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ON FREDHOLM GROUPS A. S.

REPRESENTATIONS

OF

DISCRETE

Mishchenko

The so-called Fredholm representations of the group zr were investigated in [1]. In particular, an invar, iant v was constructed for each Fredholm representation T of the group ~T E K(Br). The natural question arises: how large is the subgroup of those elements of the group K(B~0 which have the form ~T for some Fredholm representation T of the group w. The answer to this question was not given in [1], and an indirect method was proposed for the proof of the homotypic invariance of higher signatures for a definite class of groups. The method of constructing Fredholm representations of the group v which will permit getting rid of continuous families of Fredholm representations is considered herein. THEOREM 1. Let ~ be a discrete group and Bw homotypically equivalent to a compact Riemann manifold with metric of nonpositive curvature over all two-dimensional directions. Then for any vector stratification ~ E K(B~) there exists a Fredholm representation T such that ~ = X~T, A g 0. Solov'ev [2] first proved Theorem 1 in the case ~ = Z 2. i. Preliminary Information

Let us recall that a pair of unitary representations T1, T 2 in the Hilbert spaces HI, H2, respectively, withthe Fredholm operator F : Hi -- H2, such that for any element g E ~ the operator FT(g) - T(g)F is a compact operator, is called a Fredholm representation T of the group 7r. The element ~T E K(Bv) is constructed according to the Fredholm representation T. Hence, if the element ~T is understood as an equivariant family of Fredholm operators ¢~ : H1 -~ Ha, x E ~, then this family @x is uniquely determined, to homotopy accuracy, by the following condition: the operators ~x -- F, x ~ ~-~, are compact operators. The family ~x can :here be chosen linear in each simplex of some equivariant simplicial partition of the space Let T = (T1, F, g E 7r the inequality

T2)

be a Fredholm representation such that F is an isomorphism, and for any element

II

i - F -i

T~ (g)

F

T~ (g-i)[] < I.

(i)

is satisfied. Then the equivariant family ~x can be selected such that all the operators ¢Ik~, x ~ B-~, would be reversible. In this case ~T = 0. Furthermore, let Tu, u E X, be a continuous family of Fredholm representations parametrized by the parameter u running over the topological space X. Then the family Tu determines an element ~Tu E K(X x By) where, if the representation Tu satisfies condition (1) for u E Y C X, then we obtain the element ~Tu E K(X x B% Y x BTr). Fredholm complexes of representations, i.e., sequences of unitary representations Ti in the Hilbert spaces Hi, 1 --
HI

"

Ha

.....

H.,

Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 9, No, 2, pp. 36-41, April-June, 1975. Original article submitted March 13, 1974.

©1975 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

121


where all the operators res entations.

F~T~ (g) -- T~+a (g)Fi are compact, can be considered in place of the Fredholm rep-

The main technical difficulty we must overcome is the construction of the tensor product of two Fredholm representations. Let F I and F 2 be two Fredholm representations of the operator

FI : H~--~ H~, F~ : H~---~ I-I~2.
Let us consider the diagram

I®F~4[
It is clear that the complex of this diagram

[

I®F,

HI ® H~ ~-~; H~ ® H~.

,
is a Fredholm complex.

~ Hi® H~--+ (H~ ® H~)· (H~ ® H~)--+ H~ ® H 2 ~ 1

(2)

If a group ~ acts in the Hilbert spaces H k so that two Fredholm representations

are obtained, then complex (2) is not a Fredholm complex of representations if all four spaces H k are in~ J finite. G. G. Kasparov proposed a corrected construction of the tensor product of Fredholm representations. However, it is not at all suitable for our needs and we limit ourselves to a narrower class of Fredholm representations by introducing the following constraint. Let H be a Hilbert space and U a positive selfadjoint compact operator, where the space H ~ = ~ Im U ~ is compact everywhere in the space H. be a unitary representation of the group ~ in the space H and let the condition UU'T (g) ~ II< C, 11V'~ II, II(V'r (g) -- r (g) V') ~ ~< C~IIu'+I ~ If Let F: H --* H be an (unbounded) operator, let the (3) Let T

be satisfied for any integer s, -- co < s < co, ~ E H% space H °° lie in its domain of definition

