Construction of Fredholm representations and a modification of the Higson-Roe corona

Construction of F redholm represen tations and a mo dification of the Higso n-Ro e corona A.S.Mishc henk o and N.T eleman Abstract The F redholm representation theory is well adapted to constru ction of homotopy inv arian ts of non simply connected manifolds on t he base of gener- alized Hirzebruch form ula [ σ ( M )] = h L ( M )c h A f ∗ ξ , [ M ] i ∈ K 0 A (pt) ⊗ Q , (1) where A = C ∗ [ π ] is the group C ∗ –algebra of the group π , π = π 1 ( M ). The bundle ξ ∈ K 0 A ( B π ) is canonical A –bundle, generated by the natural repre- senta tion π − → A . In [1] a natural family of the F redholm representations was construct ed that lead to a sy mm etric vector bu ndle on completion of the fundamental group with a modification of the Higson-Ro e corona when the completion is a closed ma nifold. Here w e will discuss a homology versio n of symmetry in the case when completion with a modification of the Higson-Roe corona is a manifo ld with b oundary . The results were devel op ed during the visit of the first author in Ancona on March, 2007. The second version is supplemented by details of consideration t h e case of manifol ds with b oundary . The F re dho lm representation theory is well adapted to construction of homotopy inv a riants of non simply connected manifolds on the ba se of genera lized Hir z ebruch formula [ σ ( M )] = h L ( M )ch A f ∗ ξ , [ M ] i ∈ K 0 A (pt) ⊗ Q (2) where A = C ∗ [ π ] is the group C ∗ –algebra o f the g roup π , π = π 1 ( M ). The bun- dle ξ ∈ K 0 A ( B π ) is ca nonical A – bundle, ge nerated by the natural representation π − → A . The ma p f : M − → B π induces the isomorphism of fundamen tal gr oups. The element [ σ ( M )] ∈ K 0 A (pt) is generated by noncommutativ e signatur e of the manifold M under exchange of ring s Z [ 1 2 ][ π ] ⊂ A . Let ρ = ( T 1 , F, T 2 ) be a F redholm repre s ent ation o f the gr oup π , tha t is a pa ir o f unitary repr esentations T 1 , T 2 : π − → B ( H ) and a F redholm op er ator F : H − → H , such that F T 1 ( g ) − T 2 ( g ) F ∈ Comp( H ) , g ∈ π . (3) Changing the algebra B ( H ) for the Ca lkin alg e bra K = B ( H ) / Comp( H ), one comes to the repr e s ent ation b ρ of the group π × Z to the Ca lk in algebra : b ρ : π × Z − →K , (4) 1 b ρ ( g , n ) = T 2 ( g ) F n = F n T 1 ( g ) , g ∈ π , n ∈ Z . (5) ρ ∗ : K A ( X ) Id ⊗ β − → K A b ⊗ C ( S 1 ) ( X × S 1 ) b ρ − → K K ( X × S 1 ) . (6) Here β ∈ K C ( S 1 ) ( S 1 ) is the canonica l element gener a ted by regular r e presentation of the gro up Z . Applying (6) to the Hir z ebruch formula (2) o ne has homotopy inv ariance o f c orre- sp onding higher signature. 1 Construction of F redholm represen tation Let T be the sum of finite co pies of regula r representation of the gr o up π , Φ be the blo ck diago nal op erator that is defined as matrix v alued function F ( g ) , g ∈ π : F ( g ) : V − → V . (7) Let H = M g ∈ π V g , V g ≡ V , (8) T h : H − → H , V g − → V hg . (9) The co ndition that Φ is F redholm op er ator means tha t k F ( g ) k ≤ C , k F − 1 ( g ) k ≤ C (10) for all g ∈ π except a finite subset. The condition (3 ) means that lim | g | − →∞ k F ( g ) − F ( hg ) k = 0 . (11) So if the pair ρ = ( T , Φ) (12) satisfies co nditions (10), (1 1 ), then ρ is F redholm