Prospects in Mathematics. (AM-70), Volume 70 9781400881697

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Annals of Mathematics Studies Number 70

PROSPECTS IN MATHEMATICS BY

F. HIRZEBRUCH LARS HORMANDER JOHN MILNOR JEAN-PIERRE SERRE I. M. SINGER

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1971

Copyright © 1971, by Princeton University Press ALL RIGHTS RESERVED

LC Card: 72-155007 ISBN: 0-691-08094-1 AMS 1970: 15A63, 18H10, 20H 05, 35S05, 47G 05, 58G 10

Published in Japan exclusively by University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

PREFA CE “ P rosp ects in M ath em atics/’ a symposium, was held in Princeton on March 16-18, 1970 on the occasion of the dedication of the university’s new Fine Hall. Invited addresses were delivered by F . E . P . Hirzebruch, L. Hormander, J . Milnor, J .- P . Serre, and I. M. Singer. The lecturers under­ took to describe some current trends in mathematics in the perspective of the recent past and in terms of some of the expectations for the future. Among subjects discussed were algebraic groups, quadratic forms, topologi­ cal asp ects of global analysis, variants of the index theorem, and partial differential equations.

This volume presents the written texts of those

lectures.

THE EDITORS

CONTENTS P reface .................................................................................................................................

v

The Signature Theorem: Rem iniscenses and Recreation by F . H irzebruch.......................................................................................................

3

The Calculus of Fourier Integral Operators by L ars Hormander................................................................................................

33

Symmetric Inner Products in C h aracteristic 2 by John Milnor

....................................................................................................... 59

Cohomologie Des Groupes D iscrets by Jean -Pierre S erre.................................................................................................

77

Future Extensions of Index Theory and E llip tic Operators by I. M. Singer............................................................................................................171

vii

PROSPECTS IN MATHEMATICS

THE SIGNATURE THEOREM: REMINISCENCES AND RECREATION F . Hirzebruch In the years immediately following 1952, when I came to the United States for the first time, I learned a lot of my mathematics in Princeton, at the Institute and at Fine Hall. It may be a proper occasion to remember these old days.

I shall recall

(§1, §2) how the general Riemann-Roch theorem for complex manifolds is related to the signature theorem for differentiable manifolds.

The signature

theorem could be proved by Thom’s cobordism theory ([25], [26]) and was then the b asic tool for proving the Riemann-Roch theorem for algebraic manifolds [13].

This Riemann-Roch theorem (for complex manifolds) is

nowadays a sp ecial c a s e of the Atiyah-Singer index theorem for elliptic operators ([5], [6], [7], [8]):

namely, the index theorem is applied to the

Cauchy-Riemann equation. Atiyah-Bott-Singer generalized the index theo­ rem to the equivariant ca s e ([2], [3], [5], [7], [8]).

This is a fantastic

generalization of the L efsch etz fixed point theorem. The equivariant index theorem can be specialized to the signature operator ([8], p. 582).

This

gives a fixed point theorem involving the signature and generalizing the old signature theorem. The old signature theorem involves Bernoulli numbers, and has many relations to number theory and applications in topology. E xo tic spheres were discovered using it [21].

(We shall mention this in §3).

The equi­

variant signature theorem has many more number theoretical connections. In the second half of this lecture we shall point out some rather elemen­ tary connections to number theory obtained by studying the equivariant signature theorem for four-dimensional manifolds. Perhaps these connec­ tions still belong to recreational mathematics because no deeper explana3

F . HIRZEBRUCH

4

tion, for example of the occurrence of Dedekind sums both in the theory of modular forms and in the study of 4-dimensional manifolds, is known. As a theme (familiar to most topologists) under the general title “ Prosp ects of m athematics” we propose “ More and more number theory in topology. ” Remark The paper follows the original lecture rather closely, except for §6, §7, and §8, which were added later. §1.

