Fortschritte der Physik / Progress of Physics: Band 26, Heft 3 1978 [Reprint 2021 ed.] 9783112519066, 9783112519059

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A K A D E M I E

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B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR-an eine Buchhandlung oder an den Akademie-Verlag, DDR - 108 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, DDR - 701 Leipzig, Postfach 160, oder an den Akademie -Verlag, DDR - 108 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Lösche, Prof. Dr. Rudolf Ritachl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie -Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 2236221 und 2236229; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 74 Altenburg, Carl-von-Ossietzky-StraBe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik*' erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 1 8 0 , - M zuzüglich Versandspesen (Preis filr die D D R : 120,— M). Preis je H e f t 15,— M (Preis f ü r die D D R : 1 0 , - M). Bestellnummer dieses H e f t e s : 1027/26/3. © 1978 b y Akademie -Verlag Berlin. Printed in the German Democratic Republic. AN (EDV) 57 618

ISSN 0015 - 8208 Fortschritte der Physik 26, 1 4 3 - 1 7 3 (1978)

Structural Stability Theory and Phase Transitions Models V . DE A L F A B O

Istituto di Fisica Teorica dell'Università

di Torino, Italy and I.N.F.N.,

sez. di

Torino

M . RASETTI

Istituto di Fisica del Politecnico di Torino,

Italy

Abstract Global stability theory is introduced as a tool allowing the classification of mathematical models of phase transitions. The point of view is that a topological structure whose stability controls the transition, can be identified in the process of computation of the partition function. In particular we discuss mean field theories and the two dimensional Ising model. Interesting features are disclosed concerning the classification of the instabilities, such as the number of parameters and possible approximations.

1. Introduction A phase transition is in an intuitive sense an elementary phenomenon, characterized b y the fact that the system undergoing a phase transition is "stable" both below and above the critical temperature, in the sense that variations of the temperature inside a phase do not change the gross properties of the state. The system is stable, except at the temperature of phase transition. On the theoretical side, however, the treatment of the field is a branch of mathematical physics of extreme sophistication, starting from the assignment of a Hamiltonian describing the interaction of the (infinitely many) degrees of freedom of the system, then proceeding to compute the partition function and investigating its singularities (if there is a finite number of degrees of freedom then no phase transition can occur). So classification of phase transition theoretical models is not yet satisfactory; the classical Landau-Ehrenfest classification, based on the discontinuous behaviour of different thermodynamical functions, is of little a priori help for a theoretical model, unless one can compute the thermodynamical functions; however there is a long way from the knowledge of the lagrangian to the knowledge of the existence of a transition (except for one-dimensional systems where one possesses beautiful theorems). Pushed by the situation outlined above, in the last years there have been beautiful developments starting from considerations of a different kind; K A D A N O F F [1] and WIDOM [2] with deep physical intuition introduced the point of view that a system near to a phase transition is essentially scale invariant. To give a very rough idea, suppose 11

Zeitschrift „Fortschritte dsr Physik", H e f t 3

144

V . DE ALFABO a n d M . RASETTI

one is given a Hamiltonian describing interaction between sites depending upon parameters K, ... (the parameter K can be though as ft = 1/T times a coupling constant in the Hamiltonian). Consider blocks of sites of side L. Near the transition the system can be described also by a block interaction of the same form with new constants KL, ...; the free energy per block and the correlation length will be given by F(Kl,...)

= L*F(K,...),

(1.1)

• t{KL,...)

= L-H{K,...),

(1.2)

which express the independence of F and £ from the dimension of the block. If K is near to the transition point Kc then the block system is also near to the transition, KL «a Kc. So, (dKL/dL)Kc = 0 and the simplest guess compatible with scale invariance is J!7

L-I£

= A(KL-KC).

