The relative Mishchenko--Fomenko higher index and almost flat bundles II: Almost flat index pairing

This is the second part of a series of papers which bridges the Chang--Weinberger--Yu relative higher index and geometry of almost flat hermitian vector bundles on manifolds with boundary. In this paper we apply the description of the relative higher index given in Part I to provide the relative version of the Hanke--Schick theorem, which relates the relative higher index with index pairing of a K-homology cycle with almost flat relative vector bundles. We also deal with the quantitative version and the dual problem of this theorem.


Introduction
This paper is a sequel of [Kub18]. In this part II we apply the Mishchenko-Fomenko description of the Chang-Weinberger-Yu relative higher index developed in the part I to the index pairing with almost flat bundles on manifolds with boundary. Here we also make use of the foundations of almost flat (stably) relative bundles prepared in [Kub19].
The notion of almost flat bundle is introduced as a geometric counterpart of the higher index theory by Connes-Gromov-Moscovici [CGM90] for the purpose of proving the Novikov conjecture for a large class of groups. It also plays a fundamental role in the study of positive scalar curvature metrics in [GL83,Gro96]. Its central concept is the almost monodromy correspondence, that is, the rough one-to-one correspondence between almost flat bundles and quasi-representations of the fundamental group. In [Kub19], the author consider the relative and stably relative vector bundles on a pair of topological spaces as a representative of the relative K 0 -group and introduce the notion of almost flatness for them. Moreover, the almost monodromy correspondence is generalized to this relative setting.
The relation between the role of almost flat index pairing in geometry and the C*-algebraic higher index theory is clearly understood in the work of Hanke and Schick. In [HS06,HS07], they proved that the higher index α Γ ([M ]) of the K-homology fundamental class [M ] ∈ K * (M ) of an enlargeable closed spin manifold M with π 1 (M ) = Γ does not vanish without any assumption on the fundamental group concerned with the Baum-Connes conjecture. As is reorganized in [Han12], this is essentially a consequence of the fact that α Γ ([M ]) = 0 if M admits an almost flat bundle with non-trivial index pairing. The idea of Hanke and Schick relies on the fact that the dual higher index is related to the monodromy correspondence of flat bundles of Hilbert C*-modules.
The Mishchenko-Fomenko higher index map α Γ is given by the Kasparov product with the KK-class ℓ Γ ∈ KK(C, C(M ) ⊗ C * Γ) represented by the Mishchenko line bundleM × Γ C * Γ. For a unitary representation π : Γ → B(P ) to a finitely generated projective Hilbert A-module, the Kasparov product with ℓ Γ over C * Γ (which is called the dual higher index map in this paper) maps the element [π] ∈ KK(C * Γ, A) to the associated bundle The idea of Hanke-Schick is to construct a nice flat Hilbert C*-module bundle from a family of almost flat bundles.
The first purpose of this paper, studied in Section 3, is to establish a relative version of the result of Hanke-Schick. There are two ingredients of it. One is the basic theory of almost flat bundles on manifolds with boundary (particularly the almost monodromy correspondence) developed in [Kub19], and the other is the relative version of index pairing (1.1). Here the higher index is replaced with the Chang-Weinberger-Yu relative higher index map [CWY15], which is a homomorphism α Γ,Λ : K * (X, Y ) → K * (C * (Γ, Λ)), defined for a pair of CW-complexes (X, Y ) with π 1 (X) = Γ and π 1 (Y ) = Λ (for more details, see Subsection 2.1). It is proved in [Kub18] that this map is given by the Kasparov product with an element ℓ Γ,Λ ∈ KK(C, C 0 (X • ) ⊗ C * (Γ, Λ)). The key observation is Theorem 3.3, corresponding to the fact ℓ Γ ⊗ C * Γ [π] = [P] in the above paragraph. Roughly speaking, here we show that the Kasparov product with ℓ Γ,Λ maps a relative representation of (Γ, Λ), i.e. a pair of representations of Γ which is identified on Λ, to the associated relative bundle. To realize the concept in full generality, we employ the equivalence relation generated by unitary equivalence, stabilization and homotopy as the 'identification' of a pair of representation. Then we get the result corresponding to (1.1) by the same argument using the associativity of the Kasparov product. The relative Hanke-Schick theorem (Theorem 3.5) is now obtained in the same way as [HS06,HS07] with the help of the relative almost monodromy correspondence.