Conjecturing the Riemann-Roch theorem L et X be a compact complex manifold and W a holomorphic vector

bundle over X. L et Q(W) be the sheaf of germs of holomorphic section s of W. The cohomology groups H1(X,Q(W)) vanish for i > dim^ X = n and are all finite-dimensional complex vector sp aces. We define the “ Euler number’ ’

y(X,W ) =

n 2 ( - 1 ) 1 dim H1(X,Q(W)). i=0

Serre conjectured in a letter to Kodaira and Spencer (Sept. 29, 1953) that y(X,W ) is (for algebraic manifolds X) expressible in terms of the Chern c la s s e s of X and those of W. How does one come from this general conjecture to an explicit one? This will be answered here in a very fast way using modern terminology (including the functor K which was introduced much later ([1], [4], [9])). For holomorphic vector bundles

over the compact complex mani­

fold X we have (1)

y(X ,W x ©W2) = y(X ,W x) + y(X,W 2). For a holomorphic vector bundle

over the compact complex mani­

fold X 1 and a holomorphic vector bundle W2 over the compact complex manifold X 2 , we have

THE SIGNATURE THEOREM (2)

y (( X 1 x X 2 ,

5

® W2) = Y (X lf Wx) • y (X 2 ,W2).

Serre’s conjecture implies that y(X,W ) depends only on the topological c la s s of the bundle. Therefore we introduce the ring K(X), constructed from the semi-ring of isomorphism c la s s e s of topological complex vector bundles over X, and, because of (1), wish to construct an additive homo­ morphism Y : K(X) - Q such that y(X,W ) = y([W]). where [W] is the element of K(X) represented by W. Suppose ch : K(X) -» Hev(X,Q) is an additive homomorphism from K(X) to the ring of even-dimensional cohomology c la s s e s and that td(X) e Hev(X,Q) is a fixed cohomology c la s s .

Then

(3)

f - (ch (£ )-td (X )) [X],

is a candidate for y .

Here a[X ] means the evaluation of the cohomology

cla s s a on the fundamental cy cle of X.

K(X),

Condition (1) is satisfied ; condi­

tion (2) will be satisfied if ch is a ring homomorphism defined for all X and having the usual naturality properties, and if td(X) is defined for all X and satisfies (4)

td (X x x X 2) = td(X 1)® td (X 2).

How can one find such a td(X)?

Assuming it depends only on the tangent

bundle of X, we try to define td(E) eHev(B,Q ) for any complex vector bun­ dle E with base B, satisfying the usual naturality properties and such that

F . HIRZEBRUCH

6

td (E t © E 2) = tdCEj) • td(E 2)

(5) if E j , E 2

are complex vector bundles over the same base space B.

is defined as the value of td for the tangent bundle of X, then(4) is a

If td(X) con­

sequence of (5). We normalize by requiring (6)

td(E) = 1, if E is the trivial line bundle.

To get a precise conjecture for the Riemann-Roch theorem in the form (7)

y(X,W ) = (ch [W] • td(X)) [X],

we must specify ch and td. We choose for X the complex projective space P n(C) and put W= F^, where F is the line bundle associated to the hyper­ plane of X which has ch aracteristic cla s s x eH2(P n(C ),Z ). Here x is the so-called positive generator of H2(P n(C ),Z ); see [13], p. 138. We have (8)

y ( P n(C ),F k) = (n£ k). In view of the desired properties of ch and td considered above, the

sp ecial example (8) determines ch and td completely. We recall this for td.

The tangent bundle of P n(C) plus the trivial line bundle equals

F © ... © F = (n + l)F.

Therefore by (5) and (6), td(Pn(C » = (td (F))n+1

with td(F) = 1 + bjX + b2x 2 + ... = f(x), b| oo. This is the ch aracteristic power series for td. Since the arithmetic genus of P n(C) equals 1, (this is the ca s e k = 0 in (8), where F ° is the trivial line bundle), we must have, by (7),

THE SIGNATURE THEOREM (9)

7

td(P n(C)) [P n(C)] = 1.

For the ch aracteristic power series f, the formula (9) means the co efficien t of x n in (f(x))n+1 equals 1 for all n.

(10)

The only power series with constant term 1 satisfying (10) is oo f(x) = - + * _ = 1 + 5 . + 2 (—1) i_ e “ x 2 k= l

B k 5 ----k (2k )!

Here Bernoulli numbers show up in topology. The final result for ch (Chern character) and td (Todd c la s s ) is the following:

L et E be a complex vector bundle (fibre Cn) with base space

B, and c(E ) its total Chern c la s s , c(E ) = 1 + c x(E ) +

c 2 (E )

+ ... + c n(E ) 6 Hev(B,Q).