(1.3)

The above equation is the starting point of the Widom-Kadanoff scaling laws of the thermodynamical functions. Very roughly, introducing t = In L, we have KL

A

and from eqs. (1.1) and (1.2) we see that free energy and correlation length are singular for KL = KC; their behaviour can be easily obtained. W I L S O N . [ 5 ] stressed the primary role of equations like eq. ( 1 . 3 ) , proposing that the most important facts about a phase transition are embodied in equation,0 of the form L

^

L

=

U

I {

K...)

|(1.5)

where the «j are analytic functions showing simple zeros at the critical points. Obviously the Widom-Kadanoff scaling follows. The problem of a phase transition is then no more connected to a single Hamiltonian, rather to a succession of block "temperature dependent effective Hamiltonians" whose coupling converges to a critical one which gives a fixed point (in the space of parameters of the Hamiltonian) under a rescaling operation of which the previous Kadanoff block construction is a simple example. What is important in this phenomenological description is not so much the original law of force among components, as rather the scaling equation and the existence of the fixed point [4, S\. These developments display a case in structural stability, in the sense of global analysis. Now it is an appealing conjecture, and correspondent to intuition, that exact models of phase transitions present at the transition a break of structural stability, and proposals have been made in this sense [6, 7]. In the present paper we elaborate on this point of view, showing, through the examples we examine, that structural stability and catastrophe theory are the frame in which to discuss and classify the mathematical models. The idea is that a transition is identified once the structure that becomes unstable has been identified and classified. We shall see in the examples that this is so; in particular, we are able to treat the two dimensional Ising model and to show that its phase transition is due to a catastrophic process in the mathematical sense. It is rather difficult to guess a priori which structure becomes unstable; it has no direct connection, in our opinion, with the number of dimensions or the symmetry of the carrier

Structural Stability Theory and Phase Transitions Models

145

space. One important point is however that, if the structure has been identified, then one can draw some consequences without solving for the thermodynamical functions, and can find structures of a finite number of degrees of freedom that approximate very well some characteristics of the problem. The Ising model exemplifies this situation. For a finite Ising system, where there is technically no phase transition, the stable-unstable structure remains, and gives the correct expression for the critical point; one has just to discuss the crossing of eigenvalues of a 4 X 4 matrix. So one can guess in general that an approximation to a phase transition can be obtained by approximating the discussion of its structural stability. While we leave to the next chapter the precise mathematical statements about structural stability, let us sketch briefly here, as a warming up exercise, the case of a vibrating string, which can serve as an elementary illustration of the point of view, showing also the possibility, outlined above, of a very simple heuristic discussion. We take a Hamiltonian of the form [4, 5] PH

+

(1.6).

and calculate [4] the partition function T r {e~PH}

{

J

0

11*

146

V. DE ALFABO a n d M . RASETTI

obviously non analytic in a. We can recover most easily a Wilson type equation: indeed =

(1.12)

Let us notice that ¡j, is the inverse of the correlation length, so that the parameter t introduced in eq, (1.4) is proportional to the correlation length. This elementary toy model shows in its transition the presence of a structure (the geometrical manifold V = V{ 0 and a < 0, while the transition point is a boundary between two different phase stability regions. The stability behaviour of such systems has been investigated, and in particular T H O M [9] has given a classification of structural instability patterns for systems whose control space has dimension 4. The string case discussed above is nothing else than Thorn's cusp catastrophe. Indeed, for a potential quartic in

. 8V

82V

d = 0

(1.14)

which is just the unfolding of the cusp catastrophe. Of course a more rigorous treatment of this case would lead to the methods outlined in sect. 2.IV, however the elementary considerations above help to understand the point of view of the present paper. We conjecture that in any mathematical model showing phase transition there is a structure whose stability is controlled by a point p in the parameter (control) space. The parameter space is separated into stability basins, whose boundaries are instability points, while a stable basin represents a definite phase. If the structure is identified, then one can discuss all the possibilities given by the model, e.g. determine the number of control parameters required in order to obtain the complete unfolding of the instability. Having this general frame in our mind, we shall collect in the next section a short review discussion of structural stability needed in the following discussion of. mathematical models, which will be separated in a section dedicated to the mean field models and another dedicated to the Ising model. These models display indeed the features pointed above. 2. Structural stability [10]

In the kind of questions we are dealing with, the typical situation to be handled may be described in the following general ways. There is a family S of "geometrical" — where the word is to be taken in its broader sense — objects Sp. The family is continuously defined over some space P, in the sense that every element 8P € S is parametrized by a point p £ P of a space of parameters (control space). The latter, in most cases is either an euclidean space or a differentiable manifold of finite dimension; in principle it could have a more complex structure and also possibly be infinite dimensional, however this is not typical in physical problems. The important feature is that one can define a topology in P.