U F~ II< C. IIU~I~II, u' UU' (FT (g) -- T (g) F) ~ IJ~ C. []U'~ jJ,
and let there exist an operator G such that
U US (FG -- i) ~ Ir< C, ~ U'+I~ ~,

(4) (5)
(6) (7) To do this let

II u' (GF -- t) ~ ]] < C, 11U'+I~ I1,

~ U'G ~ ~ ~ C, If U'+I~ ]1.

We easily determine the Fredholm representation of the group ~ in the set (U, F, T). us introduce a norm scale in the space H ~ by assuming

(8)
and let H s denote the completion of the space H °° in the norm Jt ]i Then H ° = H, and since } s.

II~ IL, < c II~ II.
the identical mapping of the space H ~° is continued to the imbedding i : H' C H '-1. Condition (4) means that the operator F is continued to a continuous operator

(9)

(10)

F : H" --*-H s-l,
and it follows from conditions (6) and (7) that the operator (11) is a Fredholm operator. us set Furthermore,

(11) let

T~ = U~TU-'.
The operators T s are bounded, and [I r,~ II, = II ,~ II.

(12) (13)

122


and the commutator

FTs-Ts_

~ F: H s -* H s-I is compact.

2. Finite Representations Let us consider the case of the family Tx of finite representations of a group ~, parametrized by points x of a compact manifold M. Let ~ E K(M x B~) be the stratification corresponding to the family T x. Let ch~ = ~,a~ ® bi, where {ai} is the basis in the cohomologies H*(M; Q), bi E H* (Blr; Q). Let D be an elliptic pseudodifferential operator acting on the manifold M in the stratification sections 71 and 72, i.e., D : F(71) -* I'(72). Then the symbol M is the projection of T'M, ~ ~ 0, then ~(D)~ parametrized by points a(D) of the operator D is the homomorphism ~ (D) : q* (~i) --~ q* (~h), where q : T*M the cotangential stratification to the manifold M in the manifold M. Here, if ~ E is an isomorphism. Let D · ~ denote the family of pseudodifferential operators of the space B~, whose symbo ! equals (D) Q i : q* (~h Q ~)--~ q* (~ Q ~). The family It follows directly family of elliptic by a lesser order -~

(14)

D · ~ is restored uniquely by its symbol (14) to the accuracy of lesser order operators. from the construction of the operator by means of its symbol that we obtain an equivalent operators over the universal covering B~, which differ from each other at diverse points operator.

Therefore, family (14) defines an element of the group K(BTr), which is interpreted as a generalized analytical index inda (D ® ~) of the family from the elliptic operator viewpoint. It follows from [3] that ind~ (D Q ~) = p~ (~ (D) Q l) ~ g (Bzt), where p: T*M â B~ -- B~ is a natural projection. On the other hand, by selecting the operator D such that the element ch(~(D)) would be a dual element ai E H*(M), i.e., ch (~ (D)) a~ ~- k61ju, k ~ 0 , where u E H2n(T*M; Q) is a fundamental cocycle, we obtain chind~(D Q~) = chp!(a(D) Q l) = chp1(~(D) ~) = = p, (ch (g (D) ~) (15)

T(M))

=p, (ch (or (D)).ch~.r (M)) = kp, (u) Q b~ = Xbi.

We can thus formulate the following assertions. X. THEOREM 2. Let T x be a family of finite representations of the group zr, parametrized by the space Let ch ~T~ = ~al ®hi {al} be a basis in the cohomologies H*(X; Q). Then there are Fredholm repreT i = (T~, Fi, T~) of the group lr such that ch ~Ti = ~bi, kl v e 0.

sentations

THEOREM 3. Under the conditions of Theorem 2 let the family Tx satisfy condition (1) for x ~ Y C X, (a'~ be a basis in H* (X, Y; Q),Ch~T~ = ~ai (~ bi. Then there exist Fredholm representations Ti of the group 7r such that ch ~r~ = k~b~, k~ ve 0.