Suppose there is a space B / and a map

77

: B'

B such that

77 *

is injec­

tive for rational cohomology and 77*E is a direct sum of line bundles E j(l < i £ n) where E^ has the ch aracteristic c la s s x^ f H2(B ',Z ).

Such a

B ' always e xists (splitting principle). We regard x^ as element of H2 (B',Q). Then 77*c (E ) = (1 + X l) ... (l + x n), X

(11)

X

77*ch(E) = e 1 + ... + e n,

n 77* td (E ) -

n

1= 1

x ------ 1

1—e

-X ;

After ch and td have thus been found, (7) is the explicit form of the Riemann-Roch conjecture.

F . HIRZEBRUCH

8 §2.

The signature theorem L et X be a compact oriented manifold of dimension 4k without bound-

ary. Then H

(X ,R ) is a finite dimensional real vector space over which

we have a bilinear symmetric non-degenerate form B defined by (12)

B (x,y ) = ( x u y) [X], for x,y f H2k(X ,R ).

The signature of this form B, i.e ., the number of positive entries minus the number of negative entries in a diagonalized version, is called sign (X). It was first introduced by H. Weyl [27].

The signature theorem ([13], p. 86)

claims that (for a differentiable manifold X) the signature of X is a uni­ versal linear combination of Pontrjagin numbers. For example, we have for a 12-dimensional manifold (13)

sign(X 12) = — I — (62 p3—13 p ,Pl + 2P l3) [X] 3 .5 .7

(p^ « (£ ,S3),

The func­

f ^ 1> can he used for the classificatio n of lens

sp aces [3]. We have, for the free action of p

def(p;q,r) = —

S

on S2 given by q,r,

a (f , S3).

The summands in this sum are exactly the summands in (27). By summing over f we lose information, like passing from a character to its degree. N evertheless, the “ degrees” def(p;q,r) are quite interest­ ing — as we shall see. The equivariant signature theorem and some number theory.

§5.

It is amusing that the signature defects,

(29)

d ef(p;q ,l) = -

k=l

c o t-^ -* cot P

P

occur in the c la s sica l literature. P R O P O S IT IO N .

to p. (30)

L et p > 1 be a natural number and q an integer prime

Then def(p;q ,l) = -

| (q ,p ),

where (q,p) is the D edekind symbol (also ca lled D edekind sum) introduced in [11], formulas (11) and (12).

The D edekind symbol is always an integer.

THE SIGNATURE THEOREM

17

The proof of (30) is in Rademacher [23], and uses the function (( )) : R -> R defined by ((x)) = x — [x] — ((x)) = 0

if x is not an integer, , if x is an integer.

Rademacher proves ([23], pp. 276-277)

(31)

l' £ 1 c o t • c o t i i s f = 4p ‘' i 1 ( £ ) ) • ( A . k=l P P k=o p P [Dedekind also uses a function (( )).

It is the above function (( ))

used by Rademacher with a shift of ~ in the independent variable. We use the (( )) of Rademacher.] We have by [11], formula (32),

(q,p) = 6p P1 ((£)) • ( A . k=0 P P

(32)

Formulas (31) and (32) imply (30).

Dedekind proves that (q,p) is an integer.

L et us recall how Dedekind [11] defined his symbol and why he was interested in it.

Consider in the upper half plane H the function

F (z ) = e 2" lz

n (1—e 277inz)24 n=l

It is a modular function of weight 12 (compare e.g. [24], p. 154). Since H is simply connected we can define a holomorphic function f(z) in H with e f(z ) = F (z ) and

lim

(f(x-hiy) — 27 ri(x+iy)) = 0 .

y -> 00

y> 0 We have f(z+b) = f(z) + 27rib, for b e Z .

18

F . HIRZEBRUCH

Dedekind investigates the behaviour of f(z) under the modular group. His result is the following. F or ( a k j € S L (2,Z ) there is a number (d ,c) depending only on d and c \c d / such that (33)

c •f(||-^ ) = c •f(z) + 6c •log {-(c z + d )2 i + 27ri (a+d - 2(d ,c)),

where the function c log {—(cz+d)2 S is defined in the upper half plane with­ out ambiguity by requiring that the absolute value of its imaginary part is < Ittc |. Of course, for c ^ 0 the number c _1 • (a+d — 2(d ,c)) is an integer. Dede­ kind uses (33) as definition of his symbol (d ,c) for any pair of coprime in­ tegers.

