Almost flat K-theory of classifying spaces
AAlmost flat K-theory of classifying spaces
Jos´e R. Carri´on ∗ Department of MathematicsPenn State UniversityUniversity Park, PA, 16803United [email protected] Marius Dadarlat † Department of MathematicsPurdue UniversityWest Lafayette, IN, 47907United [email protected] 30, 2018
Abstract
We give a rigorous account and prove continuity properties for the correspondence betweenalmost flat bundles on a triangularizable compact connected space and the quasi-representationsof its fundamental group. For a discrete countable group Γ with finite classifying space B Γ, westudy a correspondence between between almost flat K-theory classes on B Γ and group ho-momorphism K ( C ∗ (Γ)) → Z that are implemented by pairs of discrete asymptotic homomor-phisms from C ∗ (Γ) to matrix algebras. Connes, Gromov and Moscovici [3] developed and used the concepts of almost flat bundle, almostflat K-theory class and group quasi-representation as tools for proving the Novikov conjecture forlarge classes of groups. For a compact manifold M, it was shown in [3] that the signature withcoefficients in a (sufficiently) almost flat bundle is a homotopy invariant. Moreover, the authorsindicate that they have a reformulation of the notion of almost flatness to bundles with infinitedimensional fibers which allows them to show that if Γ is a countable discrete group such that allclasses of K ( B Γ) are almost flat (up to torsion), then Γ satisfies the Novikov conjecture [3, Sec.6]. The problem of constructing nontrivial almost flat K-theory classes is interesting in itself. Sup-pose that the classifying space B Γ of a countable discrete group Γ admits a realization as a finitesimplicial complex. Using results of Kasparov [13], Yu [24] and Tu [23], the second-named authorshowed in [5] that if Γ is coarsely embeddable in a Hilbert space and the full group C*-algebra C ∗ (Γ) is quasidiagonal, then all classes in K ( B Γ) are almost flat.Inspired by [3], in this paper we investigate the correspondence between between the almostflat classes in K ( B Γ) and the group homomorphisms h : K ( C ∗ (Γ)) → Z that are implemented bypairs of discrete asymptotic homomorphisms { π ± n : C ∗ (Γ) → M k ( n ) ( C ) } ∞ n =1 , in the sense that h ( x ) ≡ ( π + n ) (cid:93) ( x ) − ( π − n ) (cid:93) ( x ) for x ∈ K ( C ∗ (Γ)); see Definition 8.1. It turns out that this correspondence ∗ Partially supported by NSF Postdoctoral Fellowship † Partially supported by NSF grant a r X i v : . [ m a t h . OA ] M a r s one-to-one (modulo torsion) if the full assembly map µ : K ( B Γ) → K ( C ∗ (Γ)) is bijective. SeeTheorem 8.7 and its generalization with coefficients Theorem 3.1.We take this opportunity to give a self-contained presentation of the correspondence (and itscontinuity properties) between almost flat bundles on a connected triangularizable compact spaceand the quasi-representations of its fundamental group; see Theorems 3.1, 3.3. As far as we can tell,while this correspondence was more or less known to the experts, it has not been well documentedin the literature. We rely on work of Phillips and Stone [19, 20]. These authors studied topologicalinvariants associated with lattice gauge fields via a construction which associates an almost flatbundle to a lattice gauge field with controlled distortion and small modulus of continuity.The following terminology is useful for further discussion of our results. If k = ( k ( n )) ∞ n =1 isa sequence of natural numbers, we write Q k = (cid:81) ∞ n =1 M k ( n ) ( C ) / (cid:80) ∞ n =1 M k ( n ) ( C ). Recall that aseparable C*-algebra A is MF is it embeds as a C*-subalgebra of Q k for some k . In other words,there are sufficiently many ∗ -homomorphisms A → Q k to capture the norm of the elements in A ,[1]. By analogy, let us say that a separable C*-algebra A is “K-theoretically MF” if there existsufficiently many ∗ -homomorphisms A → Q k to capture the K-theory of A in the following sense:For any homomorphism h : K ( C ∗ (Γ)) → Z there exist k and two ∗ -homomorphisms π ± : A → Q k such that π + ∗ ( x ) − π −∗ ( x ) = h ∞ ( x ) for all x ∈ K ( A ). Here we identify K ( Q k ) with a subgroupof (cid:81) ∞ n =1 Z / (cid:80) ∞ n =1 Z , and h ∞ ( x ) is the coset of element ( h ( x ) , h ( x ) , h ( x ) , . . . ) ∈ (cid:81) ∞ n =1 Z . With thisterminology, Theorem 8.7 reads as follows: Theorem.
Let Γ be a discrete countable group whose classifying space B Γ is a finite simplicial com-plex. If the full assembly map µ : K ( B Γ) → K ( C ∗ (Γ)) is bijective, then the following conditionsare equivalent: (1) All elements of K ( B Γ) are almost flat modulo torsion; (2) C ∗ (Γ) is K-theoretically MF. By a result of Higson and Kasparov [11], the assumptions of this theorem are satisfied by groupsΓ with the Haagerup property and finite classifying spaces.We note that a separable quasidiagonal C*-algebra that satisfies the UCT is K-theoreticallyMF by [5, Prop. 2.5]. We suspect that a similar result holds for MF algebras satisfying the UCT.
Let X be a connected compact metric space that admits a finite triangulation. This means that X is the geometric realization of some connected finite simplicial complex Λ, written X = | Λ | .Let Γ = π ( X ) be the fundamental group of X . Let A be a unital C ∗ -algebra. We establish acorrespondence between quasi-representations of Γ into GL( A ) and almost flat bundles over X with fiber A and structure group GL( A ). To be more accurate, this correspondence is betweenalmost unitary quasi-representations and almost unitary principal bundles. It is convenient to workin a purely combinatorial context. Following [19], we call the combinatorial version of an almostflat bundle an almost flat coordinate bundle . See Definition 2.5. We prove the equivalence of theseconcepts in Proposition 7.2.For our combinatorial approach we begin with a finite simplicial complex Λ endowed with someadditional structure, as follows. The vertices of Λ are denoted by i, j, k with possible indices. Theset of k -simplices of Λ is denoted by Λ ( k ) . We fix a maximal tree T ⊂ Λ and a root vertex i .2 he edge-path group and quasi-representations It will be convenient to view the fundamental group of the geometric realization of Λ as the edge-path group E (Λ , i ) of Λ. The groups π ( | Λ | ) and E (Λ , i ) are isomorphic and we will simply writeΓ for either group.Once Λ and T are fixed so is the following standard presentation of Γ in terms of the edges ofΛ (see e.g. [22, Sections 3.6–3.7]): Γ is isomorphic to the group generated by the collection of alledges (cid:104) i, j (cid:105) of Λ subject to the following relations: • if (cid:104) i, j (cid:105) is an edge of T , then (cid:104) i, j (cid:105) = 1; • if i, j, k are vertices of a simplex of Λ, then (cid:104) i, j (cid:105)(cid:104) j, k (cid:105) = (cid:104) i, k (cid:105) .We should point out that in the second relation i , j , and k are not necessarily distinct; consequently (cid:104) i, j (cid:105)(cid:104) j, i (cid:105) = (cid:104) i, i (cid:105) . Since (cid:104) i, i (cid:105) = (cid:104) i (cid:105) belongs to T one has (cid:104) i, j (cid:105) − = (cid:104) j, i (cid:105) as expected. Let F Λ be the free group generated by the edge set of Λ and let q : F Λ → Γ be thegroup epimorphism corresponding the presentation of Γ just described.Write F Λ for the image under q of the edge set of Λ; this is a symmetric generating set for Γ.Write γ ij = q ( (cid:104) i, j (cid:105) ) for the elements of F Λ . Let R ⊂ F Λ be the collection of all relators: R := {(cid:104) i, j (cid:105) | (cid:104) i, j (cid:105) is an edge of T } ∪ {(cid:104) i, j (cid:105)(cid:104) j, k (cid:105)(cid:104) i, k (cid:105) − | i, j, k are vertices of a simplex of Λ } . Choose a set-theoretic section s : Γ → F Λ of q that takes the neutral element of Γ to the neutralelement of F Λ . This section s will remain fixed for the rest of the paper.We introduce one last notation before the definition. If A is a unital C*-algebra and δ >
0, setU( A ) δ = { v ∈ A : dist( v, U( A )) < δ } . Note that U( A ) δ ⊆ GL( A ) if δ < Let A be a unital C*-algebra.(1) Let F ⊂
Γ be finite and let 0 < δ <
1. A function π : Γ → GL( A ) is an ( F , δ ) -representation of Γ if(a) π ( γ ) ∈ U( A ) δ for all γ ∈ F ;(b) (cid:107) π ( γγ (cid:48) ) − π ( γ ) π ( γ (cid:48) ) (cid:107) < δ for all γ, γ (cid:48) ∈ F ;(c) π ( e ) = 1 A , where e is the neutral element of Γ . (2) Define a pseudometric d on the set of all bounded maps Γ → A by d ( π, π (cid:48) ) = max γ ∈F Λ (cid:107) π ( γ ) − π (cid:48) ( γ ) (cid:107) . We may sometimes refer to an ( F , δ )-representation as a “quasi-representation” without speci-fying F or δ . In most of the paper we will take F = F Λ .3 jk c σi c σj c σk (a)
000 01 234 214 3 (b)
Figure 1: (a) Dual cell blocks in a simplex σ = (cid:104) i, j, k (cid:105) . (b) A triangulation of T with the dual cellstructure highlighted. The dual cover and almost flat coordinate bundles
Once Λ is fixed, so is a cover C Λ of | Λ | , called the dual cover. We recall its definition (borrowingheavily from the appendix of [20]). Let σ = (cid:104) , . . . , r (cid:105) be a simplex of Λ. For i ∈ { , . . . , r } , the dual cell block c σi ,dual to i in σ , is defined in terms of the barycentric coordinates ( t o , . . . , t r ) by c σi = { ( t , . . . , t r ) | t i ≥ t j for all j } ⊂ | Λ | . The dual cell c i , dual to the vertex i , is the union of cell blocks dual to i : c i = ∪{ c σi | i ∈ σ } . The dual cover C Λ is the collection of all dual cells. (See Figure 1.) We usually write c ij for the intersection c i ∩ c j , c ijk for c i ∩ c j ∩ c k etc. The barycenterof a simplex σ is denoted ˆ σ . Note that (cid:104) i, j (cid:105) ˆ ∈ c ij . Recall that we have fixed a unital C ∗ -algebra A .(1) An ε -flat GL( A ) -coordinate bundle on Λ is a collection of continuous functions v = { v ij : c ij → GL( A ) | (cid:104) i, j (cid:105) ∈ Λ (1) } satisfying:(a) v ij ( x ) ∈ U( A ) ε for all x ∈ c ij and all (cid:104) i, j (cid:105) ∈ Λ (1) ;(b) v ij ( x ) = v ji ( x ) − for all x ∈ c ij and all (cid:104) i, j (cid:105) ∈ Λ (1) ;(c) v ik ( x ) = v ij ( x ) v jk ( x ) for all x ∈ c ijk and all (cid:104) i, j, k (cid:105) ∈ Λ (2) ; and(d) (cid:107) v ij ( x ) − v ij ( y ) (cid:107) < ε for all x, y ∈ c ij and all (cid:104) i, j (cid:105) ∈ Λ (1) .42) Define a metric d on the set of all GL( A )-coordinate bundles on Λ by d ( v , v (cid:48) ) = max (cid:104) i,j (cid:105)∈ Λ max x ∈ c ij (cid:107) v ij ( x ) − v (cid:48) ij ( x ) (cid:107) . We may sometimes refer to an ε -flat coordinate bundle as an “almost flat coordinate bundle”without specifying ε . We think of an almost flat coordinate bundle v as a collection of transitionfunctions defining a bundle over | Λ | , with fiber A , that has “small” curvature. We substantiate thispoint of view in Section 7 where we show that there are positive numbers ε , ν, r that depend onlyon Λ such that for any ε -flat GL( A )-coordinate bundle v ij on Λ, with ε < ε , there is a 1- ˇCechcocycle (cid:101) v ij : V i ∩ V j → U ( A ) rε that extends v ij to prescribed open sets and (cid:101) v ij is rε -flat in the sensethat (cid:107) (cid:101) v ij ( x ) − (cid:101) v ij ( x (cid:48) ) (cid:107) < rε for all x ∈ V i ∩ V j . Here V i = { x ∈ | Λ | : dist( x, c i ) < ν } . Since the sets V i are open, the usual gluing construction based on (cid:101) v ij defines a locally trivial (almost flat) bundle.These objects are closely related to almost flat K -theory classes; see Section 8 for details. We state the main results on this topic. The proofs are given in subsequent sections.
