bijective by (iv) and (v).

Then

If Y is

endowed with the transported topology, 3.14 is satisfied, (a) by (i).

Since the presheaf R satisfies the Serre conditions, R(U)

is the set of continuous sections U ~ Y. is continuous by (ii).

In particular r :U X

X

~

Y

The uniqueness of the topology of Y with

these properties is obvious.

Finally, t:F • Y is continuous by

hypothesis (a).

IV

PROOF OF THEOREMS 1 AND 2 We use the notations of section III and the multiplicative

hypothesis (1.2.1) = (3.1). ( 4. 1 )

Define for x EX

y(x) = G(F,x)

= {g

E G; x E gF}

Note: (4.2)

y(gx) = g y(x) 11

by ( 1. 1. 3). The hypothesis of the following lemma generalizes the extra hypotheses of both Theorems 1 and 2 (see 4.4). 4.3

LEMMA.

Ux such that (y(x)

Assume that every point x E F has a neighbourhood

n y(y))F is a neighbourhood of y for every y E Ux.

Then there is a set of sections r

PROOF.

X

as in 3.16.

We may assume that U is open. X

r X ( z) = [ z' s (g) J for

contained in E, so g is defined. Suppose g1,g 2 E y(x)

X

n y ( z) •

n y(z) is non empty, y(x) is

By our assumptions the set y(x)

defined:

g E y (X)

Define for z E U :

We first show that r (z) is well X

n y(z).

We have to show

[z, s(g 1 )] = [z, s(g 2 )], i.e. s(g 1 )- 1 s(g 2 ) E s(E) (see 3.6). X

E

gl n g2F n F,

since g1,g2 E y(x), so s(g1)

-1

But

= s(g1 -1

s(g2)

g2)

E s (E) by (3. 1 ) •

3.16 [z, s(e)]

(a):

For z E F we have e E y(x)

n y(z),

so r (z) = X

= [z,e] = t(z).

3.16 such that x2

(b):

= s(g)

Suppose x1 ,x 2 are points ofF, h

= hx 1 = gx 1 •

have the same germ at x2 •

We have to show that r

x2

and

E s(E),

r s(g) x1

We actually show that the two maps

coincide where both are defined, namely on V = U n gU • Suppose x2 x1 z E V, g2 E y(x 2 ) n y(z). Then g- 1g 2 E y(x 1 ) n y(g- 1 z) by 4.2. So rx2(z) x2 E gF

= [z,s(g2)],

s(g)rx~z)

= [z,

s(g)s(g-1g2)].

But

n g2F n F, so 3.1 implies r (z) = r (z) x s (g) x •

2 1 1 3.16 (c): Let x1 E F, x E U , h- = s(g), where x1 1 g E y(x1) n y(x), so hx = g- x = x 2 E F. We have to show that

rx 12

2

and hr x have the same germ at x or th a t ( )r 2 1 s g x2

have

the same germ at x. V C (y(x)

Let V be a neighbourhood of x such that

n y(x 1 ))F and V c U , V c gU • We show that the two

sections coincide on V:

x1 Let z E V.

x2 By definition there is a

g1 E y(x) n y(x 1 ) n y(z). So rx (z) = [z, s(g 1 )]. On the other -1 1 1 1 hand g g1 E y(x 2 ) n y(g- x1 )n y(g- z). So ( )r (z) = s g x2 1 [z, s(g)s(g- g1 )]. But x1 E gF n g1F n F, so s(g-1 g1 ) = s(g)- 1 s(g 1 ) 4.4

LEMMA.

Either one of the following conditions implies

the hypothesis of lemma 4.3: Every point x E X has a neighbourhood U such that

(1)

¢

f

X

y(y)

(2)

n y(x) for every y E U • X

The set {gF; g E G} is a locally finite cover of X and F is

closed. 0

(3)

GF = X.

PROOF.

(1 ) :

We may assume that U is open. X

implies: y

E uX :

The hypothesis

y(x)F is a neighbourhood of x for every x E X. (y(y) (2)

Now for

n y(x))F = y(y)F is a neighbourhood of y.

is a special case of (1):

neighbourhood u such that gF n u implies

Every point x EX has a X

E gF, which is

equivalent to (1). (3)

is obvious.

Putting everything together, we obtain the following Theorem which implies Theorems 1 and 2: 4.5

THEOREM.

Suppose (1.2.1) through (1.2.4) hold, X is

connected, and any point of x E X has a neighbourhood Ux such that (y(x)

n y(y))F is a neighbourhood of y for every y E Ux (e.g. if

one of the conditions in 4.4 holds).

Then q:H

~

G is surjective.

13

There is a regular connected covering space p:Y

~X

with the

following properties: ~X

(1)

H acts on Y and p:Y

is an H-map.

(2)

K = ker q is the group of covering transformations of p.

The

group K acts transitively on the fibres of p. There is a continuous section for p ~ F.

(3)

PROOF.

The surjectivity of q was proved in 2.1.

Note that

the hypothesis of 4.3 implies that EF is a neighbourhood of F. p:Y

~

X be as in section III.

surjective.

Let

Since p is an H-map and GF = X, p is

Endow Y with the topology from 3.16.

Then pis a

covering map enjoying the properties (1) through (3) -except for the definite article in (2).

We show that Y is connected:

Since

F is connected, so is t(F), hence also s(E)t(F), sin.ce s(g)t(F) t(F) -f ¢forgE E (see 3.11).

n

Inductively, s(E)nt(F) is connected

soY= Ht(F) = Us(E)nt(F) is connected.

If Y is connected, the

group of covering transformations acts freely on Y.

This justifies

the definite article in (2). 4.6

EXAMPLE.

[10]

LetT be a simplicial complex, let the

group G act simplicially on T.

Suppose there is a subcomplex T' of

T such that for every simplex s E T there is a simplex s' E T' and a g E G with gs' = s. T, F = IT' I c III.

Let X= III be the geometric realization of

If n 0 (F) = n0 (X) = n1 (X) =

o,

q:H ~ G is an

isomorphism. PROOF.

We show that 4.4 (1) holds.

The geometric

realization III is the set of functions A from the set r 0 of vertices ofT to [0,1] such that the support of A, supp(A) = {i E T0 14

;

A(i)

-f

0}, is a simplex ofT and 2:: A(i)

= 1,

(i E T0 ).

By

definition

g~

=

~og- 1 :

~ [0,1].

T0

for every ~ with supp(~) ~ supp(~).

So

g~

E JT'j implies

g~

E IT' I

Now U~ = {~ E T; supp(~) ~

supp(~)} is an open neighbourhood of ~ in the weak topology and 4.4 (1) holds for these neighbourhoods.

V PROOF OF THEOREM 3 We need two preparatory sections. 5.1

FUNDAMENTAL GROUPOID OF G-SPACES.

Let X beaG-space.

The group G acts on the fundamental groupoid n1 (X) of X.

We can

define the semi-direct product.

which is again a groupoid (cf. [3]): Ob(n 1 (X))= X.

Its objects are Ob(n 1 (G,X)) =

For a,b EX define the set of morphisms from a to b:

n 1 (G,X)(a,b) = {(q,o:); g E G, o: E n1 (X)(a,b)} so a is a homotopy class of paths from a to b. morphisms (g,o:):

a~

band

(h,~)

where

+

(h,~):

b

~cis

Composition of

defined as

(g,o:) = (h g, ~ + hoo:),

is composition of homotopy classes of paths. Obviously n1

is a functor from G-spaces to groupoids.

If f 0 , f 1 are G-homotopic

G-maps X ~ Y, there is a natural equivalence of the functors f : n (G X)~ n (G,Y), here the groupoids are regarded as 0*' 1* 1 ' 1 categories. f

Let p be a point of X.

Let n1 (G,X;p) = n 1 (G,X)(p,p) be

the group of morphisms from p to p.

This is the fundamental group

of a transformation group considered by Rhodes [7], cf [4; §3]. 15

Projection to the first factor in n 1 (G,X;p) yields an exact sequence

if X is pathconnected.

A G-map f:X • Y induces a map E(X,p) •

E(Y,f(p)) of the resp. extensions of G, for any P EX.

If f 0 ,f 1

are G-homotopic G-maps X ~ Y, there is a horizontal isomorphism of extensions in the commutative diagram

5.2 of

NERVES OF COVERS.

This is almost literally a part

[8], for a more detailed version see [2; pp.224-227].

Let X be

a topological space, U = {ua}a E ~be a cover of X by subsets Ua. If o is a subset of

~

define U = 0

n

U•

aE o

a

Let RU be the category ~

whose objects are the nonempty U for finite subsets o of 0

whose morphisms are their inclusions.

and

The nerve NRU of the

category RU is the barycentric subdivision of what is ordinarily called the nerve of U. There is also another category XU associated to U. a topological category whose objects are the pairs (x

U ) with '

x E U0 , and whose morphisms (x, U0 i

U

0

2

Spin . -Spin( / ) 1n On BZ 2 are multiples (over 0* ) of [RP 4n+3 ,~J.

2.

PROOF OF MAIN THEOREM Since the Spin structures on the stable normal bundle are

equivalent to those on the tangent bundle, we will work with manifolds with a Spin structure on the normal bundle. Spin manifold, it is KO orientable.

If Mn is a

Let [Mn] be the fundamental

class in KO (M) corresponding to the fundamental class in K0°(M) n

of [5] via Poincar~ duality.

Let nr, r a positive integer,

denote the KO-Pontryagin classes defined in [2]. For any t1 t2 ts L L = (t 1 , t 2 , ••• ,ts) let n = n n n and n° = 1. For any element [M,f] E O~pin(BZ/2) define the KO-Pontryagin number c(nL)[M,f] as the image of [M] under the composition nL f)* KO (BSpin x BZ/2) KO (M) - - - n n (\) X

KO n (BZ/2).

This gives a homomorphism

,Z/2) and nL by x E H*(BSpin,Z/2) gives the Stiefel-Whitney numbers. THEOREM.

(~,f 1 ) and (M;,f 2 ) represent the same element

of O~pin(BZ/2) if and only if (M~,f 1 ) and (M;,f 2 ) have the same KO-Pontryagin and

Stiefel~hitney

numbers.

71

Since both numbers are cobordism invariants, one direction is clear.

To prove the other, some information about the structure

of o~pin(BZ/2) is needed, as well as some facts about KO*(RPn). (a)

LEMMA 1.

(Z[ 1/2])/z Ko (BZ/2) = { Z/2 q 0

= =

if q 3, 7 mod 8 if q 1,2 mod 8 otherwise

The proofs are elementary and/or well known, and left to the reader. Let ~:RP 4 n+ 3 ~ BZ/2 be the inclusion.

LEMMA 2.

Then

~*[RP 4 n+ 3 ] has order 22n+ 4 if n is even and 22 n+ 5 if n is odd. PROOF. KO

Consider the homology exact sequence

4n+3

(RP4n+3) ~ KO

4n+3

(RP4n+4) ~ KO

4n+3

(S4n+4)

= 0.

Then the image of [RP 4 n+ 3 ] must be a generator of K0 4 n+ 3 (RP 4 n+ 4 ) and hence it has the order given above.

Since the inclusion

. d uces a monomorp h.1sm 1n . K0 h omology, the lemma RP 4 n+ 4 ~ ~ BZ/2 1n follows. COROLLARY.

The order of [RP 4 n+ 3 ,~J in O~~!~(BZ/2) is at

least 22 n+ 4 if n is even, and 22 n+ 5 if n is odd. PROOF.

c (n

o) (

RP

4n+3

,~)

= ~*( [ RP 4n+3 ]) has the given order.

To complete the proof of the theorem, the structure of o;pin(BZ/2)

72

is needed.

In [3], Anderson, Brown and Peterson computed the group

structure, and their methods also give much information about the o;pin module structure. From [2], o;pin

= n*(MSpin) = (V

B0(8k)) V (V B0(8k+2)) V K,

where K is a wedge of suspensions of K(Z/2).

Then ~pin(BZ/2)

(MSpin A BZ/2) breaks up into direct summands.

= rei*

In [3] the Adams

spectral sequence for each summand was computed.

Since the elements

in filtration 0 are detected by Stiefel-Whitney numbers, we list those elements in filtration greater than 0.

For each summand of

the form B0(8k) A BZ/2 there are elements corresponding to

h~ ~' j > O, i

= 1,2

giving elements

i~ dimension 8j + i + Sk.

Also there are elements x. in filtration 0 with dimension J

4j - 1 + 8k, j > 0, such that h~ xj 8

=1

if j is even, 0 if j is odd.

f

0, 0 ~ q ~ 2j +

€,

where

For each summand of the form

B0(8k+2) A BZ/2 there are elements h0q y., y. with dimension J J 4j + 3 + Sk, 0

~

q

~

2j +

8

-

1, and classes corresponding to

tk wj h~ in dimension Sk + 4 + Sj + i, i J

=

Let

(j 1 , ••• ,js) be a sequence of integers with j 1 2: j 2 2:

and n (J) = j 1 + ••• + j s. [2].

= 1,2.

••• ::::

js>1,

Let MJ be the Spin manifolds defined in

For n(J) odd, Let MJ be the manifold of dimension 4n(J) + 4

with (rcJ rc 1 , [MJ]) = 1.

From computation of Stiefel-Whi tney numbers

x j and y j can be represented by MJ X (RP

4j -1

) , z- •

To complete the

proof, we must evaluate some charateri stic numbers in KOi* (BZ/2). For n(J) evenJsince (rcJ, MJ)

= 1,

we have

1)

J 4n+3 4n+3] ( / ) c(rc )[MJ x (RP ,z-)] = z-*[RP E K0 4 n+ 3 BZ 2

2)

c(rcJ(rc1)2j[MJ

X

wj

X

(s\ z-)]

= z-iJS1J E K01(BZ/2)

73

For n(J) odd, equations 2) and 3) hold with MJ replacing M3 . for n(J) odd, we have the following, where J

J

c(n )[MJ

X

(RP

4n+3

Also

= (J,1):

,~)J = 2~*

[

RP

4n+3]

.

Thus the KO-characteristic numbers distinguish those classes not distinguished by Stiefel-Whitney numbers, and the theorem is proved.

3.

CONSEQUENCES Let h* be the homology theory KO*(

) $ H*(

,Z/2).

A

reformulation of the main theorem gives the following corollary, COROLLARY.

The Hurewicz map K: ~pin(BZ/2~- h*(MSpin

A BZ/2) is a monomorphism. The theorem can also be lifted to the equivariant case. COROLLARY.

O~pin(Z/2, free) is detected by eguivariant

KO and Stiefel-Whitney numbers. The isomorphism

c:

OSpin- 0Sp 1in(BZ/2) gives a method for n n+ detecting the elements of O~in. For an element [MJ E rfin n

simply evaluate the characteristic numbers of

C[M].

'

Thus OPin n

is also detected by KO and Stiefel-Whitney characteristic numbers.

74

REFERENCES 1.

D. W. Anderson, E. H. Brown, and F. P. Peterson, Spin Cobordism, Bull. Amer. Math. Soc. _72 (1966), 256-260.

2.

D. W. Anderson, E. H. Brown, and F. P. Peterson, The Structure of the Spin Cobordism Ring, Ann. of Math. (2) 86 (1967)' 271-298.

3.

D. W. Anderson, E. H. Brown, and F. P. Peterson, Pin Cobordism and Related Topics, Comm. Math. Helv. 44 (1969), 462-468.

4.

M. F. Atiyah and R. Bott, A Lef?chetz Pixed Point Formula for Elliptic Complexes II.

Ann. of Math. 88 (1968), 451-

491. 5.

M. F. Atiyah, R. Bott, and R. Shapiro, Clifford Modules, Topology 3 (1964) Suppl. 1, 3-38.

6.

M. F. Atiyah and F. Hirzerbruch, Spin Manifolds and Group Actions, Essays on Topology and Related Topics, Memoires d~di~s

a Georges de Rham, pp. 18-28.

Springer, Berlin and

New York, 1970.

UNIVERSITY· OF CONNECTICUT STORRS, CT 06268 U.S. A.

75

EQUIVARIANT K-THEORY AND CYCLIC SUBGROUPS STEFAN JACKOWSKI

The works of Atiyah and Segal show that cyclic subgroups are distinguished in equivariant K-theory.

The aim of this paper

is to prove the theorem which also illustrates this phenomenon: THEOREM.

Let G be a finite group.

If the eguivariant map

f : X ~ Y between compact G-CW-complexes induces isomorphisms

for every cyclic subgroup S c G, then f*

is also

an isomorphism. This fact is a consequence of the generalized completion theorem in equivariant K-theory.

With every family F of subgroups

of a compact Lie group G one can associate certain topology, called the F-topology, in the representation ring R(G), and thus also

* in KG(X), where X is a G-space.

On the other hand for every family

of subgroups there exists its classifying space EF.

The Completion

Conjecture states that for every compact G-CW-complex X the projection X X EF

~

X defines an isomorphism

where ~ denotes completion in the F-topology.

For the family F

consisting of just the trivial subgroup this is the Atiyah-Segal Completion Theorem [3]. 76

In [5] we prove the conjecture in many

other cases.

In this paper we will consider the completion theorem

in case of the finite group and the family of all its cyclic subgroups.

It turns out that in this case the topology defined in the

representation ring is discrete. We discuss also the relationship between the orbit structure of a compact G-space X and the properties of the F-topology in

* KG(X).

It is described by the following theorem: THEOREM.

For a compact G-space X the following conditions

are eguivalent: a)

every cyclic subgroup S c G for which the fixed point > ~

set XS is non-empty belongs to the family F, b)

* is discrete in the F-topology, KG(X)

c)

* is complete and Hausdorff in the F-topology. KG(X)

We restrict ourselves here to the case of a finite group, but many of the results are valid also for an arbitrary compact Lie group. I am grateful to Professor Tammo tom Dieck for helpful conversation.

1.

FAMILES OF SUBGROUPS AND EQUIVARIANT COHOMOLOGY Let G be a finite group and let F be a family of subgroups

of G (see [4; Def. 1]).

For a given family F, a G-space X is

called F-free iff all isotropy subgroups occurring on X belong to

F. Throughout the paper we shall assume that a G-space X is a G-CW-complex (in the sense of Matumoto [7]).

The final object

77

in the G-homotopy category of F-free G-CW-complexes is called the classifying space for the family F (see [4; Def. 3]).

For every

family F the classifying space exists and can be obtained as an infinite join of orbits ([4; Satz 1]) with compactly generated topology.

We will denote it by EF.

In the sequel we will need the following properties of classifying spaces. Remark that for a subgroup H c G and a family F the set of subgroups F 1.1

n H :=

{K ~ HJK E F} is a family of subgroups of H.

PROPOSITION.

If H

~

G then for arbitrary family of

subgroups F of a group G there exists an H-homotopy equivalence of cla~sifying

spaces EF

1.2

PROPOSITION.

= E (F n H).

If F1, F2 are families of subgroups of

G then there exists a G-homotopy equivalence

where the topology on the cartesian product is the smallest compactly generated topology containing Tychonoff topology.

* and every pair For every equivariant cohomology theory hG of families of subgroups F1

~

*

F2 of the group G tom Dieck [4]

defined a new theory hG[F 1 ,F 2 ].

~~

Theories hG[F 1 , F2 ] "detect"

at most those orbits whose isotropy subgroups belong to F1, F2 • ~~

The properties of theories hG[All,F] were investigated by tom Dieck in [4].

Here we will study theories of the form h~[F]. ~~

We assume that the theory hG is defined on the category of all G-CW-complexes and is additive, so Milnor's lemma holds. 78

We will

need also the existence of the natural isomorphisms

for every subgroup H

~

G and H-space X.

* it In order to describe the properties of the theory hG[FJ is convenient to introduce the following definition:

1.3

A G-map F

DEFINITION.

:X~

Y is called an F-

cohomology equivalence iff it induces isomorphisms

for every subgroup H 1.4

~

F.

PROPOSITION.

Let E be

an F-free space and let

f : X ~ Y be an F-cohomology equivalence between compact G-spaces.

* x Y) ~ hG(E * X X) is Then the induced homormophism (id x f) * :hG(E an isomorphism. PROOF.

Let us assume that E is a compact G-space and

consider the. diagram over E/G: id

X f

---~--~EX

Y

1

p'

E --1t-~

The map id

E/G ~-1t-

E

x f induces the homomorphism of Segal's spectral

sequences [9] for the maps

1t

p and

1t

p' and it is easy to verify

that this homormorphism is an isomorphism on the E2-terms.

To

prove the proposition for a non-compact space Ewe apply Milnor's lemma. This is a theorem of Vietoris-Begle-type in the equivatiant case.

See also Kosniowski [6; Theorem 2.14].

79

1.5

THEOREM.

* The natural transformation hG* ~ hG[FJ

has the following properties: i~

a)

If

b)

If f :

""'

*

X is an F-free space then hG(X) ~ hG[F](X), X~

Y is an F-cohomologv equivalence between

* (Y) ~ hG[F] * (X) is an compact G-spaces then fi~ : hG[F] isomorphism. PROOF.

The first property-follows easily from tom Dieck•s

results [4; Satz 6], while the second one is a direct consequence of the Proposition 1.6. It can be proved that these two properties give a functorial characterisation of the theory hG[F] * on the category of G-CW-complexes.

2.

TOPOLOGIES IN THE REPRESENTATION RING We will describe how with every family of subgroups of

the group G one can associate a certain topology in the representation ring R(G). For a given subgroup H ~ G, I(H) will denote the kernel of the restriction homomorphism R(G) ~ R(H).

The set of ideals

I (F) := {I (H) • • • • • I (H ) IH. E F} n

1

defines the basis of neighbourhoods of zero in the representation ring.

The corresponding topology will be called the F-topology in

R(G).

Because we have assumed that the group G is finite this

topology is defined by the ideal I(H1 ) • F

80

= {H1, ••• , Hk}.

• I(Hk) where

For the given family F we will denote by I'(F) the kernel of the restriction homomorphism R(G) ~

TI

R(H)

HE F 2.1

PROPOSITION.

The F-topology in R(G) coincides with

I'(F)-adic topology. PROOF.

Because obviously I'(F)

= I(H 1 ) n... n I(Hk)

and the

ring R(G) is noetherian then the proposition follows from the fact that the radicals r(I'(F))

= r(I(H 1 )

• ••• • I(Hk)) are equal.

Observe that the last proposition implies that for every family F the F-topology coincides with th~ Fc-topology, where Fe is the family of all cyclic subgroups contained in F.

Moreover

if the family F contains all cyclic subgroups of G then the F-topology in R(G) is discrete.

In view of the next proposition

this gives a complete description of discrete F-topologies in R(G). 2.2

PROPOSITION.

The following conditions are

equivalent: a)

the family F contains all cyclic subgroups,

b)

the

F-topolog~

in R(G) is discrete,

c)

the

F-topolog~

in R(G) is complete and Hausdorff.

PROOF. (c)

~

(a)

~

(b)

~

(c) is clear.