Structural Stability Theory and Phase Transitions Models

147

Now, for a given point p 6 P let us consider the set {5} of points q 6 Qp, Qp being a suitable neighbourhood of p, and the corresponding set of objects in S. The problem is, given a rigorous meaning to the concept of "form" of an object Sq £ S, to classify the form of SQ with respect to that of Sp for all q € Qp. One then says that 8P is a structurally stable element of S if the form — which is roughly the topological type of the geometric object itself, to be thoroughly discussed in the following — of all Sg with q € ¿2p is the same as that of Sp. Of course in general the set of points q 6 P such that Sg has the same structure as Sp is an open set in P. Its boundary K = 8QP or more precisely the set K, complement of Qp in P; K = P — Qp, is called the closed set of bifurcation points. The main problem of structural stability theory is to decide whether the set K is anywhere dense. In the cases when the answer to this question is positive, the analysis proceeds by studying the topological nature of K; i.e. classifying its structure and its singularities. Having the previous global picture of the system in mind, we will review now briefly the few explicit "forms" which appear in the statistical mechanics systems we have investigated, and specify in more detail the ways in which it is realized and can be recognized. I . Geometrical realization [22] This most intuitive case of catastrophe appears often in structures related to physical systems: we encounter it in the heuristic discussion of the string and in forthcoming (Sect. 3) mean field examples. A system of algebraic equations is given, of the form Qj(xi,pk)

= 0

(2.1)

where {x,} and {pk} are some coordinate systems in the euclidean spaces R™ and P respectively, and Qj are — to begin with — polynomials in both x, and pk. If one keeps the coordinates pk fixed, the system (1) defines the algebraic set S({pk}) of points in R™ which are solution of the system for the given choice of the parameters. Now the general questions about the structural stability of S can be answered in a precise way. In fact, first of all there exists a way of giving a natural topology — in the sense of homology theory — to the ensemble of all the sets S. Moreover, there exists in P a proper algebraic subset K such that if pk and qk are two points belonging to the same connected component of the complementary set Qp = P — K, the corresponding sets S(\qk)), S{{qk}) are homeomorphic. Such an homeomorphism is assumed as a definition of the topological equivalence we are looking for. Indeed, due to the isotopy of the configuration space R", it is continuously deformable to the identity. The field of coefficients may be either complex or real. In the latter case the geometrical picture may be made more evident by constructing the graph defined by the system ( 1 ) in R ™ X P. Then — following THOM [12] — consider the projection n \ R" X P -> P. n is obviously a mapping, n: IS —> P. K is identified by the decomposition of 'S and P into a disjoint union of differentiable submanifolds, the strata, in such a way that the image of a stratum of by n gives a stratum of P ; and besides the rank of the mapping is constant in each stratum. Technically speaking, if a is any stratum of P, the projection map n'.7ir1{a) a is a fibration. The above situation is fully recognizable in the universally known classical example here reported (cuspidal catastrophe) already found in the introductory discussion of the string (see (Fig. 2.1): n = 1

148

V. de Alfako and M. Rasbtti

Thorn's main theorem on elementary catastrophes (more thoroughly discussed in the Appendix) says: let M cz R" X P be given by VXG = 0 and therefore K:M P be induced by the projection n ; i.e. let Qj be the components of a gradient vector field.

Fig. 2.1

• Then:

i) The dimension of the manifold M is the same as that of P . ii) Any singularity of K is equivalent to an elementary catastrophe. iii) K is stable under small perturbation of G.

Moreover the number of elementary catastrophes depends only upon the dimension of P and not on n : dimP 1 2 3 4

5

6

number of elementary catastrophes 1 2 5 7 11 (oo). I t is worth remarking here that, even though the above table apparently becomes useless for dim P > 6, it may indeed be extended to any finite dim P provided the concept of elementary catastrophe is suitable implemented in the following way. Classifying the singularities of K is indeed [12] a sequence of algebraic problems, dealing with linear actions of Lie groups on finite dimensional spaces; and the classification is discrete up to the above order because of the simplicity of the structure. F o r higher order degeneracies A r n o l ' d [13] showed one has to introduce the concept of modality, or number of moduli TO, i.e. the minimum number to such that a finite number of w-parameter families of orbits of the group is necesary to cover some neighborhood of a singular point; and could show that only up to dim P = 5 the singular points are simple, i.e. their modality is zero. F o r modality higher than zero, the classification may therefore be continued, but it involves also the "quantum number" m. W e will not enter further into details here, because for physical applications it is sufficient to consider simple strata of P , i.e. consisting of a single leaf, and one has not — at least up to now — to worry about more complicated strata. Even though in some particular cases the above results about the structural stability of the sets S(p) can in some measure be generalized to the case when in the system (1) the polynomials Qj(x, p) are replaced by analytic functions F(x, p) in most cases — specifically over the fields of real numbers — the bifurcation set is badly known, and a characterization in terms of stratification has not yet been given.