Let Λ be a finite connected simplicial complex with fundamental group Γ . Thereexist positive numbers C , δ , and ε such that the following holds.If A is a unital C ∗ -algebra, then there are functions (cid:26) ε -flat GL( A ) coordinate bundles on Λ (cid:27) α −−−→←−−− β (cid:26) ( F Λ , δ ) -representationsof Γ to GL( A ) (cid:27) such that: (1) if < ε < ε and v is an ε -flat GL( A ) -coordinate bundle on Λ , then α ( v ) is an ( F Λ , C ε ) -representation of Γ to GL( A ) ; and (2) if < δ < δ and π : Γ → GL( A ) is an ( F Λ , δ ) -representation, then β ( π ) is a C δ -flat GL( A ) -coordinate bundle on Λ .Moreover: (3) if < ε < ε and v and v (cid:48) are ε -flat GL( A ) -coordinate bundles, then d ( α ( v ) , α ( v (cid:48) ))
Let Λ be a finite connected simplicial complex with fundamental group Γ . Thereexist positive numbers C , ε and δ such that the following holds for any unital C ∗ -algebra A . (1) If < ε < ε and v is a normalized ε -flat GL( A ) -coordinate bundle on Λ , then d (cid:0) ( β ◦ α )( v ) , v (cid:1) ≤ C ε. (2) If < δ < δ and π : Γ → GL( A ) is an ( F Λ , δ ) -representation, then d (cid:0) ( α ◦ β )( π ) , π (cid:1) ≤ C δ. In this section we construct the map α announced in Section 3. It is a combinatorial versionof a construction due to Connes-Gromov-Moscovici [3] involving parallel transport on a smoothmanifold.Let v = { v ij : c ij → GL( A ) } be an ε -flat coordinate bundle on Λ. We will define a quasi-representation α ( v ) : Γ → GL( A ) with properties described in Proposition 4.6. Recall that the barycenter of a 1-simplex (cid:104) i, j (cid:105) ∈ Λ (1) is written (cid:104) i, j (cid:105) ˆ. For such a1-simplex, let ˚ v ij = v ij ( (cid:104) i, j (cid:105) ˆ) ∈ GL( A ). For a path I = ( i , . . . , i m ) of vertices in Λ, let˚ v I = ˚ v i i . . . ˚ v i m − i m . Define a group homomorphism ˜ π = ˜ π v : F Λ → GL( A ) as follows. If (cid:104) i, j (cid:105) ∈ Λ (1) ,let I = ( i , . . . , i ) be the unique path along T from i to i and J = ( i , . . . , j ) be the unique pathfrom i to j . Set ˜ π (cid:0) (cid:104) i, j (cid:105) (cid:1) = ˚ v I ˚ v ij ˚ v − J . (4.1)Finally, set α ( v ) = ˜ π ◦ s : Γ → GL( A ) , where s is the set theoretic section of q : F Λ → Γ that was fixed in Notation 2.1.
Let ν > and < ε < . If x , . . . , x m ∈ A , u , . . . , u m ∈ U( A ) and (cid:107) x i − u i (cid:107) < νε for all i , then (cid:107) x . . . x m − u . . . u m (cid:107) < (1 + ν ) m ε . In particular, if x , . . . , x m ∈ U( A ) ε , then x . . . x m ∈ U( A ) m ε .Proof. For i ∈ { , . . . , m } , let u i ∈ U( A ) be such that (cid:107) x i − u i (cid:107) < ε . One checks that (cid:107) x . . . x m − u . . . u m (cid:107) < m (cid:88) i =1 (cid:107) x i − u i (cid:107) (1 + νε ) m − i < (1 + ν ) m ε. (cid:3) For g ∈ F Λ , let (cid:96) ( g ) ∈ Z ≥ be the word length of g with respect to the generatingset Λ (1) . We denote by L the length (number of edges) of a longest path in Λ that starts at theroot i and does not repeat any edge. If g ∈ F Λ , then dist(˜ π ( g ) , U( A )) < L + (cid:96) ( g ) ε .Proof. Equation (4.1) implies that ˜ π ( (cid:104) i, j (cid:105) ) is a product of at most 3 L elements of U( A ) ε for anyedge (cid:104) i, j (cid:105) of Λ. It follows from Lemma 4.3 that dist(˜ π ( (cid:104) i, j (cid:105) ) , U( A )) < L ε . Another applicationof Lemma 4.3 ends the proof. (cid:3) .6 Proposition. There is a constant C (cid:48) > , depending only on Λ , T , i , and s , such that if v isan ε -flat GL( A ) -coordinate bundle on Λ , then α ( v ) is an ( F Λ , C (cid:48) ε ) -representation of Γ on GL( A ) .Proof. Write π := α ( v ). First we define a few constants so the proof will run more smoothly.Let (cid:96) = max { (cid:96) ( s ( γ )) : γ ∈ F Λ ∪ F Λ · F Λ } .If γ, γ (cid:48) ∈ Γ, then s ( γ ) s ( γ (cid:48) ) s ( γγ (cid:48) ) − belongs to the kernel of q : F Λ → Γ, that is, to the normalsubgroup generated by the set of relators R . (See Notation 2.1.) For each pair γ, γ (cid:48) ∈ F Λ chooseand fix a representation s ( γ ) s ( γ (cid:48) ) s ( γγ (cid:48) ) − = m ( γ,γ (cid:48) ) (cid:89) n =1 x n r n x − n (4.2)with { x n } ⊂ F Λ and { r n } ⊂ R . Let m = max { m ( γ, γ (cid:48) ) : γ, γ (cid:48) ∈ F Λ } and let (cid:96) be the maximumof the lengths (cid:96) ( x n ) of all the elements x n that appear in equation (4.2) for all pairs γ, γ (cid:48) ∈ F Λ .Finally, let C (cid:48) = 2 (15 L +2 (cid:96) ) m · L + (cid:96) .For any γ, γ (cid:48) ∈ F Λ , we show that (cid:107) π ( γ ) π ( γ (cid:48) ) − π ( γγ (cid:48) ) (cid:107) < C (cid:48) ε . Using Lemma 4.5 we note that (cid:107) ˜ π ( s ( γγ (cid:48) )) (cid:107) < L + (cid:96) ε < L + (cid:96) and hence (cid:107) π ( γ ) π ( γ (cid:48) ) − π ( γγ (cid:48) ) (cid:107) ≤ (cid:107) ˜ π ( s ( γ ) s ( γ (cid:48) ) s ( γγ (cid:48) ) − ) − (cid:107)(cid:107) ˜ π ( s ( γγ (cid:48) )) (cid:107) (4.3) ≤ t + (cid:96) (cid:107) ˜ π ( s ( γ ) s ( γ (cid:48) ) s ( γγ (cid:48) ) − ) − (cid:107) . Now, r n ∈ R implies that either r n = (cid:104) i, j (cid:105) is an edge of T in which case ˜ π ( r n ) = 1 or r n = (cid:104) i, j (cid:105)(cid:104) j, k (cid:105)(cid:104) i, k (cid:105) − for some vertices i, j, k , and hence˜ π ( r n ) = ˚ v I ˚ v ij ˚ v − J · ˚ v J ˚ v jk ˚ v − K · ˚ v K ˚ v − ik ˚ v − I = ˚ v I · ˚ v ij ˚ v jk ˚ v − ik · ˚ v − I . Let t be the barycenter (cid:104) i, j, k (cid:105) ˆ. Since v ij , v jk , and v − ik are ε -constant with norm ≤ ε , we getthat (cid:107) ˜ π ( r n ) − (cid:107) ≤ (cid:107) ˚ v I · ˚ v ij ˚ v jk ˚ v − ik · ˚ v − I − ˚ v I · v ij ( t ) v jk ( t ) v − ik ( t ) · ˚ v − I (cid:107) < (cid:107) ˚ v I (cid:107)(cid:107) ˚ v − I (cid:107) (1 + ε ) ε< L +4 ε ≤ L ε. By Lemma 4.5 (cid:107) ˜ π ( x n ) (cid:107) , (cid:107) ˜ π ( x − n ) (cid:107) ≤ L + (cid:96) ( x n ) ε < L + (cid:96) . Therefore (cid:107) ˜ π ( x n )˜ π ( r n )˜ π ( x − n ) − (cid:107) < (cid:107) ˜ π ( x n ) (cid:107)(cid:107) ˜ π ( x − n ) (cid:107)(cid:107) ˜ π ( r n ) − (cid:107) ≤ L +2 (cid:96) ε. Because ˜ π ( s ( γ ) s ( γ (cid:48) ) s ( γγ (cid:48) ) − ) = m ( γ,γ (cid:48) ) (cid:89) n =1 ˜ π ( x n )˜ π ( r n )˜ π ( x − n ) , applying Lemma 4.5 again we get (cid:107) ˜ π ( s ( γ ) s ( γ (cid:48) ) s ( γγ (cid:48) ) − ) − (cid:107) < (1 + 2 L +2 (cid:96) ) m ε ≤ (15 L +2 (cid:96) ) m ε. (4.4)Combined with (4.3), this proves that for all γ, γ (cid:48) ∈ F Λ (cid:107) π ( γ ) π ( γ (cid:48) ) − π ( γγ (cid:48) ) (cid:107) < L + (cid:96) · (15 L +2 (cid:96) ) m ε = C (cid:48) ε. We must also prove that π ( γ ) ∈ U( A ) C (cid:48) ε if γ ∈ F Λ . This follows immediately from Lemma 4.5since π ( γ ) = ˜ π ( s ( γ )) and (cid:96) ( γ ) ≤ (cid:96) . (cid:3) There exists
K > , depending only on Λ , T , i , and s , such that the following holds.Suppose v is an ε -flat GL( A ) -coordinate bundle on Λ . Let ˜ π = ˜ π v be as in Definition 4.2. If (cid:104) i, j (cid:105) ∈ Λ (1) , then g ij := s ( γ ij ) · (cid:104) i, j (cid:105) − ∈ ker q and (cid:107) ˜ π ( g ij ) − A (cid:107) < Kε. Suppose v = { v ij } is an ε -flat GL( A ) -coordinate bundle on Λ . Then there exista constant C > , depending only on Λ , and elements λ i ∈ GL( A ) such that the coordinate bundle w = { w ij } defined by w ij = λ i v ij λ − j is normalized and Cε -flat; v and w yield isomorphic bundleson | Λ | .Proof. Recall that L is the length of the longest path in T that starts at the root i and hasno backtracking. We show that C := 4 L +1 verifies the statement. For each vertex i of Λ, let I = ( i , . . . , i ) be the unique path along T from i to i and using notation as in 4.1 and 4.2 set λ i := ˚ v I . Thus w ij = ˚ v I v ij ˚ v − J . Then clearly w ij ( (cid:104) i, j (cid:105) ˆ) = 1 A for all (cid:104) i, j (cid:105) ∈ T (1) . Moreover (cid:107) λ i (cid:107) , (cid:107) λ − i (cid:107) < (1 + ε ) L < L since (cid:107) ˚ v i,j (cid:107) < ε by hypothesis. If (cid:104) i, j (cid:105) is a 1-simplex of Λ and x, y ∈ c ij , then (cid:107) w ij ( x ) − w ij ( y ) (cid:107) = (cid:107) λ i ( v ij ( x ) − v ij ( y )) λ − j (cid:107) < L · ε · L < Cε. Moreover, since λ i , λ − j ∈ U( A ) L ε by Lemma 4.3 and v ij ( x ) ∈ U( A ) ε it follows immediately that w ij ( x ) ∈ U( A ) L +1 ε . Therefore, w is Cε -flat.By [12, Theorem 3.2] v and w yield isomorphic bundles on | Λ | . (cid:3) In this section we describe the map β announced in Section 3. The idea is to extend a quasi-representation of Γ to a quasi-representation of the fundamental groupoid of Λ and reinterpretthe latter as a lattice gauge field. This enables us to invoke a construction of Phillips and Stone[19, 20] that associates an almost flat coordinate bundle to a lattice gauge field with controlleddistortion and small modulus of continuity. For the sake of completeness we give a full account ofthis construction.Let 0 < δ < / L , with L as in 4.4, and let an ( F Λ , δ )-representation π of Γ be given. Wewill define an almost flat coordinate bundle β ( π ) on Λ.The following notation will be used in the definition. We fix a partial order o on the vertices of Λ such that the set of vertices of anysimplex of Λ is a totally ordered set under o . One may always assume that such an order exists bypassing to the first barycentric subdivision of Λ: if ˆ σ and ˆ σ are the barycenters of simplices σ and σ of Λ, define ˆ σ < ˆ σ if σ is a face of σ (cf. [20]).When we write σ = (cid:104) i , . . . , i m (cid:105) it is implicit that the vertices of σ are written in increasing o -order. Following [20], we re-parametrize the dual cell blocks c σi using “modified barycentriccoordinates” ( s , . . . , s r ). These are defined in terms of the barycentric coordinates by s j = t j /t i .In these coordinates c σi is identified with the cube { ( s , . . . , s i , . . . , s r ) | s i = 1 and 0 ≤ s j ≤ j (cid:54) = i } . See Figure 2. 8 yz (a) x yz (b) Figure 2: (a) The identification of a dual cell block in a 2-simplex as given by barycentric coordinates.(b) The identification of a dual cell block in a 2-simplex as given by modified barycentric coordinates.The construction of β ( π ) is inspired by (and borrows heavily from) the work of Phillips andStone [20]. It is somewhat involved, but we outline the procedure in the following definition beforegoing into the details. Let i and j be adjacent vertices of Λ. We will define β ( π ) = { v ij : c ij → GL( A ) } by defining v ij on all the dual cell blocks c σij such that σ contains i and j .(1) Let u ij := ˘ π ( γ ij ) , where ˘ π : Γ → U( A ) is the perturbation of π provided by Proposition 5.6.(2) Suppose σ is a simplex in Λ containing i and j and that i < j . Write σ (in increasing o -order)as σ = (cid:104) , . . . , i, . . . , j, . . . , r (cid:105) .For an o -ordered subset of vertices I = { i = i < i < i < · · · < i m = j } , set u I := u i i u i i . . . u i m − i m , (where it is understood that if I = { i < j } , then u I = u ij ).(3) Define v σij : c σij → A , using modified barycentric coordinates on c σij (see 5.2), as follows. For s = ( s , . . . , s i = 1 , . . . , s j = 1 , . . . , s r ) ∈ c σij let v σij ( s ) := (cid:88) I λ I ( s ) u I , where λ I ( s ) = λ σI ( s ) = (cid:89) i ≤ k ≤ j s (cid:48) k with s (cid:48) k = s k if k ∈ I and s (cid:48) k = 1 − s k if k / ∈ I .The sum above is over the subsets I of { i, . . . , j } ⊆ σ (0) that contain both i and j as above.One can identify the subsets I with ascending paths from i to j that are contained in σ . Letus note that (cid:80) I λ I ( s ) = 1 for s ∈ c σij since (cid:81) i
The construction described in Definition 5.3 is an attempt to have v ij be “as constantas possible” and equal to u ij at the barycenter of (cid:104) i, j (cid:105) (cf. [19, Sec. 2]). The cocycle condition onemight hope for would force relations of the form u ij u jk = u ik , which do not necessarily hold since π (and therefore ˘ π ) is only approximately multiplicative. The definition of v σij uses successive linearinterpolation to account for this. For example:(1) If σ = (cid:104) , , (cid:105) , then v σ = u and v σ = u , but v σ ( s = 1 , s , s = 1) = s u u + (1 − s ) u . (2) If σ = (cid:104) , , , (cid:105) , then v σ = u , v σ = u , v σ = u ,v σ ( s = 1 , s , s = 1) = s u u + (1 − s ) u ,v σ ( s = 1 , s , s = 1) = s u u + (1 − s ) u , and v σ ( s = 1 , s , s , s = 1) = s s u u u + s (1 − s ) u u ++ (1 − s ) s u u + (1 − s )(1 − s ) u . Notice that in the modified barycentric coordinates ( s , . . . , s r ) the barycenter of (cid:104) i, j (cid:105) is givenby s i = s j = 1 and s k = 0 for all k (cid:54)∈ { i, j } . Therefore, v σij ( (cid:104) i, j (cid:105) ˆ) = u ij as desired.To start the construction, we first perturb π slightly so that we can deal with unitary elementsinstead of just invertible ones when convenient. Given an ( F Λ , δ ) -representation π : Γ → GL( A ) , < δ < / , there exists afunction ˘ π : Γ → U( A ) such that (1) ˘ π ( e ) = 1 A ; (2) ˘ π ( γ ) ∈ U( A ) for all γ ∈ F Λ ; (3) ˘ π ( γ − ) = ˘ π ( γ ) ∗ for all γ ∈ F Λ ; (4) (cid:107) ˘ π ( γγ (cid:48) ) − ˘ π ( γ )˘ π ( γ (cid:48) ) (cid:107) < δ for all γ, γ (cid:48) ∈ F Λ with γγ (cid:48) ∈ F Λ ; and (5) (cid:107) ˘ π ( γ ) − π ( γ ) (cid:107) < δ for all γ ∈ F Λ . The next lemma will be used in the proof.
Let ω : GL( A ) → U( A ) be given by ω ( v ) = v ( v ∗ v ) − / . If v ∈ U( A ) δ , < δ < / ,then (cid:107) ω ( v ) − v (cid:107) < δ . roof. Observe that for z ∈ A (cid:107) z − (cid:107) ≤ θ < / ⇒ (cid:107) z − − (cid:107) < θ, (5.1)using the Neumann series.Assume v ∈ U( A ) δ . Then there is u ∈ U( A ) such that (cid:107) v − u (cid:107) < δ , so (cid:107) vu ∗ − (cid:107) < δ . Let w = vu ∗ . Then ω ( w ) = vu ∗ ( uv ∗ vu ∗ ) − / = vu ∗ u ( v ∗ v ) − / u ∗ = ω ( v ) u ∗ and thus (cid:107) ω ( w ) − w (cid:107) = (cid:107) ω ( v ) − v (cid:107) .Now (cid:107) w (cid:107) < δ , so (cid:107) w ∗ w − (cid:107) ≤ (cid:107) ( w ∗ − w (cid:107) + (cid:107) w − (cid:107) < δ (1 + δ ) + δ < δ/
7. Therefore,by (5.1), (cid:107) ( w ∗ w ) − / − (cid:107) ≤ (cid:107) ( w ∗ w ) − − (cid:107) < δ/
7. Finally, (cid:107) ω ( v ) − v (cid:107) = (cid:107) w ( w ∗ w ) − / − w (cid:107) < (1 + δ )30 δ/ < δ. (cid:3) Proof of Proposition 5.6.
The idea is simple: define ˘ π = ω ◦ σ where ω is as in Lemma 5.7 and σ : Γ → A is the function σ ( γ ) = π ( γ ) + π ( γ − ) ∗ . We check the required properties.Let γ ∈ F Λ . First we prove that (cid:107) π ( γ − ) − π ( γ ) − (cid:107) < δ and (cid:107) π ( γ ) − − π ( γ ) ∗ (cid:107) < δ .Because π ( γ ) ∈ U( A ) δ , we can write π ( γ ) = uv for some u ∈ U( A ) and v ∈ GL( A ) with (cid:107) v − (cid:107) < δ . Then (cid:107) π ( γ ) − − u ∗ (cid:107) = (cid:107) v − − (cid:107) < δ by (5.1) and hence (cid:107) π ( γ ) − (cid:107) < δ .Therefore, (cid:107) π ( γ − ) − π ( γ ) − (cid:107) = (cid:107) (cid:0) π ( γ − ) π ( γ ) − (cid:1) π ( γ ) − (cid:107)≤ (cid:107) π ( γ − ) π ( γ ) − π ( γ − γ ) (cid:107)(cid:107) π ( γ ) − (cid:107)≤ δ (1 + 2 δ ) < δ. It is just as plain to see that (cid:107) π ( γ ) − − π ( γ ) ∗ (cid:107) = (cid:107) ( uv ) − − ( uv ) ∗ (cid:107) = (cid:107) v − − v ∗ (cid:107) ≤ (cid:107) v − − (cid:107) + (cid:107) v ∗ − (cid:107) < δ, as claimed. Using these bounds we see that (cid:107) σ ( γ ) − π ( γ ) (cid:107) = 12 (cid:107) π ( γ − ) ∗ − π ( γ ) (cid:107) = 12 (cid:107) π ( γ − ) − π ( γ ) ∗ (cid:107)≤ (cid:107) π ( γ − ) − π ( γ ) − (cid:107) + 12 (cid:107) π ( γ ) − − π ( γ ) ∗ (cid:107) < δ/ . (5.2)Thus dist( σ ( γ ) , U( A )) < δ/ π ( γ ) , U( A )) < δ/ < / . (5.3)In particular σ ( γ ) ∈ GL( A ). Items (1) and (2) in the statement of the proposition are immediate.For (3) observe that if z ∈ GL( A ), then ω ( z ∗ ) = ω ( z ) ∗ . It follows that ˘ π ( γ − ) = ω ( σ ( γ − )) = ω ( σ ( γ ) ∗ ) = ω ( σ ( γ )) ∗ = ˘ π ( γ ) ∗ .We deal with (5). From (5.3) and Lemma 5.7 we obtain (cid:107) ˘ π ( γ ) − σ ( γ ) (cid:107) < δ/
2. Together with(5.2) this gives (cid:107) ˘ π ( γ ) − π ( γ ) (cid:107) ≤ (cid:107) ˘ π ( γ ) − σ ( γ ) (cid:107) + (cid:107) σ ( γ ) − π ( γ ) (cid:107) < δ.
