Almost flat relative vector bundles and the almost monodromy correspondence
aa r X i v : . [ m a t h . K T ] A ug ALMOST FLAT RELATIVE VECTOR BUNDLES AND THEALMOST MONODROMY CORRESPONDENCE
YOSUKE KUBOTA
Abstract.
In this paper we introduce the notion of almost flatness for(stably) relative bundles on a pair of topological spaces and investigatebasic properties of it. First, we show that almost flatness of topologicaland smooth sense are equivalent. This provides a construction of an al-most flat stably relative bundle by using the enlargeability of manifolds.Second, we show the almost monodromy correspondence, that is, a cor-respondence between almost flat (stably) relative bundles and (stably)relative quasi-representations of the fundamental group.
Contents
1. Introduction 1Acknowledgment 32. Relative and stably relative bundles 33. Almost flatness for (stably) relative bundles 44. Comparing topological and smooth almost flatness 105. Enlargeability and almost flat bundle 166. Relative quasi-representations and almost monodromycorrespondence 19References 241.
Introduction
The notion of almost flat bundle provides a geometric perspective on thehigher index theory. It was introduced by Connes–Gromov–Moscovici [CGM90]for the purpose of proving the Novikov conjecture for a large class of groups.The original definition is given in terms of curvature of vector bundles, andhence requires a smooth manifold structure for the base space. Anotherdefinition of almost flat bundle is given in [MT05, Section 2], which makesense for bundles on simplicial complexes. The equivalence of these twodefinitions is studied in [Hun16].Its central concept is the almost monodromy correspondence , that is, thecorrespondence between almost flat bundles and quasi-representations ofthe fundamental group. This almost one-to-one correspondence has beenstudied in various contexts such as [CH90, MM01, Dad14, CD18]. It plays
Mathematics Subject Classification.
Primary 19K56; Secondary 19K35, 46L80,58J32.
Key words and phrases.
Almost flat bundle, quasi-representation, almost monodromycorrespondence. an important role in the work of Hanke–Schick [HS06, HS07], which bridgesthe C*-algebraic and geometric approaches to the Novikov conjecture andthe Gromov–Lawson–Rosenberg conjecture.The aim of paper is to consider a similar problem for manifolds withboundary. For a pair of topological spaces (
X, Y ), we introduce the notionof almost flatness for representatives of the relative K -group K ( X, Y ).Our definition is inspired from the one suggested in [Gro96] and [Lis13],but slightly different (a major difference is to treat stably relative bundlesinstead of relative bundles).There are two main conclusions of this paper. The first, Theorem 4.10,is the comparison of topological and smooth almost flat bundles, a relativeanalogue of the result of [Hun16]. This theorem has an application to theindex theory of (area-)enlargeable manifolds. Gromov–Lawson [GL83] andHanke–Schick [HS07] constructs an almost flat bundle of Hilbert C*-moduleswith non-trivial index on an enlargeable spin manifold. In this paper weconsider a relative counterpart of this idea for a Riemannian manifold M with boundary ∂M such that the complete Riemannian manifold M ∞ := M ⊔ ∂M ∂M × [0 , ∞ ) is area-enlargeable. We construct a stably relativebundle of Hilbert C*-modules on ( M, ∂M ) with non-trivial index pairing byapplying the construction of Gromov–Lawson and Hanke–Schick (Theorem5.1).The second, Theorem 6.12, is the almost monodromy correspondencein the relative setting. For a pair (Γ , Λ) of discrete groups with a homo-morphism φ : Λ → Γ, we introduce the notion of (stably) relative quasi-representation as two quasi-representations on Γ whose composition with φ are stably unitary equivalent up to small ε >
0. Following the work ofCarri´on and Dadarlat [CD18], we establish an almost monodromy correspon-dence between almost flat relative bundles and relative quasi-representationsof the pair of fundamental groups. This correspondence plays an importantrole in the paper [Kub19], which bridges the index pairing with almost flatstably relative bundles and Chang–Weinberger–Yu relative higher index. Inparticular, the almost flat stably relative bundle constructed in Theorem5.1 is used in [Kub19, Section 3.2] to show the non-vanishing of the Chang–Weinberger–Yu relative higher index through the almost monodromy corre-spondence.In this paper we consider not only relative vector bundles (or Karoubitriples) but also its refinement, stably relative vector bundles , as a repre-sentative of the relative K -group and sometimes compare them. A stablyrelative vector bundle on ( X, Y ) is a pair of vector bundles on X identifiedby a stable unitary isomorphism on Y (for a more precise definition, seeDefinition 2.1). There are two reasons to consider stably relative bundles.The first is related with the enlargeable manifolds. What is obtained fromthe enlargeability of M ∞ is not a relative but a stably relative bundle. Thesecond is related with the almost monodromy correspondence. As is pointedout in Remark 6.3, relative quasi-representation of (Γ , Λ) is the same thingas that of (Γ , φ (Λ)). That is, relative quasi-representation does not captureany information of ker φ . LMOST FLAT RELATIVE VECTOR BUNDLES 3
This paper is organized as follows. In Section 2, we introduce the notion ofstably relative bundle and show that it represents an element of the relativeK -group. In Section 3, we introduce the definition of the almost-flatness forstably relative bundles. In Section 4, we compare the topological and smoothalmost flatness and applies this to enlargeable manifolds. In Section 5, weapply the result of Section 4 to a construction of an almost flat sequence ofstably relative bundles on a enlargeable manifold with boundary. In Section6, we define the relative analogue of group quasi-representations and showsthe almost monodromy correspondence.Throughout this paper, we treat bundles of (finitely generated projec-tive) Hilbert C*-modules. This general treatment is useful for generalizingHanke–Schick theorem for a generalized notion of enlargeability introducedin [HS07] by using infinite covers. Acknowledgment.
The author would like to thank Yoshiyasu Fukumotofor introducing him to this topic. This work was supported by RIKENiTHEMS Program.2.
Relative and stably relative bundles
In this section we introduce the definition of stably relative vector bundlesand bundles of Hilbert C*-modules as a representative of relative K -group.Throughout this section A denotes a unital C*-algebra and P, Q denotefinitely generated projective Hilbert A -modules.Let ( X, Y ) be a pair of compact Hausdorff spaces. The relative K-groupK ( X, Y ) is defined as the Grothendieck construction of the monoid of equiv-alence classes of triples ( E , E , u ), where E and E are vector bundles on X and u is an isomorphism E | Y → E | Y ([Kar08, Chapter II, 2.29]). Inthis paper we call such triple a relative vector bundle . Now we modify thisdescription of the group K ( X, Y ). For a unital C*-algebra A , we define therelative K -group with coefficient in A by K ( X, Y ; A ) := K ( C ( X ◦ ) ⊗ A ),where X ◦ denotes the interior X \ Y . Definition 2.1. A stably relative bundle on ( X, Y ) with the typical fiber(
P, Q ) is a quadruple ( E , E , E , u ), where E and E are P -bundles on X , E is a Q -bundle on Y and u is a unitary bundle isomorphism E | Y ⊕ E → E | Y ⊕ E .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 say that stably relative bundles ( E , E , E , u ) and ( E ′ , E ′ , E ′ , u ′ )are isomorphic if there are unitary isomorphisms U i : E i → E ′ i for i = 0 , , U | Y , U ) u = u ′ diag( U | Y , U ). Let Bdl s ( X, Y ; A ) denote theset of isomorphism classes of stably relative bundles of finitely generatedprojective Hilbert A -modules. We consider the equivalence relation ∼ onBdl s ( X, Y ; A ) generated by • ( E , E , E , u ) ∼ ( E ′ , E ′ , E ′ , u ′ ) if they are homotopic, that is, thereis a stably relative vector bundle ( ˜ E , ˜ E , ˜ E , ˜ u ) on ( X [0 , , Y [0 , X × { } , Y × { } ) and ( X × { } , Y × { } ) areisomorphic to ( E , E , E , u ) and ( E ′ , E ′ , E ′ , u ′ ) respectively, YOSUKE KUBOTA • ( E , E , E , u ) ∼ (0 , , E | Y ⊕ E , ( v | Y ⊕ E ) ∗ u ) if v is a unitaryisomorphism from E to E , and • (0 , , E , E ) ∼ E , E , E , u ] + [ E ′ , E ′ , E ′ , u ′ ] := [ E ⊕ E ′ , E ⊕ E ′ , E ⊕ E ′ , u ⊕ u ′ ] makes the set Bdl s ( X, Y ; A ) / ∼ into an abelian monoid. Moreover,[ E , E , E , u ] has the inverse [ E , E , E , u ∗ ]. Lemma 2.2.
The group
Bdl s ( X, Y ; A ) / ∼ is isomorphic to the relative K -group K ( X, Y ; A ) .Proof. In the proof, we write as ¯K ( X, Y ; A ) := Bdl s ( X, Y ; A ) / ∼ . Let ρ : ( C ( X ◦ ) ⊗ A ) + → C denote the quotient. We define the map κ : K ( X, Y ; A ) → ¯K ( X, Y, A ) by κ ([ p ] − [1 n ]) = [ p ( A NX ) , A nX , , n ]for a projection p ∈ M N (( C ( X ) ⊗ A ) + ) with ρ ( p ) = 1 n .For a compact space X , let K ∗ ( X ; A ) := K ∗ ( C ( X ) ⊗ A ). Let i ∗ : C ( X ) → C ( Y ) denote the restriction and let j : C ( X ◦ ) → C ( X ) denote the inclusion.Consider the homomorphisms¯ ∂ : K ( Y ; A ) → ¯K ( X, Y ; A ) , ¯ ∂ [ u ] = [0 , , A nY , u ] , ¯ j ∗ : ¯K ( X, Y ; A ) → K ( X ; A ) , ¯ j ∗ [ E , E , E , u ] = [ E ] − [ E ] . Actually, the equivalence relation ∼ is defined in such a way that ¯ ∂ and ¯ j are well-defined and the second row of the commutative diagramK ( X ; A ) i ∗ / / K ( Y ; A ) ∂ / / K ( X, Y ; A ) j ∗ / / κ (cid:15) (cid:15) K ( Y ; A ) i ∗ / / K ( X ; A )K ( X ; A ) i ∗ / / K ( Y ; A ) ¯ ∂ / / ¯K ( X, Y ; A ) ¯ j ∗ / / K ( X ; A ) i ∗ / / K ( Y ; A )is exact (for the exactness at K ( Y ; A ), note that [0 , , A nY ,
1] = [ A nX , A nX , ,
1] =[0 , , A nY , u ] if u ∈ U( C ( Y ) ⊗ A ⊗ M n ) is extended to a unitary in C ( X ) ⊗ A ⊗ M n ). Now the lemma follows from the five lemma. (cid:3) Almost flatness for (stably) relative bundles
In this section we introduce the notion of ε -flatness for stably relativebundles of Hilbert A -modules. Let us recall the definition of almost flatbundle on a topological space introduced in [MT05]. Definition 3.1.