Structural Stability Theory a n d Phase Transitions Models

149

The last point in the discussion gives then rise to two major questions, which are the key to the classification of structural stability. i) can the topological type of a function F — in a neighbourhood of an isolated bifurcation point — be determined by a finite truncation of its Taylor series. ii) When question i) can be answered positively, is there a finite algorhythm to detèrmine the topological type. I t has been shown [14, 15] t h a t both in the smooth and in the finitely differentiable cases the answer to the above two questions is positive for isolated critical points. I t is however negative for non-isolated critical points, i.e. when two or more critical points coalesce into one, (situation that Thom defines as a generalized catastrophe). II. Dynamical systems [16] All the models based upon a self-consistency procedure, as the mean field cases of Sect. 3, can be reconducted to the following structure. Let M be a smooth manifold, i.e. such t h a t the overlap maps between local charts are O2, t h a t is differentiable and continuous with their partial derivatives up to the order r-th; and let T(M) denote the tangent bundle of M. A vector field X on M is any cross section of the tangent bundle, i.e. a map X:M -> T(M) such that, if n is the projection n\T{M) ->• M, n o X = identity. X is said to be smooth if the map itself is CT. If r ig i the system of ordinary differential equations m = X(m),

m£M

(2.2)

admits solutions which are locally unique, in the sense t h a t for each y 6 M there is a unique orbit, i.e. a curve to which X is everywhere tangent. In (2.2) the dot denotes differentiation with respect to some independent variable t £ R parametrizing the orbit. I n such case the smooth map M defined by X is called a flow or a dynamical system. When M is compact, there exists a one-parameter (t) group of diffeomorphisms of M, ht:M -> M such t h a t

and hsoht

= hs+t.

(2.4)

Now, if M = ht(m) denotes the orbit through TO £ M, let us designate by 0 such that any e-perturbation of X is equivalent to X. This is obviously a global requirement, implying that perturbation of the whole system let the global quality preserved-allowing for arbitrary initial conditions. A major problem for structurally stable systems is deciding whether they are open dense in % and classifying them accordingly. The classification leads to the concept of basin of attraction. For gradient systems, i.e. dynamical system determined by a potential function V:M -> R such that X = — grad V the boundary of the attraction basin is just the hypersurface constituted by set of critical points of V. For more general situation the topological structure of the basin may be very complicated. Now let us consider again % (or the subspace of % of gradient fields). By the previous definitions we may write % = S uE (2.5) where S is the open-dense set of stable systems, and E the complementary space of unstable systems (for gradient fields, E is just the set of singular points of the map V, and the discussion is reconducted to that of the previous subsection). So far we have discussed the problem of defining, and possibly classifying the points of S. The next step is to consider a controlled system, i.e. a system which is varying within the type of motion being controlled by the position of a point p £ P, the control space. Our system is represented by a point in and as p moves through P, such a point describes an arc in %. If the are crosses E, then at that moment the system abruptly changes in quality. Attractors may bifurcate, or different components of the basin boundary coalesce. The striking point here is that all what can happen depends on the topological structure of E. In general E is stratified into strata of various codimensions. An arc in % parametrized by a point p of dimension np would then meet only strata of codimension rip, and the analysis of such strata will indicate the generic ways a system can bifurcate. It is just the bifurcation theory of gradient systems that has led to Thorn's theory of catastrophe [9]. Passing from gradient to non-gradient systems, the problem of bifurcation becomes much harder, and it is virtually unexplored. III. Families of matrices We shall see that a class of interesting models of phase transitions (e.g. the two dimensional Ising model) can be reduced to a discussion of the situation where multiple eigenvalues appear, generally speaking, for a certain matrix (transfer matrix) or generalized kernel. The state space is in this case the set of eigenvectors of a family of matrices smoothly depending on a parameter. The matter has been clarified in a series of papers by V . I . A R N O L ' D [15, .77J. We shall try here to resume briefly some main results. They will help in understanding how the identification of the structure undergoing a catastrophe can be used in classifying a phase transition. Let a family of matrices be a holomorphic map A: L -> Cn'' in a neighbourhood of p = 0, the origin in the parameter space P. In such parameter space we will look for the catastrophic set (i.e. the points at which multiple eigenvalues appear), and we will discuss how to obtain the most general unfolding of a catastrophic set. The germ of the family at p = 0 (i.e. the class of equivalence of A at p = 0) is called a deformation of A0 =A(p = 0).