F irs t consequences of (33) are

i) (1,0) = 1 ii) ( - d , - c ) = —(d,c). Replacing in (33) the variable z by —z leads to (d ,c) = (d,—c), so by ii) we have iii) (—d ,c) = —(d ,c), which implies (0 ,1 ) = 0. Replacing in (33) the variable z by z+1 or by ~

respectively, leads

to the equations iv) (d,c) = (d ',c) for d=d" mod c v) 2d(d,c) + 2c(c,d ) = l+ c 2+d2-3|cd|. Formula v) is the reciprocity law for Dedekind symbols. Clearly, i) — v) allow calculating (d,c) for any coprime pair.

T here is

exactly one and only one real valued symbol defined for pairs of coprime integers satisfying i) — v).

THE SIGNATURE THEOREM

19

The reciprocity law v) and the other properties of the Dedekind symbol imply for c > 0. (34)

( l , c ) = (c - l H c - 2 ) , (2 ,c ) = (c ~ 1) 4 (c ~ 5) ;

for the first formula compare (28).

In the second formula c has to be odd.

Formula (30) shows that our signature defect d ef(p;q ,l) equals the Dedekind symbol (up to a factor).

This followed from formulas (31) and (32)

of Rademacher and Dedekind. We might try to prove (30) alternatively by establishing properties i) — v) for the signature defect. We only defined d ef(p;q ,l) for p > 1 and q relatively prime to p. Therefore we do not have i) and ii), but we have (35)

d e f (l;0 ,l) = 0 def(p ;q ,l) = d ef(p ;q ',l) if q=q' mod p.

Therefore, we only have to prove the reciprocity law for d ef(p;q ,l) corre­ sponding to v) because, clearly, (35) and the reciprocity law establish a l­ ready an inductive process for the calculation of def(p;q ,l), and this will prove again what we want: d ef(p;q ,l) = -| (q ,p ). P R O P O S IT IO N (R e c ip r o c ity la w ).

numbers > 1.

L et p ,q ,r be pairwise coprime natural

Then 2

(36)

2

2

qr def(p;q,r) + pr def(q;p,r) + pq def(r;p,q) = pqr

This uses the notation of (27). law corresponding to v). macher ([23], p. 272).

For r=l, formula (36) is the reciprocity

Formula (36) is essen tially a formula due to Rade­

In the next paragraph we shall prove (36) by using

the equivariant signature theorem for suitable 4-dimensional manifolds and group actions. where we have

It is pretty clear that (36) must be related to a situation

F . HIRZEBRUCH

20

qr points with isotropy group cy clic of order p and induced (37)

representation in the tangent space given by q,r, and co rres­ pondingly for p,q,r cy clic permuted. Before constructing an example for (37) we make some number theoreti­

cal remarks. The right side of (36) vanishes if and only if (38)

p2+q2+r2 = 3 pqr.

A triple (p,q,r) satisfying (38), for example (1 9 4 ,1 3 ,5 ), is called a Markoff triple; compare [23] and [12].

For a Markoff triple the signature defects of

the qr + pr + pq points in a situation (37) do not give a contribution to the total signature defect (20), since their sum is zero. Indeed, each one is zero: If (p,q,r) is a Markoff triple, then def(p;q,r) = def(q;p,r) - def(r;p,q) = 0. Proof. q2+r2 = 0 mod p implies ^

~ mod p. The result follows

from iv) and v) in §4. We clo se this section by pointing out the connection between the Dede­ kind symbol and the quadratic reciprocity law. For coprime integers p,q with p odd, p > 1, (39)

3d ef(p;q ,l) = 2 ( l ) - p - l (mod 8),

i .e ., ( 1 ) + (q,p) = E±L mod 4, where (3.) is the Ja co bi-L egend re symbol. From this fact and the Dedekind reciprocity v) the quadratic reciprocity theorem follows easily. Presumably, the congruence (39) ex ists in the c la s s ic a l literature. Don Zagier told me a very nice proof for (39). §6.

Proof of the Rademacher reciprocity theorem by the equivariant G-signature theorem. Consider the non-singular algebraic surface Vn in P 3(C),

Vn : 2 0n+ 2 ln+z2n+Z3n = 0.