Let X be a locally compact space with a finite open cover U := { U µ } µ ∈ I . For a finitely generated Hilbert A -module P , a U( P )-valuedˇCech 1-cocycle v = { v µν } µ,ν ∈ I on U is an ( ε, U ) -flat bundle on X with thetypical fiber P if k v µν ( x ) − v µν ( y ) k < ε for any x, y ∈ U µν := U µ ∩ U ν .We write Bdl ε, U P ( X ) for the set of ( ε, U )-flat bundles with the typical fiber P . For v ∈ Bdl ε, U P ( X ), we write E v for the underlying P -bundle. Remark . For the latter use we realize the bundle E v as a subbundleof the trivial bundle X × P n . Let { η µ } µ ∈ I be a family of positive con-tinuous functions on X such that P µ ∈ I η µ = 1 and let e µν ∈ M I denotes LMOST FLAT RELATIVE VECTOR BUNDLES 5 the matrix element, i.e., e µν e σ = δ ν,σ e µ where e µ is the standard basis of C I ∼ = Hom( C , C I ). Let p v ( x ) := X µ,ν η µ ( x ) η ν ( x ) v µν ( x ) ⊗ e µν ∈ C ( X ) ⊗ B ( P ) ⊗ M I ,ψ v µ ( x ) := X ν η ν ( x ) v νµ ( x ) ⊗ e ν ∈ C b ( U µ ) ⊗ B ( P ) ⊗ C I . Here we regard ψ v µ ( x ) as a bounded operator between Hilbert A -modules P and P ⊗ C I and consider its adjoint ψ v µ ( x ) ∗ = P η ν ( x ) v νµ ( x ) ∗ ⊗ e ∗ ν ,where { e ∗ ν } ν ∈ I ⊂ Hom( C I , C ) is the dual basis of { e ν } . Then we have p v ( x ) ψ v µ ( x ) = ψ µ ( x ) for x ∈ U µ and ψ v µ ( x ) ∗ ψ v ν ( x ) = v µν ( x ) for x ∈ U µν .That is, p v is a projection with the support E v and ψ v µ is a local trivializationof E v . Definition 3.3.
For two ( ε, U )-flat bundles v = { v µν } and v = { v µν } , a morphism of ( ε, U ) -flat bundles is a family of unitaries u = { u µ } µ ∈ I ∈ U( P ) I such that sup µ,ν ∈ I sup x ∈ U µν k u µ v µν ( x ) u ∗ ν − v µν ( x ) k < ε. We write Hom ε ( v , v ) for the set of morphisms of ε -flat bundles. Moreover,for u ∈ Hom ε ( v , v ) and δ >
0, let G δ ( u ) := n { ¯ u µ : U µ → U( P ) } µ ∈ I | ¯ u µ ( x ) v µν ( x )¯ u ν ( x ) ∗ = v µν ( x ) , k ¯ u µ ( x ) − u µ k < δ o . For ¯ u ∈ G δ ( u ), we use the same symbol ¯ u for the induced unitary isomor-phism ¯ u : E v → E v . Lemma 3.4.
There is a constant C = C ( U ) > depending only on theopen cover U such that the following hold. Let < ε < (3 C ) − , v , v ∈ Bdl ε, U P ( X ) and u ∈ Hom ε ( v , v ) . (1) The set G C ε ( u ) is non-empty. (2) The inclusion G C ε ( u ) → G C ε ( u ) is homotopic to a constant map.Proof. By replacing v with u · v := { u µ v µν u ∗ ν } µ,ν ∈ I , we may assume that u µ = 1 for any µ ∈ I , that is, k v µν ( x ) − v µν ( x ) k < ε . Let p v and ψ v µ be asin Remark 3.2.Set C := | I | + 1 (then | I | ε < / k p v − p v k ≤ sup x ∈ X X µ,ν η µ ( x ) η ν ( x ) k v µν ( x ) − v µν ( x ) k < | I | ε (3.5)and hence k p v p v p v − p v k = k p v ( p v − p v ) p v k < | I | ε. (3.6)Let us regard p v p v p v as an element of the corner C*-algebra p v ( C ( X ) ⊗ B ( P ) ⊗ M I ) p v . Then the above inequality implies that σ ( p v p v p v ) ⊂ [1 − | I | ε, | I | ε ] ⊂ [2 / , / p v p v p v is invertible.Now we consider the polar decomposition of the bounded operator p v p v : p v ( C ( X ) ⊗ P ⊗ C I ) → p v ( C ( X ) ⊗ P ⊗ C I ) , YOSUKE KUBOTA which is invertible since so is ( p v p v ) ∗ p v p v = p v p v p v . Then the unitarycomponent w := p v p v ( p v p v p v ) − / satisfies w ∗ w = p v and ww ∗ = p v .We remark that k ( p v p v p v ) − / − p v k ≤ | I | ε holds. Indeed, we have | ( t − / − ′ | = | − t − / / | ≤ / , / | t − / − | ≤ | I | ε on [1 − | I | ε, | I | ε ]. Therefore we obtain that k w ( x ) − p v p v k = k p v p v (( p v p v p v ) − / − p v ) k < | I | ε. (3.7)Now we define the family { ¯ u µ } µ ∈ I as¯ u µ ( x ) := ( ψ v µ ( x )) ∗ w ( x ) ψ v µ ( x ) . This { ¯ u µ } is contained in G C ε ( u ) since¯ u µ ( x ) v µν ( x )¯ u ν ( x ) ∗ = ψ v µ ( x ) ∗ w ( x ) ψ v µ ( x ) ψ v µ ( x ) ∗ ψ v ν ( x ) ψ v ν ( x ) ∗ w ( x ) ∗ ψ v ν ( x )= ψ v µ ( x ) ∗ ψ v ν ( x ) = v µν ( x )and k ¯ u µ ( x ) − k ≤ k ( ψ v µ ( x )) ∗ ( w ( x ) − p v p v ) ψ v µ ( x ) k + k ( ψ v µ ( x )) ∗ ψ v µ ( x ) − k < | I | ε + (cid:13)(cid:13)(cid:13) X µ η µ ( x ) ( v νµ ( x ) ∗ v νµ ( x ) − (cid:13)(cid:13)(cid:13) < ( | I | + 1) ε = C ε. To see (2), let us fix ¯ u = { ¯ u µ } ∈ G C ε ( u ). Let B denote the C*-algebra n { h µ } µ ∈ I ∈ Y µ ∈ I C b ( U µ , B ( P )) | v νµ ( x ) h µ ( x ) v µν ( x ) = h ν ( x ) ∀ x ∈ U µν o and let B sa ,r := { b ∈ B | b = b ∗ , k b k < r } for r >
0. Set δ := 4 sin − ( C ε/ e ( { h µ } ) := { ¯ u µ e ih µ } µ ∈ I gives a continuous map e : B sa ,δ → G C ε ( u ).Moreover, since any ¯ u ′ ∈ G C ε ( u ) satisfies k ¯ u µ − ¯ u ′ µ k < C ε , we have¯ u ′ = e ( − i log(¯ u ∗ µ ¯ u ′ µ )). That is, we obtain G C ε ( u ) ⊂ e ( B sa ,δ ) ⊂ G C ε ( u ) . Now we get the conclusion since e ( B sa ,δ ) is contractible. (cid:3) For an open cover U of X and a closed subset Y ⊂ X , we write U | Y forthe open cover { U µ ∩ Y } µ ∈ I Y of Y , where I Y := { µ ∈ I | U µ ∩ Y = ∅} . Fora ˇCech 1-cocycle v on U , we write v | Y for the restriction { u µ | U µ ∩ Y } µ ∈ I Y . Definition 3.8.
Let (
X, Y ) be a pair of compact spaces with a finite opencover U = { U µ } µ ∈ I . An ( ε, U ) -flat stably relative bundle on ( X, Y ) with thetypical fiber (
P, Q ) is a quadruple v := ( v , v , v , u ), where • v and v are ( ε, U )-flat P -bundle on X , • v is a ( ε, U Y )-flat Q -bundle on Y and • u ∈ Hom ε ( v | Y ⊕ v , v | Y ⊕ v ).We write the set of ( ε, U )-flat stably relative bundles on ( X, Y ) with thetypical fiber (
P, Q ) as Bdl ε, U P,Q ( X, Y ).In the particular case that Q = 0, we simply call a triple v = ( v , v , u )an ( ε, U ) -flat relative bundle and write as v ∈ Bdl ε, U P ( X, Y ). Our primaryconcern is a ( ε, U ) -flat stably relative vector bundle , that is, a ( ε, U )-flatstably relative bundle of Hilbert C -modules with the typical fiber ( C n , C m ). LMOST FLAT RELATIVE VECTOR BUNDLES 7
Definition 3.9.