(a) if the ring R(G) is complete and Hausdorff in

the F-topology then the ideal I'(F) is contained in the Jacobson radical of R(G) (cf. [2; prop. 10.15]).

The ring R(G) is finitely

generated z-module thus its Jacobson radical coincides with its nilradical (cf. [2; ex. 5.24]) - but the last one is zero because there are no nilpotent elements in R(G).

Therefore we know that 81

I'(F)

=0

and the restriction homomorphism R(G) ~ TIR(H) is a

monomorphism, which implies that all cyclic subgroups of G are contained in F. Let H be a subgroup of G and let F be a family of subgroups of G.

We will now compare two topologies in R(H):

the

F n H-topology and the F-topology which is determined by the F-topology in R(G) and the restriction homomorphism R(G) ~ R(H). 2.3

PROPOSITION.

The F-topology on R(H), when it is

regarded as an R(G)-module, coincides with its F PROOF.

n H-togology.

From the description of prime ideals in the

representation ring given by Segal [8] it follows that for the inclusion i : H ~G and a prime ideal p E Spec R(H) whose support is S ~ H, the same group S is a support for the ideal p p i*(p) E Spec R(G). This fact shows that the radicals of the ideals I'(F

n H) and i*(I'(F))•R(H) coincide. Using Atiyah's methods [1] one can describe the kernel of

the completion homomorphism in F-topology:

R(G)

~

R(G)

"

•

We

will say that an element g E G belongs to F if and only if the cyclic subgroup generated by g is contained in F. 2.4

THEOREM.

the F-topology R(G)

The kernel of the completion homomorphism in

~

R(G)

/\

consists of characters

X E R(G)

vanishing on those elements g E G, for which there exists a prime r

number p such that gP

82·

E F for some integer r.

3.

COMPLETION OF EQUIVARIANT K-THEORY Let F be a family of subgroups of the group G.

We have

associated with that family the F-topology in the ring R(G), which will be now regarded a~ coefficients of equivariant K-theory.

For

an arbitrary G-space X the projection X~ pt and the F-topology in R(G) determine F-topology in the ring K~(X).

Homomorphisms

induced by equivariant maps are continuous in the F-topology.

Let

us also remark that the canonical isomorphism KG(G/H) = KH(pt) = R(H) * * allows us to regard the ideals I(H), used in the definition of the

* * F-topology, as kernels of the homomorphisms. KG(pt) ~ KG(G/H) induced by the projections G/H ~ pt. The following proposition gives the basic connection between the orbit structure of a G-space X and the F-topology in KG(X). * PROPOSITION.

3.1

If the compact G-space X is F-free then

* the F-topology in KG(X) is discrete. The space X is compact thus X= U U ••• U X , x1 xk Gx. are the tubes around the orbits Gx .• From the

PROOF. where U

X.

~

1

1

1

definition the images of the ideals I(G

* K* (pt) ~ KG(U G

X.

) are zero.

*

X.

) under the homomorphisms

1 Thus the image of I (G

* also is zero. by the homormorphism KG(pt) ~ KG(X) 1

x, ) • ••• •

Let EF be a classifying space for the family F and let {Y} denote the collection of its G-invariant compact subsets.

Following

Atiyah and Segal [3] we will formulate the next proposition using pro-rings. 3.2.

PROPOSITION.

projection X x EF

~

For an arbitrary compact G-space X the

X defines the homomorphism of pro-rings

83

For any y E {Y} let a y:X x Y ~ X be a projection.

PROOF.

* X Y) is discrete From the previous proposition we know that KG(X in the F-topology, so there exists an ideal Iy E I(F) such that

The family of homomorphisms{~} gives the desired homomorphism of pro-rings. We will now formulate the Completion Conjecture, whose discussion can be found in [5]. 3.3

CONJECTURE.

The homomorphism a induces the natural

isomorphism of cohomology theories on the category of compact G-CW-complexes

where A

denotes the completion in the F-toplogy. In this paper we restrict ourselves to the case when F

is a family of all cyclic subgroups.

To prove the completion

theorem in this case it will be necessary to consider families determined by appropriate representations. Let V be a complex representation of the group G. We will denote FV the family of subgroups of F defined by the representation V (see [4; §4]).

We will seek for description of the FV-topology

in R(G) in terms of the Euler class e(V) of the representation V (see [4; §4]).

Consider the family of principal ideals e(V)nK~

and the suitable topology, which will be called e(V)-topology. 3.4

84

LEMMA.

The e(V)-topology coincides with the FV-topology.

PROOF. G/H ... v . . . . {o}.

For every H E FV there exists an equivariant map This implies that e(V) E I(H) and therefore the

FV-topology is weaker than the e(V)-topology. Conversely:

we will consider the Gysin sequence for the

representation V:

n* * _....__,. KG(s(V)) ...... where s(V) denotes the sphere in V with respect to some equivariant metric.

The sphere s(V) is compact thus s(V) = U U ••• U U x1 xk ~ Gx. are tubes around the orbits. Hence ~

and UX. ~ * I(G )· ... ·I(G ) c kern* 2 e(V)•KG whicbrfinishes the proof. x1 xk 3.5 PROPOSITION. If X is compact G-space such that KG(X) * is a finite generated R(G)-module then the homomorphism

is the isomorphism of pro-rings. PROOF.

The classifying space EFV is the sphere in the

infinite dimensional representation:

EFV =lim s(V $ .. $ V).

Now

we apply the Gysin sequence and using Lemma 3.4 we proceed as in the proof of Lemma 3.1 of Atiyah and Segal [3]. 3.6

COROLLARY.

are fulfilled then

If the assumptions of the Proposition 3.5

a defines

an isomorphism

""""

* 1':::1 KG[FV](X) * KG(X)-+ where "

denotes the completion in the FV-topology. PROOF.

cf. Atiyah and Segal [3; proposition 4.2].

The above corollary can be regarded as a statement dual to tom

Dieck's result ([4; Satz 5]) dscribing in homotopy terms

85

the localisation of the homology theory with respect to -Euler classes.

4.

THE FAMILY OF CYCLIC SUBGROUPS We will now prove the completion theorem for the family

Allc of all cyclic subgroups of the group G.

We have observed

that the Allc-topology in the representation ring is discrete so we have to prove that the projection EAllc ~ pt induces an isomorphism of cohomology theories

We will denote by All

p

the family of all proper subgroups

of a given group. 4.1

LEMMA.

projection EAll

p

~

If the group G is non-cyclic, then the pt induces the isomorphism of cohomology theories K* ( • ) ~ K* [All G G p

J(· )

on the category of all G-CW-complexes. PROOF.

For an arbitrary group G the family All

p

is

determined by the representation which is the sum of all nontrivial irreducible representations of G. then All

p

discrete.

If G is non-cyclic,

~ Allc and (cf. Proposition 2.2) the All -topology is p

From the Corollary 3.6 we obtain the natural isomorphism

on the category of finite G-CW-complexes.

Notice, that the theory

* KG[Allp] is additive and thus the above isomorphism holds also for infinite G-CW-complexes.

86

We recall that for finite group G there exists a chain of families of subgroups {e} = F c F1 c ... c F =All such that o n Fi" Fi_ 1 = (Hi) consists of the subgroups conjugate to a fixed subgroup H..

Such families are called adjacent.

1

4.2

THEOREM.

The projection EAll c

~

pt induces the

isomorphism of cohomology theories

on the category of all G-CW-complexes. PROOF.

We will use the induction method introduced by

Stong and tom Dieck.

= Fo c

F1 c ... c F =All be a sequence of n adjacent families in the group G. For F0 we have the isomorphism Let {e}

* * c Suppose that the homomorphism Ti_ 1 : KG[Fi_ 1 ] ~ KG[All an isomorphism.

n Fi_1 J is

We will prove that T. is an isomorphism. 1

To

prove this assertation it is sufficient to prove that the relative groups

where Fek ··- Allc

n Fk'

are isomorphic.

We will distinguish two cases: the subgroup (H) = Fi" Fi_ 1 is not cyclic. Then c c F~_ 1 ](X) is zero because F.1 =F.1-1 • We must show that i)

Fi-1](X)

= 0.

T. tom Dieck's result [4; Satz 4] provides a

useful description of the relative groups:

87

where E := G

x NHE(NH/H) and CEFi_ 1 denotes the cone over EFi_1 •

On the other hand we have:

because from the definition of adjacent families it follows that F. 1 1-

nH=

All •

From Lemma 4.1, and the long exact sequence we

p

The only orbit type occurring

-~~

deduce that KH[All, Allp](X) = 0.

on E is G/H, so for arbitrary finite subcomplex Z c E we have

* This equality and Milnor's lemma imply that KG[Fi' Fi_ 1 ](X)= 0. ii)

the subgroup (H)

= F." 1

F.1- 1 is cyclic.

We argue

similarly as is (i):

From the induction hypothesis we have that the inclusion

* F.c 1 ~ F. 1 induces an isomorphism in KG. 11-

Thus we have

* i' F i-1 J(X) = KG[F * ci' F ci-1 J (X) KG(F which finishes proof of the theorem. The corollary which we will derive from the completion theorem for the family of all cyclic subgroups is another tllustration of the fact that cyclic subgroups are distinguished in equivariant K-theory. 4,3

COROLLARY,

If the map f:X

~

Y between compact G-CW-

complexes is an Allc-cohomological equivalence in KG-theory, then 88

l• t

. d uces an 1somorphism . 1n f:KG*( Y)

PROOF.

~ ~

* KG(X) •.

Follows from Theorem 4.2 and 1.5.

* The analogous corollary for KG-theory tensored with rationals follows from the Atiyah's conjecture proved by R. Rubinsztein (unpublished): 4.4

THEOREM.

For a compact Lie group G and an arbitrary

finite G-CW-complex X the restriction homomorphism

is a monomorphism.

* We can obtain Corollary .4.3 for KG(·)® Q from this Theorem replacing X by a cone of a map f.

However, Theorem 4.4 is not

true without tensoring with Q even for the trivial G-spaces.

The

Corollary 4.3 can be regarded as the strongest version of Atiyah's conjecture without tensoring with 0.

5.

COMPLETENESS AND ORBIT STRUCTURE We will discuss what information about the orbit structure

* of a compact G-space X can be obtained from the F-topology in KG(X). 5.1

THEOREM.

Let F be a family of subgroups of the finite

For ''a compact G-space X the following conditions are

group G.

equivalent: a)

every cyclic subgroup S c G for which the fixed point set XS is non-empty, belongs to the family F,

b)

* KG(X) is discrete in the F-topology,

c)

KG(X) is complete and Hausdorff in the F-topology.

i~

89

(a)~ (b).

PROOF.

Let us denote by FX the smallest family

containing all isotropy subgroups occurring on X.

Then from the

* is discrete in the Proposition 3.1 we deduce that KG(X)

FX-topology. c The assumption about the orbit structure of X means that FX c F. The F~-topology coincides with the FX-topology (cf. section 2) so

* is discrete in the FX-topology c the ring KG(X) and hence in the F-topology. ~

(b)

(c) is obvious.

s

(c) ~ (a) Let S be a cyclic subgroup such that X

* Consider the homomorphism KG(X) map G/S ~ X.

~

I¢.

R(S) induced by an equivariant

This homomorphism imposes on R(S) a structure of a

* finitely generated KG(X)-module.

* is complete Therefore if KG(X)

and Hausdorff in the F-topology the same holds for R(S).

From

Proposition 2.3 it follows that the F-topology on R(S) coincides with the F

n S-topology

so we deduce (cf. Proposition 2.2) that

S E F.

REFERENCES 1.

M. F. Atiyah, Characters and cohomology of finite groups. Publ. Math. IHES 9 (1961 ), 23-64.

2.

M. F. Atiyah, I. Macdonald, Introduction to commutative algebra.

3.

Addison-Wesley Reading, Mass., 1969.

M. F. Atiyah, G. B. Segal, Equivariant K-theory and completion. J. Diff. Geom. 3 (1969), 1-18.

4.

T. tom Dieck, Orbittypen und aequivariante Homologie. Arch. Math.

90

XXIII (1972), 307-317.

5.

S. Jackowski, Families of subgroups and completion.

To

appear. 6.

C. Kosniowski, Equivariant Cohomology and Stable Cohomology. Math. Ann. 210 (1974) 83-104.

7.

T. Matumoto, On G-CW-complexes and theorem of J.H.C. Whitehead.

J. Fac. Sci. Univ. Tokyo Sect. 1. 18

( 1971 ) ' 363-374. 8.

G. B. Segal, The representation ring for the compact Lie group.

9.

Publ. Math. IHES 34 (1968), 113-128

G. B. Segal, Equivariant K-theory.

Publ. Math. IHES 34

(1968), 129-151.

WARSAW UNIVERSITY, PKin IXp, 00-901 WARSZAWA, POLAND. 91

'll/p MANIFOLDS WITH LOW DIMENSIONAL FIXED POINT SET CZES KOSNIOWSKI

1.

INTRODUCTION This paper contains a description of the generators of

9.1* ~p (r), where ql* 2('p (r) is the subring of the unitary ring

9.l*~p

generated by

~p

~p

bordi sm

manifolds with fixed point set of

Only the case p is prime and r ~ p-1 will

dimension less than 2r.

be considered throughout this paper.

A corollary of the main

result obtained verifies a conjecture of the autho_r in [ 6], (for the case p prime and r The result for

~

p-1).

9.l*~P(o)

consists of free ~p manifolds.

is well known since this ring

An "algebraic" result for r=1

(the case of isolated fixed points) was given in [4].

Furthermore,

for the special case of p=2, 3 explicit geometric generators of

9.l*~p(1)

were given in [4] as Theorems 2.2, 2.6.

[3] describes geometric generators of analogue of

If p=3 or 5 then

o*~p(1) -the oriented

9.l*~p(1).

The generators of 9.l*Z/p(r) are some "obvious" generators -

~p, Riemann surfaces and complex projective spaces (with z,!p action) - together with some ~p manifolds obtained from these via a "A. construction".

Suppose M is a ~p manifold with an action

of the torus (S1 )t which commutes with the action of 2/p X (S 1 )t on M).

92

~paction

(i.e. an

The A. construction produces a

~p

ip

manifold A£(M, ~) of dimension 2£

+

dim M.

Under suitable

restrictions on M, £and~ the 2{p manifold A£(M, ~)has no higher dimensional fixed point set component than M has.

Recall that in

[5] the "f construction" was used to obtain generators of ()j*7¢P. If the ~p action on M extends to an

action and if ~\S: (where

s1

l

~ s:) coincides with this action fori= 1,2, ... ,£then i=1 l A£(M, ~) = f£(M). For the purpose of this paper the f construction (s 1 )£ =

is of little value since r£(M) has a codimension 2 fixed point set Component' namely r £- 1 (M).

Sec t•1on 2 conta1ns . f urther details.

If p=2 then the condition r The ring

~*2(p(O)

~ p~1

is well understood and

adequately in [4].

means that r = 0 or 1.

m*~P(1)

is described

Therefore, throughout this paper pis an odd

prime. The proof of the main result, i.e. that a certain list of

~p

manifolds generates m*2(p(r), is roughly as follows.

map v :

~ ?¢p(r) ~

A natural

2[x 1 , x2 , ..• , xrp- 1 ] is defined with kernel

precisely the set of free

~p

manifolds.

Thus Ill* Z/p (r )/1!.1/i'P (0)

is isomorphic to a subring of the polynomial ring 2[x 1, x2 , ... ,xrp- 1 ]• Rationally the rings are isomorphic.

~*~P(r)/m*2(p(O)

The polynomial ring 2[x 1 , x2 , .•. , xrp- 1 ] is graded

by saying that deg xj = [ (j + p-1)/p]. of degree n form a v of mir:'P(r).

® Q and Q[x 1, x2, ... , xrp- 1 ]

z module

The homogeneous elements

m(n, r) which contains the image under

Let J(n, r) denote this image in m(n, r).

of generators" give an upper bound for \7J1(n, r)/J(n, r)\ finite by the remark on rationarity above).

The "list (which is

Various integrality

conditions give a lower bound for \7J1(n, r)/J(n, r)\.

These bounds

93

coincide and hence show that the list of generators do indeed generate

~*2/p(r). These last steps in the proof may remind some readers of

the last steps in the proof of the main theorem of [4].

n

of j?n(n, r)/c9(n, r)\ is

The value

pq(I) where the product is over unordered

sequences of integers I = (i(1), i (2)'

... '

i (k))' n

~

k

~

o,

satisfying (i)

1

~

i (j)

(ii)

i (j)

f.

~

for

p-1

~~= 1 [ (i(j)

(iii)

rp-1

+

for

j = 1, 2,

j = 1, 2,

... '

... '

k,

k and

p-1)/p] ~ n

while q(I) = [ (n + p-2- ~i(j))/(p-1)]. The next section, section 2, gives details of the generators. Sections 3 and 4 contain the essential ingredients to give the upper bound for

\?n(n~

r)/c9(n, r)\ while section 5 contains the ingredients

to give the lower bound. section 6.

2.

The final details are collected in

Section 7 describes some very interesting consequences.

THE GENERATORS This section contains a description of the generators of

~*zyp(r).

It is well known that

IU*~P(o),

(the subring of

~*~p

consisting of free 2/p manifolds), is precisely the subring 2/p • ~* of

~*2/P.

Elements of

~p • ~*

are of the form Zjp x M with M E 1).1*.

The action of 2/p on M being trivial while on 2/p it is the natural free action.

If we write~* as Z [M1, M2 , ••• ] and if M1 means

~ Mi(j) where I= (i(1 ), i(2), ••• ,i(k)) is some unordered k-tuple

then lll*Z/P(o) is the subring of

94

m*~P

generated by the following

set of

11P

manifolds {2{p

X

Mr;

THEOREM 2.1.

I is an unordered k tuple,

~*~p(1)

k ~ o}.

is the subring of m*2/p generated

by the following ~p manifolds {generators of

~*~P(o)}

U{point} U{Rj; (p+1)/2 ~ j ~ p-1}

U{~£CP 1 (p-j);

U{~iCP 2 (j,

~

j

~

1); (p+1 )/2

THEOREM 2.r.

~ ~p

1

(2

(p-1)/2, 0

~

~

i

~

~ p-1,~0 ~

j

..

p-1-2j} i

~

2j-p-1} .

~ r ~ p-1). ~*l(p(r) is the subring of

generated by the following Z(p manifolds {generators of m* ~p (r-1)

l

U{Mr-1}

U{~iCPr- 1 (0, 0, ••. ,0); 0


'

2

~j ( 2 ) •

'::>

• zk ~j ( k) J

'· • • '

Finally, we need to explain AiCPk(J).

'::>

More generally we

shall define Ai(M, ~) where M is a 2{p manifold and ~ is an action of the torus (s 1 )ion M commuting with the 2(p action on M.

As a

manifold Ai(M, ~) is the quotient of M x (s 3 )i by? free (s 1 )i

TI~

d S. where s~ = sd and d = 1 or 3, J=1 J J 3 then an element of M X (S )i may be written as m X TI(u., v.) with J J 3 2 m E M and ( u.' v.)ES.cC. An element Tit. E TIS: acts on J J J J J (53 )i MX by action.

If we write (Sd)i as

m x TI(u., v.) ... ~(Tit.' m) J J J

X

TI(t. u., t.-1 v.). J J J J

This action of (s 1 )i on M x (S3)i is free and so the quotient (M

X

(S3)i)/(S 1 )i is a manifold.

The group

~p

acts on M x S3 as

follows m X I1 ( U • , v . ) .,.. sm J J where s E

~P

X ( u .,

J

S- 1 v . ) J

c s 1 and sm denotes the action of s on mE M.

This

induces an action on the quotient and the resulting 2(p manifold is denoted by Ai (M, ~). REMARK. 96

If i

=i

and ~ is an extension of the 2{p action

on

Mthen A1 (M,

~) =

f(M) as defined in [5] etc.

not an extension of the 2/p action on

Mthen A1 (M,

If t = 1 and ~ is ~) =

r(M) - f(M')

up to 2/p bordism where M' is a manifold diffeomorphic to M but with a (possibly) different ~p action. So, to define AtCPk(J), we need to define some torus actions k

on CP •

Suppose L = (t(1 ), t(2), ••• ,t(k)) is a k-tuple of non-

negative integers such that t(1) + t(2) + ••• + t(k) = t. Now write 1 t k t(i) 1 1 1 (S ) as II II S . . where S . . = S and define ~L to be the i=1 j=1 l,J l,J action of

(s 1 )t on CPk given by t(k) t 2 .; ••• ;zk II 'J j=1

where t. . E S1..• l,J

l,J

k

k

Denote At(CP (J), ~L) by A1 CP (J).

Furthermore, if

L = (t(1), t(2), ••• ,t(k)) is given by p-1 t(j)

=

t

if

1 ~ j ~ [t/(p-1 )]

(p -1 )[ t/ (p -1 ) J

if

j

= [ t/ (p -1 ) J

+

1

otherwise

0

k k k then denote At(CP (J), ~L) = A1 CP (J) by AtCP (J).

3.

FIXED POINT SETS AND A. In this section we shall describe a natural map v from

~*~p

to the polynomial ring Z [x 1 , x2 , .•• ].

This map is defined

in terms of the equivariant normal bundle of the fixed point set. Restricted to

~*21P(r),

this map v factors through the polynomial

ring ~ [ x1' x2' .•• ' xrp-1 J.

Order the monomials in this ring by

97

ordering the variable {xi} so that x1 < x2 < 1 lots d enote ~ or der t erms.

< xrp- 1 , and let

By Usl. ng the "generators of 1).1*7/p (r)"

we shall prove the following result in this section. THEOREM 3.1.

For each pair of integers (i,

~)

satisfying

p ~ i ~ rp-1 and 0 ~ ~ ~ i - [ (i+p-1)/p] there is a 2/p manifold, obtainable from the "generators" of 1).1* 7/p (r), whose image under v is (x 1 )~x. + lots. p1 A corresponding result for the case 1

~

i

~

p-2 is proved

in the next section. Let M be a unitary 2(p manifold. the fixed point set.

Let F be a component of

The normal bundle v(F, M) ofF in M has a

natural complex structure and splits into a direct sum

The action of ~p in each fibre of

of (complex) vector bundles.

v.(F) is multiplication by exp(2nij/p).

Classifying each of these

J

bundles for each component of the fixed point set gives a 1).1* module homomorphism v :

~*2/p ~

e

{d.::!: o} J

mis

This ring

then deg x=n+ r,2d .• J

element xE7!1.

~*(IT~:~ -

BU(d.)) = J

graded by saying that if

X

m.

E ~(IT BU(d.)) n

This is defined by saying that if xE~ (IT BU(d.)) n

then x = n + E2(d.-1 ). J

The ring 711 is isomorphic to the polynomial ring l).l*[{nk ,J.; k::!: 1, 1 ~ j ~ p-1}], where nk . denotes the equivariant normal bundle of 'J

98

J

There is also the notion of dimension of an J

CP k-1 = { [ z o'• z 1 ,• ••• ·z • · CP k (o, , k- 1 ,o]} ln notation the grading is deg

~k

o, ... ,o,

· J).

