Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/~asmish/Lichnaja-2010/Version2010-11-20/Trudy/Publications/2008/Mishchenko-Teleman-2008-ArXiv-0709.2324v2.pdf Äàòà èçìåíåíèÿ: Mon Oct 19 12:06:52 2009 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:53:42 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 63

arXiv:0709.2324v2 [math.KT] 16 Feb 2008

Construction of Fredholm representations and a modification of the Higson-Roe corona
A.S.Mishchenko and N.Teleman
Abstract The Fredholm representation theory is well adapted to construction of homotopy invariants of non simply connected manifolds on the base of generalized Hirzebruch formula [ (M )] = L(M )chA f , [M ]


K

0 A

(pt)

Q,

(1)

where A = C [ ] is the group C ­algebra of the group , = 1 (M ). The 0 bundle KA (B ) is canonical A­bundle, generated by the natural representation -A . In [1] a natural family of the Fredholm representations was constructed that lead to a symmetric vector bundle on completion of the fundamental group with a modification of the Higson-Roe corona when the completion is a closed manifold. Here we will discuss a homology version of symmetry in the case when completion with a modification of the Higson-Roe corona is a manifold with b oundary. The results were develop ed during the visit of the first author in Ancona on March, 2007. The second version is supplemented by details of consideration the case of manifolds with b oundary.

The Fredholm representation theory is well adapted to construction of homotopy invariants of non simply connected manifolds on the base of generalized Hirzebruch formula 0 [ (M )] = L(M )chA f , [M ] KA (pt) Q (2) where A = C [ ] is the group C ­algebra of the group , = 1 (M ). The bun0 dle KA (B ) is canonical A­bundle, generated by the natural representation -A . The map f : M -B induces the isomorphism of fundamental groups. 0 The element [ (M )] KA (pt) is generated by noncommutative signature of the manifold M under exchange of rings Z[ 1 ][ ] A. 2 Let = (T1 , F, T2 ) be a Fredholm representation of the group , that is a pair of unitary representations T1 , T2 : -B (H ) and a Fredholm operator F : H -H , such that F T1 (g ) - T2 (g )F Comp(H ), g . (3)

Changing the algebra B (H ) for the Calkin algebra K = B (H )/Comp(H ), one comes to the representation of the group â Z to the Calkin algebra: : â Z-K, 1 (4)


(g , n) = T2 (g )F n = F n T1 (g ), Id : KA (X ) - K
1 b AC (S 1 )

g ,
b

n Z.

(5) (6)

(X â S 1 )-KK (X â S1 ).

Here KC (S 1 ) (S ) is the canonical element generated by regular representation of the group Z. Applying (6) to the Hirzebruch formula (2) one has homotopy invariance of corresponding higher signature.

1

Construction of Fredholm representation

Let T be the sum of finite copies of regular representation of the group , be the block diagonal operator that is defined as matrix valued function F (g ), g : F (g ) : V -V . Let H=
g

(7) (8) (9)

Vg , Vg V ,

Th : H -H, Vg -Vhg . The condition that is Fredholm operator means that F (g ) C, F
-1

(g ) C

(10)

for all g except a finite subset. The condition (3) means that
|g |-

lim

F (g ) - F (hg ) = 0.

(11)

So if the pair = (T , ) (12) satisfies conditions (10), (11), then is Fredholm representation of the group .

Consider universal covering B of classifying space B endowed with left action of the group . In correspondence to the construction by [2] the vector bundle generated by the representation on the space B can be represented as an equivariant continuous family of Fredholm operators on the space E = B . The property of equivariance corresponds to diagonal action on Cartesian product Th : E â H, (x, )-(hx, Th ( )). (13)

Namely, let the spaceB be endowed with a structure of simplicial space and E = B be endowed with the structure of simplicial structure derived from the covering E = B -B 2
p

(14)


Let {xi } be the family of vertices of E = B , one from each of orbits of the action of . Then each simplex of E = B is defined completely by their vertices = (h0 xi0 , . . . , hn xin ), h0 , . . . , hn . Any point x is uniquely defined as a convex linear combination of vertices
n

(15)

x=
k =0

k hk xik

(16)

Then the equivariant family of Fredholm operators which corresponds to the Fredholm representation (12) one can define by the formula x = x () =
n n

=
k =0 n

k

hk x

ik

=
k =0 -1 hk

k Thk .

x

ik

T

-1 hk

= (17)

=
k =0

k Thk T

Hence (x )g =

n

k Fh
k =0

-1 k

g

.

(18)

It is clear that the family (17) is equivariant. Indeed,
n

hx =
k =0

k hhk xik .