11e are left with (4). Suppose γ, γ (cid:48) ∈ F Λ are such that γγ (cid:48) ∈ F Λ . Then (cid:107) ˘ π ( γ )˘ π ( γ (cid:48) ) − ˘ π ( γγ (cid:48) ) (cid:107) ≤ (cid:107) ˘ π ( γ ) − π ( γ ) (cid:107)(cid:107) ˘ π ( γ (cid:48) ) (cid:107) + (cid:107) π ( γ ) (cid:107)(cid:107) ˘ π ( γ (cid:48) ) − π ( γ (cid:48) ) (cid:107) ++ (cid:107) π ( γ ) π ( γ (cid:48) ) − π ( γγ (cid:48) ) (cid:107) + (cid:107) ˘ π ( γγ (cid:48) ) − π ( γγ (cid:48) ) (cid:107) < δ + (1 + δ )20 δ + δ + 20 δ< δ. (cid:3) The next proposition allows us to use induction on the number of vertices of σ in the proofsthat follow. If σ ⊂ (cid:101) σ are simplices of Λ and i < j are vertices of σ , then the restriction of v (cid:101) σij to c σij is equal to v σij .Proof. We may assume that (cid:101) σ = σ ∪ { l } is a simplex of Λ that has σ as one of its faces and l / ∈ σ .If s = ( s , . . . , s l , . . . , s r ) ∈ c (cid:101) σij , then s i = s j = 1 and moreover s ∈ c σij precisely when s l = 0. Let I be a subset of { i, . . . , j } that contains both i and j as above. If either l < i or j < l , then σ and (cid:101) σ have exactly the same set of increasing paths from i to j and hence v (cid:101) σij ( s ) = v σij ( s ) for s ∈ c σij byDefinition 5.3. (In fact, v (cid:101) σij ( s , . . . , s l , . . . , s r ) = v σij ( s , . . . , s l = 0 , . . . , s r ), again by Definition 5.3.)Suppose now that i < l < j . Let I = { i = i < i < i < · · · < i m = j } be an increasing path in (cid:101) σ . If l / ∈ I and s ∈ c σij = { s ∈ c (cid:101) σij : s l = 0 } , then λ (cid:101) σI ( s ) u I = (1 − s l ) λ σI ( s ) u I = λ σI ( s ) u I . On the otherhand if l ∈ I , then λ (cid:101) σI ( s ) = 0 since s l is one of its factors. The statement follows now immediatelyfrom by Definition 5.3 since v (cid:101) σij ( s ) := (cid:88) l ∈ I λ (cid:101) σI ( s ) u I + (cid:88) l / ∈ I λ (cid:101) σI ( s ) u I . (cid:3) If i < l < j are vertices of a simplex σ of Λ , then v σij ( s ) = v σil ( s ) v σlj ( s ) for all s ∈ c σilj = c σi ∩ c σl ∩ c σj .Proof. Let I = { i = i < i < i < · · · < i m = j } be an increasing path in σ .If l / ∈ I , then λ σI ( s ) = 0 since 1 − s l = 0 is one of its factors. On the other hand, if l ∈ I , say I = { i = i < · · · < i k − < i k = l < i k +1 < · · · < i m = j } , then letting I (cid:48) = { i < · · · < i k − < i k } and I (cid:48)(cid:48) = { i k < i k +1 < · · · < i m } we see that λ σI ( s ) u I = λ σI (cid:48) ( s ) s l λ σI (cid:48)(cid:48) ( s ) u I (cid:48) u I (cid:48)(cid:48) = λ σI (cid:48) ( s ) u I (cid:48) · λ σI (cid:48)(cid:48) ( s ) u I (cid:48)(cid:48) . The statement now follows from Definition 5.3 since v σij ( s ) := (cid:88) l ∈ I λ σI ( s ) u I + (cid:88) l / ∈ I λ σI ( s ) u I . (cid:3) The family of functions { v σij | i, j ∈ σ } yields a continous function v ij : c ij = (cid:83) σ c σij → GL( A ) such that v il ( s ) v lj ( s ) = v ij ( s ) for all s ∈ c ilj = c i ∩ c l ∩ c j .Proof. Proposition 5.8 shows that if two simplices σ and σ (cid:48) contain { i, j } , then v σij = v σ (cid:48) ij = v σ ∩ σ (cid:48) ij on c σ ∩ σ (cid:48) ij , so that v ij is well-defined. The cocycle condition follows from Proposition 5.9. It remainsto show that v ij takes values in GL( A ). This will follow from the estimate (cid:107) v σij ( s ) − u ij (cid:107) < Lδ < / I = { i = i < i < i < · · · < i m = j } and u I := u i i u i i . . . u i m − i m are as in Definition 5.3,then show that (cid:107) u I − u ij (cid:107) < mδ by induction on m . This is trivial if m = 2, since in that case u I = u ij . For the inductive step, we use the estimate (cid:107) u i k i k +1 u i k +1 i k +2 − u i k i k +2 (cid:107) < δ proved inProposition 5.6(4). Because v σij ( s ) := (cid:80) I λ σI ( s ) u I , the estimate (5.4) follows since (cid:80) I λ σI ( s ) = 1for s ∈ c σij . (cid:3) Proof of Proposition 5.4.
Let δ = 1 / L .Corollary 5.10 all but implies Proposition 5.4. To complete the proof, we need to verify thealmost flatness condition. Assume i < j are vertices as in the proof of Corollary 5.10. We haveseen that (cid:107) v ij ( x ) − u ij (cid:107) < Lδ for all x ∈ c ij . Since u ij is a unitary and since 0 < δ < / L byhypothesis, we can apply (5.1) to see that (cid:107) v ji ( x ) − u ji (cid:107) = (cid:107) ( v ij ( x )) − − u − ij (cid:107) ≤ (cid:107) v ij ( x ) − u ij (cid:107) < Lδ.
We conclude that β ( π ) = { v ij } is C (cid:48)(cid:48) δ -flat where C (cid:48)(cid:48) = 280 Lδ , completing the proof of Proposi-tion 5.4. (cid:3) Most of the work needed to prove Theorems 3.1 and 3.3 was done in Sections 4 and 5. What is leftis basically bookkeeping related the various constants defined so far, but it is somewhat technicaldue to the nature of the definitions of α and β . Proof of Theorem 3.1.
The definitions of α and β are given in Sections 4 and 5. Propositions 4.6and 5.4 show the existence of δ , ε > { C (cid:48) , C (cid:48)(cid:48) } satisfying parts (1) and (2) ofthe theorem. We will actually set C = max { C (cid:48) , C (cid:48)(cid:48) + 40 , L +1 ( K + 1) } where K is provided byLemma 4.7.We prove (3). Let π = α ( v ), π (cid:48) = α ( v (cid:48) ). Let (cid:104) i, j (cid:105) ∈ Λ (1) be such that (cid:107) π ( γ ij ) − π (cid:48) ( γ ij ) (cid:107) = max γ ∈F Λ (cid:107) π ( γ ) − π (cid:48) ( γ ) (cid:107) = d ( π, π (cid:48) ) . As in Definition 4.2, let I = ( i , . . . , i ) be the unique path along T from i to i and J = ( i , . . . , j )be the unique path from i to j . Then d ( π, π (cid:48) ) ≤ (cid:107) ˜ π ( s ( γ ij )) − ˜ π ( (cid:104) i, j (cid:105) ) (cid:107) + (cid:107) ˜ π (cid:48) ( (cid:104) i, j (cid:105) ) − ˜ π (cid:48) ( s ( γ ij )) (cid:107) ++ (cid:107) ˜ π ( (cid:104) i, j (cid:105) ) − ˜ π (cid:48) ( (cid:104) i, j (cid:105) ) (cid:107)≤ (cid:107) ˜ π (cid:0) s ( γ ij ) (cid:104) i, j (cid:105) − (cid:1) − (cid:107) · (cid:107) ˜ π ( (cid:104) i, j (cid:105) ) (cid:107) ++ (cid:107) ˜ π (cid:48) (cid:0) s ( γ ij ) (cid:104) i, j (cid:105) − (cid:1) − (cid:107) · (cid:107) ˜ π ( (cid:104) i, j (cid:105) ) (cid:107) + (cid:107) ˚ v I ˚ v ij ˚ v − J − ˚ v (cid:48) I ˚ v (cid:48) ij (˚ v (cid:48) J ) − (cid:107) . From Lemma 4.7 we get that (cid:107) ˜ π (cid:0) s ( γ ij ) (cid:104) i, j (cid:105) − (cid:1) − (cid:107) ≤ Kε , where K > T , i , and s . The same bound holds with ˜ π (cid:48) instead of ˜ π . Using this and the estimates (cid:107) ˚ v kl (cid:107) < ε , (cid:107) ˚ v kl − ˚ v (cid:48) kl (cid:107) < d ( v , v (cid:48) ) for (cid:104) k, l (cid:105) ∈ Λ (1) , we see that d ( π, π (cid:48) ) ≤ · Kε · (1 + ε ) L +1 + (1 + ε ) L d ( v , v (cid:48) ) . Since (1 + ε ) L < L ε and d ( v , v (cid:48) ) < ε < Kε (1 + ε ) L +1 + (1 + ε ) L d ( v , v (cid:48) ) < L +2 Kε + 2 L +2 ε + d ( v , v (cid:48) ) . d ( π, π (cid:48) ) < C ε + d ( v , v (cid:48) ).For part (4), recall that d ( π, π (cid:48) ) = max γ ∈F Λ (cid:107) π ( γ ) − π ( γ ) (cid:107) . Let v = β ( π ) and v (cid:48) = β ( π (cid:48) )(these are C (cid:48)(cid:48) δ -flat GL( A )-coordinate bundles by Proposition 5.4). Recall that their definition (seeDefinition 5.3) makes use of the maps ˘ π and ˘ π (cid:48) (given by Proposition 5.6) respectively, and that u ij = ˘ π ( γ ij ) = ˚ v ij etc. For (cid:104) i, j (cid:105) ∈ Λ (1) and x ∈ c ij we estimate (cid:107) v ij ( x ) − v (cid:48) ij ( x ) (cid:107) ≤ (cid:107) v ij ( x ) − u ij (cid:107) + (cid:107) u (cid:48) ij − v (cid:48) ij ( x ) (cid:107) + (cid:107) u ij − u (cid:48) ij (cid:107) < C (cid:48)(cid:48) δ + C (cid:48)(cid:48) δ + (cid:107) u ij − π ( γ ij ) (cid:107) + (cid:107) π (cid:48) ( γ ij ) − u (cid:48) ij (cid:107) ++ (cid:107) π ( γ ij ) − π (cid:48) ( γ ij ) (cid:107) < C (cid:48)(cid:48) δ + 20 δ + 20 δ + d ( π, π (cid:48) ) < C δ + d ( π, π (cid:48) ) . It follows that d ( v , v (cid:48) ) < C δ + d ( π, π (cid:48) ). (cid:3) Proof of Theorem 3.3.