representation o f the group π . Consider universal cov ering f B π of classifying space B π endow ed with left a ction of the group π . In cor resp ondence to the constr uction by [2] the vector bundle g ener- ated by the r epresentation ρ on the s pace B π ca n b e repre sented as an equiv ar iant contin uous family o f F redholm op erato r s o n the space E π = f B π . The prop er ty of equiv aria nce co rresp onds to diago nal action on Cartesian pro duct T h : E π × H , ( x, ξ ) − → ( hx, T h ( ξ )) . (13) Namely , let the spa ce B π b e endowed with a structure o f simplicial space and E π = f B π b e endow ed with the structur e of simplicia l structure derived from the covering E π = f B π p − → B π (14) 2 Let { x i } b e the family of vertices of E π = f B π , one from each of or bits of the action of π . Then each simplex σ of E π = f B π is defined co mpletely by their vertices σ = ( h 0 x i 0 , . . . , h n x i n ) , h 0 , . . . , h n ∈ π . (15) An y p oint x ∈ σ is uniquely defined a s a co nv ex linear combination of vertices x = n X k =0 λ k h k x i k (16) Then the eq uiv ariant family o f F r edholm op erato rs which corres p o nds to the F r ed- holm r epresentation ρ (12) one can define by the formula Φ x = Φ x ( ρ ) = = n X k =0 λ k Φ h k x i k = n X k =0 λ k T h k Φ x i k T − 1 h k = = n X k =0 λ k T h k Φ T − 1 h k . (17) Hence (Φ x ) g = n X k =0 λ k F h − 1 k g . (1 8) It is c le ar that the fa mily (17) is equiv a riant. Indeed, hx = n X k =0 λ k hh k x i k . (19) Hence Φ hx = n X k =0 λ k T hh k Φ T − 1 hh k = T h n X k =0 λ k T h k Φ T − 1 h k ! T − 1 h = T h Φ x T − 1 h . (20) Also it is cle a r that the op era tors (17) ar e F redholm by (18) , (11 ) and (1 0). On the other side the op erator s (7) genera te the co nt inuous family F x : V − → V , x ∈ E π (21) using formula F x = n X k =0 λ k F ( h − 1 k ) . (22) 3 This family one can cons ide r a s a linea r mapping of the trivial bundle: F x : E π × V − → E π × V . (23) Consider the universal cov ering p : E π − → B π . (24) Denote K i ( E π ) = lim ← K i c ( p − 1 ( X )) , (25) where the inv er se limit takes by the family of a ll compact s ubs ets X ⊂ B π . Theorem 1 The map (23) defines the element F ( ρ ) ∈ K 0 ( E π ) . (26) Consider the direct ima ge of the bundle (23) ov e r B π : A − → B π , (27) where the fibre is the direct sum of the fib ers of the bundle (23) ov er each orbit of the actio n of the gr oup π in the space E π . The total space A is defined a s A = { ( u, ξ ) : u ∈ B π , ξ ∈ M x ∈ u ( x × V ) } . (28) Let e A − → E π (29) be the inv e rse image o f the bundle (27). The total space e A is defined as e A = { ( x, ξ ) : x ∈ E π, ξ ∈ M y ∈ [ x ] ( y × V ) } = { ( x, ξ ) , x ∈ E π , ξ ∈ M g ∈ π ( g x × V ) } . (30) Define the action o f the group π on the to tal space e A by the formula f h ( x, ξ ) = ( hx, η ) , ξ = ⊕ ξ g ∈ M g ∈ π ( g x × V ) , η = ⊕ η g ∈ M g ∈ π ( g hx × V ) , η g = ξ gh . It is c le ar that A = e A/π . (31) 4 On the other side there is an isomorphism b e tw een the bundle (29) and the bundle (13): ϕ : E π × M g ∈ π V g − → e A, (32) ϕ ( x, ⊕ ξ g ) = ( x, ⊕ ξ g − 1 . (33) This isomorphism is eq uiv ariant. The map (23) go es to the map of the dire c t imag e as the mapping e F : e A − → e A, (34) e F ( x, ⊕ ξ g ) = ( x, ⊕ F gx ( ξ g )) = = x, ⊕ n X k =0 λ k F h − 