Structural Stability Theory and Phase Transitions Models

151

Two matrices, say A and A', are equivalent if there exists a deformation of the identity C, such that A '

=

G A C -

1

.

While not every matrix can be diagonalized by a similarity transformation, it can be always cast in the "Jordan form". The relevant theorem by C. Jordan is the following. Let T be a n x n matrix whose different eigenvalues are /1; ..., ).s with multiplicities S

3

ml, ..., ms, £ Wj = n, and det (AI — T) = [J (A — A,-)"1». ¡=i j=I Then T is similar to a matrix of the form J

=

i = 1

Ai

where Al is a m ; x mi matrix of the form [-^•ilrs = ^i^rs

1

where r can be either 0 or 1. In particular if T has a multiple eigenvalue with n linearly independent eigenvectors (e.g. if T is hermitean), in the block /1; every r is replaced by 0. In this case of course T can be diagonalized by a similarity transformation. It is always possible by a similarity transformation to cast any matrix A{ in the form Ai = dXj (0); 1 ^ i, j 5S n] and the co-

172

Y . DE A L F A R O a n d M . R A S E T T I

dimension of rj to be the minimal dimension of a universal unfolding of rj itself: codim (rj) = dimR

drj

8r]

8x1'

'"'

dxn

Un + r)

[where (a1} . . a s ) s denotes the ideal of the commuting ring S generated by the a,'s]. N o w T H O M ' S [ 9 ] theorem reads: L e t / £ 'C,(n + r) be an unfolding of rj oo sufficiently fast in any spatial direction.

180

L . J . BOYA, J . F . CARIÑENA a n d J . MATEOS

Hence the condition for a solution 0 of the equations of motion to represent a soliton are E(0) - E{0o) < + 0 0 (00

dt0 = 0 = /.a{n) € M

(finite energy)

(static)

(asymptotic vacuum value).

(11.3) (II.4) (IL5)

Thus the asymptotic values of the fields are not the same in different directions for a soliton wave to arise (for example, in 1 + 1 dimension this just implies 0(-f-) 4= 0( — oo)). The question of existence of solitons could then be seen as a boundary-value problem, to wit: to construct solutions of the wa;ve equations which at spatial infinite take particular, preestablished values; as this is not a standard Cauchy-data problem (rather is a Dirichlet problem), we shall not comment too much on actual construction of the solution, and in fact, very few of the topologically possible solutions are known explicitely (for some negative results, see [32]). Let us now comment on the static (time-independent) condition; we can relax this condition and look for non-static solutions but of finite energy and joining different vacuum values at infinity; we shall prefer to call them multisoliton solutions, because in the few examples which can be calculated they correspond to two or more solitons in interaction, and therefore they cannot be static. We shall see that our topological analysis will not distinguish between one — or multisoliton solutions, in other words, both are covered by it, but usually only for a few values of the topological invariant (0, ¿ 1 ) one does have a static solution of the wave equation. Because of these abnormal boundary conditions solitons should enjoy a peculiar stability, in which the "conservation law" does not come from the symmetry of the lagrangian, say, but from the topology of the manifold of solutions. (Incidentally, this possibility was recognized a long ago [33]). In fact we can also say that solitons arise because the manifold of solutions (of certain field equations) is not connected, i.e., one cannot pass continuously from a solution to the most general one. (In physical terms this is sometimes expressed by saying that these different connected pieces ("sectors") are separated by an infinite potential barrier; see e.g. D A S H E N et al., [34] R A M A R A J A N [ I I ] etc.). The splitting of the full manifold of physically acceptable solutions into sets with different spatial boundary conditions is just the separation of the different "sheets" or components of the space of solutions; this point of view is advocated for example by COLEMAN [13]. 3. Mappings of spheres Because of eq. (II.5) each possible soliton realizes therefore a continuos mapping from the "boundary" sphere 8i_2 (for d space-time dimensions) at spatial infinity to the orbit of the vacuum (vacuum manifold); and we have to classify those maps, which can be neatly done, to describe the different types of (multi-) solitons possible. It is clear how to establish an equivalence relation between solitons due to the symmetry group, in the same way that we say that all the "vacua" are equivalent; for them we suppose that the internal group sweeps the whole vacuum manifold (indeed this is what is meant by M = O/H), but this is not the case for the solitons: the symmetry group does not in general act transitively on the (multi-)soliton set: namely if a particular soliton solution 0a satisfies 0a Aa £ M for a particular direction, there is an equivalent soliton obtained by rotating it with 0, in which 0a takes whichever value A we like on the orbit of the vacuum M. So we should classify solitons in classes ( = orbits under G), and take a particular representative per class; we can for example fix the image of a