THE SIGNATURE THEOREM

21

L et nn be the group of n—th roots of unity. It a cts on Vn by

(Z0 ’Z1 ’Z2 ’Z3> -* « ~ l z 0’z l ’z 2 'z 3>' The orbit space is P 2 ^ ) which has signature 1; therefore by (24), n - sign Vn = defy

where Y is the curve in Vn given by z Q = 0. Since Y •Y = n, we obtain by (26),

(40)

sign Vn = n - n(n^ ~1) = ” C 4-n2)

Of course, (40) is well-known and could be used conversely to prove the trigonometrical formula (25). L et p,q,r be pairwise coprime natural numbers > 1.

V =

Vpqr : z 0pqr + Z lpqr + Z/ qr +

and on it the action of H =

x /xp x ^ x

jjlt

(Z 0 ’ Z1 , z 2 ’ z 3^ ^ ( C \ . a z v

Consider

= 0,

given by

f i z 2 ’ yz 3)

for (£,a,/3,y ) f H. We put /rpqr = Gx and /xp x Mq x ^ = G2 - We have V/H = (V /G j )/G 2 = (V /G 2)/G r Since V /G j is the complex projective plane whose cohomology remains invariant under the action of G2 , we have (41)

sign V/H = sign (V /G ^ /G j = 1.

We now calcu late

sign V /G 2 by applying (24) to theca s e M=V and G=G2 .

There are three connected 2-dimensional manifolds (curves) Y 1,Y 2 ,Y 3 in V given by respectively.

= 0 (i= l,2 ,3 ). The se ts G(Yj) are ^ - { l j , ^ _ j l | , ^r- {l S

22

F . HIRZEBRUCH Therefore by (26)

(42)

.2

defy

= pqr ( E L * + - E ^ 1 +

There are three sets of pqr points each.

The first set is given by z 2=z3=0,

the second by z 1=z3=0, the third by z 1=z2=0. The isotropy group (G2)x for any point x in the first set is

p

x

p x.

The action of (G2)x in the tangent space at x is the direct sum standard one-dimensional complex representation of p

of the

with that of nT.

Therefore

def

x

q“ 1 ,• r- l It = — ( X co t El) • ( X cot — ) = 0 j=i 9 k=l r

(each factor in the preceding line is 0, because

co t^ L = — cot ZZlSLzl)). q

q

The same holds for the second and third set.

ThusXx defx = 0.

If we

apply (24) to our ca s e , then by (40), (42), and (24), (43)

sign

V /G 2= 1 (p2+q2+r2- p 2q2r2+ l).

V /G 2 is non-singular. This follows immediately by looking at the action in the neighborhood of the curves Yj and the points x. We may therefore apply (24) to the action of Gj on V /G 2 - There is one curve Y, namely the cu rv e in V /G 2g iv en by z Q=0.

(44)

defy

We h av e Y •Y = 1 and

=

(see (26)).

V /G 2 — Y can be identified under t 1=z 1P ,t2=z2q,t 3=z3r with the non-singular affine surface

THE SIGNATURE THEOREM

23

W : t 1(lr + t2pr + t 3p 1 and if q1,...,q n are integers prime to p we define

(52)

def(p;q i ,...,q ) = i 1

_ P -1 7TQ. j 77q„ j 2 cot —1— ... cot — —

n

j= l

P

P

If n is odd, then the number defined in (52) vanishes.

For n=2k we have

the following formula which generalizes Rademacher’s reciprocity formula (36). L et b0,b p ..- ,b 2k be pairwise coprime natural numbers > 1. Then

e»

f„

def(bA

bi

W - 1-

*.b2t

Pk>-

H ere pf is the r-th elementary symmetric function of the numbers and

is the L -polynomial occuring in the signature theorem (§2.

1 .5).

For example,

L i(Pi> = J p v l 2(Pi-p 2) = ^ ( 7 p 2 - P i2)For the proof we consider the meromorphicdifferential form bo bl w _ t 0+ 1 . t 1 + 1 bn

t ° - i

b i

t 1 - 1

bn t n + 1 . dt bn

t n - l

1

See [13],

THE SIGNATURE THEOREM

27

on the complex projective line and use the fact that the sum of the residues of w is zero.