To prove Theorem 2 it is sufficient to realize a class of cohomologies a i by a singular smooth closed manifold M, i.e., by a mappingf : M --" X such that f*(a i) = ku ~ Hn(M), where u is the fundamental cocyele of the manifold M. The mapping f induces a family Py = Tf (y) of finite representations of the group parametrized by points of the smooth manifold M. It is clear that the mapping g =f x 1: M x BTr ~ X x B~ transfers the stratification ~Tx into the stratification ~py. Therefore, ch ~ = ~u ~ bi q- ~ v~ ~ c~, dim v~~ n.

Hence, the assertion of Theorem 2 is reduced to the case when the domain of variation of the parameter is a smooth manifold. In the case of Theorem 3, let us realize the class of eohomologies ai by a singular smooth manifold M with boundaries OM, and let us construct the elliptic operator D analogously to [4] by a symbol which is constant in the neighborhood of the boundary 3M, If ~r : =*0h) -+ n*(~h) is a homogeneous symbol of degree 0, then the pseudodifferential operator D of order 0 is restored by the symbol 0M and satisfies the following additional condition: there exists a neighborhood U ~ 0M such that iff ~ r(7~), suppf C U, then Df = f, but if suppf N OM = qh, then supp Df N 0M = ~b. If D ~ is another such operator, then the difference D D' is a pseudodifferential operator of order (-1), where supp (D - D ~) f N 3M = ~b for any sectionf. The formula for the Atiyah-Zinger index 123


ind~ D = p! ((1)

is valid for such a class of operators, where p: T*M -* denotes an element from the group K(T*M, OT*M). To tinue the operator D in the second copy of the manifold tity operator, and to note that the index of the operator

{pt} is the mapping at a point and a simultaneously prove the formula presented it is sufficient to conM glued to M along the boundary, by using an idenD is not changed in this procedure.

Using a group of characters of the free Abelian group 7r = Z n (see [5]), we obtain from Theorem 2 the following" corollary. COROLLARY. Every class of even cohomologies a 6 H2*(Tn; Q) of the n-dimensional torus T n can be represented as a = XCh}T, ~ ~ 0, for some Fredholm representation T. Solov'ev [2] first established this result for the case n = 2.

3. Infinite Representations system assume Now, let us consider the family of Fredholm representations defined at the end of Sec. i., i.e., the (U, F, T) satisfying conditions (3)-(7) and parametrized by points of the manifold M. Exactly as in Sec. 2, let us consider the elliptic operator D (of order i) in the manifold M. that the families FxTxare smooth, i.e., their derivatives also satisfy conditions (3)-(7).
Therefore, if D: r(71) -- r(72), then we obtain the diagram

Let us

r (nl ® H) ~

r (m ® H)

l~Fx I
where D ® i

I(~F~

is understood as a pseudodifferential operator with the symbol

~ (D) ® I (see [6]).

Let A be the Laplace operator in the stratification sections 71 and 72. Let us introduce a norm in the space F (~l~ ®//) by assuming

11-I~ = I ~(a'J, + u-l), u il~ d~,

(16)

where # is a smooth measure on the manifold M, induced by a Riemann lattice, and the norm under the integral sign is understood in the sense of a norm in the stratification layer 'l~ ® H. Let H~ be the completion of the space F ('lk ® H) in norm (16). Then the diagram

HI

, o®i

H~_1

I@F~CI
H;+~ ~

I(~FX
H~

(17)

consists of bounded operators and is commutative to the accuracy of compact operators. In fact, the second assertion follows from the circumstance that; the commutator of diagram (17) can be interpreted as a zero order pseudodifferential operator with a compact symbol. The symbol of diagram (17) is a Fredholm symbol for each value of the tangential vector to the manifold M since the vertical operators are Fredholm operators. If ~ E T'M, ~ ~ 0, then since Ker Fx and Coker Fx lie in H~ C Hs, the horizontal operators realize the isomorphisms (D) @ l : q'I]1 ® Ker Fx --~ q'lh ® Ker Fx, (D) ® i : q'~h ® Coker F~--> q*~h ® Coker F~. Therefore, diagram (17) forms an elliptic complex. Finally, the commutators of the operators of diagram (17) with the representation T of the group are compact operators. Just as in Sec. 2, we obtain the following assertion.