Homotopy and Solitons

181

point of the boundary sphere in advance (say the north pole, or the value at x = for 1 + 1 dimensions), and make it correspond to a particular vacuum value & 0 . Precisely this restriction is what is needed for defining homotopy groups (see Ch. I l l ) ; namely maps of pairs x £ S, S of a point of a sphere and the sphere, on pairs y £ X, X of a point and a manifold, just that X; 0(x) — y. W e have to consider of course only continuous maps under obvious topologies (which might even be discrete, see examples in Ch. V). III. Concepts in Homotopy Theory 1. Homotopic maps We review here briefly the usual concepts in homotopy; our general references, in which rigurous proofs can be seen, are HILTON [19] and HU [20]. A good reference in general topology, which also discusses homotopy, is to be found in HOCKING-YOUNG [35], W e shall talk of (the category of) topological spaces and continuous maps. L e t X be a topological space, and let us denote by I the closed unit interval [0, 1], in the real line. B y an arc in X we mean a continuos map a:I X. The points 1; however one proves easily t h a t 7in(X) = 0 (for all n) if X contractible (e.g. R"), etc. . . . 4. General properties How are nn{X, x0) and nn(X • Xj) related? If x0Rx1 then nn{X, x0) is isomorphic t o 7in(X, x-i) but there is no canonical isomorphism in general. So if X is an arc-wise connected space (i.e. n0(X) has one element) then nn(X, x0) is isomorphic to nn(X, a^) for any two points x0, xt £ X. I n these cases we write nJX). If X = 0 is topological group, n0(G) can be endowed with a group structure: in particular if 6? is a Lie group, the arc-wise connected part of the identity coincides with the connected part of the same, which is an invariant subgroup G0; JI0(G) is identified with the factor group G/G0. Also nx(G) is an abelian group for any topological group G. If (X, x0) and ( Y , y 0 ) are homotopically equivalent ( = same homotopy type) , then n„(X,x0) and nn(Y,y0) are isomorphic for each n: homotopy groups are homotopic invariants, that is, homotopic spaces have the same homotopy groups (e.g. Dn and R"). Also if we denote by Gx (x) G2 the direct product of groups Gx and G2, we have nn[X X Y, (x0, y0)) & 7tn{X, x0) (x) 7in{Y, y0).

(III.3)

One must remark also an important alternative description of the homotopy groups. If the boundary 81" of I" (the set of faces) is "pinched" to a point, we obtain the sphere Sn with a distinguished (base) point ,s0. Thus a one-to-one correspondence might be established between elements of NN(X, x0), such it was defined and the homotopy classes of continuous maps f\Sn^-X with f(s0) = x0. The cases n = 0, 1 are easy to visualize: S° has only two points, e.g. + 1 ; one has to have a fixed image, whereas the other runs through the arc-wise connected components of X, hence JZ0(X) — card (arc-wise components). And let p: I S1 be defined by p(t) = e2nit. I t / is a continuous map j'.S1 X with /(1) = x0, the map a = fp:I-^X is such t h a t c(0) = B

for a general fibre bundle with base, total space and fibre B, E, F; o-> is not a map, and ffo+P-s-jBor simply P(B, G) for a principal fibre bundle. From the extensive literature (efr. the two books STEENKOD [.37] and HUSEMOLLER [5 S U