In this

. = 2k and deg M = dim M if M E

~*

'J

while dim ~k . = 2(k-1) and dim M =dim M (the usual dimension) if 'J

M E 9..1*. Thus, if M is a Z/p connected unitary £1p manifold of dimension m then v(M) is a homogeneous polynomial of degree m, while dim v(M) is the maximum dimension, up to bordism, of the fixed point set of M. DEFINITION 3.2. x. to be~ l

If i

10

mod p then define

. where n = [i/p] + 1. and j = i-p[i/p].

n, J

If i = kp then define xi to be Mk. Thus

'111.

is isomorphic to the polynomial ring 2[ x1 , x2 , x3 , ••• ] •

Note that deg xi = [ (i+p-1 )/p] and dim xi = [i/p].

LEMMA 3.3.

v(Mn) = x np

v (CP n+ 1 (0, 0, .•• , 0, j ) )

v(R.) = x. + r. x 1 J J J pPROOF.

= l({x.; l

i ~

for

+ (x.)

n+1

J

•

(p+1)/2 s: j s: p-1.

Direct calculation.

The map v : '111.

= x np+J.

1}].

~* ~p (r)

....,

'111.

factors through z[ {xi; 1 s: is: rp-1}] c

By ordering the variables {x.} so that x. > x. l

l

if i > j we see immediately from lemma 3.3 that the image of v (restricted to X.

l

+

~*a/p(r)) l 0 t s,

for

contains 1 s: i s: p-2

or

p s: i

S:

rp-1

and· px p- 1. 99

J

Thus the image of ~ contains terms of the following form xi + lots (px

p-1

)£xi + lots

where I= (i(1), i(2), ..• ,i(k)) with i(j) and xi= xi(1) xi(2) ••• xi(k)•

f.

p-1 for j = 1,2, ••• ,k

We shall shortly show that the image

contains terms of the form

where I= (i(1), i(2), .•• ,i(k)) with i(j) and m = [ (£ + deg xi - 1 I\ + p-2)/(p-1 )]. THEOREM 3.4. J

= (j(1),

f.

p-1 for j = 1,2, ..• ,k (I II = i(1) + i(2) + ••• + i(k).)

Suppose that L = (£(1), £(2), ••. ,£(k)) and

j(2), ••• ,j(k)) are k-tuples of non-negative integers.

l.ft? is the set of subsets of the set {(i,j)EZ x~; 1~ i~k, O~j~.t(i)}

~(AL CPk(J)) =

~ SEt?

where J - S is the k-tuple (j (1)- s(1 ), j(2)- s(2), ••• , j (k)- s(k)) with s(i) denoting the cardinality of the set {(i,j); O~j~t(i), (i,j) E s.}

Finally

\SI

= s(1) + s(2) + ••• + s(k) is the

cardinality of S. PROOF.

where [z 0

; •••

/\L CPk(J) is the set of equivalence classes

;zk] x TITI(u 1. J'' v 1. J.) = [z '; ••• ;zk'J x TITI(u . . ', v . . ') '

'

0

1,

J

1,

if and only if there exists c E co~},{o} and t . . E s 1 (1 ~ i~ k 1,

J

0 ~ J. ~ t(i)) such that zi ' = czi ~ ti, j for 1 ~ i ~ k, J

100

'

J

z

I

=

0

CZ

0

'

U.

•

I

-

1,J

t

•

u. . an d v. . I = t.-1

.

1,J

1,J

The action of s E ~p c [z • z • 0'

-+ [

1'

... .' zk]

z • z ~j ( 1 ) • o' 1 '=> '

X ITIT

•

s1

i,j·

is given by

(u . . , v . . ) 1,J

zk

• • • '

v

i,j

1,J

1,J

~j (k) J

'=>

X

( -1 ) IliT u.1,J. ' S v.1,J. •

Given any element x E M, there is a set S E g such that v 1,J ..

I

0 for (i,j) E Sand u . .

1,J

I

0 for (i,j) i_ S.

If x is in the

fixed point set then t.

=s

for

( i' j) E S, and

t.

=1

for

( i' j) is.

1, j

1,j

Furthermore u.

= o,

v.

=1

for

( i, j) E S, and

u.

= 1,

V.

=0

for

(i,j) is.

1, j

1, j

1, j

1, j

The action of s E ~p on points x with { ( u. . ' v. . ) } as just given 1, J 1, J looks like [z • z • 0'

1'

... .'

(o,

1)

rrrr

x

( 1' 0)

(i,j)i_s

.... [zo; 21 sj (1 )-s(1); ••• ; zk sj (k)-s(k)J

X

rrrr

(o,

1) x

rrrr

(1 , o ) •

(i,j)i_s

(1,j)ES

Thus we can easily see that the normal bundle of the components containing such fixed points is given by

(-1)1 8 1

(x

p-1

)1 1 1 v(CPk(J-

s))

with the factor (-1)1 8 1 appearing for orientation reasons.

101

It is not too difficult to formulate and prove the corresponding result for

v(~~(M,

~))

-this is left to the reader.

The next two results follow from Theorem 3.4. COROLLARY 3.5. 0

~ ~(i) ~

If L = (~(1), ~(2), ... , ~(k)) satisfies

p-1 fori= 1, 2, ... ,k then

COROLLARY 3.6.

Suppose j is an integer such that 1

p-1.

~

j

~

I

0 mod p

If L = (~(1), ~(2), ... , ~(k+1)) satisfies 0 ~ ~(i) ~ p-1 for

i = 1,2, ... ,k and 0

~ ~(k+1)

< j then

A CPk+ 1 (0 , 0 , ... , 0 , J. ) ) = x 1 I L I xk . + 1 ot s . v ( ll.L pp+J

We can now complete the proof of Theorem 3.1.

If i

then the result follows from Corollary 3.6.

If i = kp then we know

(from Corollary 3.5) that we have elements (x

p- 1

0 ~ ~ ~ i - deg xi. + lots where ak such that

I

Now, if k < p-1 then CPk = ak Mk + lots= ak xkp

0 mod p.

If

~ ~

AL CPk(O,

~

1 then there is an integer bk

akbk,~ = 1 + ck,~ p~ for some ck,~·

v(bk,~

)~CPk +lots for

Thus

'

o, ..• ,o) - ck,~(Rp_ 1 )~ Mk)

= (xp_ 1 )~ xkp +lots where ~ = ILl.

4.

(Recall that v(R

p- 1

) = px

p- 1

.)

ISOLATED FIXED POINTS In this section we shall concentrate on the manifolds

appearing in the statement of Theorem 2.1. the following result. 102

The aim will be to prove

There exist ~p manifolds N(n, j) for

THEOREM 4.1.

n = 1, 2, ••• , p-1 and n s: j (i)

:s;;

p-1, such that

N(n, j) is obtainable from the "generators" of

~*~p(1), (ii)

dim N(n, j)

= 2n,

and

( yp_ ) n-1 yj + lots if j 1 (iii)

:s;;

p-2

'V(N(n, j)) = p(yp_ 1 )n + lots if j

= p-1.

The yi are variables such that~ [x 1 , x2, ... ,xp_ 1 ] ~ 4:[y1 , Y2, ..• ,yp_ 1 ] with a (non-obvious) ordering y p- 1 > Yp- 3 > Yp- 5 > .•. > Y2 > Yp- 2 > yp-4 >. .> y1. 0

The proof of this result will occupy the whole of this section.

The next lemma is in part Lemma 3.3 LEMMA 4.2. (i)

(ii)

(iii)

\) (R . ) = X. + r .X 1 for (p+1 )/2 :5: j :s;; p-1' where J pJ J jr. = 1 mod p and 1 < r. < p. J J \) ( CP 1 (p- j) ) = X. + X p-j -for 1 :s;; j :s;; p-1' J x1 X. + X 1 X. 1 + X • X .+ 1 for J pJp-J p-J 2 :s;; j :s;; p-1.

PROOF.

Direct calculation.

We now introduce the variables {P 1 , P2 , ..• ,P(p-1 ); 2 , Q(p+ 1 )/ 2 , ... ,Qp_ 2 , y} which up to order will be the set of variables {y 1 , y2 , ..• , Yp_ 1 }. DEFINITION 4. 3.

(i) Define P.J to be x.J + xp-J. for 1 (ii)

s: j :s;; (p-1)/2.

Let q. be the unique integer satisfying J

103

((pp-1_p)/2 + j)q. = 1 mod p p-1 and 1 < qj < p p-1 J

Define Q. to be x. + q. x 1 for (p+1)/2 ~ j ~ p-2. J pJ J (iii)

Define y to be xp_ 1 •

There are technical reasons for defining q. in the above J

way.

These reasons should become apparent towards the end of this

section. Z[x 1 , x2 , ••• ,xp_ 1 ]

REMARK 4.4.

"=

Z[P 1 , P2 , ••• ,P(p- 1 )/2 '

0 (p+1)/2' ••• , 0p-2' y]. REMARK 4.5.

r. - q. is divisible by p. J

J

LEMMA 4. 6. (i). \J(CP 1 (p-j) = \)(CP 1 (j)) = Pj for 1

~

j

~

(p-1)/2.

\J(R. - ((q. - r.)/p)R 1 ) = Q. for (p+1)/2 ~ j ~ p-2. J J J pJ \J(R 1 )=py. p-

(ii) (iii)

In other words the image of

\J

IU*~p(1)

.... Z [P 1 , ••• ,Qp_ 2 , Y]

~- 2 •

contains the subring generated by P1 , ••• ,

If we order {Pj} as P 1 > P2 > ..• > P(p- 1 )/2 then we have the following result.

If 1

LEMMA 4.7.

~ j ~ (p-1)/2 and 0 ~ t ~ p-1 - 2j then

\)(A. CP 1 (p-j)) = ytP. + lots. t J

A.tCP 1 (p-j) = AtCP 1 (p-j) and so by Theorem 3.4

PROOF. 1

\J(A.tCP (p-j)) = ~

t

(-1)s s=O

=

~

(!) yt

\J(CP1 (p-j-s))

t

(-1)s (!) yt PJ.+s + s=O

where m =min{ (p-1)/2- j,

t}.

~

£ s=m+1

(-1)s (t) y£ p . s p-J -s

The highest order term is yt P. if J

j < P- j - t i.e. if t < p-2j. 104

If j = p-j-t then t is odd (p is

The proofs of the next few lemmata are by straightforward calculation and are left for the reader.

LEMMA 4.8.

If

(p+1)/2 < j < p-1 then

v(CP 2 (j, 1)) = -(1 + q. 1 )y Q. + (1-q.)y Q. 1 JJ J J+ q. y p . + q. 1 y p . - qJ. y p1 J p-J+1 Jp-J + p 1 QJ. + ( p p-J. +1 - QJ. 1 ) (p p-J. - QJ. )

Note that 1 + q. 1 = jf(j-1) mod p Jf. 0 mod p. Note also that q.

J -

q

j -1

+ q

j -1

q.~=

0 mod p

J

p-1

.

LEMMA 4.9. 2 2 v(CP (p-1,1)) = -(2qp_ 2 + 1)y + 2y ~- 2 - Y P2

+ (1 + qp-2)y p1 + p1 (P2 - Qp-2). Observe that (4-p)(2q

LEMMA 4.10.

p- 2

If

p-1 + 1) = -p mod P h = (p+1)/2 then

2

+ (2qh - qh qh)y . Note that qh = 2, qh-1 = p

p-1

-2 and so - (2-qh+9h-1) = 2-p

105

p-1

= pp-1

while qh + qh_ 1

~(CP 2 (h, 1))

and 2qh - qh qh

= (2-pp- 1 )Y

=0 ·

In other words

Qh + (pp- 1 - 1 )Y ph-1 - 2Y P1+ p1 Qh

If 2 < j ~ (p-1)/2 then

LEMMA 4.11.

+ p (P . - Q 1

LEMMA 4.12.

.) + y p . 1 + q

p- J

J

2

. y p1 - y p .

p- J

J-

J

2

~ ( CP ( 2, 1 ) ) = ~ (CP (p-1, 1 ) ) .

By ordering the variables {y, P., Q.} so that l l Y > Qp-2 > ... > Q(p+ 1 )/ 2 > P1 > P2 > ... > P(p- 1 )/ 2 we can state the above results in the following form. Let q . - q . 1 + q . 1 q . = t . P J JJJ J ~(CP 2 (p-1, 1))-- -(2qp-2 + 1 )y2 + lots.

PROPOSITION 4.13. (i)

(ii)

If h

= -(1+q.J- 1 )y

= (p+1)/2

If 2 < j

Q. + t. pp- 1 y2 +lots. J

J

then

~(CP 2 (h,1)) (iv)

•

If (p+1)/2 < j < p-1 then

~(CP 2 (j,1)) (iii)

p-1

-

= (2-pp- 1 )y

Qh +lots.

~ (p-1)/2 then

v(CP 2 (j, 1)) = -(1+qp-j)y Qp-j+ 1 + tp-j pp- 1 y2 + lots. (v)

v(CP 2 (2,1))

LEMMA 4.14. j

= (p+1)/2, ... ,p-1

106

= -(2qp-2

+ 1)y2 +lots.

There exist 2/p manifolds M(n, j) for and n

= 2,3, ... ,2j-p+1

such that

·

(i)

M(n, j) is obtainable from the "generators" of

m* 2'/p (1). dim M(n, j) = 2n, and

(ii)

n-1

Q. + lots J \) (M(n j)) = fY ' l py n + lots if

(iii)

if

(p+1 )/2 :;; j < p-1.

j = p-1.

For brevity let us denote

PROOF. - (1+q.

J -1

)y Q. + t. pp- 1y2 J

- (2q . + 1 )y p-2 (2 - p

p-1

by

J

2

by

D., (h < j < p-1), J

Dp_ 1 , and

)y Q

h

where h = (p+1 )/2 as before.

s·l nee

· 1) -- ILJ,O A. J CP 2 ( J, A CP 2 ( J, · 1)

we have J

= L:

s=O

. ( -1 ) s (J) y J \) ( CP 2 (J-S, 1 )) s

J

= L:

s=O

s=m+1

D + lots ( -1)s (J)yJ s p-j+ s+1

where m = min{j-h,J}.

The highest term is D. if j > p-j + J+1,

i. e. if J > 2j - p-1 •

If J = 2j - p-1 then J is even and so the

J

highest term is 2D .. J

Let J. J

=2

if J = 2j - p-1 and 1 otherwise, then the

highest term is J. D.. J

J

Let mn ., kn . be integers such that ~,J

~,J

-mn .(1+q. 1 )J. = 1 + kn . pp- 1 for h < j < p-1, ..,,J JJ ..,,J

-mJ,p+ 1 (2qp_ 2 + 1)Jp_ 2 = p + kJ,p- 1 pp- 1 , and p-1 p-1 mn h(2 - P )Jh- 1 + kJ h P • ~,

'

107

The manifolds M(n,j) may now be defined as follows: M(n,p-1)

M(n,j)

= mn-2,p-1

= mn-2,j

2 _ k p-n-1( )n \n-2 CP (p- 1 , 1 ) n-2,p-1 p Rp-1 ·

\n-2 CP

2(

j,1

) _ k

. p-n(R )n-1 Q n-2,J p p-1 j

-m t p-n- 1 (R )n n-2,j i,j j p p-1 '

= Q.J

The manifolds Q. satisfy v(Q.) J J the proof of lemma 4.14.

h < j < p-1.

if

- see lemma 4.6.

This completes

To prove the main assertion of this section - i.e. Theorem 4.1 - we let N(n, p+1-2j) N(n, p-2j)

yp_ 1 _2 j Y 2. p- J

= M(n,

= \ n-1

= Qp-j

= P.J

for

p-j)

for

CP 1 (p-j)

1 < j

for 1

~

j

~

1

for

~

~

j

~

h-1,

1 ~ j ~ h-1,

h-1,

h-1.

The ordering y > Qp_ 2 > ... > Qh > Ph_ 1 > ... > P1 induces the ordering y 1 > y 3 > ... > y2 > y 2 > y 4 > ... > y1 and this ppppcompletes the proof of theorem 4.1.

5.

INTEGRALITY CONDITIONS The purpose of this section is to describe necessary

conditions for a homogeneous polynomial p 108

Em to

lie in the image

\)

• •

m

'1-li~

~p ...

'rn

An obvious inner product is defined on '171., (see

11 \•

Elements SJ E '171., (Jan

paragraph following Definition 5.4).

~

unordered k-tuple, k

0) are found such that if M E ~P(r) and

v(M) = p~R then (SJ' R) = 0 mod p~+ 1 for all J satisfying \J\ < n-~(p-1 ).

See Theorem 5.5.

Different k-tuples sometimes

produce linearly dependent elements, Theorem 5. 7 describes a set of linearly independent elements. Let M be a 2n dimensional

~p

manifold and let F be a 2d

dimensional component of the fixed point set.

Split the normal

bundle of F in M as in section 3, i.e. \J.{F, M) = 2:: v. (F). J

j, 1

~

j

~

For each

p-1, denote by z. 1 (F), j 2 (F), .•• , z. d(')(F) the J,

·'

J,

J

formal symbols whose elementary symmetric functions give the Chern classes of v.(F), (d(j) =complex dimension of v.(F)). J

J

For brevity

denote the set z 1 , 1 (F), ..• , z 1 ,d( 1 )(F), ••• , zp- 1 , 1 (F), .•• , Denote the corresponding set.

zp-1' d (p-1 ) (F) by z1' z2' •• •' z n- d.

1' 1' ••• ' 1' 2, 2, ••• '2, ••• 'p-1' p-1' •.• 'p-1 d (p-1 )

by t 1 , t 2 , ••. , tn-d·

(Note that if zk = zi,j(F) then tk = i.)

Finally, let y1 , y2 , ••• , yn-d be the formal symbols whose elementary symmetric functions give the Chern classes of F. DEFINITION 5.1. in n variables.

Let f be a symmetric homogeneous polynomial

Define f(e (F)) to be the mod p reduction of the p

integer { f ( ••• , yJ.,

••• , ••• , z . + t . , ... ) J

J

n(z J.

+ t . ) - 1 }[ F] . J

109

THEOREM 5,2.

If ME

~~P(r),

if 1

~

r

~

p-1 and if v(M) is

divisible by p£ then ~

f(ep(F)) = 0 mod p

£+1

F E Fix for all symmetric homogeneous polynomials f of degree less than n- t(p-1). PROOF.

Define f(e(F)) to be the element of Q[s], where

s = exp(2ni/p), given by {f( ..• , exp(y j ) -1 , .•. , ... , exp(z.J + 2nit./p)-1, ••• ) J TI(exp(z. + 2nit./p)-1)- 1 TI(y./(exp(yJ.)-1))}[F]. J J J We then have the well known integrality condition ~

f ( e (F) ) E :&:: [ s]

where the summation is over the components F of the fixed point set. This integrality condition may be obtained quite easily, for example, from the K theory localisation theorem [1]. Multiplying the integrality condition by (s-1)n-t(p- 1 )-d(f)' where d(f) is the degree off, and reducing modulo (s-1) produces the desired result.

The case r= 1, general t is proved in The case t=O, 0 ~ r ~ p-1 is given as

Proposition 4.1 of [4], Corollary 2.5 of [2].

See also Theorems 1.1 and 1.2 in [6].

The above result will be restated in a different form shortly.

The ring:&:: [x 1, x2 , ..• , xrp- 1 ] is graded, Z [{xi}]/=

$m(n, r) where m(n, r) consists of those homogeneous polynomials in Z [{x)J of degree n. elements in

110

mof

(Alternatively, m(n, r) consists of those

degree n and dimension less than or equal to r.)

DEFINITION 5.3.

Let R(n, r) be the set of unordered

sequences integers I= (i(1), i(2), .•• , i(k)), k ~ 0 such that 1

S:

i(j) s: rp-1 for j = 1, 2, ... ,k and deg I= z~= 1 [(i(j) + p-1)/p]=n. Thus m(n, r) is a Z module with basis {xi; IE R(n, r)} where

xi denotes, in the usual fashion, xi( 1 ) xi( 2 ) ... xi(k) i f I= (i(1), i(2), ... , i(k)). Suppose that J = (j(1), j(2), ..• ,j(k)) is an unordered sequence of positive integers of length k s: n.

Define s3 (z 1,z 2 , .•• ,zn

to be the smallestsymmetric homogeneous polynomial in the variables z1 , z 2 , ... , z n th a t con ta1ns . th e monom1a . 1, -z j( 1 ) z j( 2 ) . • • zk j(k). 1 2 DEFINITION 5.4.

Define

following vector in m(n, r) ® l:

s3

E m(n, r) to be any lift of the

~p:

I ER.(n,r)

s 3 [ XI ]xi

where sJxi] means L: s3 (ep(F)), FE Fix(xi). By defining the obvious inner product on ?n(n, r), i.e.

O. 1 mod p

s.[x.] J 1

=

if

(-1)n mod p (j- 1 )kj mod p n

0 mod p

i

= jp

if j = 0 if

and

i = np+k

i = np+k and

j > n

otherwise.

Observe that 113

= {0

mod p if n ( -1) mod p

so[Xnp+k]

k = 0

if

I o

k

and

=

In other words s

[X.] deg(X.) s [X.], which explains why the p-1 J J 0 J condition j(i) I p-1 was included in the definition of S(n,r) in

5.6, (i.e. if J =(j(1), j(2), ••• ,j(i,)) and J' = (j(1) + (p-1) a(1), ••• ,

j(i,) + (p-1) a(.t), (p-1) a(i,+1), ••• , (p-1) a(m)) then

s3 and

S/

are linearly dependent). The set of vectors of the form ~:p 1- 1 s.[X.]X.

THEOREM 5. 12.

l=

J

l

E (£¢p)[x 1, X2, ... ,Xrp- 1 ] for 0 s: j s: rp-1 and j l_p-1 are linearly

independent. PROOF.

Define f .[X.] E ~p by J

f .[x.] = ~b J

where j

= bp+c

.t=O

1

l

(-1)b-.t (b) i,

[x]

s(b-i,)(p-1)+b+c

i

with 0 s: c < p.

We shall now prove the following "claim" 1

if

kb+c 0

l?

if

b = n if

and

b = n

c = k and

=0

c > 0

b>n

otherwise

If k = 0 then it is clear that fbp+c[Xnp] = 1 in case b+c = n and 0 otherwise.

114

For k

I

0 we have

l

b

f

bp+c

[x

np+k

]

= L:.t=O (-1 )b-.t(b) s [x ] l, (b-.t)(p-1)+b+c np+k

if b+c > n

where Pn_ 1 (.t) is a polynomial in n-1.

of d:g!ee less than or equal to

l,

The claim will follow from the following fact b

L: (- 1 )b-l,(b).tn .t=O l.

b:

if

n

=b

0

if

0

~

={

n < b.

/

To prove this fact note that the result is true i f n

= 0.