For 0 < ε < (3 C ) − , we associate the K-theory class[ v ] := [ E v , E v , E v , ¯ u ] ∈ K ( X, Y ; A )to v = ( v , v , v , u ) ∈ Bdl ε, U P,Q ( X, Y ), where ¯ u is an arbitrary element of G C ε ( u ).This definition is independent of the choice of ¯ u by Lemma 3.4 (2). Remark . The associated K-theory class in Definition 3.16 depends onlyon unitary the equivalence class of v . For v ∈ Bdl ε, U P ( X ) and u ∈ U( P ) I ,we say that u · v := { u µ v µν u ∗ ν } µ,ν ∈ I is unitary equivalent to v . Since v and u · v are cohomologous as ˇCech1-cocycles, E v and E u · v determine the same K-theory class. Similarly, wesay that v ∈ Bdl ε, U P,Q ( X, Y ) is unitary equivalent to u · v := ( u · v , u · v , u · v , u · u ) for u := ( u , u , u ) ∈ U( P ) I × U( P ) I × U( Q ) I , where u · u := { diag( u ,µ , u ,µ , u ,µ ) u µ diag( u ,µ , u ,µ , u ,µ ) ∗ } µ ∈ I Y . Then u induces an isomorphism of the underlying stably relative bundles.In particular we have [ v ] = [ u · v ] ∈ K ( X, Y ; A ).Next, we define the (resp. stably) almost flat K -group K ( X, Y ; A ) (resp.K ( X, Y ; A )) as subgroups of K ( X, Y ; A ) and study their permanenceproperty with respect to the pull-back. The discussion is inspired from thework by Hunger [Hun16].Let us fix a point x µν ∈ U µν for each µ, ν ∈ I with U µν = ∅ in the waythat x µν = x νµ . Lemma 3.11.
Let µ, ν, σ ∈ I such that U µνσ := U µ ∩ U ν ∩ U σ = ∅ . Then,for v ∈ Bdl ε, U P ( X ) , we have k v µν ( x µν ) v νσ ( x νσ ) − v µσ ( x µσ ) k < ε. Proof.
Let us choose a point x ∈ U µνσ . Then, k v µν ( x µν ) v νσ ( x νσ ) − v µσ ( x µσ ) k < k v µν ( x µν ) v νσ ( x νσ ) − v µν ( x ) v µσ ( x ) k + k v µσ ( x µσ ) − v µσ ( x ) k < ε + ε = 3 ε. (cid:3) Lemma 3.12.
Let X be a locally compact space with π ( X ) = 0 and let U beits finite good open cover. Then, there is a constant C = C ( U ) dependingonly on U such that Hom C ε ( , v ) is non-empty for any v ∈ Bdl ε, U P ( X ) .Proof. Let N U denote the nerve of U . For µ, ν ∈ I with U µν = ∅ , we write h µ, ν i for the corresponding 1-cell of N U whose direction is from ν to µ .Let us fix a maximal subtree T of N U and a reference point µ ∈ I . Then,for each µ ∈ I there is a unique minimal oriented path ℓ µ in T from µ to µ . Since U is an good open cover, X is homotopy equivalent to N U and inparticular we have π ( N U ) = 0. Therefore, the closed loop ℓ − µ ◦ h µ, ν i ◦ ℓ ν is written of the form C µν Y i =1 ℓ − ν i ◦ h µ i , σ i i ◦ h σ i , ν i i ◦ h ν i , µ i i ◦ ℓ ν i , (3.13) YOSUKE KUBOTA where each µ i , ν i , σ i ∈ I satisfies U µ i ν i σ i = ∅ (that is, { µ i , ν i , σ i } is a 2-cellof N U ).For each µ ∈ I , let µ , . . . , µ k ∈ I be the 0-cells of T such that ℓ µ := h µ k , µ k − i ◦ · · · ◦ h µ , µ i and set u µ := v µ k µ k − ( x µ k µ k − ) v µ k − µ k − ( x µ k − µ k − ) . . . v µ µ ( x µ µ ) . (3.14)By Lemma 3.11 and (3.13), we get k u µ v µν u ∗ ν − k < C µν ε. Now the proof is completed by choosing C ( U ) := 3 max µ,ν ∈ I C µν . (cid:3) Proposition 3.15.
Let U = { U µ } µ ∈ I be a finite good open cover of X .Assume that there is a subset J ⊂ I such that V := { U µ } µ ∈ J also covers X .Then there is a constant C = C ( U , V ) depending only on U and V suchthat the following hold. (1) For any v ∈ Bdl ε, V P ( X ) there is ˜ v = { ˜ v µν } µ,ν ∈ I ∈ Bdl C ε, U P ( X ) suchthat ˜ v µν = v µν for any µ, ν ∈ J . (2) Let v , v ′ ∈ Bdl ε, V P ( X ) with ˜ v , ˜ v ′ ∈ Bdl C ε, U P ( X ) constructed in (1).For u ∈ Hom ε ( v , v ′ ) , there is ˜ u ∈ Hom (4 C +1) ε (˜ v , ˜ v ′ ) such that ˜ u µ = u µ for any µ ∈ J .Proof. For σ ∈ I \ J , let U σ be the open cover { U σ ∩ U µ } µ ∈ I of U σ . Let C σ := C ( U σ ) C ( U σ ), where C ( U σ ) and C ( U σ ) are the constants as inLemma 3.4 and Lemma 3.12 respectively. Let C ( U , V ) := 2 max σ ∈ I \ J C σ .First we show (1). For σ ∈ I \ J , we apply Lemma 3.12 to the restriction v | U σ = { v σµν := v µν | U µσ } to get a morphism u σ ∈ Hom C ( U σ ) ε ( , v | U σ ). Let¯ u ∈ G C ( U σ ) C ( U σ ) ε ( u ). Then, ˜ v := { ˜ v µν } µ,ν ∈ I defined by˜ v µν ( x ) := v µν ( x ) if µ, ν ∈ J , u µν ( x ) if µ ∈ J and ν J , u σν ( x ) ∗ u σµ ( x ) if µ, ν J . is a desired ˇCech 1-cocycle.Next we show (2). For each µ ∈ I \ J , we fix σ µ ∈ J such that U µσ µ = ∅ .Let ˜ u µ := ˜ v ′ µσ µ ( x µσ µ ) u σ µ ˜ v µσ µ ( x µσ µ ) ∗ . Then, k ˜ u µ ˜ v µσ µ ( x )˜ u ∗ σ µ − ˜ v ′ µσ µ ( x ) k < k ˜ u µ ˜ v µσ µ ( x )˜ u ∗ σ µ − ˜ u µ ˜ v µσ µ ( x µσ µ )˜ u ∗ σ µ k + k ˜ v ′ µσ µ ( x µσ µ ) − ˜ v ′ µσ µ ( x ) k < C ε and hence k ˜ u µ ˜ v µν ( x )˜ u ∗ ν − ˜ v ′ µν ( x ) k≤k ˜ u µ ˜ v µσ µ ( x )˜ u ∗ σ µ − ˜ v ′ µσ µ ( x ) k + k ˜ u σ µ ˜ v σ µ σ ν ( x )˜ u ∗ σ ν − ˜ v ′ σ µ σ ν ( x ) k + k ˜ u σ ν ˜ v σ ν ν ( x )˜ u ∗ ν − ˜ v ′ σ ν ν ( x ) k < (4 C + 1) ε. (cid:3) LMOST FLAT RELATIVE VECTOR BUNDLES 9
Let (
X, Y ) be a pair of finite CW-complexes. In this paper we call U a good open cover of the pair ( X, Y ) if it is a good open cover of X suchthat U | Y is also a good open cover of Y . Such an open cover exists because( X, Y ) is homotopy equivalent to a pair of finite simplicial complexes. For apair of simplicial complexes, the family of open star neighborhoods of 0-cellssatisfies the desired property.
Definition 3.16.
Let (
X, Y ) be a pair of finite CW-complex and let U bea finite good open cover of ( X, Y ). An element ξ ∈ K ( X, Y ; A ) is (resp.stably) almost flat with respect to U if for any ε > ε, U )-flat(resp. stably) relative vector bundle v of finitely generated projective Hilbert A -modules such that x = [ v ]. Corollary 3.17.
The subgroup consisting of all (resp. stably) almost flatelements of K ( X, Y ; A ) is independent of the choice of good open covers. We write K ( X, Y ; A ) (resp. K ( X, Y ; A )) for the subgroup of (resp.stably) almost flat elements. Proof.
Let U and V be two open covers and W := U ∪ V . Assume that ξ ∈ K ( X, Y ; A ) is represented by an ( ε, U )-flat stably relative vector bundle v = ( v , v , v , u ). By Proposition 3.15 (1), we get ( C ε, W )-flat bundles w , w and w . Moreover, by Proposition 3.15 (2), u can be extended to˜ u ∈ Hom (4 C +1) ε ( w | Y ⊕ w , w | Y ⊕ w ). Finally, its restriction to V is a((4 C + 1) ε, V )-flat stably relative bundle representing ξ . (cid:3) Corollary 3.18.
Let f be a continuous map from ( X , Y ) to ( X , Y ) . If ξ ∈ K ( X , Y ; A ) is almost flat, then so is f ∗ ξ ∈ K ( X , Y ; A ) . In particular,the subgroups K ( X, Y ; A ) and K ( X, Y ; A ) are homotopy invariant.Proof. Let v = ( v , v , v , u ) ∈ Bdl ε, U P,Q ( X, Y ) be a ( ε, U )-flat representativeof ξ . Let us choose a good open cover V = { V ν } ν ∈ J of ( X, Y ) which isa subdivision of f ∗ U . Let ¯ f : J → I be a map with the property that V ν ⊂ f ∗ U f ( ν ) . Then, f ∗ v := ( f ∗ v , f ∗ v , f ∗ v , f ∗ u ) defined as f ∗ v i := { f ∗ v ¯ f ( µ ) , ¯ f ( ν ) } µ,ν ∈ J for i = 0 , , f ∗ u := { u ¯ f ( µ ) } µ ∈ J is a ( ε, V )-flat bundleon ( X , Y ) representing f ∗ ξ . By Corollary 3.17, f ∗ ξ is almost flat withrespect to an arbitrary good open cover of ( X , Y ). (cid:3) Finally we define the infiniteness of (C*)-K-area for a relative K-homologycycle as a generalization of non-relative case introduced in [Gro96, Han12],which is also independent of the choice of good open cover U by Proposition3.15 in the same way as (the proof of) Corollary 3.17. Definition 3.19.