2

^

S03(R)

0

(III.5)

(which is an exact sequence of groups, that is Z2 is invariant in SU2, and SU2/Z2 is the group S03), implies that S U2 is a "covering space" for S03; in fact this is an universal covering: for each connected Lie group G there is a unique connected and simply connected group 6 such &jC = G, where C is a discrete central subgroup of if; for details see PONTBIAGIN [36]. G is called the universal covering group for G, and n^G) =

C.

iii) homogeneous

spaces

Let ( ? b e a Lie group and let H be a closed subgroup; then if X = G/H denotes the set of left cosets (in general X is not group because H needs not be invariant), we have an important principal fibre bundle H

G^G/H

= X;

X is called an homogeneous space; the group of the bundle is H, the base is X = A typical example useful in computations is for orthogonal groups 80n-x

(III.6) G/H.

o - SOn -> S«-1

which the reader can easily check: if S0n operators in V = R™, the unit sphere Sn_1 = \x | Ex? = 1} is invariant and transitive under S0n; the little group of a vector (e.g. the north pole) is the rotation group in the orthogonal space, i.e. S 0 n ^ . i v ) vector bundles

The case i'-vector space is very important, because these vector bundles are ame-n able to homotopic and cohomologic description and classification. For example, if V is a manifold, the union of all tangent spaces to all points in V is a vector bundle (the

Homotopy and Solitons

tangent bundle),

185

z:Rn o+ T(V) -> V.

(III.7)

A "section" in this bundle is nothing else than a (global) vector field. Tensor fields can be described similary as cross sections in associated tensor bundles. f i n a l l y we quote some "functional" properties of fiberings: if h: B'

B' |

h

F ^ E - + B

F -+F'

implies

B is a map

-+B'

||

j"

F-+EJ1+B

conmutative, i.e. hon' = n o h; here E' is the subset (e, b') of E X B' with n(e) = h(b') £ 6 B. For prolongation on the other sense, if P(B, G) is principal, and G F indicates "operation of G on F as a topological transformation group", g fg

Go>P-+B J

F

implies

G

P

|

|»

—B ||

Fo>E

n

commutative, n — n oh, here E is defined as a factor space of F X P under G equivalences: (/, p) « (/', p') if (/', p') = (fg, gp^1). This E(F,B) is equivalent to the general definition of associated bundle. The functorial properties of a fibering are analogous to t h a t of exact sequence of groups, which appear for example in group extensions; see e.g. MICHEL [39].

IV. Homotopy Groups of Manifolds 1. Description of some manifolds I n this chapter we shall calculate some homotopy groups of usual spaces, all related to the classical groups. Most of our examples will in fact be homogeneous spaces of Lie groups, and therefore they are differentiable, i.e. they carry a differentiable structure. The simplest manifolds are the classical groups on their own, then spheres S n , projective spaces KP n , (K = field of numbers), etc. From the general linear group GLn(C) of complex invertible n Xn matrices, we distinguish some important subgroups which are the simple Lie groups : they come from the classical Cartan series of simple Lie algebras which include four types of simple algebras whose compact representative groups are (BACRY [41]) : A¡: unitary unimodular group SU ! + 1 ;

dim = (I + I) 2 — 1

Bi : orthogonal odd-dimensional S02i+i{R) ; C\ : symplectic unitary group Sp2i ;

dim = 1(21 + 1)

dim = (21 + 1)1

D['. orthogonal even-dimensional S02i(R) ; dim = 1(21 — 1) and with the five exceptional algebras ®6, E7, Es, Fi and G2 they exhaust the simple Lie algebras. Here I = 1, 2, 3, . . . denotes the rank of the group (dimension of the maximal abelian subgroup, number of algebraically independent "Casimir" operators, etc.), and

186

L . J . BOYA, J . F . CARINENA a n d J . MATEOS

we have the identifications A1

=

B1

=

Cl-,

B2

=

C2;

A3

=

B

3

,

(IV.L)

D1 is abelian, and D2 — Al © A1; with these qualifications, the Cartan classification exhausts all the simple Lie groups.

The symplectic groups are unimodular; for the others we have 1 -> SUn

Un

- 5 S 1 + U,

On

-^-V

-> 1

(IV.2.