For n > 0 the result follows by induction by noting that dn (x-1 )b dxn

= b(b-1)

••• (b-n+1) (x-1 )b-n

and

where Q 1 (.t) is a polynomial of n-

degree~

n-1.

Putting x

=1

produces the desired result. Consider fbp+c[Xnp+k] as the coefficients of the matrix

M= (Mbp+c,np+k ) •

This matrix M has a "block upper triangular

form" - the blocks are { m } where (m ) bp,np bp,np c,k The blocks mb

p,np

= Mbp+c,np+k •

with b > n are zero while the block Mb

b has p, p

the form 115

= k =0

.1

if

c

?

if

c =

o,

k > 0

0

if

c >

o,

k

if

c >

o,

k > 0

(mbp, bp) c, k =

l

b+c

k

=0

Therefore M is non-singular and

which is clearly non-singular. hence the vectors I: f .

l

.[X. ]X.

J

l

l

are linearly independent and hence so are the vectors

~ s .[x. ]x. J

l

since 2: f .[x.]x. i a l l

E

= 2: i

s.[X.]X. + J

l

l

2:

l

~ I II then define D = (d(1 ), d(2), ••• , d(.t)) by d( J·) = 1•(·) J - d eg 1·(·) J f or J· - 1 , 2 , ••• , ;vn •

Consider the following

element (p(yp_ 1 )P- 1 + lots)a (p(yp_ 1 )

b

.t

+lots) j~ 1 ((yp_ 1 )

where a = [ (n - I I I )/ (p-1)] and b = n Note in particular, that b ~ p-1.

II I -

d(j)

(p-1 )[ (n-

Yi(j) +lots)

II I )/ (p-1 ) ] •

This element lies in J(n,r) and

has the form pq(I) ( yp_ 1 )n-deg I yi + lots since a+b

= q(I)

and a(p-1) + b + IDI

= n-

deg I.

This completes

the proof of Theorem 6.1. THEOREM 6.2.

There exist 9 1 , 9 2, .•• ,9rp- 1 in

~[y 1 , Y2 , ••• ,yrp- 1] together with some ordering such that J(n,r)

is contained in the

z

submodule of m(n,r) with basis

Pq(I)(ep-1)n-deg I 9I + lots for IE S(n,r) with deg I~ n, where q(I) = [(n- \II+ p-2)/(p-1)]. PROOF.

From Theorem 5.5 (restatement of Theorem 5.2) we

know that if P E J(n,r) and if P is divisible by pk then

118

18.

Then FN(M) cannot fall into the following range: 122

0.

s-1 0.

dim G > m2/4 - m/2 > (m/4 + u/4) r

Then

.

By making use of Lemma 2.4, G contains a normal factor G1 which is isomorphic to one of the following groups. ( i) S p (n ) , n > (m+ u) /2 - 2 ,

(ii) SU(n), n > (m+u)/2 - 1 , (iii) Spin (n), n > (m+u) /2 •

(i) can easily be eliminated.

If

2 (m+u) - 12 < 4n - 4

~ot,

we have

= m(Sp (n) )


dim G1/H 1 >dim SU(n)/S(U(n-k) X U(k)) = 2nk- 2k 2 • By Lemma 2.4, we have dim SU(n) + dim SU(k) + 1 ~dim SU(n) +dim N(H1 ,G1 )jH1 > (m/4 + u/4) dim SU (n) /S (U (n-k) x U(k)) > ((2nk - 2k 2 )/4) (2nk - 2k 2 ) This implies that there is an inequality k4 - 2nk 3 + (n2 - 1 ) k 2 - n 2 + 1 < 0 , which is possible only when k non empty, H1

= SU(n-1).

= 1.

Since the fixed point set is

Thus

m + u - 2 ~ 2n - 1

= dim

G1/H1 ~ m - 2 - u •

It follows that u = 0 and 2n - 1 = m - 2 = r. Hence G acts transm-2 • But G :::> SU (n), hence G = SU ( (m-1) /2) or U( (m-1) /2) i ti vel y on S (cf. (b)).

Note that dim SU((m-1)/2) = m2/4 - m/2- 3/4 < m2/4- m/2.

Consequently,

G=

U((m-1)/2).

This gives the possibility

0).

We can show as in case (ii) by using Lemma 2.4 that if G1

= Spin(n),

then H1 = Spin(n-1), where H1 is the principal iso-

tropy subgroup of the action of G1 • standardly in Spin(n) by [4].

Since m ~ 20, Spin(n-1) imbedded

Hence we have possibly (e) of the

Main Lemma. '

To show that FN(CPm/2) =dim U(m/2), it is enough to show that the manifold CPm/2 cannot satisfy

~).

Otherwise, we consider

129

;s p1n · (n )•

For each x e CPm/2 /Spin (n)'

2 the projection n:CP m/2 - CPm/

· t , a 11 of which are acyclic over n -1 (x) is Sn-1 , RP n-1 or f.1xe d po1n

Q up to and including dimension n-2.

It follows from the Vietoris-

Begle mapping theorem that n*:H 2 (CPm/2/Spin(n) ;Q) Hence dim CPm/2/Spin(n) REMARK 3.1.

= m which

~

H2 (CPm/2 ;Q)

is clearly impossible.

In the Main Lemma, if we assume that

dim G > m2/4 + m/2, m > 17 , and dim G

I

(n-1), then only the possi-

bility ~) survives with n > m/2 + 2. Let us recall the definition of w(m), m a positive integer. It is defined as the largest integer j satisfying the inequality (m- j - 1) + (j - 1) < (m- j) • It is easy to show that (5)

m>(j-1)+j

j = 1 , 2, ••• , ~ (m)

,

and (6) for j

(m - j - 1) :::: (m - ~ (m) - 1) > m2/4

= 1,2, ••• ,~(m)

+

m/2 ,

if m > 17.

THEOREM 3.2.

Let G be a compact connected Lie group acting

effectively and differentiably

QQ ~

connected differentiable

m-manifold with non-empty fixed point set, m > 17. of G cannot fall into (7)

~

of the following ranges:

(m- k - 1) + (k - 1) (m- ~(m) - 1) > m2 /4 + m/2. Hence we must have possibility (e) of the Main Lemma by Remark 3.1.

130

Thus G contains a normal factor G1 = Spin(n), n > m/2 + 2.

Let G(x)

By Main Lemma, r ~ m- 2.

be a principal G-orbit of dimension r. Now

(n - 1) = dim G1 ~dim G < (m- k) • Consequently, if we let t 1 = n- 1, then t 1 ~ m- k- 1.

As

t 1 > m/2 + 1, it follows from Lemma 2.2 that tj ~ t 1 for 2 ~ j ~ v. Apply Corollary 2.3 to the action of G on G(x), we get dim G ~ (m - k - 1 )

+ (r -

(m - k - 1 ) ) ~ (m - k - 1-) + (k - 1

which is a contradiction.

>'

This complete.s~the proof of the theorem.

Now let G be a compact connected Lie group acting effectively on a topological manifold M with principal orbit of dimension t.

Then G acts transitively and

G-orbi t, say M.

a~most

effectively on a principal

The principal Tq-isotropy subgroup is trivial.

Hence the dimension of M0 = M/Tq is equal to t - q.

The group

G0 = G/Tq acts almost effectively and transitively on the compact manifold M0 •

Let V be a direct product of G.'s such that V acts 1

almost freely on M0 and V is of maximal dimension.

Following the

proof of [ 13, Theorem 1] the group G can be decomposed as (8)

G = Tq X G = Tq X H X K X V, 0

where H,K and V are each direct products of G.'s (see (2) for 1

notation) and (9) K and V act almost freely on M0 with dim K dim v = w, H =G. J1 1 dim G. ~ (t. J.1 J.1

>'

THEOREM 3.3.

G. acts almost effectively on M1 = Mofv, Jk k k, and L::. 1 t. = dim M0 - dim v = t - q - w. 1= J.1 X

X ~

~

i

~

Let G

be~

effectively and differentiably

compact connected

~~differentiable

Lie~

acting

m-manifold M with 131

non-empty fixed point set.

Suppose k. (i = 0,1, ••• ,s+1) is any l

sequence of positive integers satisfying the following condition: 18. ko = m, k.l+ 1 ::; w(k.l ) ' 0 ::; 1 ::; s, and k s > Then dim G cannot fall into the following range: (1 0)

s-1 L:i=O (k.l - ki+1 - 1) + (k s - ks+1

(11 )

PROOF.

-

1) + (ks+1

The proof will be by induction on s.

-

1)

The assertion

is certainly true when s = 0 which is precisely Theorem 3.2.

If

s > O, then we may assume by induction that the assertion is true for s- 1.

Suppose now that the dimension of G satisfies (11).

Then dim G > m2/4 + m/2 by (6).

It follows from Remark 3.1 that G

contains a normal factor G1 = Spin(n 1 ), n1 > m/2 + 2, and t 1 = n1 - 1.

Suppose dim G1

~

(k 0 - k1 ).

Then we have

s-1

L:. 0 (k.-k. 1 -1) + (k -k 1 ) l= l l+ s s+

s-1 ::; (k 0 - k1 -1 > + (L:. 1 (k. - k. 1 -1 ) + k - k 1 > l= l l+ s s+

::; dim G1 ::; dim G, which contradicts (11).

Hence dim G1 ::; (k 0 -k 1 -1), and t 1 ::; k 0 -k 1 -1.

Now we proceed to show that t 1 = k0 -k 1 -1. t 1 = k0 - k1 - u, u ::': 2.

If not,

Since t 1 > kof2 + 1, hence 2 ::; u



r + 1

l i q :s: 2r, then the last condition is reduced just to

the cocycle condition PROOF.

A

L:1i 2.

Then at most one component of F is of type Pt(p) with 2p > q. (t > 0).

Assume q > 4.

Then one component ofF is of the above

type only if F is connected. PROOF.

Let j* :

~(X) ~ ~(F)

be the homomorphism induced

from the inclusion of F in X, and let F = F1 + n. F ...... p 1 (q.) as above. Assume that n1 > 0. Then 1

"* (e ) --

J

1

••• + f 1 ® b1 + w1 +

a generator and b1 E RT).

...

+ F with s

n 1 (F ) is E H 1 + ws ' (f1

+ w2 +

By modifying the lifting e of e0 , if

necessary, we may assume that w1 = 0.

Let Ui be the submodule of

H*(F) generated by H0 (F 2 ), ••• , H0 (Fi_ 1 ), H0 (Fi+ 1 ), •.• , H0 (Fs) and

all cohomology in higher dimensions, and let U = Ui ffi H0 (Fi)' ( i=2' .•• ' s) •

= (w.) is a principal ideal, 1

in that case Theorem 4 implies that w.1 splits as a product of

215

linear factors. 2 wj E H (BT)' j=1, .•. ,m.

h 4 it follows From the proof ofT eorem

))n RT =Ann (UT H/(Ui)T H.)= wj ' J ' J where H. = w~ is the corank one subtorus of T determined by

that the localization Ann (UT/(U.)T( 1

( hj) wj ' J w.. Hence F(H.) J

2q 1 > q.

~

J

pk(2hJ.) + ..• with F1 + F 2 ; Pk(2hj). Assume J m h = q, it follows that m=1. Since 4hJ. ~ 2q 1 > q, and j=1 ~ 2 J.

·-2 , ..• , s, q= 2h. . 1 h 1Hence we may now wr1te w.1 = q.a., 1 1 We may assume n > 1, otherwise the theorem is trivial. 1

We

n

For if s = 2, F ( a 2 ) "' P ( q ) ,

claim that s = 2 is impossible.

contradicting the assumption that the action os cohomology effective. e.g. a3 is 3, and one of the a., 1 applying the same linearly independent of a2 • If also 2q 2 > q, then 1 h 1 h 1 h argument around F2 , we would also have q 2 a 2 - q3a3 = q a for some

Thus, if F is disconnected, s

a E H2 (BT).

~

h Let pi= qi fori= 2,3, ... ,s and let e 1 , ... ,eh be the

h-th roots of unity.

Then (p 2eia2 )

h

- (p 3 a 3 )

h

= n(p 2eja2 - p 3 a 3 ),

j = 1, ..• ,h)= q1 ah, by unique factorization in RT ® Q C, p 2 e 1 a 2 p3 a3 and p 2e 2 a 2 - p3 a 3 are then both complex multiples of a, which clearly contradicts linear independence of a2 and a3 if h > 1.

This

concludes the proof of the first statement of the theorem. In general, if we do not assume that 2q 2 > q, we only know . that q 12 a h2 - q 31 ah3 must spl1t as the product of the linear factors

in RT, since it generates tne annihilator ideal of a quotient module as above,

But this is possible only if h = 2 ( €2 --- -e 1 ) '

hence if q > 4 the assumption s > 1 leads to a contradiction. THEOREM 7.

Q.E.D.

Let the rank of T be at least four and q > 4.

Then the fixed point set F is either connected or consists of n+1

216

acyclic components. Suppose F = F1 + ••• + Fs with s > 1 and F1 ~ Pt(d),

PROOF. t > 0.

From the proof of Theorem 6 we have:

0 + ••• + w +

j*(e) = .•• + f 1 ® b1 +

ws ' where each of thew.l 's splits.

2

Let V be the

submodule of H*(F) generated by H*(Fi)' i ~ 2 and Hc(F 1 ), c > d; and let W-- V $ Hd (F 1 ). splits.

Then Ann (WT/V1 ) = (b1 ), hence b1 also

Let a 1 be a weight in w2 with multiplicity h1 , then we know k

1

k

that F(a 1 ) ~ P (2h 1 ) + ••• for some k and F1 + F2 ~ P (2h 1 ).

By k1 localization: Ann (WT/VT)(a1 ) n RT =Ann (WT H /VT H)= (a1 ), .. ' 1 ' 1 where k 1 is the multiplicity of a1 in we f., 1 h1 hr f3 u can now write: w2 = q 2 a 1 ••• ar, b1 u with d + 2k. = 2h., i=1, ••. , r, and the a's and f3's pair wise l

l

Here r > 1, since if r = 1: F(a~)- PK(q)+ •••

linearly independent.

which contradicts Theorem 6 unless F(a~) is connected, i.e.

F(a~) ~ Pn(q), which again contradicts the assumption that T acts cohomology effectively.

Let a. and a. be two weights in w2 and l J

let H.. be the corank two subtorus annihilated by a. and a.. lJ

l

localization:

Ann (UT/U2 )T)(a.,a.)

h

h

(ag( 1 ) g( 1 )

J

n RT =Ann (UT,H. _/(Ui)T,H .. ) =

J

lJ

J

l

Ann(WT/VT)(a.,a.) (ag( 1 )

1

g( 1)

lJ

ag(v) g(v)), where ag( 1 )' ..• , ag(v) are the a's

contained in the linear span of a. and a., Lin (a., a.).

k

By

J

l

J

Similarly:

n RT = Ann(WT,H. /VT,H .. ) =

J

••• ag(v)

k

g(v)

m lJ

f3p( 1 )

lJ

p(1)

f3p(w)

m

p(w))

, where those a's

a.). and f3's occur which are in Lin (a.' l J w v v k Now F(H .. ) ~ P (a) with a = r; 2h (m) = 2: 2k (m) + 2: 2m (m) lJ m=1 p m=1 g m=1 g + d. Here 2hg(m) = 2kg(m) + d, and v ~ 2· hence w ~ 1. Thus we have proved that for any a.l with a. with i J

t

'

j, there is at least one

217

{3

p

Le t K. J. be the corank two subtorus annihilated 1

E Lin ( a., a.J ) . 1

by ex. and {3 .• J

1

By a similar argument the localized annihilator

ideals have the same expressions as above, this time with the a's · L'1n (rv...... , {3 . ) • an d {3 ' s wh'1c h occur 1n 1

J

In the above equation for a

w ~ 1, hence it follows that v

we have this time:

~

2, and we have

~ i proved that for any pair (exi' {3J.)' there 1·s at least one ak, k T

contained in Lin (a., {3.). 1

LEMMA.

J

We can assume that the subtorus K annihilated by

(ex1, •.• , exr) is trivial (if necessary by renumbering F2 , PROOF. dimension.

... '

F )• s

We wish to prove that Lin (ex1 , ••. ,ar) has full

By localizing, it is clear that F(K) ~ Pk(q)

+ •••

for

some k, by Theorem 6 K has rank zero for one. Let L be the subtorus annihilated by all the weights in b1 , by a similar localization argument as above, all those weights are in Lin (ex1, ••• , exr).

Let H be the linear span of those weights,

if H has full dimension, we are done.

Otherwise H is a hyperplane.

If K has rank one, it then follows that H =Lin (a1 , ••• , exr). But then F(K) cannot be connected, let F. be a component ofF which J

is not included in the Pk(q) of F(K). , m1 mb Let w. = qjy1 ••• yb, then none of the yi 's are contained J in Lin (a.' 1

... '

ex ). r

On the other hand Lin (ex1, ••• , ar) = H ~Lin (y 1 , ••• , yb)' consequently Lin (yi' ••• , yb) has full dimension.

, Q.E.D.

We can now conclude the proof of theorem 7 by applying Sylvester's Theorem; S. Hansen [9]:

218

we use here a generalized version due to

Let 0 be a finite set of points which spans the

projective n-space.

Then there is a hyperplane P, spanned by 0

and a hyperplane P' in P, spanned by 0 only one point outside P'. [ ex1 J'

... '

[ex.] and [ex. J in P' ].

(0

[~k]'

nP

contains

~

4, there are at least two points

hence there is a [~k] in pI •

Let [ex ] be in m P)-P', then there is another [op] on the line through [ex J and m J

n

such that 0

In our case, let 0 be given by

Then, since rk T

[ exr].

n P'

n P,

'

in contradiction to the above theorem.

REMARK.

Q.E.D.

Sylvester's theorem was first applied to the

cohomology theory of transformation groups by Skjelbred [16], who If T acts cohomology effectively on a Poincare duality

proved:

space X such that dim Hl* (X) = dim

H~~

1 (F), F has two components F and

F2 such that the restriction homomorphism H~~ (X ) ~ H* ( F1 ) is onto, and dim F1

f

dim F2, then rk T < 4.

DEFINITION.

The action ofT on x~ Pn(q) is of spherical

type if either: 1)

the fixed point set F is connected, or

2)

F is the disjoint union of acyclic components, and the local

geometric weight systems are identical around each component Obviously these are the actions with the simplest cohomological orbit structure.

In case 1) we have F- Pn(h), j*(e)

=

+

f ® b, where f is the cohomology generator of F and b splits, i.e. h

h

r q' E Q, wi E H2 ( BT. ) b = q'w 11 ••• wr, are given by F(~)- Pn(h + 2hi).

The corank one F0 -varieties

In case 2) it is clear that all

].

Hence we have j*(e) = q 1n + h h1 r , with qi E Q, i= 1, ••• ,n+1 and b = q'w1 wr ' q E Q, F 0 ~varieties are connected.

+ ••• +

219

DEFINITION.

For an action of spherical type the geometric

weight system is the above cohomology class b. The geometric weight system completely determines the whole cohomological orbit structure of the action, in fact the following theorem is completely analogous to the case of a cohomology sphere (Hsiang [ 12]). h1 h THEOREM 8. Let b = q ' w1 ••• wr r be the geometric weight of a torus action on X""' Pn(q) of spherical type with F""' Pn(h), (if. h = 0, F is the disjoint union of n + 1 acyclic components).

Then a

subtorus H is the connected component of an isotropy if and only if His of the form (wi( 1 ), ••• ,wi(k))l, and then F(H)= Pn(h + 2hi( 1 ) + ••• + 2hi(k) + 2hi(k+ 1 ) +

+ 2hi(k+m)), where

wi( 1 ), ••• , wi(k+m) are the weights which are in Lin(wi( 1 ), ••• ,wi(k)). Our main result for torus actions on cohomology projective spaces now follows quickly from Theorem 7. COROLLARY 1.

Let I be a torus group of rank at least 5 which

act cohomology effectively on X""' Pn(q), q > 4.

Then the action

is of spherical type. PROOF. +

Suppose F is not connected.

Fn+ 1 with each F.1 acyclic.

weight around F1•

j_

By Theorem 7, F = F1 +

Let w.1 be a local geometric

Then H. = w. has rank at least four. l

l

Since F(H.)

has at least one component which is not acyclic, it follows again by Theorem 7 that F(Hi) is connected, hence wi is a local 'weight with the same multiplicity around all the componen t s F , ••• , F • 1 n+1

220

l

3.

ORBIT STRUCTURE FOR ACTIONS OF COMPACT CONNECTED LIE GROUPS ON COHOMOLOGY PROJECTIVE SPACES. ' By constructing James' reduced products of spheres

equivariantly, we obtain examples of actions of spherical type

X~ Pn(q) with q even and n arbitrary.

Let A be a Hausdorff space

with base point a, and let An be the set of sequences in A- {a} with no more than n terms.

There 1s a projection p

n

from An to A

n

which excludes those terms of the n-tuple which equal a; and we give An the identification topology from the product An.

If a group G

acts on A with a as a fixed point, then A is a G-space in an n

obvious way. Recall that H*(A00 ) is isomorphic to the cohomology of the loop space of the suspension of A (James q even, we have A ~ Pn(q). n

[14], [15]).

If A = Sq

'

Thus, if G is a torus which acts

linearly on Sq, the equivariant reduced product construction (with one of the fixed points chosen as base point) gives an action on (Sq)

n

~ Pn(q) of

spherical type.

Obviously, this is of case 2)

if and only if the given action on Sq has two isolated fixed points. The geometric weight system defined in the last section, coincides with the usual geometric weight system defined by Hsiang [12] for the original action on Sq.

Thus it follows from Corollary 1 that

for actions of tori of rank at least five on higher cohomology projective spaces, there is perfect correspondence between theory and examples. It is also clear that once we have defined the geometric weight system as above, the more detailed study of the cohomological orbit structure of such actions of compact, connected Lie groups of

221

rank at least five on X~ Pn(q) can be reduced to the corresponding problem for cohomology spheres.

A systematic study of this case

has been carried out by Hsiang [12].

The main idea is to restrict

the action to the maximal torus and use Weyl invariance of the geometric weight system.

We only sketch a few of the results which

are obtained when those methods are applied to our case. Let T be the maximal torus of a compact, connected Lie group of rank at least five, and assume that G acts cohomology effectively on

X~

Pn(q), q > 4.

observations:

We have the following straightforward

RT is a W-module and RG

acts on the fixed point set F(T).

= RTw•

The Weyl group W

If x E F(T) and w E W, the system

of local geometric weights around wx are obtained from those around h1 h x by conjugation with w. On the other hand, if b = q'w 1 ••• wrr is the geometric weight system, those systems are both given by {(w1 ;h1 ), ••• , (wr;hr)}. THEOREM 9.

Hence this is Weyl group invariant.

Let G

= SU(k),

Qrr X~ Pn(q), with k(k-1) > q > 4.

k > 5, act cohomology effectively Then all orbits are finitely

covered by complex Stiefel manifolds. PROOF.