We prove (1) from the statement of the theorem first. Let ε = 1 / LC , δ = 1 / LC , and C = 70 KC . ( C is provided by Theorem 3.1, K by Lemma 4.7 and L byNotation 4.4). Let 0 < ε < ε and suppose v = { v ij : c ij → GL( A ) } is an ε -flat GL( A )-coordinatebundle on Λ. Let π = α ( v ) and v (cid:48) = { v (cid:48) ij : c ij → GL( A ) } = β ( π ). Observe that π is an ( F Λ , C ε )-representation and C ε < / L so that the construction of β ( π ) from Section 5 may be used. Wewant to prove that d ( v , v (cid:48) ) = max (cid:104) i,j (cid:105)∈ Λ (1) max x ∈ c ij (cid:107) v ij ( x ) − v (cid:48) ij ( x ) (cid:107) < C ε. Recall the notation ˚ v ij = v ij ( (cid:104) i, j (cid:105) ˆ) from 4.1. Since v is ε -flat and v (cid:48) is C ε -flat, it follows that d ( v , v (cid:48) ) < max (cid:104) i,j (cid:105)∈ Λ (1) ( ε + (cid:107) ˚ v ij − ˚ v (cid:48) ij (cid:107) + C ε ) . (6.1)Let (cid:104) i, j (cid:105) ∈ Λ (1) and set g ij := s ( γ ij ) · (cid:104) i, j (cid:105) − as in Lemma 4.7. Applying the definition of ˜ π (seeEquation (4.1)) and the fact that v is normalized (Definition 3.2), we obtain π ( γ ij ) = ˜ π ( s ( γ i,j )) = ˜ π ( (cid:104) i, j (cid:105) )˜ π ( g ij ) = ˚ v ij ˜ π ( g ij )The definition of β ( π ) shows that ˚ v (cid:48) ij = ˘ π ( γ ij ) and Proposition 5.6 implies (cid:107) π ( γ ij ) − ˘ π ( γ ij ) (cid:107) < C ε .Thus (cid:107) ˚ v ij − ˚ v (cid:48) ij (cid:107) < (cid:107) ˚ v ij − π ( γ ij ) (cid:107) + (cid:107) π ( γ ij ) − ˚ v (cid:48) ij (cid:107) = (cid:107) ˚ v ij (1 − ˜ π ( g ij )) (cid:107) + (cid:107) π ( γ ij ) − ˘ π ( γ ij ) (cid:107) < (cid:107) ˚ v ij (cid:107)(cid:107) − ˜ π ( g ij ) (cid:107) + 20 C ε< (1 + ε ) (cid:107) − ˜ π ( g ij ) (cid:107) + 20 C ε. (6.2)Lemma 4.7 guarantees that (cid:107) A − ˜ π ( g ij ) (cid:107) < Kε . In combination with (6.1) and (6.2) this provesthat d ( v , v (cid:48) ) < ε + C ε + (1 + ε ) Kε + 20 C ε < C ε. We prove (2) from the statement of the theorem. Let 0 < δ < δ and suppose π : Γ → GL( A ) isan ( F Λ , δ )-representation. Let v = { v ij } = β ( π ) (this is a C δ -flat GL( A )-coordinate bundle) andlet ˘ π : Γ → U( A ) be given by Proposition 5.6. Let also π (cid:48) = α ( v ) (this is an ( F Λ , C δ )-representationof Γ to GL( A )). We want to prove that d ( π, π (cid:48) ) = max γ ∈F Λ (cid:107) π ( γ ) − π (cid:48) ( γ ) (cid:107) < C δ. (cid:104) i, j (cid:105) ∈ Λ (1) . As above, we may write s ( γ ij ) = (cid:104) i, j (cid:105) · g ij where by Lemma 4.7, (cid:107) ˜ π (cid:48) ( g ij ) − A (cid:107) < KC δ .First notice that ˚ v ij = ˘ π ( γ ij ) by the definition of β ( π ) = v (Definition 5.3). Let I be the uniquepath along T from i to i and J be the unique path from i to j . Observe that v is normalized since˘ π ( e ) = 1 A ; hence ˚ v I = 1 A = ˚ v J because I and J are paths in the tree T . Then, by the definition of α ( v ) = π (cid:48) , π (cid:48) ( γ ij ) = ˜ π (cid:48) (cid:0) s ( γ i,j ) (cid:1) = ˜ π (cid:48) ( (cid:104) i, j (cid:105) ) · ˜ π (cid:48) ( g ) = ˚ v I ˚ v ij ˚ v − J · ˜ π (cid:48) ( g ) = ˚ v ij · ˜ π (cid:48) ( g ) = ˘ π ( γ ij ) · ˜ π (cid:48) ( g ) . Therefore (cid:107) π (cid:48) ( γ ij ) − π ( γ ij ) (cid:107) ≤ (cid:107) π (cid:48) ( γ ij ) − π ( γ ij )˜ π (cid:48) ( g ) (cid:107) + (cid:107) π ( γ ij )˜ π (cid:48) ( g ) − π ( γ ij ) (cid:107)≤ (cid:107) ˘ π ( γ ij ) − π ( γ ij ) (cid:107)(cid:107) ˜ π (cid:48) ( g ) (cid:107) + (cid:107) π ( γ ij ) (cid:107)(cid:107) ˜ π (cid:48) ( g ) − A (cid:107) < δ (1 + KC δ ) + (1 + δ )( KC δ ) < C δ. (cid:3) The goal of this section is to connect the notion of almost flat coordinate bundle from Definition 2.5,which is defined using simplicial structure and involves cocycles defined on closed sets, with thenotion of almost flat bundle over a compact space from Definition 7.1 below.Almost flat bundles and K -theory classes appeared in the work Gromov and Lawson [9], ofConnes, Gromov, and Moscovici [3, 18, 21]. In these references a vector bundle over a Riemannianmanifold is called ε -flat if there is a metric-preserving connection with curvature of norm less than ε . Almost flat K -theory classes have been studied in different contexts in [2, 4, 5, 10, 16, 17]. Weadapt the definition to bundles over topological spaces as in [5] and connect this with the versionfor simplicial complexes by proving Proposition 7.3.Let X be a compact space and let V = { V i } be a finite open cover of X . A ˇCech 1-cocycle { v ij : V i ∩ V j → GL( A ) } satisfies v ij ( x ) = v ji ( x ) − for all x ∈ V i ∩ V j and v ik ( x ) = v ij ( x ) v jk ( x ) forall x ∈ V i ∩ V j ∩ V k . Let ε ≥ { v ij : V i ∩ V j → GL( A ) } is ε -flat , if(a) v ij ( x ) ∈ U( A ) ε for all x ∈ V i ∩ V j ; and(b) (cid:107) v ij ( x ) − v ij ( y ) (cid:107) < ε for all x, y ∈ V i ∩ V j .(2) A principal GL( A )-bundle E over X is ( V , ε )- flat if its isomorphism class is represented byan ε -flat cocycle { v ij : V i ∩ V j → GL( A ) } .It is clear that if V (cid:48) is an open cover that refines V , then the restriction of { v ij } to V (cid:48) is also ε -flat.We now establish a result that connects the notion of ε -flat ˇCech 1-cocycles from Definition 7.1with the notion of ε -flat coordinate bundle in the simplicial sense as given in Definition 2.5. Supposethat X = | Λ | is the geometric realization of a finite simplicial complex Λ. Recall that X hasa (closed) cover C Λ given by dual cells c i ; see Section 2. Let d be the canonical metric for thetopology of X obtained using barycentric coordinates. Fix a sufficiently small number ν > V i = { x ∈ X : dist( x, c i ) < ν } , then for any finite intersection V i ∩ V i ∩ · · · ∩ V i k (cid:54) = ∅ ⇔ c i ∩ c i ∩ · · · ∩ c i k (cid:54) = ∅ . (7.1)15ote that if { v ij } is as in Definition 7.1 and the cover { V i } satisfies (7.1), then the restrictionof { v ij } to c i ∩ c j ⊂ V i ∩ V j is an ε -flat coordinate bundle. Proposition 7.2 below allows us reversethis operation. There are numbers ε > and r > , depending only on Λ , such that for any < ε < ε , any ε -flat GL( A ) -coordinate bundle { v ij : c i ∩ c j → U ( A ) ε } on Λ , and any ν > satisfying (7.1) , there is an rε -flat cocycle { (cid:101) v ij : V i ∩ V j → U ( A ) rε } that extends v ij . Proposition 7.2 is a direct consequence of Proposition 7.3 below, whose content is of independentinterest in connection with extension properties of principal bundles.Let Y be a closed subspace of a compact metric space X and let { U i } ni =1 be a closed cover of Y . For ν > i ∈ { , . . . , n } let U νi := { x ∈ X : dist( x, U i ) ≤ ν } and set (cid:101) Y = (cid:83) ni =1 U νi . Fix ν > U νi ∩ U νi ∩ · · · ∩ U νi k (cid:54) = ∅ ⇔ U i ∩ U i ∩ · · · ∩ U i k (cid:54) = ∅ (7.2) (1) For any cocycle v ij : U i ∩ U j → GL ( A ) on Y there exist ν > satisfying (7.2) and a cocycle (cid:101) v ij : U νi ∩ U νj → GL( A ) that extends v ij , i.e. (cid:101) v ij = v ij on U i ∩ U j . (2) There exist ε ∈ (0 , and a universal constant r = r n that depends only on n such that forany < ε < ε , any ε -flat cocycle v ij : U i ∩ U j → U ( A ) ε on Y , and any ν > satisfying (7.2) , there is an rε -flat cocycle (cid:101) v ij : U νi ∩ U νj → U ( A ) rε on (cid:101) Y which extends v ij . Proof.
We begin with the proof of (1) and will explain subsequently how to adapt the argumentto prove (2) as well. We prove (1) by induction on the cardinality n of the cover. Suppose that thestatement is true for any integer ≤ n −
1. Let v ij : U i ∩ U j → GL( A ) be given with 1 ≤ i, j ≤ n .Set Y n − := (cid:83) n − i =1 U i . By the inductive hypothesis, there exist ν > (cid:101) v ij : U νi ∩ U νj → GL( A ), 1 ≤ i, j ≤ n −
1, which extends v ij . Thus the following condition, labelledas ( n − (cid:101) v rs = (cid:101) v rt (cid:101) v ts on U νr ∩ U νt ∩ U νs for all 1 ≤ r ≤ t ≤ s ≤ n − . ( n − n − n we proceed again by induction on increasing k ∈ L n , where L n is theset of those integers 1 ≤ k ≤ n with the property that U k ∩ U n (cid:54) = ∅ . The inductive hypothesis thatwe make is that the functions { v in : i ≤ k, i ∈ L n } extend to functions (cid:101) v in : U νi ∩ U νn → GL ( A ) suchthat the following conditions (depending on k ) are satisfied: (cid:101) v in = (cid:101) v ij (cid:101) v jn on U νi ∩ U νj ∩ U νn for i, j ≤ k with i, j ∈ L n . (1 , k ) (cid:101) v in = (cid:101) v ij v jn on U νi ∩ U j ∩ U n for i ≤ k ≤ j with i, j ∈ L n . (2 , k )If L n reduces to { n } , then we simply define (cid:101) v nn = 1 and we are done. Assume L n contains morethan one element.Let (cid:96) be the smallest element of L n (so (cid:96) < n ). To construct (cid:101) v (cid:96)n we first define an extension v (cid:48) (cid:96)n of v (cid:96)n on suitable closed subsets of U ν(cid:96) ∩ U νn as follows: v (cid:48) (cid:96) n = (cid:101) v (cid:96) j v jn on U ν(cid:96) ∩ U j ∩ U n for all (cid:96) ≤ j < n, j ∈ L n . (0 (cid:48) )16et us observe that v (cid:48) (cid:96) n is well-defined since if (cid:96) ≤ i ≤ j < n , i, j ∈ L n , then (cid:101) v (cid:96) i v in = (cid:101) v (cid:96) j v jn on U ν(cid:96) ∩ U i ∩ U j ∩ U n if and only if v in = (cid:101) v i,(cid:96) (cid:101) v (cid:96)j v jn on the same set. In view of condition ( n − v in = (cid:101) v ij v jn on U i ∩ U j ∩ U n , which holds true