In addition, there is another application of Theorem 3.3 to the index theoretic refinement of the Hanke-Pape-Schick codimension 2 index obstruction for the existence of positive scalar curvature metric [HPS15], which is discussed in Subsection 3.3. Here, the higher index of a codimension 2 submanifold W of M (with a condition on homotopy groups) is related to the relative higher index of the manifold M \ U , where U is a tubular neighborhood of W .
In the rest part of the paper we discuss in-depth problems related to relative index theory of almost flat bundles. In Section 4, we study the quantitative version of Theorem 3.5. A key idea of [HS06] is to treat an infinite family of almost flat bundles simultaneously and relate the asymptotics of the index pairings with the higher index. On the other hand, the ℓ 1 (Γ)-valued higher index, instead of the usual C * (Γ)-valued one, is mapped to a projection up to a small correction by a single quasi-representation. This map is studied in [CGM90] and compared with the index pairing with the associated almost flat bundle. In [Dad12], Dadarlat gives an alternative approach using Lafforgue's Banach KK-theory. Here, we reformulate the result of [Dad12] in terms of the quantitative K-theory introduced in Oyono-Oyono-Yu [OOY15] instead of Banach KK-theory. After that, we generalize the result of Connes-Gromov-Moscovici to the relative setting.
In Section 5, we study the dual problem of Theorem 3.5, in other words, the relative version of the problem proposed by Gromov in [Gro96, Section 4 2 3 ]. It is a consequence of the almost monodromy correspondence that any almost flat bundle is obtained by pull-back from the classifying space BΓ. Then it is a natural question whether any element of K 0 (BΓ) (or K 0 (BΓ) ⊗ Q) is represented by an almost flat bundle. This question is first considered in [Gro96,Section 8 14 15 ] geometrically for the fundamental group of a Riemannian manifold with non-positive sectional curvature. After that, Dadarlat gives a KK-theoretic approach to this problem in [Dad14]. Here we follow this idea to study the almost flat K-theory class of the pair (BΓ, BΛ). The celebrated Tikuisis-White-Winter theorem [TWW17] in the theory of C*-algebras enables us to include a large class of residually amenable groups to the range of our discussion. We show that any element of the range of the dual higher index map β Γ,Λ : K 0 (C * (Γ, Λ)) → K 0 (X, Y ), i.e. the Kasparov product with ℓ Γ,Λ over C * (Γ, Λ), is represented by an almost flat stably relative vector bundle. Moreover, we also show that such elements are represented by an almost flat relative vector bundle if φ : Λ → Γ is injective.
Notation 1.2. Throughout this paper we use the following notations.
• For a C*-algebra A, let A + denote its unitization A + C · 1.
• For a C*-algebra A, let M(A) denote its multiplier C*-algebra and Q(A) := M(A)/A. • For a C*-algebra A and a < b ∈ R∪{±∞},let A(a, b) := A⊗C 0 (a, b).
Similarly we define A[a, b) and A [a, b]. For a Hilbert A-module E, let E(a, b) denote the Hilbert A(a, b)-module E ⊗ C 0 (a, b). • For a * -homomorphism φ : A → B, let Cφ denote the mapping cone C*-algebra defined as • For a Hilbert A-module E, let B(E) and K(E) denote the C*-algebra of bounded adjointable and compact operators on E respectively. Let U(E) denote the unitary group of B(E). • For a compact space X and a Hilbert A-module P , let P X denote the trivial bundle X × P on X.
Acknowledgment. The author would like to Martin Nitsche and Thomas Schick for their careful reading and helpful comments on a previous version of this paper. This work was supported by RIKEN iTHEMS Program.

Prelimilaries
In this section we summarize the results of [Kub18] and [Kub19] which will be used in this paper. Throughout this paper we focus on the complex coefficient K-theory, C*-algebra, vector bundle and so on.