This generalizes Rademacher’s residue proof ([23], p. 277)

of Dedekind’s reciprocity formula (see v) in §5). How does (53) relate to the signature theorem? We shall connect it with a result which Bott presented in this conference. L et na be the group of a-th roots of unity and G = ^

x ...x ^

. Let

G act on the complex projective space P n(C) (homogeneous coordinates v

- ’zn> by a (z 0 , . . . , z n) = (a 0z 0, . . . , a nzn), a t G ,

Bott calcu lates the rational Pontrjagin c la s s e s in the sense of Thom for the rational homology manifold P n(C )/G (compare [17]).

He does this for

arbitrary natural numbers bj > 1. If the bj are pairwise prime, then his formula reduces to the following result. L et bQ,b 1,...,b n be pairwise coprime natural numbers > 1 and G = jzb x ...x a c to n P n(C) as explained above. L et p 1 and G = jLtb x ...x pfo ^

Then the total Pontrjagin cla ss p of P 2^(C)/G is

given by the equation n*p = (l+ b 02x 2) (1 + b ^ x 2) ... (l+ b 2k2x2), if and only if the bj satisfy the Diophantine equation (55)

bQbj ... b2k = L k(P l ,...,p k).

H ere pr is the r-th elementary symmetric function of the numbers b02 , . . , b2k2 . For k=l, the Diophantine equation (55) is satisfied if and only if (bQ,b 1,b2) is a Markoff triple (§5).

For k > 2 , we do not know any solu­

tions (for bj pairwise prime and positive) except ( 1 ,1 ,...,1 ) and ( 1 ,1 ,...,1 ,2 ) and permutations. Are there other solutions?

For k=2, we have the Dio­

phantine equation 45 b0blb 2b3b4 = 7( 2 . b ^ b j2) - ( ? bi2)2 .

Find all solutions! A dded in Proof: Using the IBM 7090 computer at Bonn, Don Zagier found the solution (2, 7, 19, 47, 59). It is the only (non-trivial) relatively prime solution in integers < 100. Mathematisches Institut der Universitat Bonn (Sonderforschungsbereich Theoretische Mathematik) Bonn, West Germany

THE SIGNATURE THEOREM

29

R EFER EN C ES

[1]

Atiyah, M. F .:

K-Theory, W. A. Benjamin, Inc. New York, Amsterdam

1967. [2]

Atiyah, M. F ., and R. Bott: A L efsch etz fixed point formula for elliptic complexes: I. Ann. of Math. 86 (1967), 374-407.

[3]

Atiyah, M. F ., and R. Bott:

A L efsch etz fixed point formula for elliptic

complexes: II. Applications. Ann. of Math. 88 (1968), 451-491. [4]

Atiyah, M. F ., and F . Hirzebruch: Vector bundles and homogeneous sp aces.

Proceedings of Symposia in Pure Mathematics. Vol. 3, 7-38,

Am. Math. Soc. 1961. [5]

Atiyah, M. F ., and G. B. Segal:

The index of elliptic operators: II.

Ann. of Math. 87 (1968), 531-545. [6]

Atiyah, M. F ., and I. M.

Singer: The

index of elliptic operators on

compact manifolds. Bull. Amer. Math. Soc. 69 (1963), [7]

Atiyah, M. F ., and I. M.

Singer: The

422-433.

index of elliptic operators: I.

Ann. of Math. 87 (1968), 484-530. [8]

Atiyah, M. F ., and I. M.

Singer: The

index of elliptic operators: III.

Ann. of Math. 87 (1968), 546-604. [9]

Borel, A., and J .-P . Serre: Le theoreme de Riemann-Roch (d'apres Grothendieck).

[10] Brieskorn, E .:

Bull. Soc. Math. Fran ce 86 (1958), 97-136. Beispiele zur Differentialtopologie von Singularitaten.

Inventiones Math. 2 (1966), 1-14[11] Dedekind, R .:

Erlauterungen zu zwei Fragmenten von Riemann. Rie-

mann’s gesammelte Mathematische Werke 2. Aufl. 1892. (Dover Publi­ cations, New York, 466-478).

Dedekinds gesammelte Mathematische

Werke, erster Band, Friedrich Vieweg Braunschweig 1930, 159-173.

30

F . HIRZEBRUCH

[12] Frobenius, F . G.: Uber die Markoffschen Zahlen.

Sitzungsber. d.