(19)

Hence
n n



hx

=
k =0

k T

hh

k

T

-1 hhk

=T

h k =0

k Thk T

-1 hk

T

-1 h

= T h x T

-1 h

.

(20)

Also it is clear that the operators (17) are Fredholm by (18) , (11) and (10).

On the other side the operators (7) generate the continuous family Fx : V -V , x E using formula Fx =
k =0 n

(21)

k F (h

-1 k

).

(22)

3


This family one can consider as a linear mapping of the trivial bundle: Fx : E â V -E â V . Consider the universal covering p : E -B . Denote
i Ki (E ) = lim Kc (p -1

(23)

(24) (X )), (25)

where the inverse limit takes by the family of all compact subsets X B . Theorem 1 The map (23) defines the element F () K0 (E ). (26)

Consider the direct image of the bundle (23) over B : A-B , (27)

where the fibre is the direct sum of the fibers of the bundle (23) over each orbit of the action of the group in the space E . The total space A is defined as A = {(u, ) : u B ,
x u

(x â V )}.

(28)

Let A-E be the inverse image of the bundle (27). The total space A is defined as A = {(x, ) : x E , (y â V )} = {(x, ), x E ,
y [x ] g

(29)

(g x â V )}. (30)

Define the action of the group on the total space A by the formula fh (x, ) g It is clear that A = A/ . 4 (31) = (hx, ), = g
g

(g x â V ), (g hx â V ),
g

= g = gh .


On the other side there is an isomorphism between the bundle (29) and the bundle (13): : E â
g

Vg -A,
-1

(32) (33)

(x, g ) = (x, g

.

This isomorphism is equivariant. The map (23) goes to the map of the direct image as the mapping F : A-A, (34)

F (x, g ) = =

(x, Fgx (g )) =
n

x,
k =0

k Fh

-1 -1 kg

(g ) . (35)

F (x, g ) = (x, Fgx (g ) It is clear that the map (34) goes to (17) under the isomorphism (32). So the following theorem holds:

Theorem 2 Consider the Fredholm representation of the group of the form (12). Let K(B ) be the element defined by the mapping (17) . Then p! (F ()) = where is the direct image in K ­theory.

K0 (B ),
0

(36) (37)

p! : K0 (E )-K (B )

Consider the action of the group on the Cartesian product E â V as the left action on the first factor and identical on the second one. Consider on the space E a metric with the property r(xg , y g )-0, |g |-. (38)

Let E be the completion of the space E (with respect to the metric r). Then any continuous mapping f : (E , E \E )-(B (V ), U (V )) defines the continuous family of the Fredholm representations (x), x E . By the theorem 1 the family (x) generates the equivariant family Fx
,y

(39)

: E â E â V -E â E â V . 5

(40)


and therefore the element F ((x)) K0 ((E â E )/ ) . Let be the direct image in K ­theory. Then p! (F ((x))) = ( The symmetric property of the element ( (1 u)(
x) x) x)

(41) (42)

p! : K0 ((E â E )/ ) -K (B â B )

0



K0 ().

(43)

holds: u K0 (B ). (44)

= (u 1)(

x)

K0 (B â B ),

2

Symmetric cohomology classes in H (M â M )

In the case when the space B is a compact manifold and the space E is compactified to the disk with extension of the action of , we obtain new proof of the Novikov conjecture in the case [3]. For that consider a closed orientable compact manifold M and a cohomology class w H (M â M ; Q). Assume that w satisfies a symmetric condition: w · (1 x) = (x 1) · w, x H (M ; Q). (45)

Our aim is to describe such symmetric elements w. Let xi , 0 i N be a (homogenious) basis in H (M ; Q), x0 = 1 H 0 (M ; Q), xN H n (M ; Q), dim M = n, xN , [M ] = 1. Then the multiplication tensor kj is defined by the formula i xi · xj = kj xk , i
k k0 = ki = i , i 0

(46) (47) (48)

N ij

= xi · xj , [M ] .

Associativity of the multiplication means that (xi · xj ) · xk = xi · (xj · xk ), that is lj sk xs = (lj xl ) · xk = i il = (xi · xj ) · xk = xi · (xj · xk ) = = xi · (l k xl ) = sl l k xs , j ij that is lj sk = sl l k . il ij 6 (51) (50) (49)


Represent the element w in the form w = µij xi xj . Then the condition (45) can be written as µil xi xl · xk = µlj xk · xl xj or or Assume that we have the case µN j = µj Then from (55) one has µil N = µlN i l . k lk or
i l µil N = 0 i l = i 0 = k . k k lk N j = 0 .