Here we give a remark on a realization the relative Mishchenko line bundle ℓ Γ,Λ , which is an element of the K 1 -group K 1 (C 0 (X • 2 ) ⊗ Cφ), as a unitary of C 0 (X • 2 ) ⊗ Cφ. Let U := {U µ } µ∈I be a finite open cover of X such that the restriction ofX to each U µ is a trivial bundle. We choose a local trivialization θ µ :X| Uµ ∼ = U µ ×Γ and let γ µν denote the transformation function Let {η µ } µ∈I be a family of continuous functions such that supp(η µ ) ⊂ U µ , 0 ≤ η µ (x) ≤ 1 and η 2 µ = 1. We write M I for the matrix algebra on C I and let {e µν } µ,ν∈I denote the matrix unit. Then, is a projection whose support is isomorphic to V as Hilbert C * Γ-module bundles on X. This means that ℓ Γ = [P V ].
2.2. Rational surjectivity of the dual relative higher index map. The rational injectivity of the relative higher index map and the rational surjectivity of its dual are studied in [Kub18, Section 6]. In Section 5 we apply the latter for the existence of almost flat relative vector bundle representing an arbitrary element of relative K 0 -group of the pair (BΓ, BΛ).
2.3. Almost flat relative bundles. Here we briefly review the notion of almost flat (stably) relative vector bundles and its almost monodromy correspondence. Let X be a finite CW-complex with a good open cover U := µ∈I . Then the fundamental group Γ := π 1 (X) is generated by G := {γ µν } µ,ν∈I once we fix a translation function {γ µν } µ,ν∈I of the Γ-Galois coveringX.
Definition 2.13. Let X, U , Γ, G be as above. Let A be a unital C*algebra, let P be a finitely generated projective Hilbert A-module and let T be a maximal subtree of the 1-skeleton N (1) It is said to be normalized on T if v µν − 1 < ε for any µ, ν ∈ T . • A map π : Γ → U(P ) is a (ε, G)-representation of Γ on P if π(e) = 1 and π(g)π(h) − π(gh) < ε for any g, h ∈ G.
Definition 2.17. Let (X, Y ) be a pair of compact spaces. A stably relative bundle on (X, Y ) with the typical fiber (P, Q) is a quadruple (E 1 , E 2 , E 0 , u), where E 1 and E 2 are P -bundles on X, E 0 is a Q-bundle on Y and u is a A stably relative bundle of Hilbert C-modules with the typical fiber (C n , C m ) is simply called a stably relative vector bundle of rank (n, m). We simply call a stably relative bundle of the form (E 1 , E 2 , 0, u) a relative bundle.
Remark 2.18. A stably relative bundle associates an element of the relative K 0 -group K 0 (X, Y ; A) := K 0 (C 0 (X • 1 )⊗A) in the following way. Let f 1 (r) := min{1, max{0, 1 − 3r}} and f 2 (r) := min{1, max{0, 3r − 2}}. The inverse of κ is given by mapping (E 1 , E 2 , E 0 , u) to Let (X, Y ) be a pair of finite CW-complexes. We say that a good open cover is a good open cover Definition 2.19. Let (X, Y ) and U be as above and let P be a finitely generated Hilbert A-module. Let T be a maximal subtree of the 1-skeleton of N (U ) such that T | N (U | Y ) is also a maximal subtree.
• An (ε, U )-flat stably relative bundle on (X, Y ) with the typical fiber (P, Q) is a quadruple v : . It is said to be normalized on T if v 1 , v 2 are normalized at T and v 0 is normalized at T ∩ Y . We write the set of (ε, U )-flat stably relative bundles on (X, Y ) normalized at T with the typical fiber (P, Q) as Bdl ε,U P,Q (X, Y ) T . We define the metric on Bdl ε,U P, 20. An (ε, U )-flat stably relative bundle v = (v 1 , v 2 , v 0 , u) associates an element of the relative K 0 -group K 0 (X, Y, A). We give some remarks on this element.
(1) If ε > 0 is sufficiently small and v, v ′ ∈ Bdl ε,U P, where C > 0 is a constant depending only on U . This family {ū µ } µ∈I induces a bundle mapū : (3) Let {η µ } and e µν be as in Remark 2.14. The element That is, w is identified withū in (2) under the canonical isomorphism We remark that thisw satisfies We say that an element ξ ∈ K 0 (X, Y ; A) is (resp. stably) almost flat with respect to a good open cover U if for any ε > 0 there is a (ε, U )-flat (resp. stably) relative vector bundle v of finitely generated projective Hilbert Amodules such that (1) We write K 0 af (X, Y ; A) (resp. K 0 s-af (X, Y ; A)) for the subgroup of (resp. stably) almost flat elements.