1 k g − 1 ( ξ g ) ! . e F ( x, ⊕ ξ g ) = ( x, ⊕ F gx ( ξ g ) (35) It is c le ar that the ma p (34) go es to (17) under the isomor phism (32). So the following theo rem holds: Theorem 2 Consider the F r e dholm r epr esentation of the gr oup π of the form (12). L et ξ ρ ∈ K ( B π ) b e the element define d by the mapping (17) . Then p ! ( F ( ρ )) = ξ ρ ∈ K 0 ( B π ) , (36) wher e p ! : K 0 ( E π ) − → K 0 ( B π ) (37) is the dir e ct image in K –the ory. Consider the a ction of the gr oup π o n the Ca rtesian pro duct E π × V as the left action on the firs t factor and identical on the seco nd one. Consider o n the spa ce E π a metr ic with the pr op erty r ( xg , y g ) − → 0 , | g |− →∞ . (38) Let E π b e the completion o f the spa c e E π (with resp ect to the metric r ). Then any contin uous mapping f : ( E π , E π \ E π ) − → ( B ( V ) , U ( V )) (39) defines the contin uous family of the F redholm represe ntations ρ ( x ), x ∈ E π . By the theorem 1 the family ρ ( x ) genera tes the equiv a riant family F x,y : E π × E π × V − → E π × E π × V . (40) 5 and ther efore the element F ( ρ ( x )) ∈ K 0 (( E π × E π ) /π ) . (41) Let p ! : K 0 (( E π × E π ) /π ) − → K 0 ( B π × B π ) (42) be the direct image in K –theory . Then p ! ( F ( ρ ( x ))) = ξ ρ ( x ) ∈ K 0 () . (43) The sy mmetr ic prop erty of the element ξ ρ ( x ) holds: (1 ⊗ u ) ξ ρ ( x ) = ( u ⊗ 1) ξ ρ ( x ) ∈ K 0 ( B π × B π ) , u ∈ K 0 ( B π ) . (44) 2 Symmetric cohomology classes in H ∗ ( M × M ) In the ca se when the spa c e B π is a co mpact manifold and the space E π is com- pactified to the disk with extensio n of the a ction of π , we o bta in new pr o of o f the Novik ov conjecture in the case [3]. F or that consider a clo sed or ie ntable compact ma nifold M a nd a coho mology c lass w ∈ H ∗ ( M × M ; Q ). Ass ume that w sa tisfies a s ymmetric condition: w · (1 ⊗ x ) = ( x ⊗ 1) · w , x ∈ H ∗ ( M ; Q ) . (45) Our aim is to descr ibe such symmetric elements w . Let x i , 0 ≤ i ≤ N b e a (homogenious) ba sis in H ∗ ( M ; Q ), x 0 = 1 ∈ H 0 ( M ; Q ), x N ∈ H n ( M ; Q ), dim M = n , h x N , [ M ] i = 1. Then the multiplication tensor λ k ij is defined by the formula x i · x j = λ k ij x k , (46) λ k i 0 = λ k 0 i = δ k i , (47) λ N ij = h x i · x j , [ M ] i . (48) Asso ciativity o f the multiplication means that ( x i · x j ) · x k = x i · ( x j · x k ) , (49) that is λ l ij λ s lk x s = ( λ l ij x l ) · x k = = ( x i · x j ) · x k = x i · ( x j · x k ) = = x i · ( λ l j k x l ) = λ s il λ l j k x s , (50) that is λ l ij λ s lk = λ s il λ l j k . (51) 6 Represent the element w in the for m w = µ ij x i ⊗ x j . (52) Then the condition (4 5 ) can b e wr itten as µ il x i ⊗ x l · x k = µ lj x k · x l ⊗ x j (53) or µ il x i ⊗ ( λ j lk x j ) = µ lj ( λ i kl x i ) ⊗ x j , (54) or µ il λ j lk = µ lj λ i kl . (55) Assume that we have the ca se µ N j = µ j N = δ j 0 . (56) Then fro m (55) one has µ il λ N lk = µ lN λ i kl . (57) or µ il λ N lk = δ l 0 λ i kl = λ i k 0 = δ i k . (58) This mea ns that the matrix k µ ij k is the inv erse matrix of the matrix k λ N ij k : k µ ij k = k λ N ij k − 1 . (59) The r e st of rela tions from (5 5) are the co nsequence from asso ciativity (51): λ N i ′ i µ il λ j lk = λ N i ′ i µ lj λ i kl , δ i ′ l λ j lk = λ N i ′ i µ lj λ i kl , λ j i ′ k = µ lj λ i kl λ N i ′ i , λ N j j ′ λ j