We must prove that the connected component G0 of any X

isotropy group is a standardly embedded SU(k-p) for some p. +

bk

=0

be the usual coordinates on T,

then W is the symmetric group on (b1, ••• , bk. ) nkbk with ni integers. of b occurs if w

= bi

Let

Let b

= n 1b 1 +

••• +

It is easy to see that the shortest'W-orbit for some i and the second

shortest if

w = b.l + b., J the length of the 1 atter (modulo sign) is k(k-1 )/2. Hence, by the dimension restrictions of the Theorem, it is clear that only the shortest orbit can occur in the geometric weight

222

system, which must then coincide with the weight system for a direct sum of copies of the standard representation of SU(k) and a trivial representation; i.e. the geometric weights are given by

= {(b1 ;p), ••• , (bk;p)}

B

for some p.

When choosing a suitable

point x on an arbitrary orbit of X, one may assume that the maximal torus T1 of G~ is included in T; i.e. there exist weights bi( 1 ), ••• , bi(p) such that T1 0

that TX

= T~ = (bi( 1 ), ••• ,b(p))~, ~

= (b1 , ••• ,bp )

•

one may as well assume

Let D(G) be the weight system of the adjoint

representation of G and D(G~T 1 the restriction of this to T1 • action of G along the orbit G/G X

X

has

.·

w~ight

The 0

system D(G)!T 1 -D(Gx);

hence D(G~) ~ D(G)!T 1 - B!T 1 •

Direct substitution then gives

D(SU(k-p)) c D(G~) c D(G)!T 1 •

A Lie algebra computation then gives

G0X

""'

SU(k-p). REMARK.

groups; e.g.:

Q.E.D. There are analogous theorems for other classical If G

= SO(k),

k ~ 13 and k(k-3)/2 > q > 4, then all

orbits are finitely covered by real Stiefel manifolds. G

= SP(k),

If

k ~ 6 and k(k-1 )/2 > q > 4, all orbits are finitely

covered by quaternionic Stiefel manifolds.

For more details and

classification of principal orbit types, we refer to Hsiang [12]. This also contains a discussion of the important case when the geometric weight system is modelled on the adjoint action, and the Weyl group acts as a group generated by topological reflections on F (T).

223

REFERENCES

:.

c.

Allday and T. Skjelbred, The Borel formula and the

topological splitting principle for torus action on a Poincar~ duality space.

2.

.

N Bourbaki

'

Elements of Mathematics.

Hermann, Paris. 3.

A. Borel, et al.

G. Bredon,

Seminar on Transformation Groups.

Princeton University Press 1960.

Academic Press 1972.

T. Chang and M. Comenetz, Group Actions on Cohomology Projective Planes and Products of Spheres.

6.

Bull. Amer. Math. Soc. 78 (1972) 1024-1026.

T. Chang and T. Skjelbred, The topological Schur lemma and related results.

8.

Preprint.

T. Chang and T. Skjelbred, Group Actions on Poincar~ Duality Spaces.

7.

Ann.

Introduction to Compact Transformation Groups.

New York-London. 5.

Commutative Algebra.

1972.

of Math. Studies 46. 4.

Ann. of Math. 100 (1974) 322-326.

Ann. of Math. 100 (1974) 307-321.

T. Chang and T. Skjelbred, Lie Group actions on a Cayley projective plane and a note on orientable homogeneous spaces of prime Euler characteristic.

To appear in Amer. J. of

Math. 9.

S. Hansen, A generalization of a theorem of Sylvester on the Lines Determined by a finite point set.

Mathematica

Scandinavica 16 (1965), 175-180. 10.

W. Y. Hsiang, On characteristic classes and the topological Schur lemma from the topological transformation groups viewpoint.

224

Proc. Symposia in Pure Math. XVII

105-115 (1971 ).

11.

W. Y. Hsiang, On some fundamental theorems in cohomology theory of topological transformation groups.

Taita J. Math.

2, (1970)' 61-87. 12.

W. Y. Hsiang, On the splitting principle and the geometric weight system of topological transformation groups I. Proc. 2nd Conf. on Compact Transf. Groups. 1971.

Lecture Notes in Math. 298, 334-402.

Heidelberg-New York. 13.

Amherst. Mass. Berlin-

Springer 1972.

W. Y. Hsiang and J, C. Su, On the geometric weight system of topological actions on cohomology quaternionic projective spaces.

14.

Invent. Math. 28 (1975), 107-127.

I, M. James, Reduced Product Spaces.

Ann,of Math. 62 (1975),

170-197. 15.

I. M. James, The Suspension Triad of a Sphere.

Ann, of Math.

63, (1956), 407-429. 16.

T. Skjelbred, Torus actions on manifolds and affine dependence relations.

Preprint.

IHES, Bures-sur-Yvette.

1975. 17.

P. Tomter, On the geometric weight system for transformation groups on cohomology product of spheres. report,

18.

Universitgt Bonn.

Prliminary

1972.

P, Tomter, Transformation groups on cohomology product of spheres.

Invent, Math. 23 (1974), 79-88.

UNIVERSITY OF OSLO, OSLO 3, NORWAY.

225

ON THE EXISTENCE OF GROUP ACTIONS ON CERTAIN MANIFOLDS STEVEN H. WEINTRAUB

In this note, we show how a relatively straight-forward application of surgery theory yields existence and non-existence theorems for certain kinds of group action,

The most extreme

results in either direction may be summarized as follows: THEOREM,

Let M be a simply-connected manifold whose hom-

ology is torsion-free and concentrated in even dimensions, a)

Then

For any odd p, if M admits a semi-free Z/p action with

isolated fixed points, so does any PL-manifold homotopy equivalent to M b)

If M admits a semi-free locally smooth

s1

action, and

M', _a manifold homotopy equivalent toM, also admits one with the same fixed-point data, then there are only finitely many possibilities for the PL homeomorphism type of M'.

("Fixed-point data"

is defined in Definition 1.3). This argument was developed for highly-connected M in [8]. Among the other manifolds to which it applies are products of spheres, and, more interestingly, to projective spaces (complex and quaternionic).

For these latter cases, in addition to the fact

that they admit interesting actions, there is a wealth of information known about their fixed-point data (see [2, Chapter VII]). In these cases we may apply an idea of Hsiang [3] to conclude

226

THEOREM.

Let M

= CP 2n

smoothly on M with dim(MZ/p)

or HPn, n > p-1, and let Z/p act

~

4n - 2p.

Then infinitely many

manifolds, homotopy equivalent to M, admit such an action with the same fixed-point data.

1.

GENERAL RESULTS Let G be a cyclic group of odd order.

Throughout this

section M2 n will denote a simply-connected manifold whose homology is concentrated in even dimensions and has all of its torsion prime to the order of G.

M will be the total space of a semi-free G-

action with fixed-point set F. neighbourhood of F, let T of T.

If N(F) is an equivariant tubular

=M -

N(F) and let X be the orbit space

We also suppose that T is simply-connected (which will be

the case when codim(F) > 2).

Let n : T

~X

be the projection.

Recall from [5] that in this situation we have the following commutative diagram of exact sequences: k

L2n-1 (G,

! L2n-1 ({ 1 },

u

i=1 k

u

T1

G)

.... hT(X) ~ [X,G/PL] ~ n*

~ ni~ T1

{1}) ... hT(T) _.T [T,G/PL]

i=1

...

...

k

L2n (G,

t L2n ({ 1},

u

G)

i=1 (I) k

u

{ 1} )

i=1

jr hT(M) (As usual, the "G" in "G/PL" has nothing to do with the group G that is acting.) Here the union is over the boundary components of T.

hT(T)

denotes the homotopy triangulations of T where the boundary is

227

The map from hT(M) to hT(T) is that induced from

allowed to vary.

deleting Int(N(F)).

k

k-1

Also, note that L*(G, U G) ,.. EB L~~- 1 (G), and L2 k_ 1 (G)= 0 i=1 i=1 if G has odd order (by [1]), and similarly for {1}.

Also,

L*(G) ~ L*({1}) is onto. Now, if M' E hT(M) admits a G-action with fixed point set F' such that M'- F' is homotopy-equivalent toM- F, then M'- N(F') = n~~((M' - N(F')/G)) in hT(T), i.e., M' E Im(r- 1n* : hT(X) ~ hT(M)).

Thus, our first goal will be to

determine Im(r- 1 ni~). LEMMA 1.1. an isomorphism.

[(X, oX), G/PL] ® Q ~ [(T, aT), G/PL] ® Q is

Also, if F is a union of isolated points,

[X,G/PL] ~ [T,G/PL] is onto while [(T, oT), G/PL]/Im([(X, oX), G/PL])= Gx(M)- 1 • PROOF.

Since (X,oX) is rationally equivalent to (T,oT)

(the latter being a finite cover of the former) the first part is immediate.

Similarly, [X,G/PL] (q) = [T,G/PL](q) for any q prime

to the order of G. Thus to prove the second statement, we need only consider the situation at a prime q dividing the order of G.

Now

[X,G/PL](q) = K0°(X)(q)' [T,G/PL](q) = K0°(T)(q)' and KO may be computed from the Atiyah-Hirzebruch spectral sequence, which collapses for both for dimensional reasons (since except ~n the top dimension their cohomology is concentrated in even dimensions). Hence, it suffices to show that Heven(X : z(q)) ~ Heven(T : z(q)) is onto.

228

T even (X : Z(q))' we use the spectral sequence of o compute H the cover.

Now, as T is homotopy equivalent to M - a finite set of

points (in fact, x(M)

H~*(T) Ep, 2 q

=

~

i ~ 2n

dim H. (M) points, by Smith theory), 1

= W~(M) for~~~ 2n-2, and H2n-1 (I) = zX(M). In particular, = 0 i f e1" ther p or q lS · od d, except for q = 2n-1, so E0 ' q 2

survives to E00 for q < 2n-1, so

H~'(X) ~

H*(T) is onto for*< 2n-1.

Then, by considering the cohomology ladder of the pairs (X, oX) and

(I, oT), i t is easy to see that H~~(T, oT)/Im(H~'(X, oX)) = Gx(M)- 1 and the last statement follows. THEOREM 1.2. of isolated points.

Let M be as above, and suppose F is a union Then, any PL-manifold homotopy equivalent to

M also admits a semi-free G-action with isolated fixed-point set. PROOF.

By 1.1, n~~ : [X, G/PL] ~ [I, G/PL] is onto.

chasing (I), i t follows that

11:~~ :

hT(X)-+ hT(T) is onto.

By

But

r : hT(M) -+ hT(T) is an isomorphism, since T is obtained from M by deleting the interiors of disks, while M is obtained from T by coning off the boundary spheres (a well-defined process in the PL category) and these constructions are inverses of each other. DEFINITION 1.3.

Let G act semi-freely on M.

The fixed-

point data of the action consists of the equivariant PL homeomorphism class of N(F) and the homotopy type of the pair THEOREM 1. 4.

Suppose H4 i (M : Q)

f

(I, oT).

0 for some i with

0 < 4i < dim M, and M admits a G-action with given fixed-point data.

Then infinitely many manifolds of the homotopy type of M

do not admit semi-free G-actions with the given fixed-point data. PROOF.

The condition on

H~'

(M) insures that there are

229

infinitely many manifolds of the homotopy type of M. Suppose first that F is a union of isolated points. To investigate actions with the same fixed-point data as the given action on M, we must fix oX (and hence aT) and so must consider the analogue of diagram (I) where X (I) is replaced by

(X, oX) (and (I, aT), respectively) and the Wall groups are the Then n* : [(X, oX), G/PL]- [ (T, aT), G/PL]

absolute Wall groups. is not onto, so

ni~ :

hi (X) .... hi (I) cannot be either, so neither

can r -1 ni~. If F is more complicated, Im(r

-1

n*) can only get smaller

as r -1 may fail to be defined.

COROLLARY 1.5. action.

Suppose M, as above, admits a semi-free

s1

Then, if M' homotopy equivalent to M admits one with the

same fixed-point data, there are only finitely many possibilities f or th e h omo t opy t ype o f M'. If, in addition, H4 i+ 2 (M

Z/2)

=0

for all i with

4i + 2 < dim(M), and H4 i(M) has no 2-torsion, M' must be PL homeomorphic to M.

PROOF.

s1 contains every cyclic group,

so the image of

[(X, oX), G/PL] in [(I, aT), G/PL] may be made arbitrarily large by choosing X

= M/G

for G sufficiently large.

In fact, not only is

the cokernel arbitrarily large but it must be isomorphic to Gx (M) -1 so no element of infinite order can be in all the cokernels.

Hence, any element in all the cokernels must be a 2-torsion element. This proves the first claim, and the second follows since in that case there are no 2-torsion elements.

230

2.

PROJECTIVE SPACES Among the manifolds to which 1.5 applies are complex and

quaternionic projective spaces.

In this section, we show that,

as opposed to the situation of circle actions, infinitely many smooth manifolds of the homotopy type of cp 2 n or HPn, n > 2, admit Z/p actions.

If M = CP 2n or HPn, recall that Hsiang showed

in [3] that there is a subgroup of finite index in KO(M) such that any S in this subgroup satisfies the conditions: s is fibre-homotopically trivial pi (s) =

o

for

o
p-1, semi-freely and smoothly and

2n

n or HP ,

dim(~/p) ~ 4n - 2p. Then

infinitely many manifolds of the homotopy type of M admit a Z/p action with the same fixed-point data. PROOF.

Let m =max (2n, 4n - 2p).

Then

S = S n KO(M,

is a free abelian group of rank greater than (p-1)/2.

Also,

KO(T, oT) .... S is onto, and KO(X, oX) ® Q .... KO(T, oT) ® Q is an isomorphism since T is a finite cover, so KO(X, aX) is onto a 231

Mm)

· d ex -S o f S • subgroup of finite 1n

Hence [(X, oX)/(X, oX)m, G/0]

contains an abelian group of rank greater than (p-1 )/2

= rank

(L 4 n_ 1 (Z/p)), and as it is easy to see the surgery obstruction map

is a homomorphism on this subgroup of [(X, oX), G/0] it must have infinite kernel.

Then this kernel gives rise to infinitely

many homotopy smoothings of (X, oX) and hence to actions on infinitely manifolds of the homotopy type of M.

REMARK.

Our proof in this section could have been

rephrased to look more like section 1.

For finding a fibre homo-

topically trivial bundle is the same as finding a map of

M

into

BO which is homotopically trivial when mapped into BG, so pulls back to a map of

M

into G/0.

surgery obstructions on X and smoothings of

3.

M

Then our conditions ensure that the M

vanish so we obtain homotopy

covering those of X.

CONCLUDING REMARKS Similar questions to the ones we consider here have been

investigated by other people.

Petrie has conjectured [4] that if

a homotopy CPn admits a smooth

s1 -action

must preserve the ~

the homotopy equivalence

characteristic class.

if the fixed point set of an

s1

Wang [6) proved that

action on a homotopy CPn, n > 3,

is homotopy equivalent to CPk U CP£

'

the underlying manifold must

be, in fact, tangentially homotopy equivalent to CPn, and announced that there are infinitely many such distinct Z/p actions (on distinct homotopy CPn's). Also, Petrie constructed a v er Y 1n · t eres t"1ng s1 action ( in [4]) on CP 3 where the representations at the fixed-points do not

232

agree with those of any linear action.

In light of the condition

in 2.1, we ask if there can arise actions on homotopy CPn's with fixed-point data not agreeing with those of any action on the standard CPn.

(If not, that would show many homotopy CPn's admit

no smooth Z/p actions.) Finally, in [7], we considered in detail the question of the existence of smooth Z/p actions on homology HP 2 's. We conclude with an interesting number-theoretic question. Consider the case of actions with isolated fixed points, so the fixed-point data reduces to the normal tepresentations at the fixed-points.

For actions on spheres, with two fixed points,

Atiyah-Bott showed the normal representations at the two fixed points must agree (except for sign), i.e., that they must be the normal representations of a linear action.

On the other hand,

Petrie's example on CP 3 with four fixed points, is distinguished from any linear action by the fact that the normal representations at the fixed points differ from those of any linear action.

In

[7], we found a collection of representations that could algebraically arise from an action on HP 2 , with three fixed points, for the prime 11, which are distinct from those of any linear action, but such do not exist for smaller primes.

Thus, at the

moment, we have an anomaly, and we ask in general what is the situation for the case of three fixed points.

233

References 1.

A. Bak, The Computation of Surgery Groups of Odd Torsion Groups.

2.

Bull. Amer. Math. Soc. 80 (1974), 1113-1116.

G. Bredon, Introduction to Compact Transformation Groups. Academic Press, New York, 1972.

3.

W. C. Hsiang, A note on free differentiable actions of and

s3

on homotopy spheres.

s1

Ann. of Math. 83 (1966), 266-

272. 4.

T. Petrie, Smooth

s1

actions on homotopy complex projective

spaces and related topics.

Bull. Amer. Math. Soc. 78 (1972),

105-153. 5.

C. T.

c.

Wall, Surgery on Compact Manifolds, Acadmic Press,

New York, 1970. 6.

K. Wang, Differentiable Circle Group Actions on Homotopy Complex Projective Spaces.

7.

Math. Ann. 214 (1975), 73-80.

S. Weintraub, Group Actions on Homology Quaternionic Projective Planes, to appear.

B.

S. Weintraub, Semi-free Z/p actions on Highly-Connected Manifolds, Math.

z.

LOUISIANA STATE UNIVERSITY BATON ROUGE, .LA 70803

u.s.A.

234

145 (1975), 163-185.

PART TWO

(Summaries and Surveys)

PROPER TRANSFORMATION GROUPS HERBERT ABELS

This is a survey. torical remarks.

No proofs will be given.

Also no his-

Both can be found in the references cited at the

respective places.

The talk has three parts:

I Definition, examples,

II Results, III H* , problems. c

I.

DEFINITIONS, EXAMPLES Our standing assumptions will be:

G is a locally compact

Hausdorff topological group, X a locally compact, paracompact, locally connected, connected Hausdorff space.

We look at continuous

actions of G on X, G X X ~ X sending (g,x) to g • x. action is g • x active actions:

=x

for every x

One very lazy.

E X and every g E G.

I consider

let g. be a net in G, divergent in G, i.e. every 1

compact subset of G contains only a finite number of points of the net.

Then g.x diverges in X. 1

every compact subset K of X.

We need a bit more: g.K diverges for 1

This is equivalent to the following

property: DEFINITION.

The action of G on X is proper if and only if

for every pair K, L of compact subsets of X the set

f g E G;

gK n Lf

has compact closure in G.

237

¢J

Equivalently:

for every pair x,y of points in X there are

· hb our h oo d s Ux' Uy of x andy respectively such that fg ne1g gU

X

E G;

n Uy i ¢j has compact closure. EXAMPLES. E 0)

G compact.

Every action is proper.

One goal of the

theory is to reduce the study of proper actions of a group to that of its compact subgroups (cf. following it). E 1)

Theorem 5 below and the remarks

Henceforth we assume that G is not compact. Let

X~

Y be a covering space and let G be the group

of decktransform3tions.

G endowed with the discrete topology acts

proper 1y on X. E 2)

Let X be a metric space.

The group G of isometries

endowed with the compact open topology acts properly on X. examples are

Sub-

let X be a Riemannian manifold, G its group of iso-

metries, or let X be a bounded domain in en (more generally in a Stein manifold), G its group of biholomorphisms.

In each of these

cases G acts properly on X, when endowed with the compact open topology. E 3)

Let G

locally connected). action is proper.

=X

(this time X need not be connected nor

G acts on X by left translations. This example can be varied:

subgroup of G, K a compact subgroup of G.

This

let H be a closed

The action of H on G/K

induced by left translations is proper. E 4) generators.

Let G be a discrete group with a finite set E of Copsider the Cayley diagram C(G,E), i.e. the graph

whose vertices are the elements of G and whose edges are the pairs

238

(g,gh), g E G, hE E U E- 1 •

The geometric realization X of

C~,E)

is a proper G-space under the action induced by left translation. There are some questions arising naturally Q 1:

Given X.

compact) G on X?

Is there a proper action of some (non

If so, X is not compact, trivially.

A necessary

condition on X is given by Theorem 1 below. Q 2:

Given G and X.

If yes, classify them.

Is there a proper action of G on X?

An answer to this question is given in

Theorem 5 and the discussion following it for G almost connected.

Q 3:

Any such gives a necessary condition for yes to Q 2.

G and X?

II.

Let G act properly on X. rAre there relations between

RESULTS Some of the nicest results are obtained in terms of the

ends of X. A compactification Y of X is called 0-dimensional if y,x has (Menger-)dimension zero, equivalently if Y'X is totally disconnected. The endpoint compactification (or Freudenthal compactification) X+ of X is a 0-dimensional compactification of X such that 1)

If Y is a 0-dimensional compactification of X, there is a

. cont1nuous map X+ - Y rna k ing t h e d.1agram

X+

y

'~X/ co~utative.

239

Or equi va 1 entl y: 2)

X+'X does not disconnect X+ locally, i.e. given a neighbour-

. + hood V 1n X of a point y

open subsets u1' u2 of v

E X+'X there do not exist two disjoint

nX

such that u1

u

u2 = v

n X and

(E.g. if y has a neighbourhood base of V's such that V X+ exists and is essentially unique.

nected).

We

y EU1

n X is

n u2.

con-

ca 11 E (X) = X+'X

the space of ends of X and e(X) = #E(X) the number of ends of X. EXAMPLES. R+ = (0, 1) + = [ 0, 1 ] by 2) •

E 1)

E 2)

(Rn) + = Sn, n > 1.

So e (R) = 2.

So e (Rn) = 1 for n > 1.

Rn is a

group manifold so a proper Rn-space, n? 1. E 3)

More generally:

let M be a compact co~nected manifold

without boundary of dimension> 1, let E be a finite subset of M or a convergent sequence of points plus limit point. +

we have M = X •

Then for X= M'E

E. g. C'(Z +iz ) has infinitely many ends and is a

proper Z + iZ -space. In the examples we have spaces X admitting proper actions of a non-compact group with e(X)

= 1,2,oo.

For all our theorems the hypothesis will be: HYPOTHESIS.

Let X be a proper G-space for a non-compact

group G. THEOREM 1 [1].

e(X)

~

e(X)

= 1,2

QE

If

G is connected,

2.

Now we turn to the question: cases e(X) because:

240

=.

= 1,2 if X •

and= resp?

what are the groups G in the

In case e(X)

=1

nothing can be said

1s a proper G-space, so is X X X and e(X x X)= 1.

For e(X) = 2 there is THEOREM 2 [1].

Then the orbit space G~

Suppose e(X) = 2.

is compact and G contains

.§.

cocompact discrete

subgroup

The structure of these groups G is known:

::5

z.

G contains a

compact normal subgroup K such that G/K is an extension of R or Z by a group of

~

2 elements.

If e(X) = ~, we cannot expect much in general:

let G be a

discrete infinite group acting properly on a manifold X of dimension

> 1.

Then X minus the orbit of one point is a proper G-space Y with _.c

e(Y) = oo, byE 3.

points:

So we have to pay special attention to the limit

let X be a proper G-space.