(2) We say that a K-homology class ξ ∈ K * (X, Y ) has infinite (resp. stably) relative K-area if there is an (resp. stably) almost flat Ktheory class x ∈ K 0 (M, N ) such that the index pairing x, ξ is non-zero. (3) We say that ξ ∈ K * (X, Y ) has infinite (resp. stably) relative C*-Karea if for any ε > 0 there is a C*-algebra A ε and a (resp. stably) relative (ε, U )-flat bundle v of finitely generated projective Hilbert A ε -modules such that the index pairing [v], ξ ∈ K 0 (A ε ) is non-zero.
In particular, we say that a spin manifold M with the boundary N has (stably) relative infinite (C*-)K-area if so is the K-homology fundamental class [M, N ] ∈ K * (M, N ). Finally we review the almost monodromy correspondence in the relative setting.

Relative index pairing with coefficient in a C*-algebra
In this section, we establish an obstruction for the relative higher index to vanish arising from an index pairing with coefficient in a C*-algebra. It has two applications; a relative version of the Hanke-Schick theorem [HS06,HS07] and the non-vanishing of relative higher index in the setting of Hanke-Pape-Schick [HPS15].
3.1. Index pairing with stably h-relative representations. For a representation of the fundamental group Γ = π 1 (M ) of a closed spin manifold M on a finitely generated projective Hilbert A-module P , i.e., a homo- coincides with the index pairing with the flat P -bundle associated to π. Here we develop its relative version. The relative counterpart of group representation is a pair of representations of Γ whose restriction to Λ are identified 'up to stabilization and homotopy' in the following sense.
In particular,Ū is a U -connection. By Lemma 3.4 we obtain that On the other hand, letP i : This concludes the proof since ι * • f * :  (1) If M has infinite stably relative C*-K-area, then the relative higher Proof. First we show (1). By the assumption, for each n ∈ N there is a C*-algebra A n , a pair of finitely generated projective Hilbert A n -modules (P n , Q n ) and a ( 1 n , U )-flat stably relative bundle v n : We define the stably relative bundle is a stably relative flat bundle. Let Π ∈ KK(C * (Γ, Λ), D) denote the Kasparov bimodule associated to the stably relative representation α(τ * v) as in Theorem 2.24. By Theorem 3.3 we obtain that It is non-zero because ker τ * is identified with The claim (2) is proved in the same way. We only remark that in this case Π is a relative representation of (Γ, Λ), which is actually a relative representation of (Γ, φ(Λ)) by [Kub19, Remark 6.3].
Together with Theorem 2.22, Theorem 3.5 implies the following relative version of the result of [HS06,HS07]. Theorem 3.7. Let M be an n-dimensional closed spin manifold with an embedded codimension 2 submanifold N satisfying ). We apply this theorem to M = N × D 2 ⊔ N ×S 1 M 0 , Γ 1 = π 1 (N × D 2 ), Λ = π 1 (N ) and Γ 2 = π 1 (M 0 ) in the setting of Theorem 3.7. Then the conclusion of Theorem 3.7 implies the non-vanishing of µ Γ ([M ]). In particular, we obtain that M does not admit any metric with positive scalar curvature, as is proved in [HPS15, Theorem 4.3].
As is remarked at the introduction of [HPS15], the stable Gromov-Lawson-Rosenberg conjecture proved by Rosenberg-Stolz [RS95] and [HPS15, Theorem 4.3] also implies the non-vanishing of the higher index of M if Γ satisfies the Baum-Connes injectivity. Here we give a direct proof of this fact without the assumption of Baum-Connes injectivity.