since (cid:101) v ij extends v ij .By Tietze’s theorem we can now extend the function v (cid:48) (cid:96) n defined by (0 (cid:48) ) to a continuous function (cid:101) v (cid:96) n : U ν(cid:96) ∩ U νn → A . Since GL( A ) is open in A we will have that (cid:101) v (cid:96) n ( x ) ∈ GL( A ) for all x ∈ U ν(cid:96) ∩ U νn provided that ν is sufficiently small. We need to very that (cid:101) v (cid:96)n satisfies (1 , (cid:96) ) and (2 , (cid:96) ). Condition(1 , (cid:96) ) amounts to (cid:101) v (cid:96)n = (cid:101) v (cid:96)(cid:96) (cid:101) v (cid:96)n on U ν(cid:96) ∩ U νn which holds since (cid:101) v (cid:96)(cid:96) = 1. Condition (2 , (cid:96) ) reduces to (cid:101) v (cid:96)n = (cid:101) v (cid:96)j v jn on U ν(cid:96) ∩ U j ∩ U n for (cid:96) ≤ j with j ∈ L n . This holds true in view of (0 (cid:48) ) and so the basecase for the induction is complete.Fix k ∈ L n , k < n and suppose now that we have constructed (cid:101) v in : U νi ∩ U νn → GL( A ) for all i ∈ L n with i ≤ k such that the conditions (1 , k ) and (2 , k ) are satisfied. Let (cid:96) ∈ L n be the successorof k in L n . We may assume that (cid:96) < n for otherwise there is nothing to prove. We construct a map (cid:101) v (cid:96)n on U ν(cid:96) ∩ U νn that satisfies the corresponding conditions (1 , (cid:96) ) and (2 , (cid:96) ) as follows. The first stepis to define an extension v (cid:48) (cid:96) n of v (cid:96) n on suitable closed subsets of U ν(cid:96) ∩ U νn as follows: v (cid:48) (cid:96) n = (cid:101) v (cid:96) i (cid:101) v in on U ν(cid:96) ∩ U νi ∩ U νn for all i ≤ k, i ∈ L n . (1 (cid:48) ) v (cid:48) (cid:96) n = (cid:101) v (cid:96) j v jn on U ν(cid:96) ∩ U j ∩ U n for all (cid:96) ≤ j < n, j ∈ L n . (2 (cid:48) )We need to observe that the conditions (1 (cid:48) ) and (2 (cid:48) ) are compatible so that v (cid:48) (cid:96),n is well-defined andcontinuous. There are three cases to verify. First we check that (cid:101) v (cid:96),i (cid:101) v in = (cid:101) v (cid:96)j (cid:101) v jn on U ν(cid:96) ∩ U νi ∩ U νj ∩ U νn for i, j ≤ k , i, j ∈ L n . This is a consequence of conditions ( n −
1) and (1 , k ). Second, we verify that (cid:101) v (cid:96) i v in = (cid:101) v (cid:96) j v jn on U ν(cid:96) ∩ U i ∩ U j ∩ U n for (cid:96) ≤ i, j < n , i, j ∈ L n . Note that this holds if and only if v in = (cid:101) v i,(cid:96) (cid:101) v (cid:96)j v jn on the same set. In view of condition ( n −
1) this reduces to the equality v in = (cid:101) v ij v jn on U i ∩ U j ∩ U n , which holds true since (cid:101) v ij extends v ij . Finally we need to verify that (cid:101) v (cid:96)i (cid:101) v in = (cid:101) v (cid:96)j v jn on ( U ν(cid:96) ∩ U νi ∩ U νn ) ∩ ( U ν(cid:96) ∩ U j ∩ U n ) = U ν(cid:96) ∩ U νi ∩ U j ∩ U n for i ≤ k < (cid:96) ≤ j < n , i, j ∈ L n . By( n − (cid:101) v in = (cid:101) v i j v jn on U νi ∩ U j ∩ U n . The latter equality holds dueto condition (2 , k ) which is satisfied by the inductive hypothesis. By Tietze’s theorem we can nowextend the function v (cid:48) (cid:96) n defined by (1 (cid:48) ) and (2 (cid:48) ) to a continuous function (cid:101) v (cid:96) n : U ν(cid:96) ∩ U νn → A . SinceGL( A ) is open in A we will have that (cid:101) v (cid:96) n ( x ) ∈ GL( A ) for all U ν(cid:96) ∩ U νn provided that ν is sufficientlysmall. It is clear that the functions ( (cid:101) v in ) i ≤ (cid:96) satisfy the conditions (1 , (cid:96) ) , (2 , (cid:96) ) as a consequence of(1 (cid:48) ), (1 , k ), (2 (cid:48) ) and (2 , k ). This completes the inductive step from k to (cid:96) and hence from n − n . During this step we had to pass to a possibly smaller ν but this does not affect the conclusion.(2). The proof follows the pattern of the proof of (1) with one important modification. Namelywe use the following strengthened version of Tietze’s theorem due to Dugunji [6]. Let X be anarbitrary metric space, Y a closed subset of X , A a locally convex linear space and f : Y → A acontinuous map. Then there exists an extension (cid:101) f : X → A of f such that (cid:101) f ( X ) is contained in theconvex hull of f ( Y ).Fix a point x ij in each nonempty intersection U i ∩ U j and set ˚ v ij := v ij ( x ij ). Since the cocycleis ε -flat, we have that (cid:107) v ij ( x ) − ˚ v ij (cid:107) < ε .Let us define positive numbers r ( i, j ) for 1 ≤ i ≤ j ≤ n as follows. If i = j , then r ( i, j ) = 1. If i < j , r ( i, j ) is defined by the following recurrence formula. Set r k = max { r ( i, j ) : 1 ≤ i ≤ j ≤ k } and r = 1. If 1 ≤ (cid:96) < n then we define r ( (cid:96), n ) = (3 r n − + 7) max { r ( i, n ) : 1 ≤ i < (cid:96) } with theconvention that max ∅ = 1.We only need to consider the maps (cid:101) v (cid:96)n with (cid:96) ∈ L n = { i : U i ∩ U n (cid:54) = ∅} and (cid:96) < n . We proceedas in proof of (1) by induction on n and k ∈ L n with the additional provision that(3 (cid:48) ) (cid:101) v ij ( U νi ∩ U νj ) ⊂ B (˚ v ij , r ( i, j ) ε ), for all 1 ≤ i ≤ j ≤ n − i, j ) with i < k , i ∈ L n and j = n . 17he basic idea of the proof is to observe that it follows from the equations (0 (cid:48) ), (1 (cid:48) ) and (2 (cid:48) )that v (cid:48) (cid:96) n is close to (cid:101) v (cid:96) i (cid:101) v in (if i < (cid:96) ) or (cid:101) v (cid:96) i v in (if (cid:96) ≤ i ) both of which are near ˚ v (cid:96) i ˚ v in and hence v (cid:48) (cid:96),n is close to ˚ v (cid:96)n . It will follow that the image of v (cid:48) (cid:96)n is contained in a ball B (˚ v (cid:96)n , r ( (cid:96), n ) ε ) where r ( (cid:96), n ) is a universal constant computed recursively from previously determined r ( i, j ). Thereforewe can invoke the strengthened version of Tietze’s theorem of [6] to extend v (cid:48) (cid:96),n to a continuousmap (cid:101) v (cid:96),n with values in convex open ball B (˚ v (cid:96)n , r ( (cid:96), n ) ε ).Fix k ∈ L n , k < n . By the inductive hypothesis, suppose that we have constructed (cid:101) v ij and theysatisfy (3 (cid:48) ). We need to consider two cases. The first is the case when k = min L n . Letting (cid:96) = k and 0 < ε <
1, then from condition (0 (cid:48) ), for each x ∈ U ν(cid:96) ∩ U j ∩ U n with (cid:96) ≤ j < n , j ∈ L n : (cid:107) v (cid:48) (cid:96) n ( x ) − ˚ v (cid:96) n (cid:107) ≤ (cid:107) (cid:101) v (cid:96) j ( x ) − ˚ v (cid:96) j (cid:107)(cid:107) v jn ( x ) (cid:107) + (cid:107) ˚ v (cid:96) j (cid:107)(cid:107) v jn ( x ) − ˚ v in (cid:107) ++ (cid:107) ˚ v (cid:96) j − ˚ v jn ˚ v (cid:96) n (cid:107) < r ( (cid:96), i ) ε (1 + ε ) + ε (1 + ε ) + ε (3 + 2 ε ) ≤ (3 r n − + 7) ε ≤ r ( (cid:96), n ) ε. Let (cid:96) be the successor of k in L n . We may assume that (cid:96) < n otherwise we are done. If x ∈ U ν(cid:96) ∩ U j ∩ U n with (cid:96) ≤ j < n , j ∈ L n , then using (2 (cid:48) ) it follows just as above that (cid:107) v (cid:48) (cid:96) n ( x ) − ˚ v (cid:96) n (cid:107) ≤ (3 r n − + 7) ε = r ( (cid:96), n ) ε. On the other hand, if x ∈ U ν(cid:96) ∩ U νi ∩ U n with i < (cid:96) , i ∈ L n , then using (1 (cid:48) )we have (cid:107) v (cid:48) (cid:96) n ( x ) − ˚ v (cid:96) n (cid:107) ≤ (cid:107) (cid:101) v (cid:96) i ( x ) − ˚ v (cid:96) i (cid:107)(cid:107) (cid:101) v in ( x ) (cid:107) + (cid:107) ˚ v (cid:96) i (cid:107)(cid:107) (cid:101) v in ( x ) − ˚ v in (cid:107) + (cid:107) ˚ v (cid:96) i − ˚ v in ˚ v (cid:96) n (cid:107) < r ( (cid:96), i ) ε (1 + ( r ( i, n ) + 1) ε ) + (1 + ε ) r ( i, n ) ε + ε (2 + 3 ε ) ≤ (3 r ( (cid:96), i ) + 7) r ( i, n ) ≤ (3 r n − + 7) max { r ( i, n ) : i < (cid:96), i ∈ L n }≤ r ( (cid:96), n ) ε. In view of this estimates we can extend v (cid:48) (cid:96)n to (cid:101) v (cid:96)n using the strengthened version of Tietze’s theoremso that (cid:101) v (cid:96)n ( U ν(cid:96) ∩ U νn ) ⊂ B (˚ v (cid:96)n , r ( (cid:96), n ) ε ). It follows that (cid:107) (cid:101) v ij ( x ) − ˚ v ij (cid:107) < r n ε for all x ∈ U νi ∩ U νj .This completes the proof. (cid:3) One of the motivations for this paper is the detection of nontrivial K-theory elements of a group C*-algebra, via lifting of homomorphisms K ( C ∗ (Γ)) → Z to quasi-representations C ∗ (Γ) → M m ( C ).Suppose that the full assembly map is a bijection for a discrete group Γ. Roughly speaking, ourmain result states that the quasi-representations C ∗ (Γ) → M m ( C ) which induce interesting partialmaps on K-theory are as abundant as the non-trivial almost flat K-theory classes of the classifyingspace B Γ. More generally, for a C*-algebra B we consider the connection between almost flat K-theory classes in K ( C ( B Γ) ⊗ B ) and quasi-representations C ∗ (Γ) → M m ( B ) that implement agiven homomorphism K ∗ ( C ∗ (Γ)) → K ∗ ( B ).Let A be a unital C*-algebra. A quasi-representation π : Γ → GL( A ) extends to a unital linearcontraction π : (cid:96) (Γ) → A in the obvious way. We like to think of π as “inducing” a partially definedmap π (cid:93) : K ( (cid:96) (Γ)) → K ( A ) (cf. [4, 5]). We briefly recall the definition of π (cid:93) . In the definition wewrite χ for the function ζ (cid:55)→ πi (cid:82) C ( z − ζ ) − dz, where C = { z ∈ C : | z − | = 1 / } . (c.f. [4]) . Let D , B be Banach algebras and let π : D → B be a unital contractivemap. Let p ∈ M m ( C ) ⊗ D be an idempotent and let x = (id m ⊗ π )( p ) ∈ M m ( C ) ⊗ B . Define18 (cid:93) ( p ) = [ χ ( x )] ∈ K ( B ) whenever (cid:107) x − x (cid:107) < /