A point y E X+'X is called a

limit point of G, if y is in the closure of some orbit Gx, x E X. Any limit point is in the closure of any orbit Gx, x E X.

Let L(G)

be the set of limit points. One can show that

= X+,L(G) is a proper G-space and

So again: #L(G) ~ 2 or infinite, etc.

X+= X+. 1

THEOREM 3 [2]. (a)

x1

.§_

Suppose #L(G) is infinite.

free product G1 *K G2 amalgamated ~

.§.

Then G is compact open

subgroup K of G1 and G2 , K I G1 , K I G2 , QE (b)

an HNN-extension G = G1 *a , where K is compact open sub-

group of G1 , a is and K I G1

.2.!:

.§_!}

.QQ_en continuous injective homomorphism a: K --G1,

a(K) I G1 •

The~ G

= G1

*a has the generators and

relations (G 1 ,x; x k x- 1 = a(k) fork E K). This theorem is the version for topological groups of a theorem of Stallings [11]. Under the hypotheses of Theorem 3, L(G) is a minimal G-space (i.e. the closure of any G-orbit in L(G) is L(G)) except if (b) occurs 241

In this exceptional case there is a fixed point in L(G) and any other orbit is dense. The exceptional case cannot occur if G is locally connected (see [4]). I mention in passing that there are relations between ends of spaces and ends of groups ([7,8,9,10,2]). It would be useful if one could define higher-dimensional analogues of the end-point compactification. Returning to the examples at the beginning of this section n

II, we have seen that - with the notations of E3 - for M = S are E with 1 action.

~

#E

~

2 or #E =

oo

there

such that X= M'E admits a proper

What about other manifolds M = X+? THEOREM 4.

If X+ is an n-manifold without boundary, X+ is

homeomorphic to sn. The theorem proved in [6] actually says:

if one limit point

E X+has a neighbourhood homeomorphic to an n-ball, then X+~ Sn.

y

For the next result we need the following fact. group G almost connected if the factor group G/G

0

We call a

is compact, where

G is the connected component of the neutral element. 0

FACT.

An almost connected group G contains a maximal com-

pact subgroup K. group of K.

Any compact subgroup of G is conjugate to a sub-

The space G/K is homeomorphic to a euclidean space.

The dimension of this euclidean space is unique, because any two maximal compact subgroups of G are conjugate. non-compact dimension of G: n (G). c

242

We call it the

Let X be ~proper G-space, G almost connected.

THEOREM 5 [3]. Let K be

maximal compact subgroup of G.

~

Then there is

~

G-map

f :X __. G/K.

It is easy to see that this map turns X into a locally trivial fibre bundle over G/K with fibre S group K.

= f- 1 (K/K)

and structure

Since G/K is contractible this fibre bundle is trivial. So n (G)

x~sxRc

• n (G)

So a necessary condition for G acting properly on X is that R c is a direct factor of X. actually sufficient.

It is easy to see that this condition is

More precisely:

let E(G;X) be the set of

equivalence classes of proper actions of G on X, two actions being equivalent if there is a G-homeomorphism between them. E (G; X) = U E (K; S) the disjoint union being taken over all spaces one such out of every class of homoeomorphic S.

s

Then

n (G) such that x~ S X R c , This formula reduces

the problem of determining all proper actions of G on X to two prone (G) blems: (1) determine all S such that X~ S X R up to homeomorphism, (2) determine the equivalence classes of actions of a maximal compact subgroup K on S.

III.

H* , PROBLEMS. c

By H* (. ;k) we mean sheaf theoretic cohomology with coefc

ficients in the constant sheaf with stalks a field k and compact supports.

Let us define n c (X;k)

= inf{i;Hi(X;k) i 01, c

n c (X;k)

= oo

if H*(X;k) = 0. We consider n (X;k) as the non-compact dimension c c of x. This is in keeping with our earlier notation: nc~) = nc(G;k) 243

for every k, if G is almost connected. More generally: n (G) x a! s x JR c and the K~nneth theorem imply: COROLLARY.

the formula

Hypotheses and notations ~ in Theorem 5.

Then

and n

c

(X;k) = n (G) c

compact. Note the qualitative implication:

given two proper actions

of such G on X then either both orbit spaces are compact or both are non-compact. This raises the question whether the corresponding statement for arbitrary G and X is true.

It is easily seen to be not true in

general. COUNTEREXAMPLE:

inject the free group on two generators into

itself i : G- G such that i(G) is of infinite index in G.

Then in

I Example 4 the orbit space under the natural action of G on X is compact whereas the proper action (g,x) - i(g)x has non-compact orbit space. There is the following interesting result: THEOREM 6 (H.C. Wang [12]).

Let X be~ differentiable

manifold with finite dimensional cohomology algebra H*(X;k) with coefficients

in~

finite field k.

Given~~

entiable, proper and effective actions of G on X. orbit

spaces~

G and two differThen either both.

compact or both are non-compact.

Note that the hypotheses imply that G is actually a Lie group.

244

The corresponding theorem for differentiable manifolds with boundary is false.

A counterexample can be constructed starting

with the above counterexample G,X. neighbo~rhood of X.

3

Embed X in R , take Y a regular

Extend the action of G to Y.

Y is contractible.

But the conclusion of the above theorem holds if X is a differentiable manifold with boundary and dimkH~(X;k) < ~ fork a finite field (see below).

This assumption is equivalent to Wang's for

orientable differentiable manifolds by CONJECTURE. finite field. on X.

Poincar~

Let X be a space with

Given two proper and

duality.

dimkH~(X;k)

effec~ive


0

X. by 1

Here we do not assume the X. 's are 1

If (G1 ,X 1 ), for example, has an arc of fixed points

and for i > 1, (Gi ,\) has a single fixed point, then (G,Q) has an arc of fixed points. (iv)

If G is a compact Lie group, then, as a space, G is

an ANR and cone (G) is a compact AR. (G) by left translation on levels.

G acts canonically on cone In this case (G, cone (G)) has

exactly the cone point, p, as a fixed point and cone (G)' rpJ consists of a half-open interval of invariant copies of G.

Combining

this procedure with that of (ii) above we get any compact Lie group acting on Q semi-freely with a single fixed point; in other words every orbit other than the fixed point is·a full copy of G. (v)

Any compact group which is an inverse limit of Lie

groups (e.g. a Cantor or solenoidal group) can act on Q with a single fixed point.

We write G = 1 im G.• -

For g

= (g.) l

E lim G. -

l

define (G,Q) by gx = (g.x.) > 0 cone (G.), l l l where g. acts on x. as in (iv) above. Under this action it is not l l and for x

=

l

(x.) Err. l

l

in general true that each orbit other than that of the fixed point is full.

For example any orbit in the cone(G 1 ) is reproduced in

the product by crossing such orbit with the fixed (cone) points of the other factors.

The fact that Cantor and solenoidal groups are

easily seen to operate effectively on Q ( and thus on any Q-manifold M regarded asK x Q with the identity action on K) makes the infinite-dimensional case dramatically different from that of finitedimensional manifolds. (vi)

The following example is due to Chapman.

be regarded as cone(Q) with vertex p. induces group G acting on cone (Q)

= Q'

Let Q'

Then any group G action on Q by G acting on levels of the

253

cone with each fixed point of Q under G generating an arc of fixed points in cone (Q).

If (G, Q) is semi -free with a single fixed point,

then (G,Q') is semi-free with an arc a of fixed points.

Now the

arc a is a 2-set in cone Q and it is known that if such an arc a is shrunk to a point with all other points preserved, then the resultant space Q* is homeomorphic to Q' (and to Q) and the induced action gives G acting on Q* semi-freely with a single fixed point.

Each

level of the cone Q above p produces an invariant copy of Q under (G,Q*).

Deleting the fixed point p* we have both Q*' p* and the

orbit space ofQ*'P* admitting half open interval factors.

As

noted below in Section 4, this will let us show the equivalence of appropriate finite group actions. In general, the process of shrinking 2-set arcs or other 2-sets of trivial shape in Q has produced many examples and theorems in infinite dimensional topology.

Indeed, Edward's results referred

to above as characterizing Q-factors and Q-manifold factors employ another form of the shrinking of 2-sets of trivial shape.

3.

SOME EXPLICIT EXAMPLES OF 2/3 ACTIONS We consider various 2/3 actions on Q by way of illustration.

(a) From (iv) above on Lie gro~p actions we have 2/3 acting o~ Q with a single fixed point with Q regarded as a product of triods (letters 'Y').

The action rotates each factor about its branch point.

(b) From (ii) above we may cons1"der Q as a product of discs with 2/3 rotating each disc about its centre.

(c)

From (ii) above we may

consider Q as a product of 6-ods w1"th 2/3 rotat1ng · each 6-od about 254

its branch point.

Each full orbit hits just 3 branches.

(d)

For

any ANR, X, which admits a free Z/3 action we can cone X to get an AR and a semi-free Z/3 action with a single fixed point, the cone point, as in (iv) above.

Any countable infinite product of such

AR's is Q and by (iii) we can induce a semi-free Z/3 action on Q with a single fixed point.

(e)

For any of the actions a) to d) we

can cone the action as in (vi) above, shrink the arc of fixed points to a point and get a different appearing Z/3 action on Q with a half-open interval of invariant copies of Q intersecting only in their common fixed point. It is known that all these actions are, in fact, equivalent

(up to

4.

conj~gation)

as we point out in the next section.

THE BASIC ARGUMENT. Here we show the equivalence of certain semi-free finite

group actions on Q with a single fixed point. Let Q0

= Q '- { p l

where p is the fixed point of the action.

Then we have G acting freely on Q , a contractible non-compact Qo

manifold homeomorphic to Q X [0,1). apparatus is applicable as noted. Q (2), assumed to be copies of Q • 0

Elementary covering space Let G act freely on Q (1) and '

0

We have the diagram

0

Q f1) - - - Q (2)

oi

o~

Q0 (1 )/G

Q0 (1 )/G

with the two orbit spaces (called reduced orbit spaces) being non-

255

compact Q-manifolds.

They are also Eilenberg-MacLane spaces and

since their homo t opy groups a re l· dentl·cal, they are homotopy equivalent.

When these

~nd

other) conditions are known to insure

that Q (1)/G and Q (2)/G are homeomorphic then any homeomorphism 0

0

can be lifted to a homeomorphism of Q0 (1) and Q0 (2) and with the one point compactifications of Q0 (1) and Q0 (2) and of 0 0 (1)/G and Q0 (2)/G we have the induced actions on Q(1) and Q(2) equivalent.

In general,two non-compact Q-manifolds of the same homotopy type need not be homeomorphic; we need them to be of the same infinite simple homotopy type (Chapman).

However, there are known

results applicable to the cases cited in sections (2) and (3) which give us partial results. I.

By Chapman's theorem which says that two Q-manifolds

which are of the same homotopy type and which also admit half-open interval factors are homeomorphic, we know that any two semi-free finite group actions of G on Q with a single fixed point are equivalent if the reduced orbit spaces admit half-open interval factors. Thus for a given finite group G, any two of the coned and shrunken arc actions described in Section 2(vi) (or discussed in Section 3 e) are necessarily equivalent.

Indeed, the coning of Q and shrinking

of the resultant fixed arc produces new actions of G which are necessarily equivalent regardless of the actions we start with.

If

there is an exotic action, it is tamed by the coning and shrinking process. The partial result that handles all the known cases cited above is due to Wong - who originally stated it only for special

256

cases such as involutions.

Wong defines a semi-free action with

single fixed point as trivial at the fixed point if there exist arbitrarily small invariant contractible neighbourhoods of the fixed point.

Then he proves that any two semi-free actions of a

finite group G on Q with single fixed point and trivial at that point are equivalent.

By selecting small invariant neighbourhoods

of the fixed point in finitely many factors and crossing with the other factors we get Wong's condition of triviality at the fixed point in all known examples of the types discussed above.

5.

OPEN PROBLEMS CONCERNING GROUP ACTIONS The following represent only a few of many questions

inherent in the preceding discussion. a)

The open problem which has attracted the most attention

is the problem as to whether every two semi-free actions of a finite group G on Q with a single fixed point are equivalent. has the fixed point property.

Note that Q

Wong, West, Hastings and David Edwards

have all worked on this problem using various techniques including proper homotopy equivalences and/or some pro-homotopy apparatus. The problem appears to be very delicate.

Final results are unknown

even for the Z/2 (involution)case. b)

An associated problem is whether every semi-free action

of a finite group G on Q with a single fixed point is trivial at the fixed point (in the sense of Wong as in Section 4). c)

Problems related to those of a) or b) include the more

general questions of equivalence of two semi-free actions of a

257

finite group G with an arc of fixed points or with some other elementary set of fixed points.

Such questions potentially intro-

duce different techniques since the one-point compactification of spaces and orbit spaces under free actions are no longer automatically available. d)

The inverse limit procedures of section 2(v) produce

group actions on Q which are not semi-free.

A natural question is

to ask whether Cantor or solenoidal group actions on Q can exist with the added provisions of semi-free actions with a single fixed point.

In other words, does there exist a free Cantor group (or

solenoidal group) action on Q ? 0

e)

A major area of investigation is the determination of

conditions under which (finite) group actions on Q-manifolds admit factorizations into an action on a countable locally finite polyhedron K and an action on Q.

s2

For example, does every involution on

x Q admit a factorization into an involution on

involution on Q?

by an

In this connection, it should be recalled that

Q-manifold factorizations as K x Q are not unique.

s1

s2

X Q is homeomorphic to the Mobius band X Q.

For example,

Thus the proper

framework for questions should be the existence of a factorization into a finite-dimensional factor by Q rather than the use of a particular factorization.

Also it seems likely that actions may be

describable in terms of maps on or of factors rather than rof actions on factors per se.

To date, these questions of actions on Q-

manifolds have had no serious inquiry.

It seems likely that an

interesting and useful theory can be found. LOUISIANA STATE UNIVERSITY, BATON ROUGE, LA 70803, U.S.A.

258

A NON-ABELIAN VIEW OF ABELIAN VARIETIES L. AUSLANDER, B. KOLB AND R. TOLIMIERI

INTRODUCTION The role of nilpotent groups, particularly the Heisenberg group, in the theory of Theta functions and Abelian varieties has been implicitly known for some time and Mumford.

mad~

explicit by Weil and

In this paper we will describe a way of naturally

attaching a rational nilpotent Lie group N(T) and an automorphism J of N(T) to a complex torus T.

In terms of the morphism proper-

ties of the pair (N(T),J) and certain homogeneous spaces of N(T), we will be. able to: 1.

Give a necessary and sufficient condition forT to be

an abelian variety. 2.

If T is an abelian variety, fully describe the field

of meromorphic functions of the variety. In order to make the presentation as striking as possible we will develop the theory without mentioning mer.omorphic functions, line bundles or theta functions.

At the end of the

paper we will relate the objects used by us to the classical ones.

1.

THE RATIONAL NILPOTENT LIE ALGEBRA OF CERTAIN MANIFOLDS. Since our first construction is of some independent interest

we will present it in a general setting and later see how to

259

specialize it to the cases of particular interest to us. Let M be a manifold.

Let H*(M,R) be its real cohomology

algebra and let H+ (M) be the algebra of elements of positive degree. Clearly H+(M) is a nil-associative algebra as the product of m elements, m > dim M, is zero.

Now given any real associative

algebra A we can convert the underlying vector space A into a Lie algebra L(A) by defining the Lie bracket of x, y E L(A) by the formula [x,y] = xy- yx where the multiplication on the right hand side is in the algebra A. +

We will denote the Lie algebra associated to H (M) by L(M).

Since

L(M) is nilpotent the exponential m3pping makes L(M) into a Lie group which we will denote by N(M).

One can verify easily that L(M)

is a 2-step nilpotent Lie algebra whose centre contains all elements of even order in H+ (M). We will be interested in this construction in precisely two special cases:

first, when M is a compact Riemann surface of genus

g > 0, in which case L(M) can be identified with the Heisenberg algebra L of dimension (2g+1).

v1 , ••• ,Yg,z

Explicitly Lhasa basis

x1 ••. ,Xg'

such that the Lie bracket satisfy [X.,Y.] = Z l l

i

1 ' ••• 'g

[x.,x.] -- [v.,v.] -- o l J l J [X.,Y.] l

J

=0

i

-I

j

1 ::::; i, j ::::; g

.

Clearly Z spans the centre z of L and L/z is abelian.

Further, and

this will be very important in our later considerations, the complex structure on M is classically shown, using holomorphic forms, to 260

induce an automorphism J of L.

The automorphism J restricted to

the centre is the identity mapping and J induces an automorphism on L/z whose square is minus the identity.

Second, when T =

vjo,

where V is a 2n dimensional real vector space and D is a lattice subgroup so that T is a 2n dimensional torus.

In this case the

cohomology algebra can be identified with the Grassman algebra A(V*), where V* is the vector dual of V. Let us now return to our general setting. If M has torsion + + + free cohomology, H (M,Z) c H (M,R) gives a basis of H (M,R) which +

~

is a Z submodule.

Thus H (M,Z) determine~ a rational form of the

Lie algebra L(M).

Hence the Lie group N(M) has discrete co-compact

subgroups

r and so we may naturally assign the compact homogeneous

spaces or, so called, nilmani folds N-(M) /r to the pair H+ (M, Z) cH+ (M, R). Let us return again to the torus T = V/D.

Let v1 , ••• ,v2 n

be a basis of H1 (T,Z). We can use this basis to determine a rational 1 1 structure on H (T,R). In our algebraic model for H (T,R), this reflects itself in an identification of A+ (V*) and A+ (V) and an identification of L(T) and L(A+ (V)).

Clearly the elements of

. A+ (D) c A+ (V) generate a lattice subgroup and so determine a rat1onal +

form of A (V) which we will call the D-rational form. Now assume that V has a complex structure J, i.e., an automorphism of V such that J 2 =-I, where I is the identity mapping. +

Then J induces an automorphism of A (V) and hence of L(T).

We will

denote this automorphism of L(T) also by J. The following definition is suggested by the classical facts about Jacobi varieties of compact Riemann surfaces.

261

DEFINITION.

Let L be the 2n+1 dimensional Heisenberg alge-

bra and let L(T) be defined as above.

A Lie algebra morphism

~ :L(T) ~Lis said to be of type H with respect D if (1)

~is a D-ratio~al epimorphism (i.e., the kernel K of

is a D-rational subspace). (2)

~

(3)

~

~)

If K equals the kernel of

(5)

The J induced map

I1\1 (V) I/\j (V)

is 1 - 1 = 0 for j > 2 ~ then K is J invariant

is the identity. Since the underlying vector space of L(T) and/\+ (V) coincide, we will use this identification in discussing L(T). the ce~tre of L.

Let z denote

Because /\ 2 (v) is central in L(T) and ~is an epi-

morphism, we have easily that ~(/\2 (V))= z.

Once we fix an iso2

morphism of z to the reals, we can identify ~~/\ (V) with a linear functional, i.e., an element of /\ 2 (v*) we identified with the coho2

mology group H (T,R). Let us now indicate some of the immediate consequences for T

= V/D

of the existence of an H morphism

~

:L(T)

~

T.

Let N denote

the group obtained from L by means of the exponential mapping. will call N the Heisenberg group.

We

We have, by identifying an abelian

Lie group with its Lie algebra, the following composition~ of group . homomorphism N(T) ~ N E NI z

=v

where we will use z to denote the centre of N also.

262

Consider D c V

and let v 1 , ••• ,v 2 n be a basis of D.

Let x1 , ••• ,x 2 n be elements of

N such that p (x.) 1

= v.1

= 1 , ••• , 2n

i

Let 6 0 be the subgroup of N generated by the elements x 1 , ••• ,x 2 n. Then because of our assumption about the rationality of the mapping ~ it follows that 6 0 is a discrete subgroup of N whose quotient space

is compact. DEFINITION.

We call 6D a cover of D, and use f6Dl to denote

the set of covers of D. We close this section with the easy,

but crucial observation

that

!

V/D is a principal circle bundle under the action of the group z/6G

2.

n z.

SOME ANALYTIC FACTS ABOUT N/6G In this section N will denote as usual a Heisenberg group,

z

the centre of N, and

r

a discrete subgroup of N such that

a)

r\N is compact.

b)

The co~~utator subgroup of

r, [r,r],

equals f

n z.

Clearly condition b is satisfied by covering groups 6 0 • Consider the compact homogeneous space of right co-sets of

r

in N, r"N.

Then r"N admits a unique probability measure ~

invariant under the action of N.

We get a unitary representation

R of N on £ 2 (r'\N), where we have fixed the measure ~' by setting 263

2

F E J: (r'\N)

RVF (fw) = F (fwv) , v,w E N.

It is clear that we have a representation of the Lie

algebra Lon C00 (f'\N) and, indeed, if LC denotes the complexification of L then we have a representation of LC on C00 (f"N).

We

also consider functions F on f"N as functions on N which are fixed under left translations by f,

In general, if F is a function in N

and v E N, we set

= F (v - 1w)

(L F) (w) v

Thus if L F y

= F,

wE N •

y E f, we may view F as a function on f"N

and we will call such functions r periodic functions. restrict R to z(N)/z(r)

= T1 ,

We may

the circle group, and so we get a

unitary representation, RIT 1 , of the circle group T1 •

It is then

well known that 2

S: (f"\N) = Ein;.H (f)

m

where the sum is the orthogonal direct sum and where

when sEN,

yE

REMARK.

z and ~(f) is the positive generator of z It is important to note that if F,G E H

m

F/G is, at least formally, a function on N/fz

= T where

n r. (r) then

fz is the

subgroup generated by f and z in N. Now if we have an automorphism J of L such that J

2

= -I mod z

J determines a direct sum decomposition Lrv

= V.

1

(J) E9 V • (J) E9 z

where v+1" (J) is the eigenvalue 264

-1

c

+1. su b spaces of LC with respect to

J and

zc

is the centre of LC. Let A(J) = {FE ~ (N) I Vi (J)F = Ol.

In general V. (J) deterl

mines a set of n linearly independent differential operators on N. We will be interested in F E A(N) which are also r periodic.

To fix

notation let GJ(J,r) = A(J)

n .~: 2 (r\N)

8m(J,r) = A(J)

n Hm(r).

It may happen that Gl(J,r) is trivial.

3.

STATEMENT OF MAIN RESULTS Let T = V/D be a complex torus.

Then T is called an abelian

variety if there exist enough meromorphic functions on T to separate points when T is an abelian variety.

We will use

~(T)

to denote the

field of meromorphic functions on T. THEOREM 1. there exists

.§D.

V/D = T is an abelian variety i f and only i f

H morphism cp :L(T)---+ N with respect to D such that

i f ~D is ~ cover of D then A(J, ~D) is not empty.

THEOREM 2. cover of Din N.

Let T be

~

abelian variety and let

Let F, G E Glm(J,6 0 ) and

let~

~D

be

~

be the field generated

£y 1F/GIF, G E Glm(J,6 0 ), mE Zl then~= ~(T).