Let (
X, Y ) be a finite CW-complex and let ξ ∈ K ( X, Y ).(1) We say that ξ has infinite (resp. stably) relative K -area if there is an(resp. stably) almost flat K-theory class x ∈ K ( X, Y ) such that theindex pairing h x, ξ i is non-zero.(2) Let U be a good open cover of ( X, Y ). We say that ξ has infinite(resp. stably) relative C*- K -area if for any ε > A ε and a (resp. stably) relative ( ε, U )-flat bundle v of finitelygenerated projective Hilbert A ε -modules such that the index pairing h [ v ] , ξ i ∈ K ( A ε ) is non-zero. A compact spin manifold M with the boundary N has (stably) relative(C*-)K-area if so is the K-homology fundamental class [ M, N ] ∈ K ∗ ( M, N ).4.
Comparing topological and smooth almost flatness
The notion of almost flat bundle is originally introduced in [CGM90]in terms of Riemannian geometry of connections in the following way. Let(
M, g ) be a compact Riemannian manifold with a possibly non-empty bound-ary. A pair e = ( E, ∇ ) is a smooth ( ε, g ) -flat vector bundle on M if E is ahermitian vector bundle on M and ∇ is a hermitian connection on E whosecurvature tensor R ∇ ∈ Ω ( M, End E ) satisfies k R ∇ k := sup x ∈ M sup ξ ∈ V T x M \{ } k R ∇ ( ξ ) k End( E x ) k ξ k < ε. An element x ∈ K ( M ) is said to be almost flat (in the smooth sense)if for any ε > ε, g )-flat vector bundles e =( E , ∇ ) and e = ( E , ∇ ) such that x = [ E ] − [ E ]. It is proved in[Lis13, Proposition 3] that almost flatness of an element of the K -group isindependent of the choice of the Riemannian metric g on M . Definition 4.1.
For two smooth ( ε, g )-flat vector bundles e and e on( M, g ), a morphism of smooth ( ε, g )-flat bundles from e to e is a unitarybundle isomorphism u : E → E with k u ∇ u ∗ − ∇ k Ω < ε, where k · k Ω is the uniform norm on Ω ( M, End( E )). Definition 4.2.
Let (
M, g ) be a compact Riemannian manifold with theboundary N . For n ≥ m ≥
0, a smooth ( ε, g ) -flat stably relative vectorbundle of rank ( n, m ) on ( M, N ) is a quadruple e = ( e , e , e , u ), where • e = ( E , ∇ ) and e = ( E , ∇ ) are rank n smooth ( ε, g )-flat vectorbundles on M , • e = ( E , ∇ ) is a rank m smooth ( ε, g )-flat vector bundle on N and • u : e | N ⊕ e → e | N ⊕ e is a morphism of ( ε, g )-flat bundles.In the particular case of m = 0, we simply call a triplet e = ( e , e , u ) asmooth ( ε, g )-flat relative vector bundle of rank n .We write [ e ] for the element of K ( X, Y ) represented by the underlyingstably relative vector bundle ( E , E , E , u ). Lemma 4.3.
Let ( M, g ) be a Riemannian manifold and let x, y ∈ C ∞ ( M, M n ) whose spectra (as elements of C ( M ) ⊗ M n ) are included to a domain D ⊂ C .Let γ be the boundary of a domain D ′ supset ¯ D and let f be a holomor-phic function defined on a neighborhood of D ′ . Then there is a constant C = C ( g, D, γ, f ) depending only on g , D , γ and f such that k d ( f ( x ) − f ( y )) k Ω ≤ C ( k dx k Ω k x − y k + k dx − dy k ) , where k · k Ω is the uniform norm on the space of matrix-valued -forms Ω ( M, M n ) . LMOST FLAT RELATIVE VECTOR BUNDLES 11
Proof.
The functional calculus f ( x ) is given by the Dunford integral f ( x ) = 12 πi Z λ ∈ γ f ( λ )( λ − x ) − dλ. Since • d (( λ − x ) − ) = − ( λ − x ) − ( dx )( λ − x ) − (which follows from d (( λ − x )( λ − x ) − ) = d (1) = 0), • ( λ − x ) − − ( λ − y ) − = ( λ − x ) − ( y − x )( λ − y ) − and • k ( λ − x ) − k ≤ C ′ := inf { d ( λ, x ) | λ ∈ γ, x ∈ D } ,we obtain that k d ( f ( x ) − f ( y )) k≤ (2 π ) − k f k L sup λ ∈ γ k ( λ − x ) − dx ( λ − x ) − − ( λ − y ) − dy ( λ − y ) − k≤ (2 π ) − k f k L (cid:16) sup λ ∈ γ k (( λ − x ) − − ( λ − y ) − ) dx ( λ − x ) − k + sup λ ∈ γ k ( λ − y ) − dx (( λ − x ) − − ( λ − y ) − ) k + sup λ ∈ γ k ( λ − y ) − ( dx − dy )( λ − y ) − k (cid:17) ≤ (2 π ) − k f k L (2( C ′ ) k dx kk x − y k + ( C ′ ) k dx − dy k ) , where k f k L is the L -norm of f on γ . Now the proof is completed bychoosing C as (2 π ) − k f k L ( C ′ ) · max { C ′ , } . (cid:3) Lemma 4.4.
Let X be a finite CW-complex with an open cover U . For < ε < / , let { v ′ µν } µ,ν ∈ I be a family of unitaries in B ( P ) such that k v ′ µν v ′ νσ − v ′ µσ k < ε . Let ˘ ψ µ ( x ) := X ν ∈ I η ν ( x ) ⊗ v ′ νµ ⊗ e µ ∈ C ( X ) ⊗ M n ⊗ C I ,v µν ( x ) := ( ˘ ψ µ ( x ) ∗ ˘ ψ µ ( x )) − / ˘ ψ µ ( x ) ∗ ˘ ψ ν ( x )( ˘ ψ ν ( x ) ∗ ˘ ψ ν ( x )) − / , where { η µ } µ ∈ I and { e µ } µ ∈ I be as in Remark 3.2. Then v := { v µν } µ,ν ∈ I is aˇCech -cocycle satisfying k v µν ( x ) − v ′ µν k < ε , and hence is (8 ε, U ) -flat.Proof. Firstly, ˘ ψ µ ( x ) ∗ ˘ ψ ν ( x ) − v ′ µν = X σ ∈ I η σ ( x ) ( v ′ µσ v ′ σν − v ′ µν )implies k ˘ ψ µ ( x ) ∗ ˘ ψ ν ( x ) − v ′ µν k < P σ η σ ( x ) k v ′ µσ v ′ σν − v ′ µν k < ε . In particular,we get k ψ µ ( x ) ∗ ψ µ ( x ) − k < ε , and hence k ( ˘ ψ µ ( x ) ∗ ˘ ψ µ ( x )) − / − k < k ˘ ψ µ ( x ) ∗ ˘ ψ µ ( x ) − k < ε (here we use the fact | z − / − | < | z − | for z ∈ [1 / , / k v µν ( x ) − v ′ µν k≤k ( ˘ ψ µ ( x ) ∗ ˘ ψ µ ( x )) − / − kk ˘ ψ µ ( x ) ∗ ˘ ψ ν ( x ) kk ( ˘ ψ ν ( x ) ∗ ˘ ψ ν ( x )) − / k + k ˘ ψ µ ( x ) ∗ ˘ ψ ν ( x ) kk ( ˘ ψ ν ( x ) ∗ ˘ ψ ν ( x )) − / − k + k ˘ ψ µ ( x ) ∗ ˘ ψ ν ( x ) − v ′ µν k≤ ε (1 + ε ) + ε (1 + ε ) + ε < ε. For the last inequality we use ε < / (cid:3) Lemma 4.5.
Let < ε < / . Let M be a Riemannian manifold with afinite open cover U = { U µ } µ ∈ I . Then there exists a constant C = C ( g, U ) depending only on g and U such that the following holds: For any w ∈ Bdl ε, U n ( M ) , there is v ∈ Bdl U , εn ( M ) such that • k v µν ( x ) − w µν ( x ) k < ε for any x ∈ U µν and • each v µν is smooth and k dv µν k Ω ( U µν , M n ) < C ε .Proof. Let ˘ ψ µ and v µν be as in the statement of Lemma 4.4 for v ′ µν = w µν ( x µν ). By Lemma 3.11 and Lemma 4.4 we obtain that { v µν } µ,ν ∈ I is(24 ε, U )-flat and k v µν ( x ) − w µν ( x ) k ≤ k v µν ( x ) − w µν ( x µν ) k + k w µν ( x µν ) − w µν ( x ) k < ε. Now we consider an estimate of the differential dv µν . Let κ := max µ k dη µ k .Note that k d ( η µ ) k = k η µ dη µ k ≤ κ . Then we get k d ( ˘ ψ ∗ µ ˘ ψ ν ) k = k d ( ˘ ψ ∗ µ ˘ ψ ν − w µν ( x µν )) k≤ X σ ∈ I k d ( η σ ) k · k w µσ ( x µσ ) w σν ( x σν ) − w µν ( x µν ) k < κ | I | · ε. By the assumption ε < /
6, we have that the spectrum σ ( ˘ ψ µ ( x ) ∗ ˘ ψ µ ( x )) − / )is included to the interval [1 / , / D and D ′ be the open disk of ra-dius 2 / / γ = ∂D ′ . Weapply Lemma 4.3 for x = 1, y = ˘ ψ ∗ µ ˘ ψ µ , D and γ as above and f ( z ) = z − / . Then we get a constant C = C ( g, D, γ, z − / ) and an inequality k d (( ˘ ψ ∗ µ ˘ ψ µ ) − / ) k < C · κ | I | ε . Finally we obtain k dv µν k≤k d (( ˘ ψ ∗ µ ˘ ψ µ ) − / ) kk ˘ ψ ∗ µ ˘ ψ ν kk ( ˘ ψ ∗ ν ˘ ψ ν ) − / k + k ( ˘ ψ ∗ µ ˘ ψ µ ) − / kk d ( ˘ ψ ∗ µ ˘ ψ ν ) kk ( ˘ ψ ∗ ν ˘ ψ ν ) − / k + k ( ˘ ψ ∗ µ ˘ ψ µ ) − / kk ˘ ψ ∗ µ ˘ ψ µ kk d (( ˘ ψ ∗ ν ˘ ψ ν ) − / ) k < C κ | I | ε · (3 / · κ | I | ε · · C κ | I | ε · (3 / · C + 24) κ | I | ε. The proof is completed by choosing C := (36 C + 24) κ | I | . (cid:3) Lemma 4.6.