4. In a similar manner, one defines the pushforward π (cid:93) ( u ) ∈ K ( B ) of an invertible element u ∈ M m ( C ) ⊗ D as the class of [(id m ⊗ π )( u )] , under theassumption that π is (sufficiently) approximately multiplicative on a suitable finite subset of D that depends on u .In general π (cid:93) ( p ) is not necessarily equal to π (cid:93) ( q ) if [ p ] = [ q ] in K ( D ). To bypass this nuisance,we use a discrete version of the asymptotic homomorphisms of Connes and Higson.A discrete asymptotic homomorphism from an involutive Banach algebra D to C*-algebras B n consists of a sequence { π n : D → B n } ∞ n =1 of maps such thatlim n →∞ (cid:107) π n ( a + λa (cid:48) ) − π n ( a ) − λπ n ( a (cid:48) ) (cid:107)(cid:107) π n ( a ∗ ) − π n ( a ) ∗ (cid:107)(cid:107) π n ( aa (cid:48) ) − π n ( a ) π n ( a (cid:48) ) (cid:107) = 0for all a, a (cid:48) ∈ D and λ ∈ C . The sequence { π n } n induces a ∗ -homomorphism D → (cid:81) ∞ n =1 B n / (cid:80) ∞ n =1 B n . If each B n is a matrix algebra over some fixed C*-algebra B , then this further induces a grouphomomorphism K ∗ ( D ) → ∞ (cid:89) n =1 K ∗ ( B ) / ∞ (cid:88) n =1 K ∗ ( B ) . A discrete asymptotic homomorphism gives a canonical way to push forward an element x ∈ K ∗ ( D )to a sequence ( π n (cid:93) ( x )) of elements of K ∗ ( B ), which is well-defined up to tail equivalence: twosequences are tail equivalent, written ( y n ) ≡ ( z n ), if there is m such that y n = z n for all n ≥ m .Note that one can adapt Definition 8.1 to maps which are approximately contractive (in additionto being approximately multiplicative). Let Γ be a discrete countable group with a finite set of generators F . We need thefollowing observations:(1) A sequence of ( F , δ n )-representations { π n : Γ → U ( B n ) } ∞ n =1 , with δ n → n → ∞ , induces adiscrete asymptotic homomorphism (still written ( π n ) ∞ n =1 ) from the involutive Banach algebra (cid:96) (Γ) to the C*-algebras B n .(2) A discrete asymptotic homomorphism { π n : (cid:96) (Γ) → B n } ∞ n =1 as above induces a ∗ -homomorphism π ∞ : (cid:96) (Γ) → B ∞ := (cid:81) ∞ n =1 B n / (cid:80) ∞ n =1 B n and hence a ∗ -homomorphism ¯ π ∞ : C ∗ (Γ) → B ∞ such as the following diagram is commutative. (cid:96) (Γ) π ∞ (cid:47) (cid:47) j (cid:15) (cid:15) B ∞ C ∗ (Γ) ¯ π ∞ (cid:60) (cid:60) (where j is the canonical map).(3) Let { ¯ π n : C ∗ (Γ) → B n } ∞ n =1 be a discrete asymptotic homomorphism given by some set-theoretic lift of ¯ π ∞ . If y ∈ K ∗ ( (cid:96) (Γ)), then we have the following tail equivalence: (cid:0) ¯ π n(cid:93) ( j ∗ ( y )) (cid:1) ∞ n =1 ≡ (cid:0) π n(cid:93) ( y ) (cid:1) ∞ n =1 (cid:101) X be the universal cover of X = | Λ | . Consider the dual cover C Λ = { c i } i of X and theassociated open cover V ν = { V i } i where V i = { x ∈ X : d ( x, c i ) < ν } . The Mishchenko line bundle isthe bundle (cid:101) X × Γ (cid:96) (Γ) → X , obtained from (cid:101) X × (cid:96) (Γ) by passing to the quotient with respect to thediagonal action of Γ. It is isomorphic to the bundle E obtained from the disjoint union (cid:70) V i × (cid:96) (Γ)by identifying ( x, a ) with ( x, γ ij a ) whenever x ∈ V i ∩ V j , where γ ij ∈ Γ are as in Notation 2.1; seefor example [2, Lemma 3.3]. Let { χ i } be a partition of unity subordinate to { V i } . It follows thatthe Mishchenko line bundle corresponds to the class of the projection e := (cid:88) i,j e ij ⊗ χ / i χ / j ⊗ γ ij ∈ M N ( C ) ⊗ C ( X ) ⊗ C [Γ] , (8.1)where { e ij } are the canonical matrix units of M N ( C ) and N is the number of vertices in Λ. Wehave inclusions of rings C [Γ] ⊂ (cid:96) (Γ) ⊂ C ∗ (Γ). The class of the idempotent e in K ( C ( X ) ⊗ (cid:96) (Γ))or K ( C ( X ) ⊗ C ∗ (Γ)) is denoted by (cid:96) . For an ( F Λ , ε )-representation π : Γ → U ( A ) as in Definition 2.2, we set (cid:96) π := (id C ( X ) ⊗ π ) (cid:93) ( e ) ∈ K ( C ( X ) ⊗ A ) . From Definition 5.3 we get the coordinate bundle β ( π ) associated with π . Applying Proposi-tion 7.3 to β ( π ) we obtain an almost flat cocycle { v ij : V i ∩ V j → GL( A ) } . Let E π be the bundleconstructed from the disjoint union (cid:70) V i × A by identifying ( x, a ) with ( x, v ij ( x ) a ) for x in V ij . There is ε such that, for any < ε < ε and any ( F Λ , ε ) -representation π : Γ → U ( A ) , [ E π ] = (cid:96) π ∈ K ( C ( X ) ⊗ A ) . Proof.
Let ε > π : Γ → U ( A ) be an ( F Λ , ε )-representation. Because the bundle E π isrepresented by the idempotent p = (cid:80) i,j e ij ⊗ χ / i χ / j ⊗ v ij , it follows that p − (id C ( X ) ⊗ π )( e ) = (cid:88) i,j e ij ⊗ χ / i χ / j ⊗ ( π ( γ ij ) − v ij ) . Now, by the construction of { v ij } , there is a constant C depending only on Λ such that sup x ∈ V i ∩ V j (cid:107) π ( γ ij ) − v ij ( x ) (cid:107) < Cε . Therefore, (cid:107) p − (id C ( X ) ⊗ π )( e ) (cid:107) < / ε is chosen to be sufficiently small. (cid:3) Let X be a compact connected space and let B be a unital C*-algebra. We consider locallytrivial bundles E over X with fiber finitely generated projective Hilbert-modules F over B andstructure group GL( A ), where A = L B ( F ), the C*-algebra of B -linear adjointable endomorphismsof F . The K-theory group K ( C ( X ) ⊗ B ) consists of formal differences of isomorphism classes ofsuch bundles. Let V be a finite open cover of X . A bundle E as above is ( V , ε )-flat if it admits an( V , ε )-flat associated GL( A )-principal bundle in the sense of Definition 7.1(2) . An element x ∈ K ( C ( X ) ⊗ B ) is almost flat if there is a finite open cover V of X such that for every ε > V , ε )-flat bundles E ± over X such that α = [ E + ] − [ E − ]. We saythat x ∈ K ( C ( X ) ⊗ B ) is almost flat modulo torsion if there is a torsion element t ∈ K ( C ( X ) ⊗ B )such that x − t is almost flat. 20y the UCT given in [14, Lemma 3.4], the Kasparov product KK ( C , C ( B Γ) ⊗ B ) × KK ∗ ( C ( B Γ) , C ) → KK ∗ ( C , B ) , ( x, z ) (cid:55)→ (cid:104) x, z (cid:105) , induces an exact sequenceExt( K ∗ ( B Γ) , K ∗ +1 ( B )) (cid:26) K ( C ( B Γ) ⊗ B ) (cid:16) Hom( K ∗ ( B Γ) , K ∗ ( B )) . (8.2)If K ∗ ( B ) is finitely generated and torsion free, then the torsion subgroup of K ( C ( B Γ) ⊗ B ) coincideswith the image of Ext( K ∗ ( B Γ) , K ∗ +1 ( B )). Let Γ be a discrete countable group whose classifying space B Γ is a finite simplicialcomplex and let B be a unital C*-algebra. Consider the following conditions: (1) For any x ∈ K ( C ( B Γ) ⊗ B ) there is t ∈ Ext( K ∗ ( B Γ) , K ∗ +1 ( B )) such that x − t is almostflat. (2) For any group homomorphism h : K ∗ ( C ∗ (Γ)) → K ∗ ( B ) there exist discrete asymptotic homo-morphisms { π ± n : C ∗ (Γ) → M k ( n ) ( B ) } n such that ( π + n (cid:93) ( y ) − π − n (cid:93) ( y )) ≡ ( h ( y )) for every y inthe image of the full assembly map µ : K ∗ ( B Γ) → K ∗ ( C ∗ (Γ)) . Then (1) ⇒ (2). Moreover if K ∗ ( B ) is finitely generated and if µ is split injective, then (2) ⇒ (1).Proof. (1) ⇒ (2). Let h : K ∗ ( C ∗ (Γ)) → K ∗ ( B ) be given. Then h ◦ µ ∈ Hom( K ∗ ( B Γ) , K ∗ ( B )). Bythe UCT (8.2), there is x ∈ K ( C ( B Γ) ⊗ B ) such that h (cid:0) µ ( z ) (cid:1) = (cid:104) x, z (cid:105) for all z ∈ K ∗ ( B Γ) . Note that if t ∈ Ext( K ∗ ( B Γ) , K ∗ +1 ( B )), then (cid:104) x + t, z (cid:105) = (cid:104) x, z (cid:105) . Thus without any loss of generalitywe may assume that x is almost flat. Therefore there exist a finite open cover V of B Γ, a decreasingsequence ( ε n ) of positive numbers converging to 0 and two sequences ( E ± n ) of bundles over B Γsuch that E ± n are ( V , ε n )-flat and satisfies x = [ E + n ] − [ E − n ] for all n . By passing to barycentricsubdivisions of the simplicial structure Λ of B Γ we may assume that the dual cover C Λ refines theopen cover V . By Proposition 4.8 we may arrange that the coordinate bundles underlying the ( E ± n )are normalized.Write F ± n for the fibers of E ± n ; these are finitely generated projective Hilbert B -modules andtherefore embed as direct summands of some B k ( n ) . This gives full-corner embeddings A ± n := L B ( F ± n ) ⊂ M k ( n ) ( B ). Using Proposition 4.6 and Proposition 5.6 we associate with E ± n quasi-representations π ± n : Γ → U( A ± n ) such that lim n →∞ (cid:107) π ± n ( st ) − π ± n ( s ) π ± n ( t ) (cid:107) = 0 for all s, t ∈ Γand lim n →∞ (cid:107) π ± n ( s − ) − π ± n ( s ) ∗ (cid:107) = 0 for all s ∈ Γ. The sequences ( π ± n ) induce morphisms ofgroups Γ → U ( A ±∞ ) and hence ∗ -homomorphisms π ±∞ : (cid:96) (Γ) → A ±∞ and ¯ π ±∞ : C ∗ (Γ) → A ±∞ where A ±∞ = (cid:81) ∞ n =1 A ± n / (cid:80) ∞ n =1 A ± n . Let ¯ π ± : C ∗ (Γ) → (cid:81) ∞ n =1 A ± n be a set-theoretic lifting of π ±∞ . Write¯ π ± = (¯ π ± n ) n . For a sufficiently multiplicative quasi-representation π : Γ → U( A ) δ and a sufficientlysmall δ >