We will close this announcement with an interpretation of the set 160 }. Let A E /\ 2 (v~~) which we identified with H2 (V/D,R).

A will

be called the Chern class of some line bundle over V/D if and only if A E H2 (vjD,Z), i.e. A takes integral values on D 1\ D.

Clearly

A E H2 (vjD,Z) determines N and therefore f6 0 } and vice versa. 265

THEOREM 3.

There

is·~

natural 1 - 1 correspondence between

~~DJ and isomorphism classes of holomorphic line bundles ~

V/D

with Chern class A. Let

~1

and

~2

be elements of { ~D J.

inner automorphism of N taking

~1

to 62.

* morphisms of N can be identified with V. representation of

*

D c

v*

v*

b e such that

THEOREM4. abelian variety,

C.U.N.Y NEW YORK, N.Y. 10031 U.S. A.

266

The group of inner autoThis gives us a natural

as a transitive group acting on (~ 0 ).

D* (~D )

Let

= ~D.

We~ identify

v*/n*

variety _£i V/D.

Then there exists an

V*/D * with

f~ 0 ).

_!i V/D is.§ll

is~ abelian variety, the so called dual

NON COMPACT LIE GROUPS OF TRANSFORMATION AND INVARIANT OPERATOR MEASURES ON HOMOGENEOUS SPACES IN HILBERT SPACE * M. P. HEBLE

INTRODUCTION Our approach is somewhat as follows.

By suitable assump-

tions about the space and the group acting on it, we try to ensure the existence of some (if not occur when we assume compactness.

all)~of

the ingredients which

This, admittedly, is a slight

departure from the programme indicated by Smale (cf. [10], [11], [6]).

We are able to arrive at some results on integrability of

Pfaffian systems in Hilbert and Banach spaces and on invariant operator-measures, and these results as well as their proofs, are completely non-trivial, owing to the technical difficulties involved.

The detailed proof are contained in [3], and our results

are analogous to some earlier results due to S. S. Chern ([2],

[8], [9]).

I.

Our basic space is Hn, the n-fold direct sum of H with

itself (n ~ 1) where His an infinite-dimensional real Hilbert space.

The group G of transformations acting on Hn is an

*This paper is a short summary (without proofs) of the author's paper [3] which will be published in "Advances in Mathematics" (Academic Press Inc.)

267

. Hn ont~ Hn where the para r-parameter Lie group of diffeomorp h lSms meter r-tuple a

= B ffi

Br

= (a

1 , ·· · '

a ) is a point in the "parameter space" r

••• ffi B (r ~ 1), B being some infinite-dimensional Banach

space, andrG is subject to the assumptions (A) - (D) below.

In Br

let B. be the subspace consisting of vectors (a 1 , •.• , ar) with l

For 1 ::;; h < r we r

shall denote the subspace Bh+ 1 ffi ••• ffi Br of B (A)

by r-hB.

n onto Hn Each element g E G is a diffeomorphism o f H ,

satisfying: (i) f(g; x)

= g(x)

n

n

= x' E H for x E H , g E G; (iii)

= g(x)

x'

= f(g; x) is simultaneously continuous with respect to

both x E Hn and g E G;

(iv) the "coordinates" x' =(x~, .•. ,x~) of

the transformed point are functions

x~ l

= cp.(x 1 , .•• ,x; a 1 , .•• ,a ), l n r

c1 -smooth (Frechet) with respect to the x.l s

i = 1, ••• ,n,

which are

and a.'s, and

c2 -smooth with respect to the a.'s for each fixed l

l

x

E Hn. This assumption implies that each a E Br yields an element

Ta E G and Ta maps a given a E Br into a' E Br by:

T a I = Ta T a·

The group of all these transformations T a called the parameter group, denoted P. (B)

The transformation in G corresponding to a= (0, •.• ,0)

is assumed to be the identity: •.. , n.

m T

•

l

0 1, • • • , X n ; ,

(x

• • • ,

O) =

X. , l

l·

Next we assume that if a= (a 1 , ••• ,ar) is changed-by the

addition of a vector, denoted da = (da 1 , ••• , dar)' then the diffeomorphisms T T are related by: a+da' a r e = e (a; da) = [ e 1' ••• ' e

268·

= 1,

r

J=

[ L: k=1

A1 (a) da k

= T where T- 1T a a+da 8 k' · · • '

with Aki E '9 P(Bk' Bi)' 1 /~ 1·, k /~ r, represents an element of ~(Br). the parameter point a

+

. The operator matrix ((A~))

Our assumption implies that, for

da, we have "in the first approximation",

or in other words (" ;l" means "approximately equal"); where

Dn-+.J cp., 1

a=O

is the first partial derivative of cp. with respect to a. at a= 0, J

l

and

th~

difference between the left side and the right side is We say that the

o(\\da\1) for each i = 1, ... ,n.

€· l

define the

infinitesimal transformation T €

(C)

By "frame" or "coordinate system" R. we shall mean either

the ordered n-tuple

m Hilbert

space~

H1 , H2 , ... ,Hn with

Hn = H1 ~ ••• ffi Hn' or the image of this under a transformation: X ~

9x + a, 9 being an orthogonal

transformation:

Hn onto - .. Hn. 7

,

and a a fixed point independent of x. (D)

A transformation T

a

E G transforms a certain "absolute"

or "initial" frame R0 into a frame Ita : R.a = Ta~o' shall mean, more explicitly, the following. orthonormal system {cp(i)} a

By this we

We consider a complete

in each H. in the coordinate system aE I

1

The "absolute" coordinate system R0 will then consist 0 0 of the orthogonal subspaces H1 , •.. , Hn which are images of • s Hn onto,. Hn 0 0 H under an orthogonal transformat1on Q H 1' · ' ' ' n (") {cp(i,O)} , carrying the c,o.s, {~ 1 } into the c,o,s, a a

·

l=

1,

n

••• , .

aE I

aE I

Similarly the frame ~ a will consist of the orthogonal

subspaces H~, ••• ,H~ which are images, respectively, of H1 , ••• ,Hn f"'\a ,• Hn onto Hn . th . under an orthogonal transformat1on Q , carry1ng e

269

c.o.s. {cp(i)} a:

into {cp(i,a)} o:EI

Let x E Hn

· o:EI

a:

and let x'=T x.

'

a

of x, x' with respect to R0 by

We represent the coordinates

x~, x~ 0

respectively (i

= 1, ... ,n),

and the coordinates of the same points x, x' with respect to Ra by (i,O)} Using {cp 0:

x~a, respectively. l

{cp(i,a)} a: E I

'

a:

, the o:E I

meaning of the assumption (D) can be stated as a lemma. 0 . -xi' 1 = 1 , ••• , n.

,a _

xi

LEMMA 1.

Next, the transformation carrying Ra into It a+ d a is -1

Ta+ da Ta

= T-a1

T- 1 (T T- 1 )T which, when referred to Ra becomes a a+da a a

Ta+da = Tw say, where we use the symbol w = [w1 , ••• , wr].

Since we make extensive use of exterior differential forms in the paper, it is convenient at this point to recall the basic concepts. For any Banach spaces E, F, and any integer k ~ 1, we shall denote ~

by 8k(E; F) the B-space of continuous k-linear mappings E [~ 1 (E,

in

F

F)= ~(E; F)], and by ~k(E; F) the closed linear subspace

~k(E;

F) consisting of continuous k-linear alternating mappings

E ~ F [for k = 1, ~ (E; F) = 8(E; F)]. given integer p

~

For a given set U E E, and

0 or p = oo, we shall denote by

O~p) (U;

F) the

vector space of differential forms of degree k, defined in U with values in F (or briefly, differential k-forms defined in U with values in F), of class p, 1. • e • mapp·n l gs w

U ... ~ (E; F)

which are of class cP.

If U - E, we s h a 11 Slmply · talk of

differential k-forms

E into F, of class cP.

consider the case E

QQ

= F = Bh ,

We shall only

f or a g1ven · integer h : 1

~

h

~

r.

Differential k-forms of class cP on Bh will be called cP-smooth

270

differential k-~ on Bh, and for k= 1 such forms will be called

form~

cP-smooth Pfaffian

Qll

Bh.

(For standard results and techniques

concerning exterior differential forms, one may refer to Cartan

[1], or Lang [5].) Returning to the group G, the Pfaffian forms w. (a; da) l

are called the relative components of G (or of R ) or the forms a

of Maurer-Cartan.

With the help of the transformation T

w

corresponding tow= [w1, ••• ,wr] we can relate the value of ~i for a small change ox in x to the value da in a.

of~l

for a small change

In fact,we have LEMMA 2.

In the first approximation

~. l

(x + ox; a)

= ~ l. ( x; a + da), i = 1, ... , n. Other useful properties of thew. 's are stated in the l

next lerrunas. LEMMA 3.

Thew.l 's are "independent" meaning that the -

--

operator-matrix ( (~)) 1 ~ i, k ~ r

in the above representation of

the w. (a; da) has the following properties: l

(i)

[(A~(a)]] yields ~ bijective linear mapping: Br onto Br , f or eac h f.1xe d a E Br ••

(ii)

the operator equation

for a given r-tuplet [A1 (a), ..• ,Ar(a)] with A.(a) E ~(B., Br), has a unique solution [c1 (a), .•• , J

J

C (a)] where C.(a) E S(B., Br). J

r

LEMMA 4.

J

The w. are invariant under P.

--

l

--

Any Pfaffian form

invariant under P is a linear combination with constant operatorcoefficients of the wl.• 271

In view of Lemma 4, we also use the name "forms of Maurerr

Cartan" for any linear combinations Oi

=

2::

i

Ck lllk' i

= 1, •.• ,r

k=1 The proofs of the major results of this section

~heorem

1, and Corollary 1.1 to be stated below) require a clarification . t y 1n . Br" • of the concept of a "smooth ( r-h ) -var1e

This is done

via the next few definitions. Let Q be the set of all

s

c2 -smooth

diffeomorphisms

Br onto Br with the properties: for a E Br,

S(a + da)- S(a) = [[A~(a)]]da + O[jjdojj],

where (i) for each a E Br, [[A~(a)]] 1 : ; i,k::;;r yields a linear r onto r i 1 homeomorphism B B , (ii) [[Ak(a)]] is C -smooth with respect to a E Brand (iii) for each a E Br, [[A~(a)]] satisfies the contention of Lemma 3.

Then we define a (r-h)-variety in Br

as follows (cf. [4]; [12], pp. 304-307). DEFINITION 1.

By a smooth (r-h)-variety in Br we shall mean

the image of r-hB+w where w E Br, under a diffeomorphism S : Br onto Br, where S E Q. We also need to define appropriate "coordinates" or "parameter r-tuples" for points in Br with reference to the · t·1es vh., W.r-h (defined below). var1e Also, we need to express a l J given (r-h)-variety by means of convenient vector equations.

We

proceed as follows.

wr-h are submanifolds in Br Br onto 8 r ~

272

Since S is a diffeomorphism

' the tangent space at o, T(Br) 0 i.e.

is linearly

homeomorphic under the differential dS(O) to the tangent space of S(Br) (= Br) at a T (Vh) a·

= S(O) ,

and dS (0 ) ( Bh) is linearly homeomorphic to

Hence, further T (Vh ) , T(W r-h ) , the tangent spaces a

a

Wr-h at a = S(O) are direct summands in Br (cf. [7], p. 22, Theorem and Corollary).

A further induction argument shows that

Br is diffeomorphic to the product manifold V~ x r-h X i = 1' .•• 'h, Wr-h = S(B.), X W!'- h' where v~ j l l

...

j ·= 1, ••• ,r-h.

...

x vhh x w1r-h

= S (Bh+ j ) ,

We shall denote a point in this product manifold

h r-h by (y 1 , ••• ,yh; yh+ 1 , ••• ,yr) with yi E Vi' 1 ~ i ~-h, yh+j E Wj , ~

1

j

~

r-h. DEFINITION 2.

The r-tuples (y1 , ••• ,yr) corresponding to r h r-h points y E B, where yi E Vi' 1 ~ i ~ h, yh+j E Wj , 1 ~ j ~ r-h, will be called "coordinates" or''parameter r-tuples" determined by . . vh. an d wr-h the var1et1es . • J

l

r

Next, let S E Q, and let Pi be the projection of B onto B., 1 l

~

i

~

h.

Then the (r-h)-variety V = S( r- hB + w) will be

given by vector equations: pi.

s -1 ( y ) -

where

S·l

si _== p i. 8 -1 [ ( y1, ••• ,yh,• yh+1'"""' y r )] -- o ' 1 ~ i ~ h,

= P. (w),

~

1

~

i

h, and y is a variable point on V.

We

l

can write these vector equations as: ~ ~ ) Fi (y1'"""' Yr'. '='1'"""''='h

= 0,1

~

i ~h.

These equations then have the following property E:

E:

the functions F. are l

c2 -smooth

and s

jointly with respect to

= (s1' ••• ,sh)

r

varying over B

= vh1

X •••

X Wr-h and over Bh -- B $ ... $ Bh respec t•1ve 1y;

r-h

1

273

the operator-matrix [[djFi]] 1 ~i,j~h yields a linear homeomorHh onto~Bh where Hh = T(Vh)a; and the operator-matrix . h onto h [[dr+jFi]J 1 ~i,j~h yields a linear homeomorph1sm B B •

phism:

Next let ' be a family of smooth (r-h)-varieties with the property /A: !A:

through each point of the space Br there passes one and only one variety of '·

~ ) E Bh, Then clearly' can be parametrised by points S = ( s 1 , ••• ,~h and the functions F. in the vector equations F. = 0 (1 ~ i ~h) l

l

2

which represent the varieties of ,, are C -smooth jointly with respect to y E Br and s E Bh and further have t h e property

]8

a b ove.

Thus we agree upon the CONVENTION.

A family' of smooth (r-h)-varieties V

satisfying lA will be denoted by h vector-equations. F.(y;s)=O, l

where: ';

1~i~h,

( i) s varying over Bh is taken to parametri se the family

(ii) for each fixed s E Bh, y E Br varies over the particular

variety V corresponding to the parameter-point s, and the F. have l

the property JB above. DEFINITION.

The system of vector differential euqations

o1

= o, ... ,oh = o,

where {o1, ••• ,oh} form a system of Pfaffian forms, is called completely integrable if and only if there exists a family '

of

smooth (r-h)-varieties V = Vr-h satisfying lA above and such that the equations Oi = O, 1 ~ i ~ h are identically satisfied when the

274

point moves on a variety V E call the varieties V E

o1

=

o, ... ,oh

=

~

~.

And under these circumstances we

the integral varieties of the system

o.

These troublesome details having been satisfactorily taken care of, we can now state the two major results of this section. THEOREM 1.

If the Pfaffian system o1 (a; da) = O, ••• ,~(a; da)

= 0 Qi vector equations where the Oi

~

h independent Pfaffian

forms, is completely integrable and is invariant under W,

then

it is equivalent to the system found by equating to~ (vector), h linear combinations of the wi (a; da)'s. · COROLLARY 1.1. (vector) system

o1

~necessary

condition for the Pfaffian

= O, ••• ,Oh = 0 to be completely integrable, is

that the exterior derivatives vanish:

II.

o: = o. l

Let G be an r-parameter Lie group of transformations

on Hn as in I.

We now consider a collection of submanifolds n in

Hn , and assume that G transforms the elements n transitively. Consider a fixed element n 0 and assume that the subgroup g which leaves n 0 fixed is a closed Lie subgroup depending upon a set of elements belonging to a (r-h)-variety in Br.

We denote the

subgroup as well as the (r-h)-variety related to it, by the same Then the transformations T E G or s E G, for short, such

symbol. that s

s

d ~

g, will map g = gr-h "nto sgr-h so, that these sgr-h fill 1

up all of the parameter space Br, any two being either distinct or identical.

By virtue of the results of I, the varieties sgr-h are

shown to be h-parameter integral varieties of the system formed by

275

the first h forms of Maurer-Cartan of G viz. of the system Ul1 = 0' ••• ' Ulh = 0.

We next consider a suitable field ~ (which might in special cases by a sigma-field) of subsets of these h-parameter varieties, each of which can be specified by a particular parameter h-tuple ~

': >

=

(~

~

)

'::>1 ' ••• ' '::>h '

so that the problem of finding a measure on sets

of elements n invariant under G is equivalent to that of finding a measure on sets of points

s,

invariant under F.

The differential

form Oh(z) = w1 (z) A ••• A ~(z), defined for z E ~' is invariant under Wbut may not always define a density of measure

on~.

The

major theorem of this section is: THEOREM 2. be~

~

necessary and sufficient condition for

~

to

density of measure for the elements n is that its exterior

derivative vanish:

III.

dOh = O.

In this section we assume n > 1.

We consider the

orthogonal group O(n) of orthogonal transformations T acting on Hn _onto Hn , an d f'1rst s h ow that a convenient parametrisation of O(n) is by points a= (a 1 , ••• ,a) E Br' where B =~(H), for r = n(n-1)/2.

We then consider the Grassmanian G~, i.e. the h

collection of h-subspaces of H , 1 ~ h < n, where by h-subspaces in Hn we mean an image of Hh = H m H d 1 w ... EB hun er an orthogonal transformation T E O(n).

A su1't a bl e s1gma· f'1e 1d

in Gh is formed

~

n

by considering a fixed h-subspace viz. V0 = Hh = H

m

m

1 w ••• w

the h-subspaces V' which are images of

v0

by elements of O(n) are

collected into subsets Bt,t' = {V' a h-subspace in Hnlt ~ 276

Hh' an d

d(v 0 ,

V') < t'},

0 ~ t, tl ~ 1, where

a is

the metric:

a=

k·d

d(V0 , V1 )

= max{d(v0 ,

d (v0 , vI)

=

' V1 ) , d(V 1 , v0 )},

inf VE V 1

II u-v\\,

and k is a positive constant so chosen that sup{k·d(V 0 , V1 )\V 1 a h-subspace in Hn}

= 1.

We take~ to be the sigma-field formed fre~ such sets Bt t

'

the results of I, II we prove: THEOREM 3.

1r't

1

-

By

1 •

Tt is an element of measure .2.!2 Bt t

1,

'

where

This collection {~t} is shown to have the semi-group property:

Rs •T

= lls+t'

s, t ~ O, the semi-group { ']\} ~O is tC uniformly equi-continuous: rn't = e , and the operator-jet measure t

thus determined on

~

is countably additive.

REFERENCES 1.

H. Cartan, Differential forms.

Boston:

Houghton-Mifflin,

1970.

2.

S. S. Chern, On integral geometry in Klein spaces.

Ann.

of Math. 43 (1942), 178-189. 3.

H. P. Heble, Integral-geometric measures on homogeneous spaces in Hilbert space.

To appear in Advances in Mathematics.

277

4.

N. H. Kuiper, Les Varietes Hilbertiennes.

Seminaire de

Mathematigue Superieure, Montreal, 1969. 5.

s.

Lang, Differentiable manifolds.

New York:

Addison-

Wesley, 1972. 6.

z.

Nitecki, Differentiable dynamics.

M.I.T. Press,

Cambridge, Mass., London, England, 1971. 7.

R. S. Palais, Lectures on the topology of infinitedimensional manifolds.

8.

Brandeis University, 1964-65.

L. A. Santalo, Introduction to integral geometry. Actualites Sci. Ind. No. 1198, Paris, Hermann, 1953.

9.

L. A. Santalo, Integral geometry, pp. 147-193 in "Studies in Global Geometry and Analysis", ed. S. S. Chern, MAA Studies in Maths., Vol. 4.

10.

S. Smale, Differentiable dynamical systems.

Bull. Amer.

Math. Soc. 73, 747-817, 1967. 11.

R. F. Williams, Non-compact Lie group actions. Proc. Conf. Transf. Groups, New Orleans 1967, pp. 441-445, Springer, Berlin and New York, 1968.

12.

H. Whitney, Complex Analytic Varieties. 1972.

UNIVERSITY OF TORONTO TORONTO, ONTARIO, M5S 1A1 CANADA.

278

'

Addison-Wesley,

APPROXIMATION OF SIMPLICIAL G-MAPS BY EQUIVARIANTLY NON DEGENERATE MAPS SOREN ILLMAN

*

Let G be a finite group.

A simplicial G-complex consists of ~:G

a simplicial complex X together with a G-action

x X- X such

that the map g:X -X is a simplicial homeomorphism for every g E G. We say that a simplicial G-complex X is an r,eguivariant simplicial complex if the following conditions are satisfied. 1.

For every subgroup H of G we have that if s

= (v o , ••• ,v) n

is

a simplex of X and s' = (h v , ••• ,h v ), where h. E H, i = o, ••• , n, o o

n n

1

also is a simplex of X then there exists hE H such that hv.1 = h.v., 1 1 i

= o, ... , 2.

n •

For any simplex s of X the vertices v o , ••• ,v n of scan be

ordered in such a way that we have Gv c ... c Gv n o Here G denotes the isotropy subgroup of G at x.

We call

X

G v

the principal isotropy subgroup of the simplex s and n

maximal isotropy subgroup of s.

the

~

''-'

v

0

The above conditions are purely

technical in the sense that any simplicial G-complex can be made into an equivariant simplicial complex by passing to barycentric subdivisions.

*

(A simplicial G-complex satisfying condition 1 is

Partially supported by an NSF grant at the Institute f"or Advanced

Study, Princeton, 1974-75.

279

called regular by Bre don [1 ; P• 116] • )

Let Z be an equivariant sub-

complex of the equivariant simplicial complex X.

Consider the fol-

lowing property. PROPERTY P. of s such that v G

C

v+

Let s be a simplex of X and let v be some vertex

E Z.

Then if v' is any vertex of s such that

G , we also have v' E Z. v

We say that

Z is strongly full in X if Z is full in X and

satisfies Property P.

This condition is also purely technical in

the same sense as above. We shall now define the notion of an eguivariant combinaterial manifold.

Let p:G- O(n) be an orthogonal representation.

By

Rn(p) we denote euclidean space Rn together with G-action through p. Furthermore we define on (p) = convex hull of f:t,g ei

I

g E G, i = 1, • • ·, n J '

(Here e1 ,e 2 , ••• ,en denote the standard unit vectors in~.)

The

G-spaces Sn-1 ( p) and Dn(p) can be triangulated such that they become equivariant simplicial complexes.

An equivariant simplicial complex

M is an eguivariant combinatorial manifold if the following holds. For every vertex v of M there exists an

orthogo~al

representation

r:Gv- 0~) and a Gv-equivariant p.£. homeomorphism n-1 a:Lk ( v,M)- S (r).

We say that a simplicial G-map f:X- M is eguivariantly nondegenerate if f embeds equivariant simplexes.

(An equivariant

simplex of X is a G-subset of the form Gs, where s is an ordinary

280

simplex of X.)

It is easy to see that a simplicial G-map f:X- M

which is isovariant (i.e., Gf~)

= Gx

for every x EX) and non-

degenerate in the ordinary (non-equivariant) sense (i.e. embeds ordinary simplexes) is also equivariantly nondegenerate.