i ′ k = λ N j j ′ µ lj λ i kl λ N i ′ i , λ N j j ′ λ j i ′ k = δ l j ′ λ i kl λ N i ′ i , λ N j j ′ λ j i ′ k = λ i kj ′ λ N i ′ i , λ j i ′ k λ N j j ′ = λ N i ′ i λ i kj ′ , (60) Compare with (51): λ l ij λ s lk = λ s il λ l j k . (61) As a consequence from (59) one can obtain relations for symmetric ele men ts of the form w = ( x ⊗ 1)( µ ij x i ⊗ x j )(1 ⊗ y ) = ( µ ij x i ⊗ x j )(1 ⊗ xy ) . (62) 7 3 Manifolds with b oundary Assume now that a clo s ed orientable compact manifold M has nonempt y b oundary ∂ M . Then one has the Poincare duality as a commutativ e diagr am · · · − → H k +1 ( M ) j − → H k +1 ( M , ∂ M ) δ − → H k ( ∂ M ) i − → x   D x   D x   D · · · − → H n − k ( M , ∂ M ) j ∗ − → H n − k ( M ) i ∗ − → H n − k ( ∂ M ) δ ∗ − → i − → H k ( M ) j − → H k ( M , ∂ M ) − → · · · x   D x   D δ ∗ − → 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 h x ∧ y , [ M ] i = ( x, D y ) . (64) Here x ∈ H ∗ ( M ) y ∈ H ∗ ( M , ∂ M ), or y ∈ H ∗ ( M ) x ∈ H ∗ ( M , ∂ M ) and the op eration ∧ defines the pairing ∧ : H i ( M ) × H j ( M , ∂ M ) − → H i + j ( M , ∂ M ) (65) such the pairing (65) gener ates the mo dule str ucture ov e r the r ing H ∗ ( M ) with the action on the H ∗ ( M , ∂ M ): y ∧ ( x 1 · x 2 ) = ( y ∧ x 1 ) · x 2 = ± x 1 · ( y ∧ x 2 ) , y ∈ H ∗ ( M ) , x 1 , x 2 ∈ H ∗ ( M , ∂ M ) . (66) Consider Cartesian square M × M . The b oundary ∂ ( M × M ) is the manifold which is splitted into the union ∂ ( M × M ) = ( M × ∂ M ) ∪ ( ∂ M × M ) , ( M × ∂ M ) ∩ ( ∂ M × M ) = ∂ M × ∂ M . (67) Consider a cohomolo gy class w ∈ H ∗ ( M × M , ∂ M × M ; Q ). This co homology can be descr ib e d as a tensor pr o duct H ∗ ( M × M , ∂ M × M ; Q ) ≈ H ∗ ( M , ∂ M ; Q ) ⊗ H ∗ ( M ; Q ) . (68) Assume that w s atisfies a symmetric co ndition: w · (1 ⊗ y ) = ( y ⊗ 1 ) · w ∈ H ∗ ( M × M , ∂ M × M ) , y ∈ H ∗ ( M ; Q ) . (69) The r e sult is similar to the manifolds without b o unda ry: 8 Theorem 3 L et w ∈ H ∗ ( M × M , ∂ M × M ) satisfy the symmetry c ondition (69). L et x i ∈ H ∗ ( M , ∂ M ) , y j ∈ H ∗ ( M ) b e b ases, w = µ ij x i ⊗ y j . (70) Then k µ ij k = k λ N ij k − 1 . (71) wher e λ N ij = h y i ∧ x j , [ M , ∂ M ] i . (72) T o pr ov e this consider (homogenio us) bases in the cohomo logy gr oups H ∗ ( M ; Q ) and H ∗ ( M , ∂ M ; Q ): x i ∈ H ∗ ( M , ∂ M ) , y j ∈ H ∗ ( M ) . Let y 0 = 1 ∈ H 0 ( M ; Q ) ≈ Q , x N ∈ H n ( M , ∂ M ; Q ) ≈ Q , dim M = n, h x N , [ M , ∂ M ] i = 1 . The pair ing (65) is defined by the formula y i ∧ x j = λ k ij x k . ( 73) If y i , y k ∈ H ∗ ( M ) then y i ∧ y k = ν s ik y s , (7 4) such that ν s 0 k = ν s k 0 = δ s k . The prop er t y (69) ca n b e rewrited a s µ ij x i ⊗ y j y k = µ ij y k ∧ x i ⊗ y j (75) or µ ij ν s j k x i ⊗ y s = µ ij λ s ki x s ⊗ y j . (76) Hence µ ij ν s j k = µ ls λ i kl . ( 77) In par ticular when i = N one has µ N j ν s j k = µ ls λ N kl . (78) Assume similar to manifolds without b oundar y that the element w satis fie s the condition (79): µ N j = δ j 0 . (79) Hence δ s k = µ ls λ N kl . (80) The results were partially supp orted by the gra nt of RFBR No 05-0 1-009 23-a, the grant of the s uppo rt for Sc ie n tific Schoo ls No NSh-619.2 003.1 , and the gra n t of the foundation ”Russia n Universities” pr o ject No. RNP .2.1.1.5 055 9 References [1] A. C. Mishchenko. 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