0, we will denote by E π the corresponding almost flat bundle constructed using thecocycle β ( π ) constructed in Proposition 5.4 (see Notation 8.3). For n sufficiently large we have that[ E π ± n ] = [ E ± n ]. This follows from Theorem 3.3 and Proposition 7.2 since bundles whose cocycles aresufficiently close to each other are isomorphic.Let us recall that the full assembly map µ : K ∗ ( B Γ) → K ∗ ( C ∗ (Γ)) is implemented by theMishchenko line bundle (cid:96) ∈ K ( C ( B Γ) ⊗ C ∗ (Γ)), via the Kasparov product KK ( C , C ( B Γ) ⊗ C ∗ (Γ)) × KK ∗ ( C ( B Γ) , C ) → KK ∗ ( C , C ∗ (Γ)) , (cid:96), z ) (cid:55)→ µ ( z ) := (cid:104) (cid:96), z (cid:105) . We have seen earlier (8.1) that one can represent (cid:96) by a projection e in matrices over C ( B Γ) ⊗ C [Γ]. So long as n is sufficiently large, Proposition 8.4 guarantees that [ E π ± n ] equals (cid:96) π ± n , the push-forward of e by id C ( B Γ) ⊗ π ± n in K ( C ( B Γ) ⊗ A ± n ) ∼ = K ( C ( B Γ) ⊗ B ). The latter isomorphism isinduced by the full-corner embeddings A ± n ⊂ M k ( n ) ( B ). It follows that (cid:104) x, z (cid:105) = (cid:104) [ E + n ] , z (cid:105) − (cid:104) [ E − n ] , z (cid:105) = (cid:104) (cid:96) π + n , z (cid:105) − (cid:104) (cid:96) π − n , z (cid:105) for all z ∈ K ∗ ( B Γ). Let µ (cid:96) : K ∗ ( B Γ) → K ∗ ( (cid:96) (Γ)) be Lafforgue’s (cid:96) -version of the assembly map.It is known that j ∗ ◦ µ (cid:96) = µ where j : (cid:96) (Γ) → C ∗ (Γ) is the canonical map [15]. By [4, Theorem 3.2and Corollary 3.5] we have that (cid:104) (cid:96) π ± n , z (cid:105) = π ± n (cid:93) (cid:0) µ (cid:96) ( z ) (cid:1) for each z ∈ K ∗ ( B Γ) so long as n is sufficiently large. (For z ∈ K ( B Γ) we interpret π ± n (cid:93) (cid:0) µ (cid:96) ( z ) (cid:1) as π ± n (cid:93) ( p z ) − π ± n (cid:93) ( q z ) where p z , q z are projections with µ (cid:96) ( z ) = [ p z ] − [ q z ]. There is a similarinterpretation of π ± n (cid:93) (cid:0) µ (cid:96) ( z ) (cid:1) for z ∈ K ( B Γ) obtained by replacing idempotents by invertibleelements.) Therefore, (cid:0) π + n (cid:93) (cid:0) µ (cid:96) ( z ) (cid:1) − π − n (cid:93) (cid:0) µ (cid:96) ( z ) (cid:1)(cid:1) ≡ ( (cid:104) x, z (cid:105) ) = (cid:0) h ( µ ( z )) (cid:1) for all z ∈ K ∗ ( B Γ). From Remark 8.2(3) we deduce that¯ π ± n (cid:93) (cid:0) µ ( z ) (cid:1) = ¯ π ± n (cid:93) (cid:0) j ∗ ( µ (cid:96) ( z )) (cid:1) ≡ π ± n (cid:93) (cid:0) µ (cid:96) ( z ) (cid:1)(cid:0) ¯ π + n (cid:93) (cid:0) µ ( z ) (cid:1) − ¯ π − n (cid:93) (cid:0) µ ( z ) (cid:1)(cid:1) ≡ (cid:0) h ( µ ( z )) (cid:1) for all z ∈ K ∗ ( B Γ). The discrete asymptotic homomorphisms { ¯ π ± n : C ∗ (Γ) → M k ( n ) ( B ) } n have thedesired properties.(2) ⇒ (1). Let us assume now that K ∗ ( B ) is finitely generated and that µ is split injective.Let x ∈ K ( C ( B Γ) ⊗ B ) be given. We will find an almost flat element y ∈ K ( C ( B Γ) ⊗ B ) suchthat x − y ∈ Ext( K ∗ ( B Γ) , K ∗ +1 ( B )). Since µ is split-injective by hypothesis (i.e. the image of µ isa direct summand of K ∗ ( C ∗ (Γ)) , it follows from the exactness of the sequence (8.2) that there isa homomorphism h : K ∗ ( C ∗ (Γ)) → K ∗ ( B ) such that h ◦ µ ( z ) = (cid:104) x, z (cid:105) for all z ∈ K ∗ ( B Γ). By theassumptions in (2) there are two discrete asymptotic homomorphisms { π ± n : C ∗ (Γ) → M k ( n ) ( B ) } n such that ( π + n (cid:93) (cid:0) µ ( z ) (cid:1) − π − n (cid:93) (cid:0) µ ( z ) (cid:1) ) ≡ (cid:0) h (cid:0) µ ( z ) (cid:1)(cid:1) = ( (cid:104) x, z (cid:105) ) (8.3)for all z ∈ K ( B Γ). By a standard perturbation argument we may assume that π ± n ( s ) ∈ U k ( n ) ( B )for all n and s ∈ Γ. Invoking [4, Thm. 3.2 and Cor. 3.5] we obtain that( (cid:104) (cid:96) π ± n , z (cid:105) ) ≡ (cid:0) π ± n (cid:93) (cid:0) µ ( z ) (cid:1)(cid:1) . By Proposition 8.4, we have that (cid:96) π ± n = [ E π ± n ] where the bundles E π ± n are ( V , ε n )-flat and ε n → x n := [ E π + n ] − [ E π − n ], it follows from (8.3) that for any z ∈ K ∗ ( B Γ), there is n z suchthat (cid:104) x, z (cid:105) = (cid:104) x n , z (cid:105) for n ≥ n z . Since K ∗ ( B Γ) is finitely generated there exists n such that x − x n ∈ H := Ext( K ∗ ( B Γ) , K ∗ +1 ( B )) for all n ≥
0. Since K ∗ ( B ) is finitely generated, the group H is finite. Therefore after passing to a subsequence of ( x n ) we may arrange that the sequence( x − x n ) is constant and so there is t ∈ H such that x + t = x n for all n . It follows that y := x + t is is almost flat and x − y ∈ H . (cid:3) B Γ is a finite complex and the group K ∗ ( B ) is finitely generated and torsion-free, it followsthat the group Ext( K ∗ ( B Γ) , K ∗ +1 ( B )) is finite and by (8.2) it coincides with the torsion subgroupof K ( C ( B Γ) ⊗ B ). By taking B = C in Theorem 8.6 we obtain the following. Let Γ be a discrete countable group whose classifying space B Γ is a finite simplicialcomplex. If the full assembly map is bijective, then the following conditions are equivalent (1) All elements of K ( B Γ) are almost flat modulo torsion. (2) For any group homomorphism h : K ( C ∗ (Γ)) → Z there exist discrete asymptotic homo-morphisms { π ± n : C ∗ (Γ) → M k ( n ) ( C ) } n such that ( π + n (cid:93) ( y ) − π − n (cid:93) ( y )) ≡ ( h ( y )) for every y ∈ K ( C ∗ (Γ)) . With the terminology from the introduction, condition (2) above amounts to saying that C ∗ (Γ)is K-theoretically MF. Gromov indicates in [7,8] how one constructs nontrivial almost flat K -theory classesfor residually finite groups Γ that are fundamental groups of even dimensional non-positively curvedcompact manifolds. References [1] Bruce Blackadar and Eberhard Kirchberg,
Generalized inductive limits of finite-dimensional C ∗ -algebras , Math. Ann. (1997), no. 3, 343–380. MR 1437044 (98c:46112)[2] Jos´e R. Carri´on and Marius Dadarlat, Quasi-representations of surface groups , J. Lond. Math.Soc. (2) (2013), no. 2, 501–522. MR 3106733[3] Alain Connes, Mikha¨ıl Gromov, and Henri Moscovici, Conjecture de Novikov et fibr´es presqueplats , C. R. Acad. Sci. Paris S´er. I Math. (1990), no. 5, 273–277. MR 1042862[4] Marius Dadarlat,
Group quasi-representations and index theory , J. Topol. Anal. (2012), no. 3,297–319. MR 2982445[5] , Group quasi-representations and almost flat bundles , J. Noncommut. Geom. (2014),no. 1, 163–178. MR 3275029[6] James Dugundji, An extension of Tietze’s theorem , Pacific J. Math. (1951), 353–367. MR0044116[7] Mikhael Gromov, Geometric reflections on the Novikov conjecture , Novikov conjectures, indextheorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., vol.226, Cambridge Univ. Press, Cambridge, 1995, pp. 164–173. MR 1388301[8] ,
Positive curvature, macroscopic dimension, spectral gaps and higher signatures , Func-tional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993), Progr. Math.,vol. 132, Birkh¨auser Boston, Boston, MA, 1996, pp. 1–213. MR 1389019[9] Mikhael Gromov and H. Blaine Lawson, Jr.,
Positive scalar curvature and the Dirac operatoron complete Riemannian manifolds , Inst. Hautes ´Etudes Sci. Publ. Math. (1983), no. 58, 83–196 (1984). MR 720933 2310] Bernhard Hanke and Thomas Schick,
Enlargeability and index theory , J. Differential Geom. (2006), no. 2, 293–320. MR 2259056[11] Nigel Higson and Gennadi G. Kasparov, E -theory and KK -theory for groups which act properlyand isometrically on Hilbert space , Invent. Math. (2001), no. 1, 23–74. MR 1821144[12] Max Karoubi, K -theory , Springer-Verlag, Berlin, 1978, An introduction, Grundlehren derMathematischen Wissenschaften, Band 226. MR 0488029[13] Gennadi G. Kasparov, Equivariant KK -theory and the Novikov conjecture , Invent. Math. (1988), no. 1, 147–201. MR 918241[14] Gennadi G. Kasparov and Georges Skandalis, Groups acting on buildings, operator K -theory,and Novikov’s conjecture , K -Theory (1991), no. 4, 303–337. MR 1115824[15] Vincent Lafforgue, K -th´eorie bivariante pour les alg`ebres de Banach et conjecture de Baum-Connes , Invent. Math. (2002), no. 1, 1–95. MR 1914617[16] Vladimir M. Manuilov and Aleksandr S. Mishchenko, Almost, asymptotic and Fredholm repre-sentations of discrete groups , Acta Appl. Math. (2001), no. 1-3, 159–210, Noncommutativegeometry and operator K -theory. MR 1865957[17] Alexander S. Mishchenko and Nicolae Teleman, Almost flat bundles and almost flat structures ,Topol. Methods Nonlinear Anal. (2005), no. 1, 75–87. MR 2179351[18] Henri Moscovici, Cyclic cohomology and invariants of multiply connected manifolds , Proceed-ings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) (Tokyo), Math.Soc. Japan, 1991, pp. 675–688. MR 1159254[19] Anthony Phillips and David Stone,
Lattice gauge fields, principal bundles and the calculationof topological charge , Comm. Math. Phys. (1986), no. 4, 599–636. MR 832541[20] Anthony V. Phillips and David A. Stone,
The computation of characteristic classes of latticegauge fields , Comm. Math. Phys. (1990), no. 2, 255–282. MR 1065672[21] Georges Skandalis,
Approche de la conjecture de Novikov par la cohomologie cyclique (d’apr`esA. Connes, M. Gromov et H. Moscovici) , Ast´erisque (1991), no. 201-203, Exp. No. 739, 299–320 (1992), S´eminaire Bourbaki, Vol. 1990/91. MR 1157846[22] Edwin H. Spanier,
Algebraic topology , McGraw-Hill Book Co., New York, 1966. MR 0210112[23] Jean-Louis Tu,
The gamma element for groups which admit a uniform embedding into Hilbertspace , Recent advances in operator theory, operator algebras, and their applications, Oper.Theory Adv. Appl., vol. 153, Birkh¨auser, Basel, 2005, pp. 271–286. MR 2105483[24] Guoliang Yu,
The coarse Baum-Connes conjecture for spaces which admit a uniform embeddinginto Hilbert space , Invent. Math.139