Let A be a C*-algebra and let B := B(H A ) and J := K(H A ). Let Z 1 and Z 2 be bundles of infinitely generated projective Hilbert A-modules with the typical fiber Z 1 and Z 2 respectively. Then from the right, the Q(Z i )-action from the left and the inner product are induced from the product of operators). Suppose that there is a bundle homomorphism U : Proof. First, notice that there are isometries V i : Z 2 ). Indeed, let S denote a unitary lift of 0Ū * U 0 and let W : Z 1 ⊕ Z 2 → H A be an isometry (which exists by the Kasparov stabilization theorem [Kas80, Theorem 2]). Then V 1 := W V ′ 1 and V 2 := W SV ′ 2 , where V ′ i : Z i → Z 1 ⊕ Z 2 is the embedding to the i-th direct summand, is desired isometries. Moreover, by a pull-back with respect to a deformation retract of N 0 , we may assume that P 1 = P 2 on a neighborhood O of N 0 . Let ψ be a continuous function supported on O such that 0 ≤ ψ ≤ 1 and ψ| N 0 ≡ 1 and let P ′ := ψP 1 + (1 − ψ)P 2 . Now we apply Lemma 3.9 to determine the left and right hand side of (3.11). Since (P 1 , P ′ ) is a self-adjoint lift of (q(P 1 ), q(P 2 )) ∈ M(C 0 (M Similarly, since (P ′ , P 2 ) is a self-adjoint lift of (P This completes the proof of the lemma. An essential ingredient of the codimension two obstruction theorem, which is given in the proof of [HPS15,Theorem 4.3], is the existence of a nice Λ×Z-Galois covering onM \W • . Here we restate it for our convenience. Lemma 3.12. There is a Z-Galois coveringM 0 overM 0 := (π •π) −1 (M 0 ) with the following properties: • Its restriction toπ −1 (N 0 ) ∼ =Ñ × S 1 is the universal covering.
Then the homomorphism pr Λ •r (where pr Λ : Λ×Z → Λ is the projection) is equal to j * . Indeed, both pr Λ • r and j * map [γ] to the trivial element and the induced homomorphisms from π 1 (M \W • )/ [γ] to Λ are the inverse of the composition Therefore the coveringM 0 ofM 0 associated to The equality r • i * = id Λ×Z means that the restriction ofM 0 toN 0 is the universal coveringÑ × R of N × S 1 . That is, the restriction of the Z-Galois coveringM 0 toπ −1 (N ) ∼ =Ñ ×S 1 is the universal covering. At the same time, the restriction of the Z-Galois coveringM 0 to each connected component of π −1 (π −1 (N ) \N ) is trivial because it is extended to a connected component of (π •π) −1 (W ), which is simply connected. of Hilbert A-modules overM 0 , whereM 0 is as in Lemma 3.12. We associate to them bundles of infinitely generated (by Lemma 3.13) Hilbert A-module bundles on M 0 , which are equipped with the canonical flat structures. Let Z i := π(x)=x 0 (V i )x is the fiber of Z i on x 0 and let σ i :Γ → U(Z i ) denotes the associated monodromy representation. Note that σ 2 factors through Γ. By the construction ofM 0 in Lemma 3.12, we have an isomorphism of flat A-module bundles between the restriction of V 1 and V 2 onπ −1 (N 0 ) \N 0 . It induces a partial isometry U : As in Lemma 3.10, letZ i denote the bundle B(Z i , H A )/K(Z i , H A ) of Hilbert B/J-modules and letZ i : Then σ i and U above inducesσ i :Γ → U(Q(Z i )) ∼ = U(Z i ) andŪ :Z 1 → Z 2 respectively. ThenŪ is a unitary andŪ x 0σ 1 (g)Ū * x 0 =σ 2 (g) holds for any g ∈ Λ×Z. This particularly implies thatσ 1 (γ) = 1 (where γ is the generator of Z ⊂ Λ × Z), that is,σ 1 :Γ → U(Z 1 ) factors through Γ.