In the

ordinary case, i.e., without the presence of any group action, the assumption dim X ~ dim M is enough to guarantee that a simplicial map, after introducing subdivisions, can be approximated by a nondegenerate map.

In the equivariant case a condition of the form

~

for every subgroup H of G is not enough as simple

dim XH

dim

~

examples show. below.

Our basic Lemma for the equfvariant case is Lemma 1

(We denote below

?H

=

~ x E X! H~G X ~.)

From now on X will

always denote a compact equivariant simplicial complex. LEMMA 1. that Y

nZ

Let Y and Z be eguivariant subcomplexes of X such 0

is strongly full in Y.

Let f:X - Dn(p) be ~ linear

0

G-map such that fj :Y n Z- Dn(p) is isovariant and nondegenerate.

for every subgroup H of G. 0

Let

€

> 0 be given.

~~

Then there exists

0

~ linear G-map h:X - Dn(p) such that hj :Y- Dn(p) is isovariant and nondegenerate and such that his eguivariantly g-homotopic reljZ! to f. If we assume that f is a linear isovariant map then the situation is quite different and the dimension assumptions are of the type given in'Lemma 3. LEMMA 2.

Let us first state the following.

Let f:X - Rn(p) be ~ linear isovariant map.

Then

there exists 6 > 0 such that any linear G-map h:X- Rn(p) satisfying d(f(x),h(x)) < 6, for every x EX, is isovariant and moreover isovariantly o-homotopic to f. 281

Let y and Z be --- --

LEMMA 3. that

y n Z is

eguivariant subcomplexes of X such 0

Let f:X ~ Dn(p) be~ linear iso-

strongly full in Y.

variant mdp such that fl :Y

nZ~

o

Dn(p~ is nondegenerate.

Assume that

Let € > 0 be given. Then there exists on on D (p) such that hi :Y ~ D (p) is !lQ!2:: linear isovariant map h:X

for every subgroup H of G. ~

degenerate and such that his isovariantly €-homotopic reliZI to f. Using the above lemmas one can now prove Theorems 4 and 5. The proof of this step is very similar to the proof of the corresponding step in the ordinary non-equivariant case.

(Compare with

the proof of Lemma 7.2 in Hudson [2] or the proof of Theorem 1.6.10. Part 1 in Rushing [3].)

If f:X

~ M is a G-map and~a: denotes some

component of XH we denote by ~(a:) the component of MH containing f (XH).

a:

THEOREM 4. M be

~

Let Y be

~

eguivariant subcomplex of X and let

eguivariant combinatorial manifold.

Let f:X

~

M be

~

simplicial G-map such that fl :Y ~ M is eguivariantly nondegenerate. Assume that dim (XH - (j>H U Y)) ~ dim MH ( ) - dim M>H - 1 a: a: fa: f(a:) for every subgroup H of G and each component XH of xH. a:be given.

Let € > 0

Then there exist equivariant subdivisions X' of X and M' ~

M'

be

~

of M and an eguivariantly nondegenerate simplicial G-map h:X' such that his eguivariantly €-homotopic reliYI to f. THEOREM 5.

Let X, Y and M be as above.

simplicial isovariant map such that fl :Y degenerate.

282

Assume that

~

Let f:X

~ y

M is eguivariantly

~

dim(XH a: f _£E

(i>11a:

U Y)) ~dim MH( ) fa:

every subgroup H of G and each component Xa:H of XH•

be given.

Let e > 0

Then there exist eguivariant subdivisions X' of X and M'

of M and an eguivariantly nondegenerate simplicial G-map h:X' ~ M' such that h is isovariantly e-homotopic rel IYI to f.

REFERENCES 1.

G. Bredon.

Introduction to compact transformation groups.

Academic Press. 2.

J.F. Hudson.

New York and

Lond~n,

1972.

Piecewise-linear topology.

Benjamin, New York,

1969. 3.

T.B. Rushing. New York and

Topological embedding§. Lo~don,

Academic Press.

1973.

UNIVERSITY OF HELSINKI HELSINKI 10, FINLAND.

283

EQUIVARIANT RIEMANN-ROCH TYPE THEOREMS AND RELATED TOPICS KATSUO KAWAKUBO

INTRODUCTION Let G be a compact Lie group and let hG() be an equivariant multiplicative cohomology theory.

Let M and N be closed G-manifolds

of class c2 •

~

Then for a G-map f:M

N, we shall define an 'equi-

variant Gysin homomorphism'

under certain conditions. In this talk, I would like to show that the study of our equivariant Gysin homomorphism is effective for various kinds of studies of transformation groups.

1.

DEFINITION OF AN EQUIVARIANT GYSIN HOMOMORPHISM In the non-equivariant case, a manifold has a unique stable

normal bundle, which enables us to define a unique Gysin homomorphism. However, in the equivariant case, a stable normal bundle is not unique in a similar sense.

Hence we need a device to get a unique-

ness of our equivariant Gysin-homomorphism which is essential for our later uses. Fix a property P, which will be given concretely later in respective category. 284

For example, P

= orientation

in the oriented

category.

Denote by

B a set of G-vector bundles over finite CW-

complexes and by B the set consisting of each element of with the property P.

a together

Assume that the set B is endowed with the

following properties: 1)

Lets- X be an element of Band f:Y- X beaG-map

where Y is a finite CW-complex.

Then the induced bundle f*s belongs

to 6 and has a unique property P such that the bundle map preserves the property 2)

P: \'

B Has a multiplication B x B- B which induces the

' ordinary multiplication of

a.

For a G-vector bundle ~ - X, we denote by D(s) (resp. S (s)) the total space of the disk bundle (resp. the sphere bundle) associated with s·

An element tG(s) of hG(D(s),S(s)) is called a Thorn

class ( or hG-orientation class) if for any compact G-invariant subspaceY of X, the correspondence x- tG~IY) x gives an isomorphism

We assume that we can assign to each element

~

of B a Thorn

class tG(s) such that i)

for a G-bundle map f:s ~ s' preserving P, tG (g) = f *tG (S I ) ,

ii)

tG(g x 'T)) = tG(s) x tG('Tl) for g, 'Tl E B.

When such a correspondence is given, we say that B is oriented for hG. Now we are ready to define our equivariant Gysin homomorphisn Fix a property p and a set B with the property P.

Let hG be an equi·

285

variant multiplicative cohomology theory such that B is oriented for hG.

We consider a category of G-actions such that the G-vector

bundles appearing in the category belong to the set B. will become clear later.) manifolds of class

c2 •

(This meaning

Let M and N be hG-oriented closed G-

That is to say the tangent G-vector bundles

of M and N belong to B, hence they are hG-oriented.

Then for a G-map

f:M- N, we define our equivariant Gysin homomorphism f::hG(M) - hG(N) as follows.

As is well-known, there exists a G-representation space

V and a G-embedding e:M-

v.

Regarding V as a G-vector bundle over

one point, we assume that V belongs to the set B. is also a G-embedding. the embedding f x e.

Then f x e:M-N x V

Denote by v the normal G-vector bundle of Then our equivariant Gysin homomorphism is

defined by the composition of the following three homomorphisms which will be explained in a moment: ¢1 "'

¢2 "'

¢3

hG (M)--. hG (D (v) /S (v))--+ hG (NxD (V) /NxS (V)) ----. hG (N) EXPLANATION.

hG denotes the reduced cohomology ring.

Let

tG (M) E hG (D (TM) /S (TM) ) tG (N) E hG (D (TN) /S (TN) ) tG(V) E hG(D(TV)/S(TV)) be the orientation classes respectively where TM, TN, TV denote the tangent G-vector bundles of M, N and

v

respect 1· vely.

It iseasy to

see that we can choose a canonlca · 1 orientation class tG(v) of v such that tG(M)

X

tG(v) = (f

X

e)*(tG(N)

X

tG(V)) •

The~ the homomorphism ¢1 is defined to be the Thorn

286

isomorphism by

making use of the Thorn class tG r,v).

Th e h omomorph'1sm

t1. ~

2

• th e 1s

induced homomorphism c* by the natural collapsing map: c:N

X

D(V)/N

X

S(V) __, D(v)/S(v) •

The homomorphism ¢ 3 is again defined by the Thorn isomorphism in the manner of the definition of ¢ 1 • DEFINITION.

When f:M __, pt, f, is called an index homo-

morphism and is denoted by Ind. PROPOSIT LON. dent of

~11

i) i i) iii) iv) v)

The _e_gyi:_y 2) we need

another invariant given by the isometric structure of a diffeomorphism which lies in W(- 1 )k(Z,Z), the Witt group of (-1)k-symmetric 295

isometric structures over i([4], [5]).

W(- 1 )k(Z,Z) is a subgroup

co

00 of W(- 1 )k (Z,Q) and is of the form Z $ (Z/4)= E9 (Z/2) , too.

The description of bordism groups of knots and diffeomorphisms gives an algebraic connection between these groups and the question is whether there is any geometrical connection. can be obtained by considering fibred knots. knot ~k

4

knot over

This

A fibred knot is a

Sk+ 2 together with a fibration of the complement of the

s1 •

Examples of fibred knots are the Brieskorn spheres.

The subgroup of ck represented by fibred knots is denoted by c~. . k k+2 we can construct a diffeoNow, for a f1bred knot~ ~ S morphism as follows.

Let g:W

the fibration Sk+ 2 - ~k ~

~

s1 •

W be the diffeomorphism classifying The fibre W can be considered as a

manifold with_ boundary~ and glow is the identity.

To obtain a

diffeomorphism on a closed manifold we consider the double WU~(-W) and the diffeomorphism g U Id. THEOREM 1:

Fork> 3 this construction induces a

hom~

morphism

This homomorphism is injective. The proof is given by comparison of the invariant classifying knots and diffeomorphisms;so it is not a direct proof. would be interesting to have such a proof.

It

This could give an idea

of how to prove Theorem 1 for low dimensions.

We state this as a

problem. PR03LEM.

Give a direct geometrical proof of Theorem 1 and

extend it to low dimensions.

296

As it stands Theorem 1 is not of mu.ch worth as we have not yet said anything about CF • 2k-1 THEOREM 2. CF2 k_ 1 .o.. 1· 1somorp · h"1c t o Q • ~ Q _§ 00

detected £y

Brieskor~

PROBLEM.

spheres.

Determine the torsion subgroup of

cF2 k_ 1 •

Now we want to state some problems concerning low dimensions. In

[1] Casson and Gordon have shown that the bordism classes of

classical knots ture.

s1 ~ s 3

are not determined by their isometric struc-

This fact and the connection between knots and diffeomor-

phisms dare to raise the following CONJECTURE.

conj~cture.

Bordism classes of diffeomorphisms on surfaces

are not determined by their isometric structure. REMARK.

A consequence of this conjecture would be that at

least a relative version of the h-cobordism Theorem does not hold in dimension 4.

REFERENCES 1.

A.J. Casson, C. MeA. Gordon:

Cobordism of classical knots.

Preprint 1975. 2.

M. Kervaire:

Les noeuds de dimensions superieures.

Bull.

Soc. Math. France 93 (1965), 225-271. 3.

M. Kreck:

Cobordism of odd-dimensional diffeomorphisms, to

appear in Topology (1976). 4.

M. Kreck:

Bordism of diffeomorphisms, to appear in Bull.

A.M. S. (1976). 5.

M. Kreck:

Bordismusgruppen von Diffeomorphismen.

Preprint

Bonn (1976).

297

6.

J. Levine:

Knot cobordism groups in codimension two.

Comm. Math. Helv. 7.

W.D. Neumann:

UNIVERSITY OF BONN BONN, WEST GERMANY

298

44 (1969), 229-244.

Equivariant Witt rings.

Preprint. Bonn (1976).

SOME REMARKS ON FREE DIFFERENTIABLE INVOLUTIONS ON HOMOTOPY SPHERES PETER Ll:'JFFLER

n,m Let R denote the n+m-

Let G denote the group Z/2.

dimensional real vector space with a non-trivial G-action on the first n coordinates. tM

$

€

k, t _ -

€

We call a G-manifold (n,m)-framed, if

k+n, t+m

Denote the corresponding bordism group

by nG and by nG [1] if no fixed poi~ts are allowed. n,m n,m

Let us

denote by Sn,m the sphere in Rn,m in some equivariant metric and 0

J

introduce wG = lim [sn+k,m+t,Sk,t] , where [ , ] denotes equin,m k,t G G variant basepoint preserving homotopy classes of maps, and 0

wG [1] = lim[Sn+k,m+t,Sk,t A EG+] • A well-known transversality n,m k,,e G argument assures that the Pontryagin-Thom-construction yields an . h"1sm n G 1somorp n,m

= wGn,m

and nG [1] = wG [1] at least form= -1. n,m n,m

Now if ~n·is an n-dimensional homotopy sphere with a free differentiable involution T then an easy lemma asserts that (~n,T) is (n+1, -1) -framed. If we denote by 9G(fr9G) the diffeomorphism classes of

n n n-dimensional homotopy spheres with free differentiable involution (together with a specific framing) we get a map of sets w~,- 1 [1].

fre~_ 1 ~

One can describe the image by surgery techniques and

the existence of a degree one mapping. If we denote by im JGn,- 1 the subgroup of wGn,- 1 generated by JG the standard sphere with different framings (we have im n, -1

=0 299

for n

f 0(4))

G .. G G j· G we can define en_ 1 ~ wn, -1 = wn,-1 1m 3 n,-1 • G

9 n-1

PROPOSITION 1.

~

.-G

.

t•

•

wn _1 !2 sUrJeC 1ve. , Now Bredon and Landweber (Annals of Math. 89, 1969) have

[ J computed the kernel of wG n ,- 1 1

--+

wG n, -1·

It is a cyclic subgroup

Z/b with b = 2a (4a ) with a =order of KO(RP(n-1)) if n n n n n n (n

f

0(4)

= 0(4)). x E Z/b

PROPOSITION 2.

represents a homotopy sphere if

n

n

x

= :!:_

1 (8) for n

= 2(4) f

2 (4)

As Hoo and Mahowald have computed

w~

'

_1 [1]

upton~

14

this result gives a classification of free involutions on homotopy spheres up to this dimension except for n = 9, 1 o. we denote by L::d the Brieskorn sphere of dimension 4n + 1 2 2 d + 22n+1 = O, d odd, and involution defined by zo + 21 + If

...

(z 0 ,z1 ,z 2 , ... ) ..... (z 0 ,-z 1 ,-z 2 , ..• ) then we have PROPOSITION 3.

L: d is diffeomorphic to L::d, ~ to an action

of L2 (Z/2,+), i f and only i f d = d 1 (22 n+ 2 ). Let e:w G 1 ~ wG 1 1 be the Smith-homomorphism and let n,n- ,£ = lim(wG ,_

n ,- 1

,e).

Then a recent result of Snaith says that£~ 0. T

If we define a standard involution to be a free involution on a homotopy sphere such that this element bounds in wGn,- 1 then Proposition 2 gives a classification of the standard involutions (at least for n

f

0(4)). Now£

PROPOSITION 4.

f

0 says

There exist non-standard involutions 2n the

standard sphere Sn in almost all dimensions.

" " UNIVERSITY OF GOTTINGEN, GOTTINGEN, WEST GERMANY

300

COMPACT TRANSITIVE ISOMETRY GROUPS GORDON W. LUKESH

In this note we would like to announce recent results concerning homogeneous Riemannian manifolds. elsewhere.

The proofs will appear

Theorem 1 is a classification of compact homogeneous

Riemannian manifolds having 'large' iso~etry groups.

Theorem 2 is

concerned with a conjecture and theorem due to Wu-yi Hsiang on the degree of symmetry of homogeneous manifolds.

In particular, we

show that a result of his ([1]) concerning the second Stiefel manifold Vn' 2

SO (n) /SO (n-2) is in error.

This work was completed while the author was a graduate student at the University of Massachusetts, under the direction of Professor Larry Mann.

The author would like to thank Professor Mann

for suggesting these problems, and for many useful conversations. THEOREM 1.

Let M

= K/H

be~ m-dimensional compact homo-

geneous Riemannian manifold with K an effective

Lie~

isometries of dimension at least m2/4 + m/2, (m ~ 19).

of

Then~ of

the following must hold: 1.

If M is simply connected, then either (i)

M = CP

n

(m=2n).

The metric on M is determined Q£ to

~

scale factor, and K is locally isomorphic to SU(n+1). or (i i)

M = Sk x Vm-k, k > m/2 •

· 1· some t n · c _2 t Sk .!..§

~

s t an dar d sp h er

· h omogeneous Vm-k = K2 /H 2 is a simply connected Riemann1an 301

manifold, and the metric QD M is the product metric. K is locally isomorphic to Spin(k+1)

X

K2 where

dim K2 s (m-k) (m-k+1)/2. or (iii)

M = sm = u (n) /U (n-1) (IIF2n-1).

There ~ uncountably

many homothetically distinct homogeneous metrics ~ M having U(n) 2.

~

full

isometry~·

If M is not simply connected, then either (iv)

M = CPn X s 1 (IIF2n+1).

The metric

QQ

M is the product

metric of ~ metric on CPn (determined ~ to ~ scale factor) and

~

usual metric in s 1 •

K is locally iso-

morphic to U(n+1). or

(v)

m-k where A is~ group of order two acting M= sk XA V m-k freely on sk X v ' k ~ m/2. sk is isometric to a standard sphere and the metric on M is metric.

or

(vi)

locally~

product

K is as in part (ii).

M is~ simple lens space finitely covered £y Sm=U(n)/U(n-1). Such

~

lens space possesses uncountably many homo-

thetically distinct homogeneous metrics having U(n) as full isometry group. or (vii)

M = RP

k

. .m-k , k ~ m/2.

Xv

. 1some . t r1c . Rp k 12

to~

standard

. t·1ve space, Vm-k 12 . R"1emann1an . h omogeneous, 1 pro]ec

~

and the metric on M is the product metric.

K is as in

part (ii). REMARKS.

The proof relies on two basic facts.

First,

the isometry group of a compact Riemannian manifold is compact, thus we may apply results from compact transformation groups, most

302 .

notably, Mann [4].

Second, invariant metrics on M = K/H are in one-

one correspondence with inner products on the tangent space to the coset H in M, invariant by the linear isotropy action of H.

For

example, if the linear action of H is irreducible, the invariant metrics on M are determined up to a scale factor.

The proof of this

theorem will appear in [2]. If the linear isotropy action is reducible, as in (iii), we have shown that it is possible to vary a given homogeneous metric through a family of homogeneous metrics.

By computing the curvature

tensor and sectional curvatures for these metrics, it is possible to distinguish the resulting Riemannian manifolds.

In (iii) all but

one of the spheres obtained has non-constant curvature. see

For

detail~

[3]. Another application of this 'variation' technique involves

a concept due to Hsiang:

Define the degree of symmetry of a dif-

ferentiable manifold M to be the maximal dimension of all isometry groups of all possible Riemannian metrics on M.

If M = K/H, where

K is compact, semi-simple, then there is a 'natural' metric on M, associated with the Gartan-Killing form of the Lie algebra of K. CONJECTURE.

(Hsiang,

[5]).

The natural metric on a homo-

geneous manifold is the most symmetric metric. In

[1], Hsiang tested this conjecture with the second Stiefel

. n,2 man1fold V

so(n)/SO(n-2), which has SO(n) x S0(2) as full com-

pact transitive diffeomorphism group.

He claims that the natural

metric on vn' 2 (n ~ 31, odd) alone, up to a scale factor, has SO(n) x S0(2) as isometry group, and that all other metrics have

303

isometry groups of strictly lower dimension.

However, using the

variation technique we have shown the following: The second Stiefel manifold Vn' 2 (n ~ 31, odd)

THEOREM 2.

has uncountably many homothetically distinct homogeneous metrics having SO(n) x S0(2) as isometry group. Details will appear in [3].

REFERENCES 1.

W.Y. Hsiang.

The natural metric on SO(n)/SO(n-2) is the

most symmetric metric.

Bull. Amer. Math. Soc. 73 (1967),

55-58. 2.

G. Lukesh.

Compact homogeneous Riemannian manifolds,

to appear. 3.

G. Lukesh.

Variations of metrics on homogeneous manifolds,

to appear. 4.

L.N. Mann.

Highly symmetric homogeneous spaces.

Canadian

J. Maths. 26 (1974), 291-293.

5.

P. Mostert, (Editor).

Proc. Conf. on Transf. Groups.

New Orleans, 1967 Berlin-Heidelberg-New York Springer, 1968.

UNIVERSITY OF MASSACHUSETTS AMHERST, MA 01002, USA Present address: UNIVERSITY OF TEXAS AUSTIN, TEXAS 78712, USA

304

A PROBLEM OF BREDON CONCERNING HOMOLOGY MANIFOLDS W. J. R. MITCHELL

Let L denote a principal ideal domain and X a locally compact Hausdorff topological space of finite cohomological dimension over L.

X is called a generalised n-manifold over L (n-gmL) if (i)

(ii)

V x

c

X is

X,

H*(X,X-x;L) =

~f.

if

0

if

* 'I

n

clcL •

[Here sheaf cohomology and Borel-Moorehomology are understood; under our assumptions X is clcL if and only if for U open and x

c U,

there exists V with x E V c U and im j*:H * (V) ~ H* (U) finitely c c generated.] Bredon [1] calls a space X an n-hmL if it satisfies (i). These concepts arise in the study of group actions on manifolds. The clcL condition (ii) is vital for many arguments, but can be troublesome to verify. of (i).

In [2] Bredon asks if it is a consequence

In the same paper he proves this when L is a countable

field. THEOREM.

Let X and L be

~

above.

Assume further that L

is countable, X is first countable and that for U open in X,

* Hc(U;L) is countable. for all

x

Then if the modules Hp (X,X- x;L)

~

EX and all integers p, X is clcL.

305

The proof, which will appear elsewhere, uses a considerable elaboration of the methods of [2].

* is The assumption on He()

probably unnecessary, and in any case holds if X is separable metric. The first countability assumption on X is essential to the method of proof, even if Lis a field.

Indeed, using a non-principal ultra-

filter on the integers one easily constructs an inverse system of Z/2-modules {v } such that dir lim Hom(V ,Z/2) a:

a:

= Z/2,

bonding maps having infinite dimensional images.

but with all the

This is a counter-

example to the obvious generalisation of lemma 2.1 of [2].

Note

that 'change of rings' difficulties prevent a straightforward deduction from the results of [2] of the following Corollary to our Theorem. COROLLARY.

With X, L as in the Theorem, assume X is an

Then X is an n-gm •

--

--

L

REFERENCES 1.

G. E. Bred on.

Sheaf Theory.

McGraw-Hill, 1967.

2.

G.E. Bredon.

Generalised Manifolds, Revisited, in Topology

of Manifolds (Georgia Conference, 1969) ed. J.C. Cantrell C.H. Edwards Jr., Markham 1970.

CHRIST'S COLLEGE CAMBRIDGE

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