Let < ε < C . There is a constant C = C ( U ) dependingonly on U such that the following holds: For ( ε, U ) -flat bundles v and v on X with k dv iµν k < ε (for i = 1 , ) and u ∈ Hom ε ( v , v ) , there is ¯ u ∈ G C ε ( u ) such that k d ¯ u µ k Ω < C ε .Proof. Let ψ iµ := ψ v i µ and p i := p v i for i = 1 , w and { ¯ u µ } µ ∈ I be as inRemark 3.2. As in the proof of Lemma 3.4 (1), we may assume that u µ = 1 LMOST FLAT RELATIVE VECTOR BUNDLES 13 for all µ ∈ I . As in the proof of Lemma 4.5, let κ := max µ k dη µ k . Then wehave inequalities k dψ iµ k = k d X ν η ν v iµν ⊗ e ν k≤ X ν ( k dη µν kk v iµν kk e ν k + k η ν kk dv iµν kk e ν k ) ≤ | I | ( κ + (3 C ) − ) , k d (( ψ µ ) ∗ ψ µ ) k = k d (( ψ µ − ψ µ ) ∗ ψ µ ) k = k d X ν η ν ( v µν − v µν ) ∗ v µν k≤ X ν ( k d ( η ν ) kk ( v µν − v µν ) ∗ kk v µν k + k η ν kk d ( v µν − v µν ) ∗ kk v µν k + k η ν kk ( v µν − v µν ) ∗ kk dv µν k ) ≤| I | (2 κε + 2 ε + ε ) = (2 κ + 3) | I | ε, k dp i k = k d X η µ η ν v iµν ⊗ e µν k≤ X µ,ν ( k d ( η µ η ν ) kk v iµν kk e µν k + k η µ η ν kk dv iµν kk e µν k ) < | I | (2 κ + (3 C ) − ) , k dp − dp k = k d X η µ η ν ( v µν − v µν ) ⊗ e µν k≤ X µ,ν ( k d ( η µ η ν ) kk v µν − v µν kk e µν k + k η µ η ν kk dv µν − dv µν kk e µν k ) < | I | (2 κε + 2 ε ) = | I | (2 κ + 2) ε. Let C ′ denotes the maximum of | I | ( κ + (3 C ) − ) , (2 κ + 3) | I | , | I | (2 κ + ε )and | I | (2 κ + 2).By the above inequalities together with (3.5), we get k d ( p p p ) − dp k = k dp kk p − p kk p k + k p kk d ( p − p ) kk p k + k p kk p − p kk dp k Let ( M, g ) be a Riemannian manifold possibly with a collaredboundary. Let U := { U µ } µ ∈ I be an open cover of M such that any two points x, y in each U µ is connected by a unique minimal geodesic in U µ . Then thereis a constant C = C ( g, U ) depending on g and U such that the followinghold: (1) Let e = ( E, ∇ ) be an ε -flat bundle on M . Then, there exists a localtrivialization ψ e µ : U µ × C n → E | U µ such that the ˇCech -cocycle v e := { v e µν ( x ) := ψ e µ ( x ) ∗ ψ e ν ( x ) } µ,ν ∈ I forms a ( C ε, U ) -flat bundle. (2) Let u : e → e be a morphism of ε -flat bundles. Then, u := { u µ := ψ e µ ( x µ ) ∗ u ( x µ ) ψ e µ ( x µ ) } forms a morphism of ( ε, U ) -flat bundles such that u ∈ G C ε ( u ) . For example, an open cover consisting of open balls of radius less thanthe injectivity radius of M satisfies the assumption of Lemma 4.7 (when M has a boundary, take an open cover of the invertible double ˆ M as above andrestrict it to M ). Proof. Let x, y ∈ U µ . We write [ x, y ] for the minimal geodesic connecting x and y in U µ and D µ ( x, y ) := [ z ∈ [ x,y ] [ x µ , z ] . We define the constant C as C := max µ sup x,y ∈ U µ max { d ( x, y ) , D µ ( x, y )) } < ∞ . (4.8)For a path ℓ : [0 , t ] → M , let Γ ∇ ℓ : E ℓ (0) → E ℓ ( t ) denote the parallel trans-port along ℓ . We fix an identification of E x µ with C n . Then ψ e µ ( x ) := Γ [ x,x µ ] : E x → E x µ ∼ = C n gives a local trivialization of E . Let v e µν ( x ) := ψ e ν ( x ) ∗ ψ e µ ( x ). Then v e µν ( y ) ∗ v e µν ( x )is the parallel transport along the boundary of the surface D µ ( x, y ) ∪ D ν ( x, y ).By a basic curvature estimate of the holonomy (see for example [Gro96,pp.19]), we get k v e µν ( y ) ∗ v e µν ( x ) − k < Area( D µ ( x, y ) ∪ D ν ( x, y )) · k R ∇ k < C ε. LMOST FLAT RELATIVE VECTOR BUNDLES 15 To see (2), it suffices to show that k ψ e µ ( x ) ∗ u ( x ) ψ e µ ( x ) − u µ k < C ε . Let x ( t ) denote the point of [ x, x µ ] uniquely determined by d ( x, x ( t )) = t . Since u Γ ∇ [ x µ ,x ] u ∗ − Γ ∇ [ x µ ,x ] = Γ u ∇ u ∗ [ x,y ] − Γ ∇ [ x µ ,x ] = Z d ( x,x µ )0 ( u ∇ ddt u ∗ − ∇ ddt )Γ [ x µ ,x ( t )] dt, we obtain that k Γ ∇ [ x µ ,x ] u Γ ∇ [ x,x µ ] − u k = k u Γ ∇ [ x µ ,x ] u ∗ − Γ ∇ [ x µ ,x ] k ≤ d ( x, y ) ε ≤ C ε. (cid:3) Lemma 4.9. Let ( M, g ) and U be as in Lemma 4.7. Then there is a constant C = C ( g, U ) depending only on g and U such that the following hold forany < ε < C . (1) Let v be a ( ε, U ) -flat vector bundle. Then, the underlying vectorbundle E v admits an ( C ε, g ) -flat connection ∇ v . (2) For u ∈ Hom ε ( v , v ) , there is ¯ u ∈ G C ε ( u ) such that k ¯ u ∇ v ¯ u ∗ −∇ v k Ω < C ε .Proof. By Lemma 4.5, we may assume that { v µν } is (24 ε, U )-flat and k dv µν k Let M be a compact Riemannian manifold with the bound-ary N . An element x ∈ K ( M, N ) is almost flat in smooth sense if and onlyif it is almost flat in topological sense (i.e., in the sense of Definition 3.16).Proof. By Lemma 4.7 and Lemma 4.9, we can associate from smooth or topo-logical ε -flat stably relative bundles to the other. Since this correspondencepreserve the underlying stably relative bundle, we get the conclusion. (cid:3) Enlargeability and almost flat bundle A connected Riemannian manifold ( M, g ) is said to be (resp. area-) en-largeable if for any ε > M and an (resp.area-) ε -contracting map f ε with non-zero degree from ¯ M to the sphere S n with the standard metric, which is constant outside compact subset of M .Here we say that f ε is area- ε -contracting if k Λ T x f ε k ≤ ε for any x ∈ M ∞ .Note that any enlargeable manifold is area-enlargeable. Theorem 5.1. Let ( M, g ) be a compact Riemannian spin manifold with acollared boundary N . If M ∞ is area-enlargeable, then M has infinite stablyrelative C*-K-area. Firstly we prepare some notations. For M, N as above, let M r denote thespace M ⊔ N N × [0 , r ] and N r := ∂M r for r ∈ [0 , ∞ ]. We choose an opencover of M using g as in Lemma 4.7. Let q r denote the continuous map M r → M determined by q r | M = id M and q r | N × [0 ,r ] is the projection to N .We define the open cover U k of M k as U k := { U ( µ,k ) := q ∗ r U µ ∩ V l } ( µ,l ) ∈ I × k , where V = M ◦ , V l = N × ( l − , l + 1) for l = 1 , . . . , n − V k = N × ( k − , k ]. Next, for a covering ¯ π : ¯ M → M , we write ¯ U for the opencover of ¯ M consisting of connected components of π − ( U µ )’s and ¯ I for theindex set of ¯ U . We use the same letter ¯ π for the canonical map ¯ I → I .Similarly we define ¯ U k and ¯ I k . Lemma 5.2. Let k ∈ N and let ( v , w , u ) be a ( ε, U k ) -flat relative bundlewith the typical fiber P on ( M k , N k ) . Then there is a stably relative (2 ε, U ) -flat bundle v of Hilbert A -modules on ( M, N ) such that [ v ] = [ v , w , u ] underthe canonical identification K ( M, N ; A ) ∼ = K ( M k , N k ; A ) .Proof. For l = 0 , . . . , k , we define a ( ε, U | N )-flat P -bundle v l on N by v l := { v ( µ,l )( ν,l ) | U ( µ,l )( ν,l ) ∩ N ×{ l } } µ,ν ∈ I under the canonical identification of ( N, U | N ) with ( N ×{ l } , U k | N ×{ l } ). Sim-ilarly we define w l for l = 0 , . . . , k .For l = 0 , . . . , k , fix x µ,l ∈ U µ,l ∩ N × { l + } . We define u l = { u l,µ } µ ∈ I by u l,µ := v ( µ,l +1)( µ,l ) ( x µ,l ) l = 0 , . . . , k − u ( µ,k ) l = k,w ( µ, k − l )( µ, k − l +1) ( x µ, k − l +1 ) l = k + 1 , . . . , k. Then we have u l ∈ Hom ε ( v l , v l +1 ), u k ∈ Hom ε ( v k , w k ) and u k − l ∈ Hom ε ( w l +1 , w l ) for l = 0 , . . . , k − v and ˜ v be restrictions of v and w to M with the open cover U k | M = U respectively. Let Q = P k , let ˜ v := v ⊕ · · · ⊕ v k ⊕ w k ⊕ · · · ⊕ w and let ˜ u = { ˜ u µ } µ ∈ I , where each ˜ u µ : P ⊕ Q → P ⊕ Q is determined by˜ u µ ( ξ , ( ξ , . . . , ξ n )) = ( u n,µ ξ n , ( u ,µ ξ , u ,µ ξ , . . . , u k − ,µ ξ k − ))for ξ , . . . , ξ k ∈ P . Then we have k ˜ u µ (˜ v µν ⊕ ˜ v µν )˜ u ∗ ν − ˜ v µν ⊕ ˜ v µν k < ε, LMOST FLAT RELATIVE VECTOR BUNDLES 17 that is, v := (˜ v , ˜ v , ˜ v , u ) is a stably relative (2 ε, U )-flat bundle with thetypical fiber ( P, Q ) on ( M, N ).Finally we observe that [ v , w , u ] = [ v ] in K ( M, N ; A ). Let q l : M → M l be a diffeomorphism extending the canonical identification N → ∂M l andlet E l := q ∗ l E v l , E k − l +1 := q ∗ l E w l for l = 0 , . . . , k . Note that E l | N ∼ = E v l .Let us choose { ¯ u l,µ } µ ∈ I ∈ G C ε ( u l ) by Lemma 3.4, which induces a unitarybundle isomorphism ¯ u l : E l | N → E l +1 | N . Then the K-theory class [ v ] isrepresented by [ E v , E v , E | N ⊕ · · · ⊕ E k | N , ¯ U ], where¯ U ( ξ , ( ξ , . . . , ξ k )) = (¯ u k ξ k , (¯ u ξ , . . . , ¯ u k − ξ k − )) . Let E := E v and E k +1 := E v . Now we use the equivalence relations onBdl s ( X, Y ; A ) discussed in pp.3 to obtain[ v ] =[ E v , E v , E | N ⊕ · · · ⊕ E k | N , ¯ U ]=[ E v ⊕ E ⊕ · · · ⊕ E k , E ⊕ · · · ⊕ E k ⊕ E v , , ¯ U ]= k X l =0 [ E l , E l +1 , , ¯ u l ] = [ E k , E k +1 , , ¯ u k ] = q ∗ k [ v , w , u ] . (cid:3) Let F → ¯ M → M be a (possibly infinite) connected covering and extend itto ¯ M ∞ → M ∞ . Let σ denote the monodromy representation of Γ := π ( M )on ℓ ( F ) and let A := { ( T, S ) ∈ B ( ℓ ( F )) ⊕ | S ∈ σ ( C ∗ (Γ)) , T − S ∈ K } . Then the the exact sequence0 → K ( ℓ ( F )) i −→ A pr −−→ σ ( C ∗ (Γ)) → , (5.3)where i is the embedding to the first component and pr is the projectionto the second component, splits.For a complete Riemannian manifold M with an open cover U such thateach U µ is relatively compact, a ˇCech 1-cocycle v on U is compactly sup-ported if v µν ≡ µ, ν ) ∈ I with U µ ∩ U ν = ∅ . Ifa ˇCech 1-cocycle v is supported in an open submanifold M , i.e., v µν ≡ µ, ν ) ∈ I with U µ ∩ U ν M = ∅ , we associate a relative bundle( v | M , , ) on M with the open cover { U µ ∩ M } . Lemma 5.4. Let M r , ¯ M r and A be as above. Then there is a Hilbert A -module bundle P on M and a ∗ -homomorphism θ : C ( ¯ M ∞ ) → K ( C ( M ∞ , P )) such that, for any compactly supported ( ε, ¯ U ∞ ) -flat vector bundle v ∈ Bdl ε, ¯ U ∞ n ( ¯ M ∞ ) (with the support included to ¯ M r ), the corresponding element θ ∗ [ v , , ] ∈ K ( M r , N r ; A ) is represented by an ( ε, U ) -flat bundle of finitely generatedprojective Hilbert A -modules.Proof. Let ˆ σ : Γ → U( A ) be the representation given by ˆ σ ( γ ) := ( σ ( γ ) , σ ( γ )).Let A denote the C*-algebra bundle ˜ M r × Ad ˆ σ A , which acts on the Hilbertbundle H := ˜ M r × ˆ σ ( ℓ ( F ) ⊕ ). Then C ( M r , A ) is isomorphic to K ( C ( M r , P )),where P := ˜ M r × ˆ σ A . Let p P := P η µ η ν ˆ σ ( γ µν ) ⊗ e µν ∈ C ( M r , A ) ⊗ M I as inRemark 3.2 and let τ denote the identification of C ( M r , A ) with the cornersubalgebra p P ( C ( M r , A ) ⊗ M I ) p P . The Hilbert space L ( M r , H ) is canonically isomorphic to L ( ¯ M r ) ⊕ .Moreover, the Γ-equivariant inclusion c ( F ) ⊂ K ( ℓ ( F )) ⊂ A induces θ : C ( ¯ M r ) ∼ = C ( M r , C ) → C ( M r , A ) , where C := ˜ M r × Ad ˆ σ c ( F ). We remark that it is extended to θ : C ( M r , C + ) → C ( M r , A ), where C + := ˜ M r × Ad ˆ σ c ( F ) + . Similarly we define θ µν : C b ( U µν , C + ) → C b ( U µν , A ).We fix a local trivialization χ µ : L ( U µ , ℓ ( F ) ⊕ ) → L ( U µ , H ) ∼ = L (¯ π − ( U µ )) ⊕ coming from that of the covering space ϕ µ : U µ × F → ¯ π − ( U ) as a fiberbundle with the structure group σ (Γ). Then there is γ µν ∈ Γ for each µ, ν ∈ I such that χ ∗ µ χ ν = ˆ σ ( γ µν ). Then the ∗ -homomorphism τ is writtenexplicitly as τ ( f ) := X µ,ν η µ η ν · χ ∗ µ ( f | U µν ) χ ν ⊗ e µν . For an ( ε, U ∞ )-flat bundle v ∈ Bdl ε, ¯ U ∞ n ( ¯ M ∞ ) supported in ¯ M r , let˜ v ′ µν := Y ¯ π (¯ µ )= µ, ¯ π (¯ ν )= νU ¯ µ ¯ ν = ∅ diag( v ¯ µ ¯ ν , ∈ ( C b ( U µν , C + ) ⊗ M n ) ⊕ , ˜ v µν := χ ∗ µ θ µν (˜ v ′ µν ) χ ν ∈ C b ( U µν , A ) ⊗ M n for any µ, ν ∈ I r . Then ˜ v := { ˜ v µν } µ,ν ∈ I r is a ˇCech 1-cocycle on M r takingvalue in the unitary group of A ⊗ M n . Moreover, by the construction, ( ε, ¯ U )-flatness of v implies that ˜ v := { ˜ v µν } µ,ν ∈ I is also an ( ε, U )-flat bundle ofHilbert A -modules.As in Remark 3.2, let p ˜ v := X µ,ν η µ η ν ⊗ ˜ v µν ⊗ e µν ∈ C ( M, A ) ⊗ M n ⊗ M I ,p v := X µ,ν η µ η ν ⊗ ˜ v ′ µν ⊗ e µν ∈ C ( M r , C + ) ⊗ M n ⊗ M I r ,p := X µ,ν η µ η ν ⊗ n ⊗ e µν ∈ C ( M r , C + ) ⊗ M n ⊗ M I r , Then we have [ p ] = [1 n ], p v − p ∈ C ( M ◦ r , C ) ∼ = C ( ¯ M ◦ r ) and the differenceelement [ p v , p ] ∈ K ( C ( ¯ M r )) is equal to [ v ] − [1 n ]. Therefore, the remainingtask is to show that θ ∗ ([ p v ] − [1 n ]) = [ p ˜ v ] − [1 n ].The projection( τ ◦ θ )( p v ) = X σ,τ X µ,ν η σ η τ η µ η ν ⊗ χ ∗ σ θ µν (˜ v ′ µν ) χ τ ⊗ e µν ⊗ e στ ∈ C ( M r , A ) ⊗ M n ⊗ M I ⊗ M I is equal to the projection as in Remark 3.2 associated to the ˇCech 1-cocycle { χ σ ˜ v µν χ τ } ( µ,σ ) , ( ν,τ ) ∈ I on the open cover U := { U µσ } ( µ,σ ) ∈ I and the squareroot of partition of unity { η µ η σ } ( µ,σ ) ∈ I . At the same time, if we use thesquare root of partition of unity { η µ δ µσ } (where δ µσ denotes the delta func-tion) instead of { η µ η σ } , then the corresponding projection is identified with LMOST FLAT RELATIVE VECTOR BUNDLES 19 p ˜ v . That is, the support of ( τ ◦ θ )( p v ) is isomorphic to that of p ˜ v . Thisconcludes the proof. (cid:3) Proof of Theorem 5.1. By taking the direct product with T if necessary,we may assume that n := dim M is even. Let E be a vector bundle on S n such that c n ( E ) = 1 and let us fix a hermitian connection. For ε > 0, let f ε : ¯ M ∞ → S n be an area- ε -contracting map with non-zero degree. Thenthe induced connection on f ∗ ε E with the pull-back connection is ( Cε, g )-flatin the smooth sense, where the constant C > E . Let k ∈ N such that f ε maps N × [ k, ∞ ) to the base point ∗ of S n .By Lemma 4.7, there is a local trivialization { ψ ¯ µ } ¯ µ ∈ ¯ I of f ∗ ε E such that v := { v ¯ µ ¯ ν = ψ ∗ ¯ µ ψ ¯ ν } µ,ν ∈ I is ( C ε, ¯ U r )-flat. Here we remark that the proof ofLemma 4.7 also works for the noncompact manifold ¯ M since the constant C = C ( g, ¯ U k ) given in (4.8) actually coincides with C ( g, U k ). Note thatwe also have C ( g, ¯ U k ) = C ( g, ¯ U ), that is, there