(52)

(53) (54) (55)

µil xi (jk xj ) = µlj (i l xi ) xj , k l µil jk = µlj i l . k l

(56) (57)

(58) N ij : (59)

This means that the matrix µ

ij

is the inverse matrix of the matrix µij = N ij
-1

.

The rest of relations from (55) are the consequence from asso ciativity (51): Ni µil jk = Ni µlj i l , i i k l
i l jk = Ni µlj i l , i k l


j k = µlj i l Ni , k i i N j k = N µlj i l Ni , jj i jj k i N jj


(60)

j k i

=

l j



i l Ni k i

,

N j k = i j Ni , jj i i k j k N = Ni i j , jj i k i Compare with (51): lj sk = sl l k . il ij (61) As a consequence from (59) one can obtain relations for symmetric elements of the form w = (x 1)(µij xi xj )(1 y ) = (µij xi xj )(1 xy ). (62) 7


3

Manifolds with b oundary

Assume now that a closed orientable compact manifold M has nonempty boundary M . Then one has the Poincare duality as a commutative diagram · · · - H
k +1

(M )

- H

j

k +1

(M , M ) D

-



Hk ( M ) D

-

i

· · · -

H

n-k

(M , M )

D

-

j



H

n-k

(M )

-

i

H

n-k

( M ) -





-

i

Hk (M ) D

-

j

Hk (M , M ) D

- · · ·

-





H

n+1-k

(M , M ) -

j



H

n+1-k

(M )

- · · ·

(63) The Poincare duality has the relation with multiplication in cohomology by the formula x y , [M ] = (x, Dy ) . (64) Here x H (M ) y H (M , M ), or y H (M ) operation defines the pairing : H i (M ) â H j (M , M )-H
i+j

x H (M , M ) and the (65)

(M , M )

such the pairing (65) generates the module structure over the ring H (M ) with the action on the H (M , M ): y (x1 · x2 ) = (y x1 ) · x2 = ±x1 · (y x2 ), y H (M ), x1 , x2 H (M , M ). (66)

Consider Cartesian square M â M . The boundary (M â M ) is the manifold which is splitted into the union (M â M ) = (M â M ) ( M â M ), (67) (M â M ) ( M â M ) = M â M . Consider a cohomology class w H (M â M , M â M ; Q). This cohomology can be described as a tensor product H (M â M , M â M ; Q) H (M , M ; Q) H (M ; Q). Assume that w satisfies a symmetric condition: w · (1 y ) = (y 1) · w H (M â M , M â M ), The result is similar to the manifolds without boundary: 8 y H (M ; Q). (69) (68)


Theorem 3 Let w H (M â M , M â M ) satisfy the symmetry condition (69). Let xi H (M , M ), yj H (M ) be bases, w = µij xi yj . Then µij = N ij where N = yi xj , [M , M ] . ij (72) To prove this consider (homogenious) bases in the cohomology groups H (M ; Q) and H (M , M ; Q): xi H (M , M ), yj H (M ). Let y0 = 1 H 0 (M ; Q) Q, xN H n (M , M ; Q) Q,
-1

(70) . (71)

dim M = n,

xN , [M , M ] = 1.

The pairing (65) is defined by the formula yi xj = kj xk . i If yi , yk H (M ) then such that
s 0k s yi yk = ik ys , s k

(73) (74)

=

s k0

= . The property (69) can be rewrited as µij xi yj yk = µij yk xi y
j

(75) (76) (77)

or
s µij j k xi ys = µij s i xs yj . k

Hence
s µij j k = µls i l . k

In particular when i = N one has
s µN j j k = µls N . kl

(78)

Assume similar to manifolds without boundary that the element w satisfies the condition (79):
j µN j = 0 .

(79) (80)

Hence
s k = µls N . kl

The results were partially supported by the grant of RFBR No 05-01-00923-a, the grant of the support for Scientific Schools No NSh-619.2003.1, and the grant of the foundation "Russian Universities" pro ject No. RNP.2.1.1.5055

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References
[1] A. C. Mishchenko. Metric approach to construction of Fredholm representations, (in russian). In International Conference devoted to memory of P.S.Alexandroff, Thesises,2 pages, Moscow, 1996. [2] A.S. Mishchenko. Homotopy invariants of non simply connected manifolds. 1. Rational invariants. Izv.Akad.Nauk.SSSR, 34(3):501­514, 1970. [3] F.T. Farrell and W.C. Hsiang. On novikov's conjecture for non­positively curved msnifolds, 1. Annals of Mathematics, 113:199­209, 1981.

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