For the forth equality, we use the boundary of Dirac is Dirac principle

Relative quantitative index pairing
In this section, we reformulate the index theorem for the image of the higher index under a quasi-representation developed by Dadarlat [Dad12] and generalize it to the relative setting. Instead of Lafforgue's Banach KKtheory, on which the formulation of [Dad12] is based, we use the quantitative K-theory introduced by Oyono-Oyono and Yu [OOY15]. 4.1. Quantitative K-theory and almost * -homomorphism. We start with a quick review of the quantitative K-theory. The basic reference is [OOY15]. We say that a filtered C*-algebra is a C*-algebra A equipped with an increasing family {A r } r∈[0,∞) of closed subspaces of A such that For a unital filtered C*-algebra A, 0 ≤ ε ≤ 1 4 and r > 0, let and P ε,r ∞ (A) := n∈N P ε,r n (A), U ε,r ∞ (A) := n∈N U ε,r n (A). For k ∈ N, let 1 k denote the unit of M k ⊂ A + ⊗ M k . We introduce the equivalence relation to P ε,r ∞ (A) × N and U ε,r ∞ (A) as • (p, k) ∼ (q, l) if diag(p, 1 l ) and diag(q, 1 k ) are connected by a continuous path in P ε,r ∞ (A), • u ∼ v if u and v are connected by a continuous path in U 3ε,2r ∞ (A).
Definition 4.2. Let A and D be filtered C*-algebras. A bounded linear map π : A r → D κr is a complete (ε, r, κ)- * -homomorphism if π(a * ) = π(a) * for any a ∈ A r and π n (ab) − π n (a)π n (b) ≤ ε a b holds for any a, b ∈ A r ⊗ M n , where π n := π ⊗ id Mn .
This means that π n 2 < π n + ε and hence π n < 1 + ε/2. That is, π is a completely bounded map between operator spaces (a reference on completely bounded maps and operator spaces is [BO08, Appendix B]). In particular, π ⊗ id B : A r ⊗ B → D κr ⊗ B is a well-defined completely bounded map for any nuclear C*-algebra B ([BO08, Corollary B.8]).
A C*-algebra is said to be quasi-diagonal if it admits a faithful representation π : A → B(H) with an increasing sequence p n of finite rank projections such that [π(a), p] → 0 for any a ∈ A (for more details, see for example [BO08, Section 7]).
Proof. First, π ⊗ id n∈N Mn is a complete (ε, r, κ)- * -homomorphism since A ⊗ ( n M n ) is canonically isomorphic to n (A ⊗ M n ). Since there is an isomorphism we obtain that π ⊗id Mn/ Mn is also a complete (ε, r, κ)- * -homomorphism.
Recall that a nuclear C*-algebra B is quasi-diagonal if and only if there is a faithful * -homomorphism ϕ : B → n∈N M n / n∈N M n . Since the diagram Mn commutes, π ⊗ id B is also a (ε, r, κ)- * -homomorphism.
Proposition 4.7. Let π be a self-adjoint (ε, G r Γ )-representation of Γ on P . Then π is a unital complete (|G r Γ | 2 ε, r, 1)- * -homomorphism. Proof. Let x = γ∈G r Γ a γ u γ and y = γ∈G r Γ b γ u γ be elements in C * (Γ) r ⊗ M n , where a γ and b γ are elements of M n . We remark that a γ ≤ x and b γ ≤ y for any γ ∈ Γ. Indeed, let τ : C * Γ → C denote the tracial state given by τ ( c γ u γ ) := c e . Then we have x y ε. Let X be a finite CW-complex and let Γ := π 1 (X) (note that Γ is finitely presented). Let U := {U µ } µ∈I be a good cover of X and let {γ µν } µ,ν∈I be a flat transition function of the universal coveringX → X. Let G Γ := {γ µν } µ,ν∈I . Let v = {v µν } be a U(P )-valuedČech 1-cocycle. As are mentioned in (2.4) and Remark 2.14, the projections have the support isomorphic to V and E v respectively.
Theorem 4.10 is related to the Connes-Gromov-Moscovici index formula [CGM90, Théorème 10], which is generalized in [Dad12]. Let τ be a tracial state on the C*-algebra A. For a bundle E of finitely generated Hilbert A-modules, let ch τ (E) ∈ Ω even (M ) denote the Chern character defined in [Sch05, Definition 5.1]. In particular, if A = C and τ is the identity map, then ch τ (E) is the usual Chern character.
Then it is straightforward to check that α alg Γ,Λ satisfies ι Cφ • α alg Γ,Λ = α Γ,Λ in a similar fashion to Proposition 4.8. It is also checked in the same way as Proposition 4.8 that the map α alg Γ,Λ is well-defined independent of the choice of (ϕ 1 , ϕ 2 ).