is a uniform upper boundfor C ( g, ¯ U k )’s.The remaining task is to show that the pairing h θ ∗ [ v , , ] , [ M, N ] i isnon-trivial. For an even-dimensional connected manifold X , we write β X for the image of the Bott generator in K ( X ) by an open embedding. Then[ E ] − [ C n ] = β S n ∈ K ( S n ) and hence[ v , , ] = f ∗ ε [ E ] − [ C n ] = deg f ε · β ¯ M ∞ ∈ K ( ¯ M ∞ ) . Let us choose an open subspace U of M such that ¯ π − ( U ) ∼ = U × F anda copy ¯ U ⊂ ¯ π − ( U ) of U . Then we have C ( U, A ) ∼ = C ( U ) ⊗ A and thediagram K ( C (¯ π − ( U ))) θ ∗ / / ι ∗ (cid:15) (cid:15) K ( C ( U ) ⊗ A ) ι ∗ (cid:15) (cid:15) K ( C ( ¯ M ∞ )) θ ∗ / / K ( C ( M ∞ , A ))commutes, where the vertical maps ι ∗ are induced from open embeddings.By the construction of θ ∗ , we have θ ∗ β ¯ U = β U ⊗ [ p ] ∈ K ( C ( U ) ⊗ A ), where p ∈ K ( ℓ ( F )) ⊂ A is a rank 1 projection,. Therefore we obtain that h θ ∗ β ¯ M , [ M, N ] i = h θ ∗ ι ∗ β ¯ U , [ M, N ] i = h ι ∗ θ ∗ β ¯ U , [ M, N ] i = h θ ∗ β ¯ U , [ U ] i = h β ⊗ [ p ] , [ U ] i = [ p ] ∈ K ( A ) , and hence h θ ∗ [ v , , ] , [ M, N ] i = deg( f ε ) h θ ∗ β, [ M, N ] i = deg( f ε ) · [ p ] . This finishes the proof since K ( K ( ℓ ( F ))) → K ( A ) is injective (we recallthat the exact sequence (5.3) splits). (cid:3) Relative quasi-representations and almost monodromycorrespondence Let Γ be a countable discrete group and let G be a finite subset of Γ.Recall that a map π : Γ → U( P ) is a ( ε, G ) -representation of Γ on P if π ( e ) = 1 and k π ( g ) π ( h ) − π ( gh ) k < ε for any g, h ∈ G . Let qRep ε, G P (Γ) denote the set of ( ε, G )-representations ofΓ on P . Definition 6.1. Let π and π be ( ε, G )-representations of Γ. An ε -intertwiner u ∈ Hom ε ( π , π ) is a unitary u ∈ U( P ) such that k uπ ( γ ) u ∗ − π ( γ ) k < ε .Let φ : Λ → Γ be a homomorphism between countable discrete groups.Let G = ( G Γ , G Λ ) be a pair of finite subsets G Γ ⊂ Γ and G Λ ⊂ Λ such that φ ( G Λ ) ⊂ G Γ . Definition 6.2. A stably relative ( ε, G ) -representation of (Γ , Λ) is a quadru-ple π := ( π , π , π , u ), where • π : Γ → U( P ) and π : Γ → U( P ) are ( ε, G Γ )-representations of Γ, • π : Λ → U( Q ) is a ( ε, G Λ )-representation of Λ, and • u ∈ Hom ε ( π ◦ φ ⊕ π , π ◦ φ ⊕ π ).We write qRep ε, G P,Q (Γ , Λ) for the set of stably relative ( ε, G )-representationsof (Γ , Λ) on ( P, Q ).We say that two ( ε, G )-representations π and π ′ are unitary equivalent ifthere are unitaries U , U ∈ U( P ) and U ∈ U( Q ) such that π i = Ad( U i ) ◦ π ′ i for i = 0 , , u ′ ( U ⊕ U ) = ( U ⊕ U ) u . Remark . There is an obvious one-to-one correspondence between qRep ε, G P (Γ , Λ)and qRep ε, G P (Γ , φ (Λ)). Moreover, any relative ( ε, G )-representation ( π , π , u )is unitary equivalent to ( π , Ad( u ∗ ) ◦ π , u = 1.Finally we give the almost monodromy correspondence between almostflat bundles on a pair of finite CW-complexes and quasi-representations ofthe fundamental groups.Let ( X, Y ) be a pair of finite CW-complexes with a good open cover U .We write Γ := π ( X ), Λ := π ( Y ) and φ : Λ → Γ for the map induced fromthe inclusion. Fix a maximal subtree T of the 1-skeleton N (1) U of the nerveof U such that T ∩ N (1) U| Y is also a maximal subtree of N (1) U| Y . Definition 6.4. We say that v ∈ Bdl ε, U P ( X ) is normalized on T if k v µν ( x ) − k < ε for any h µ, ν i ∈ T . We also says that v = ( v , v , v , u ) ∈ Bdl ε, U P,Q ( X, Y )is normalized on T if v , v and v are normalized on T . Let Bdl ε, U P ( X ) T (resp. Bdl ε, U P,Q ( X, Y ) T ) denote the set of ( ε, U )-flat bundles normalized on T . Lemma 6.5. Any stably relative ( ε, U ) -flat bundle v is unitary equivalent (inthe sense of Remark 3.10) to a stably relative ( ε, U ) -flat bundle normalizedon T .Proof. It suffices to show that, for any v ∈ Bdl ε, U P ( X ), there is u ∈ U( P ) I such that u · v is normalized on T . Such u is constructed inductively (indeed,an inductive construction gives a family u = { u µ } µ ∈ I with the property that u µ = u ν v µν ( x µν ) ∗ for any h µ, ν i ∈ T ). (cid:3) Now we give a one-to-one correspondence up to small correction between(resp. stably) relative quasi-representations and almost flat (resp. stably)relative bundles normalized on T . LMOST FLAT RELATIVE VECTOR BUNDLES 21 As in Lemma 3.12, a 1-cell h µ, ν i ∈ N (1) U \ T corresponds to an element γ µν := [ ℓ − µ ◦ h µ, ν i ◦ ℓ ν ] of Γ. Let G Γ := { γ µν | h µ, ν i ∈ N (1) U \ T } ⊂ Γ . Similarly we define G Λ as the set of elements of Λ of the form γ µν for h µ, ν i ∈ N (1) U| Y \ T . Let F G denote the free group with the generator { s µν | h µ, ν i ∈ N (1) U \ T } . We fix a set theoretic section τ : Γ → F G , that is, τ ( γ µν ) = s µν . Definition 6.6 ([CD18, Definition 4.2]) . For v ∈ Bdl ε, U P ( X ) T , let α ( v )( γ ) := n Y k =1 u µ k +1 v µ k +1 µ k ( x µ k +1 µ k ) u ∗ µ k for γ ∈ Γ such that τ ( γ ) = s µ ,µ · · · · · s µ k − ,µ k . Here u µ is as in (3.14).It is essentially proved in [CD18, Proposition 4.8] that there is a constant C = C ( U ) depending only on U such that α ( v ) is a ( C ε, G )-representationof Γ in P .Conversely, suppose that we have a ( ε, G )-representation of Γ. Let { η µ } µ ∈ I and { e µ } µ ∈ I be as in Remark 3.2. Let us define˘ ψ πµ := X η ν ⊗ π ( γ νµ ) ⊗ e ν ∈ C ( X ) ⊗ B ( P ) ⊗ C I ,v πµν := (( ˘ ψ πν ) ∗ ˘ ψ πν ) − / (( ˘ ψ πν ) ∗ ˘ ψ πµ )(( ˘ ψ πµ ) ∗ ˘ ψ πµ ) − / . By Lemma 4.4, we have the inequality k v πµν ( x ) − π ( γ µν ) k < ε . This impliesthat v π := { v πµν } µ,ν ∈ I is (8 ε, U )-flat bundle normalized on T . Definition 6.7. For π ∈ qRep ε, G P (Γ), we define β ( π ) to be v π ∈ Bdl ε, U P ( X ) T .We consider the distance in Bdl ε, U P ( X ) and qRep ε, G P (Γ) defined as d ( v , v ′ ) := sup µ,ν ∈ I k v µν − v ′ µν k ,d ( π, π ′ ) := sup γ ∈G Γ k π ( γ ) − π ′ ( γ ) k . Lemma 6.8. There is a constant C = C ( U ) > depending only on U such that the maps α and β satisfy d ( α ( v ) , α ( v ′ )) ≤ d ( v , v ′ ) + C ε,d ( β ( π ) , β ( π ′ )) ≤ d ( π, π ′ ) + C ε,d ( β ◦ α ( v ) , v ) ≤ C ε,d ( α ◦ β ( π ) , π ) ≤ C ε. Proof. By Corollary 3.18, we may assume that X is a finite simplicial com-plex and U is the open cover of X consisting of star neighborhoods U µ of0-cells µ . We choose x µν as the median of the 1-cell h µ, ν i .Let GL( P ) δ denote the set of T ∈ B ( P ) with d ( T, U( P )) < ε and letCrd εP ( X ) T denote the set of ε -flat coordinate bundles on X normalized on T . Here, an ε -flat coordinate bundle on a simplicial complex is a family { v µν } of ε -flat GL( P ) ε -valued functions v µν on the union of simplices of thebarycentric subdivision of X included to U µ ∩ U ν which satisfies the cocycle relation (for the precise definition, see [CD18, Definition 2.5]). It is said tobe normalized on T if v µν ( x µν ) = 1 for h µ, ν i ∈ T . We remark that