U)

and 1

-*S0n->

a s exact sequences of groups (that is Un/SUn

Z2

1

(IV.2.0)

TJx; 0n/S0n m Z2). H e r e " d e t " is the de-

terminantal map. We define first spheres as homogeneous spaces; if in R n+1 we consider a sphere S" invariant under SOn+1, the isotopy group of the sphere is SOn and therefore we have the fibering S O ^ ^ S O ^ S « - ! ;

similar arguments prove

(IV.3.0)

SU^^SU.-^S*»-\

and

Sp2(n-1) ^ Sp2n

8

(IV.3 .Sp)

.

Projective spaces are spaces of rays; namely if in F„(K) the field of quanternions), we establish the equivalence xf&y

if

y = ?.x,

A€ K*,

(IV.3.Z7)

(K* =

K" where K = R, C or H (H is K -

{0}),

then K" — (0}/«s is called the (n — l)-th projective space associated to K", and is de- •« noted KP™-1. For the three cases we have respectively the fundamental diagram

S° = Z2-> -Si"-1 - > RP«-1 ^ • II R* -h* Rn — {0}

(1VA.R)

RP"-1

which is commutative (i.e., composition of arrows with fixed start and end gives the same result); also S 1 -> S 2"- 1 - > CP"- 1 and

C *

C" -

£3

Sln-1

H

HK -

{0) - > C P * -

1

(IV.4.C)

Hp«-! (0} - > H P " - 1 .

(IV.4 J?)

In the three cases the upper row is obtained in this way: we cut each ray by the unit sphere ||a;|| = 1, in. K": each ray gives just two points (K = R, S°) or a "phase" (K = C, S 1) or a unit norm quaternion (K = H, S 3). The reduction from the lower to the upper row is referred to as "reduction of the group of the bundle to the compact subgroup" (e.g. K O B A Y A S H I - N O M I Z U [ 4 < § ] , I , pag. 8 3 ) . For n = 2 we have the beautiful Hopf dia-

187

Homotopy and Solitons gram of spheres

S° - X S 1 II

Si

S1 S3 -> S2 II

S4

II s ' _» $ 1 5 ->

(IV.5)

which is due to the four (and only four: ADAMS [42]) division algebras over the real field: the last row in (IV.5) is caused by Cayley's octonions (see e.g. [29], p. 53). Finally we mention two classes of homogeneous spaces, the Stiefel manifolds V„pk, or sets of k orthonormal vectors in the R™ space, equivalent to the factor space 80n.k

o* S0n -> V„ik

(IV.6)

which generalize the sphere (Si!~1 = VnA), and the Grassmann manifolds planes ( = ¿-dimensional subspaces) in RTC characterized by

On-k X Ok —> 0n —>• G„ik

Gnk of sets of k

(IV.7)

SOn-kxSOko+SOn^SGnik.

Sometimes amputated spaces are considered, i.e. ordinary R* without a point or a line, etc. The following facts can be easily demonstrated: R" — {x} is homotopic (not homeomorphic!) to S"-1 (in particular, R — (0) is disconnected) ; Rn — {x} \ x2 = • • • xn = 0} is homotopic to S"~2 (in particular R3 without a line is homotopic to the circle), etc. 2. The exact homotopy sequence As we said in Ch. I l l the computation of homotopy groups is of the outmost difficulty; intuition helps for TZ0 and only. One of the fundamental tools in computing higher homotopy groups is the so-called exact homotopy sequence, which we pass to explain. We said in Ch. I l l that a map f:X, x0(x0 € X ) —> Y, y0 = f(x0) induces a morphism f^:N„(X, x0) TCJY, y0). Now consider a principal fibre bundle P(B, G); a point 60 £ B determines a sequence where i is the inclusion (map) of the fibre over b0 on the total space, therefore we have 71,(0)

nn(P)

(IV.8)

7tn(B)

(we omit base points). Now let f : I n - > B represent the class [/] £ n n (B) coming from map f :In, 81" p-\ba £ B); as 8In contains /»-i(/«-i can be indentified with the first n face of I ), it induces a map d'.[f] -> [g\ £ called the boundary map (d is a group homomorphism) by composition of maps; one has therefore a long sequence -> nn{G) ^

nn(P)

nn(B)

«„_,(