Dual assembly map and almost flat bundles
In this section, we relate the dual higher index map β Γ,Λ defined in Proposition 2.3 with the almost monodromy correspondence, Theorem 2.24. The goal of this section is to show that the index pairing with elements of the subgroup K 0 s-af (X, Y ) of almost flat K-theory class has rich information enough to detect the non-vanishing of the relative higher index under a certain assumptions on the fundamental groups. Lemma 5.2. The inclusions Then there is a commutative diagram of exact sequences Lemma 5.4. The diagram commutes.
Proof. Here we write as S := C 0 (−1, 1). Apply the five lemma to the diagram of exact sequences which commutes by Lemma 5.4 and Lemma 5.5.
Lastly we consider the case that A and B are unital and φ : A → B preserves the unit. Let (σ, H) and (τ, K) are unital ample * -representations of A and B respectively and (σ,H) := (σ ⊕ τ, H ⊕ K). Then the * -representations σ + := σ ⊕ 0 H onto H + := H ⊕2 and τ + := τ ⊕ 0 K onto K + := K ⊕2 (where 0 H is the zero representation to H) extend to unital ample representations of A + and B + respectively. Here we use σ + and τ + for the definition of C*-algebras as in (5.1). We also define the C*-algebras D u By the six term exact sequence associated to the extension the K-group of B turns out to be zero. Hence the composition induces an isomorphism of K-theory. This finishes the proof since the quotient D L (φ) → D L (φ)/M 2 C also induces the isomorphism of K-theory.
Here we say that a discrete group Γ is residually amenable if for any nontrivial element γ ∈ Γ there is a homomorphism from Γ to an amenable group Γ ′ which maps γ to a nontrivial element. For example, all residually finite groups are residually amenable. In particular, all finitely generated linear groups [Mal40] and 3-manifold groups [Hem87] (thanks to Perelman's proof of the geometrization theorem) are examples of residually amenable groups. Note that they also satisfy the condition (2.6).
Since Γ is residually amenable, there is a decreasing sequence N n of normal subgroups of Γ such that Γ n := Γ/N n is amenable and n N n = {e}. Let λ n denote the left regular representation Γ → U(ℓ 2 (Γ n )) and let λ denote the left regular representation Γ → U(ℓ 2 (Γ)). Now it suffices to show that λ is weakly contained in n λ n .
Lemma 5.10. For a residually amenable group Γ, the intermediate com- Moreover, a homomorphism φ : Λ → Γ between residually amenable groups induces the * -homomorphism φ A : C * A (Λ) → C * A (Γ). Proof. Let Γ n and λ n be as in Lemma 5.9. By the Tikuisis-White-Winter theorem [TWW17], the group C*-algebra C * (Γ n ) is quasi-diagonal. Pick a dense sequence {a n } n∈N of C * A (Γ). Then, for each n > 0 there is an increasing sequence {p n,m ∈ B(ℓ 2 (Γ n ))} n≤m of finite rank projections such that [λ n (a l ), p n,m ] < 2 −m for all l ≤ m. Then, p m := p n,m is an increasing sequence of finite rank projections in ℓ 2 (Γ n ) such that [ n λ n (a l ), p m ] → 0 for all l ∈ N. Since n λ n is a faithful representation of C * A (Γ), the proof of the first part of the lemma is completed.
The second part follows from the fact that φ * (A Γ ) ⊂ A Λ since amenability is passed to subgroups.
As is remarked at the beginning of Section 5, we can choose τ as the zero representation if φ is injective. Then the projection f in the above argument is the zero projection, and hence the obtained β(π ′ ) is an (ε, U )-flat relative vector bundle on (X, Y ). Therefore, a given element x ∈ Im(β Γ,Λ • j φ (γ Γ )) is almost flat.
For a pair of (not necessarily finite) CW-complexes (X, Y ), we say that an element x ∈ K 0 (X, Y ) is (resp. stably) almost flat if f * x is (resp. stably) almost flat for any continuous map f from a pair of finite CW-complexes (Z, W ) to (X, Y ). Then Theorem 5.13, together with Theorem 2.10 (2), implies the following.
Equivalently, we characterize infiniteness of K-area by the characteristic class. Proof. It immediately follows from Corollary 5.14. We only remark that the Chern character gives an isomorphism between K 0 (BΓ, BΛ) Q and