124


THEOREM 4. Let T = (U, Fx, Tx) be a family of Fredholm representations of the group ~ parametrized by points of the space X and satisfying condition (1) in the subspace Y C X. Let {ai} be a basis in the cohomologies H* (X, Y; Q), ch ~r = ~a~ ® b~, b~ ~ H* (Bn; 0). Then Fredholm representations T i of the group ~ exist such that ch ~Ti----X~b~, X~~= 0. 4. Proof of Theorem 1

Let us take a family of Fredholm complexes constructed in See. 6 of [1] as the family of Fredholm representations Tx, x E T*Br. It has been proved in [1] that the family Tx satisfies condition (1) for sufficiently large, in the norm, cotangent vectors } E T*B~, and chTx = ~a~ ® b~, where {b~ is the dual basis to the basis in the space T*B~. It is sufficient for us to verify compliance with conditions (3)-(7). Let us recall the construction of the family of Fredholm complexes of the representations of the group r. The space ~ is a Riemann manifold, diffeomorphic to the Euclidean space in which we introduce the coordinates (x, ~), x E B"n, ~ is the cotangent vector. Let ~ be the complexification of the cotangent stratification to the manifold Bn, raised to the space T*Bn'-"-~,At,1 are the external degrees of the stratification ~?. Let us consider the complex A : A°,l °' . A',] ..... An,]

(dim B~ = n), for which the homomorphisms ak are determined at the point (x, ~) as external multiplication by the vector (i~ + co(x)), and co(x) ~- ~xLd~i/]/1 -F ]xI 2. Finally, the family of Fredholm complexes H is defined as a stratification over T*B~, obtained from a complex by using the direct image for the projection p:T'~n--~ T'Bn,

H:Ho A, .H1

.4,, ....

A~-I~ H~, Hk=pl(A~l),

A~ = Pl(ak).

As the operator U we consider an operator obtained as the direct image of the operator of multiplication by the function 1/[1 + p(x, 0)], where p(x, 0) is the range from the point 0 to the point x along the geodesic. Condition (3) follows from the fact that p (gx, 0) -<~ p (x, 0) ~- p (0, gO). The conditions

nV'A~ B< C ! V" ~D, UU'~(A~T (g) -- r (g) Ak) ~ g< C UV'+'~
are valid in place of conditions (4) and (5). Therefore, for (4) and (5) to be satisfied, the substitution Ak -* U-1Ak is necessary. Theorem 1 is proved completely. LITERATURE CITED

1.
2.

3.
4.

5. 6.

A. S. Mishchenko, "Infinite representations of discrete groups and high signatures," Izv. Akad. Nauk, Ser. Matem., 38__, 81-106 (1974). Yu. P. Solov'ev, "A theorem of Atiyah-Hirzebruch type for infinite groups," Vestnik Moskovsk. Ser. Matem., No. 6 (1975). Weishu Shih, "Fiber eobordism and the index of a family of elliptic differential operators," Bull. Amer. Math. Soc., 72, No. 6, 984-991 (1966). M. F. Atiyah and I. M. Zinger, "Index of elliptic operators, I", Usp. Matem. Nauk, 23, No. 5, 97143 (1968). G. Lusztig, "Novikov's higher signature and families of elliptic operators," J. Diff. Geom., 7_, 229256 (1971). G. Like, "Pseudodifferential operators on Hilbert bundles," J. Diff. Equations, 12, No. 3, 566-589 (1972).

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