Konigl. Preussischen Akad. der Wiss. zu Berlin (1913), 458-487. Gesammelte Abhandlungen Band III, Springer-Verlag, Berlin-HeidelbergNew York 1968, 598-627. [13] Hirzebruch, F .:

Topological methods in algebraic geometry, Third

enlarged edition, Springer-Verlag, Berlin-Heidelberg-New York 1966. [14] Hirzebruch, F .:

Involutionen auf Mannigfaltigkeiten. Proceedings of

the Conference on Transformation Groups, New Orleans 1967, SpringerVerlag, Berlin-Heidelberg-New York 1968, 148-166. [15] Hirzebruch, F .:

The signature of ramified coverings.

Global Analysis

Papers in honor of K. Kodaira, University of Tokyo P re ss , Princeton University P ress 1969, 253-265. [16] Hirzebruch, F .:

Free involutions on manifolds and some elementary

number theory. Symposia Mathematica (Istituto Nazionale de Alta Matematica, Roma), Vol. V, Academic P re ss 1971, 411-419. [17] Hirzebruch, F .:

Pontrjagin c la s s e s of rational homology manifolds

and the signature of some affine hypersurfaces. Liverpool Symposium on singularities of smooth manifolds and maps 1969-1970, to appear in Lecture Notes in Mathematics, Springer Verlag. [18] Kervaire, M. A. : A manifold which does not admit any differentiable structure.

Comm. Math. Helv. 34 (1960), 257-270.

[19] Kervaire, M. A., and Milnor, J .:

Groups of homotopy spheres I. Ann.

of Math. 77 (1963), 504-537. [20] Kuiper, N. A.: Algebraic equations for nonsmoothable 8-manifolds, Institut des hautes Etudes Scientifiques, Publications Mathematiques 33 (1967), 139-155. [21] Milnor, J .:

On manifolds homeomorphic to the 7-sphere.

64 (1956), 399-405.

Ann. of Math.

THE SIGNATURE THEOREM [22] Mordell, L. J .: kind sums.

31

L attice points in a tetrahedron and generalized Dede­

The Journal of the Indian Math. Soc. 15 (1951), 41-46.

[23] Rademacher, H.:

L ectu res on Analytic Number Theory, Notes. Tata

Institute of Fundamental R esearch, Bombay, 1954-55. [24] Serre, J .- P .:

Cours d’Arithmetique, P re ss e s Universitaires de Fran ce,

P aris 1970. [25] Thom, R.:

Varietes d iffe re n tia te s cobordantes, C. R. Acad. Sci.

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handlungen Band II, Springer-Verlag, Berlin-Heidelberg-New York 1968, 416-432.

THE CALCULUS OF FOURIER INTEGRAL OPERATORS L ars Hormander 0.

Introduction. In recent years there has been a trend in the theory

of general partial differential equations towards essen tially constructive methods.

The development of the theory of pseudo-differential operators

gives such an approach to the study of (hypo-) elliptic differential opera­ tors. We shall devote this lecture to a more general operator calculus which should play a similar role in the study of genuinely non-elliptic operators.

Only a few applications will be indicated here but in the au­

thor’s view it seem s a likely prospect that the techniques will prove to be useful in many other contexts such as the study of mixed problems and overdetermined system s. Before passing to the discussion of Fourier integral operators we shall give a brief historical survey of various techniques used in the study of general partial differential equations.

In doing so we shall of course em­

phasize recent developments which motivate the interest of Fourier integral operators. 1.

D ifferential operators with constant co efficien ts. Such an operator

in Rn can be written as a polynomial P(D) in D = (—i

d

/

i(9/dxn).

We are concerned here with existen ce and regularity of solutions of the dif­ ferential equation P(D)u = f where u and f are distributions in an open subset of Rn. A solution of this equation when f e C ^ (R n) is given at least formally by u = E f where (1 .1 )

E f(x) = (2 n T n f e i< x ^ > e (f) f (£)

Here f denotes the Fourier transform of f, 33

.

34

LARS HORMANDER f(f) = f e - i< x ^ > f(x) dx ,

and e(£) = P ( f ) ” 1 . Of course it is necessary to give an interpretation of (1 .1 ) which eliminates the divergence which could be caused by the zeros of P.

This is not hard to do, however, and fairly complete results on the

general questions of existen ce and regularity then follow by simple argu­ ments using functional analysis. The details cannot be given here (see e.g. Hormander [7, Chapters 3 and 4]) but we should note that the formal reason for the simplicity is that for the convolution operators defined by (1 .1) composition is equivalent to multiplication of the corresponding “ symbols” e(