therestriction gives a map R : Bdl ε, U P ( X ) T → Crd εP ( X ) T .Let qRep ε, G P (Γ) denote the set of ( ε, G )-representation which takes value inGL( P ) ε instead of U( P ). In [CD18], Carri´on and Dadarlat construct maps α CD : Crd εP ( X ) → qRep C ′ ε, G P (Γ) T , β CD : qRep ε, G P (Γ) → Crd C ′ εP ( X ) , which is compatible with our α and β in the sense that • d ( v , v ′ ) − ε ≤ d ( R ( v ) , R ( v ′ )) ≤ d ( v , v ′ ) for any v , v ′ ∈ Bdl ε, U P ( X ), • α CD ◦ R ( v ) = α ( v ) for any v ∈ Bdl ε, U P ( X ), • d ( R ◦ β ( π ) , β CD ( π )) < ( C ′ + 8) ε for π ∈ qRep ε, G P (Γ).Here, the second is obvious from the constructions (compare [CD18, Defi-nition 4.2] with Definition 6.6) and the third follows from β CD ( π ) µν ( x µν ) = π ( γ µν ) (which is obvious from the construction [CD18, Definition 5.3]) andthe inequality k v πµν ( x ) − π ( γ µν ) k < ε remarked above. Now, the lemmafollows from [CD18, Theorem 3.1, Theorem 3.3]. (cid:3) Lemma 6.9. Let ∆ I : U( P ) → U( P ) I denote the diagonal embedding. Thereis a constant C = C ( U ) depending only on U such that the following hods: (1) Let π , π ∈ qRep ε, G P (Λ) . If there exists u ∈ Hom ε ( π , π ) , then ∆ I ( u ) ∈ U( P ) is contained in Hom C ε ( β ( π ) , β ( π )) . (2) Let v , v ∈ Bdl ε, U| Y P ( Y ) T . If there exists u ∈ Hom ε ( v , v ) , then k u µ − u ν k ≤ C ε and u µ ∈ Hom C ε ( α ( v ) , α ( v )) .Proof. To see (1), let v i := β ( π i ). By Lemma 6.8, we have d ( v , u · v ) = d ( β ( π ) , β (Ad( u ) ◦ π )) < C ε + d ( π , Ad( u ) ◦ π ) = ( C +1) ε. This means that ∆ I ( u ) ∈ Hom ( C +1) ε ( v , v ).Next we show (2). If h µ, ν i ∈ T , we get k u µ − u ν k ≤ k u µ v µν ( x µν ) u ∗ ν − k + k u µ ( v µν ( x µν ) − k < ε and hence k u µ − u ν k < T ) ε . Therefore we have d ( π , Ad( u µ ) ◦ π ) = d ( α ( v ) , α (∆ I ( u µ ) · v )) ≤ d ( v , u · v ) + d ( u · v , ∆ I ( u µ ) · u ) + C ε ≤ (1 + 2 diam( T ) + C ) ε. Now the proof is completed by choosing C := C + 1 + 2 diam( T ). (cid:3) Definition 6.10. Let us fix µ ∈ I and let C = max { C , , C } . Wedefine two maps α : Bdl ε, U P,Q ( X, Y ) T → qRep C ε, G P,Q (Γ , Λ) , β : qRep ε, G P,Q (Γ , Λ) → Bdl C ε, U P,Q ( X, Y ) T , by α ( v , v , v , u ) = ( α ( v ) , α ( v ) , α ( v ) , u µ ) , β ( π , π , π , u ) = ( β ( π ) , β ( π ) , β ( π ) , ∆ I ( u )) . LMOST FLAT RELATIVE VECTOR BUNDLES 23 We define the metric on Bdl ε, U P,Q ( X, Y ) and qRep ε, G P,Q (Γ , Λ) by d ( v , v ′ ) := max { d ( v , v ′ ) , d ( v , v ′ ) , d ( v , v ′ ) , d ( u , u ′ ) } d ( π , π ′ ) := max { d ( π , π ′ ) , d ( π , π ′ ) , d ( π , π ′ ) , d ( u, u ′ ) } . Lemma 6.11. If v , v ′ ∈ Bdl ε, U P,Q ( X, Y ) satisfies d ( v , v ′ ) < ε , then v and v are homotopic in the space Bdl (4 C +1) ε, U P,Q ( X, Y ) .Proof. By Lemma 3.4, there are { ¯ u iµ } for i = 1 , , u iµ v iµν (¯ u iν ) ∗ =( v ′ ) iµν and k ¯ u µ − k < C ε . Since ¯ u is near to the identity, ¯ u i,sµ := exp( s log(¯ u iµ ))is a unitary-valued functions such that k ¯ u i,sµ − k < C ε . Then( { ¯ u ,sµ v µν (¯ u ,sν ) ∗ } µ,ν , { ¯ u ,sµ v µν (¯ u ,sν ) ∗ } µ,ν , { ¯ u ,sµ v µν (¯ u ,sν ) ∗ } µ,ν , u )is a continuous path in Bdl (4 C +1) ε, U ( X, Y ) connecting v with ( v ′ , v ′ , v ′ , u ).Also, u s = { u sµ := u µ exp( s log( u ∗ µ u ′ µ )) } µ ∈ I is a continuous path connecting u with u ′ such that k u sµ − u ′ µ k < ε , which makes ( v ′ , v ′ , v ′ , u s ) to a homotopyof (3 ε, U )-flat bundles. (cid:3) Theorem 6.12. Let ( X, Y ) be a finite simplicial complex and let Γ := π ( X ) and Λ := π ( Y ) . (1) For v , v ′ ∈ Bdl ε, U P,Q ( X, Y ) T , we have d ( α ( v ) , α ( v ′ )) ≤ d ( v , v ′ ) + C ε and d ( β ◦ α ( v ) , v ) ≤ C ε . (2) For π , π ′ ∈ qRep ε, G P,Q (Γ , Λ) , we have d ( β ( π ) , β ( π ′ )) ≤ d ( π , π ′ )+ C ε and d ( α ◦ β ( π ) , π ) ≤ C ε .Proof. It follows from Lemma 6.8 and Lemma 6.9. (cid:3) Corollary 6.13. If there is a continuous map f : ( X , Y ) → ( X , Y ) whichinduces the isomorphism of fundamental groups, then K ( X , Y ; A ) is in-cluded to f ∗ K ( X , Y ; A ) . In particular, if ( B Γ , B Λ) has the homotopytype of a pair of finite CW-complexes, then K ( X, Y ; A ) ⊂ f ∗ K ( B Γ , B Λ; A ) ,where f is the reference map.Proof. For sufficiently small ε > 0, let v ∈ Bdl ε, U P,Q ( X , Y ) be a representativeof ξ ∈ K ( X , Y ; A ). By Remark 3.10 and Lemma 6.5, we may assumewithout loss of generality that v is normalized on T . Here we write α X,Y and β X,Y for the map α and β with respect to the pair ( X, Y ). Then,˜ v := β X ,Y ◦ α X ,Y ( v ) is a ( C ( U ) C ( U ) ε, U )-flat bundle on ( X , Y )which satisfies d ( v , f ∗ ˜ v ) < C ( U ) ε . Hence [ v ] = f ∗ [˜ v ] by Lemma 6.11. (cid:3) Remark . Let ( X, Y ) be a pair of finite CW-complexes with π ( X ) := Γand π ( Y ) := Λ and let U be an open cover of ( X, Y ). Assume that theinduced map Λ → Γ is injective. Then the double ˆ X := X ⊔ Y Y × [0 , ⊔ Y X has the fundamental group Γ ∗ Λ Γ by the van Kampen theorem. We associatean open cover ˆ U of ˆ X to U asˆ U = { U µ,i := q ∗ i U µ ∩ X ◦ i } ( µ,i ) ∈ I ×{ , } , where X := X ⊔ Y × [0 , X := Y × [0 , ⊔ X and q i : X i → X for i = 1 , G ⊂ Γ ∗ Λ Γ denote the union of two copies of G Γ ⊂ Γ. In this setting, there is a correspondenceBdl ε, U P ( X, Y ) / / α (cid:15) (cid:15) Bdl ε, ˆ U P ( ˆ X ) α (cid:15) (cid:15) o o qRep ε, G P (Γ , Λ) / / β O O qRep ε, ˆ G P (Γ ∗ Λ Γ) , β O O o o which commutes up to small perturbations. This is a counterpart in almostflat geometry of the higher index theory of invertible doubles studied in[Kub18, Section 5]. • We fix a point x µν ∈ U µν ∩ Y for each µ, ν ∈ I with U µν ∩ Y = ∅ .For ˆ v ∈ Bdl ε, ˆ U P ( ˆ X ), let v i := { ˆ v ( µ,i )( ν,i ) | q ∗ i U µν ∩ X } µ,ν ∈ I for i = 1 , u := { u µ := ˆ v ( µ, µ, ( x µν ) } for µ, ν ∈ I with U µν ∩ Y = ∅ . Then( v , v , u ) is a relative ( ε, U )-flat bundle on ( X, Y ). • For v = ( v , v , u ) ∈ Bdl ε, U P ( X, Y ), pick ¯ u ∈ G C ε ( u ) by Lemma 3.4.Then ˆ v = { ˆ v ( µ,i )( ν,j ) } given byˆ v ( µ,i )( ν,j ) := ( q ∗ v iµν ) | q ∗ U µν ∩ X ◦ i if i = j , q ∗ ( v µν ¯ u ν ) | q ∗ U µν ∩ Y × (0 , if i = 1, j = 2 ,q ∗ ( v µν ¯ u ∗ ν ) | q ∗ U µν ∩ Y × (0 , if i = 2, j = 1 , is a (( C + 1) ε, ˆ U )-flat bundle on ˆ X . • For a ( ε, ˆ G )-representation ˆ π of Γ ∗ Λ Γ, let π and π denote itsrestrictions to the first and second copies of Γ. Then, π ( π , π , ε, ˆ G P (Γ ∗ Λ Γ) to qRep ε, G P (Γ , Λ). • For π ∈ qRep ε, G P (Γ , Λ) of the form ( π , π , ε, ˆ G )-representationˆ π of Γ ∗ Λ Γ constructed in the following way. Pick a set theoreticsection τ : Γ ∗ Λ Γ → Γ ∗ Γ and let ˆ π ( γ ) := ( π ∗ π )( τ ( γ )). Then ˆ π isa (2 ε, ˆ G )-representation of Γ ∗ Λ Γ. References [CD18] Jos´e R. Carri´on and Marius Dadarlat, Almost flat K-theory of classifying spaces ,J. Noncommut. Geom. 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