Realizing the analytic surgery group of Higson and Roe geometrically, Part III: Higher invariants
aa r X i v : . [ m a t h . K T ] J a n REALIZING THE ANALYTIC SURGERY GROUP OF HIGSONAND ROE GEOMETRICALLYPART III: HIGHER INVARIANTS
ROBIN J. DEELEY, MAGNUS GOFFENG
Abstract.
We construct an isomorphism between the geometric model andHigson-Roe’s analytic surgery group, reconciling the constructions in the pre-vious papers in the series on “
Realizing the analytic surgery group of Higsonand Roe geometrically ” with their analytic counterparts. Following work ofLott and Wahl, we construct a Chern character on the geometric model for thesurgery group; it is a “delocalized Chern character”, from which Lott’s higherdelocalized ρ -invariants can be retrieved. Following work of Piazza and Schick,we construct a geometric map from Stolz’ positive scalar curvature sequenceto the geometric model of Higson-Roe’s analytic surgery exact sequence. Introduction
This paper is the last in a series of three whose topic is the construction ofa geometric (i.e., Baum-Douglas type) realization of the analytic surgery groupof Higson and Roe [20, 21, 22]. The first paper in the series [11] dealt with thegeometric cycles, the associated six term exact sequence, and the geometricallydefined secondary invariants. In the second paper [12], the relationship betweenthese geometrically defined secondary invariants and analytically defined secondaryinvariants was developed − mainly focusing on the relative η -invariant. In particular,an alternative proof of a theorem of Keswani [30, Theorem 1.2] was developed.The current paper concerns itself with higher invariants: K -theoretic invariantsgiving the isomorphism with Higson-Roe’s original analytic model as well as Cherncharacters in non-commutative de Rham homology.Before giving the motivation for this paper and its role in the series of papers,we state the two main theorems of the current paper. Theorem.
For a discrete finitely generated group Γ and a compact metric space X with a map X → B Γ , there is an isomorphism: λ S an : S geo ∗ ( X, L X ) −→ ∼ = K ∗ ( D ∗ ( ˜ X ) Γ ) ,λ S an : ( W, ξ, f ) j X ind AP S ( W, Ξ , f ) − ρ ( W, Ξ , f ) . Moreover, λ S an is compatible with the analytic surgery exact sequence. Here K ∗ ( D ∗ ( ˜ X ) Γ ) is Higson-Roe’s analytic surgery group [20, 21, 22], explainedfurther in Subsection 1.1, and j X the natural mapping K ∗ ( C ∗ r (Γ)) → K ∗ ( D ∗ ( ˜ X ) Γ ).The triple ( W, ξ, f ) is a geometric surgery cycle for the Mishchenko bundle L X → X from [11] and ( W, Ξ , f ) denotes the cycle ( W, ξ, f ) with further analytic data fixed.A short review of geometric cycles for surgery and the construction of λ can be Date : September 21, 2018. found in Subsection 2.1. The above theorem appears as Theorem 2.5 in the mainbody of the paper; it is the main result of Section 2.
Theorem.
For a discrete finitely generated group Γ and a dense Fr´echet subalgebra A ⊆ C ∗ r (Γ) , there is a Chern character from the geometric surgery cycles for A tothe delocalized de Rham homology groups of A ch del : S geo ∗ ( X ; A ) → ˆ H del,
We will recall the relevant notions from coarse geometry and higher index theoryin this section. When proving the main results of this paper, the techniques of[40, 41] will play a major role and therefore we mainly use the notation of thosepapers. We will indicate whenever our notation strays from the notation of [40, 41].For an overview of the notions that we will make use of, see [40, Section 1]. A moredetailed exposition can be found in [44].The setup that prevails throughout this section, and the two subsequent, is thatof a Γ-presented space, see [21, Definition 2.6]. A Γ-presented space is a propergeodesic metric space ˜ X equipped with a free and proper action of Γ. We set X = ˜ X/ Γ. We will assume that X is compact and often that it is a finite CW-complex. In the notations of [40, 41], X denotes the space with a free and properΓ-action.1.1. Coarse geometry and the analytic surgery exact sequence.
We choosea Hilbert space H X with a Γ-equivariant representation of C ( ˜ X ) that is assumedto be non-degenerate and ample. We call such a Hilbert space a Γ-equivariant X -module. The particular choice of Γ-equivariant X -module will only affect theassociated invariants by a natural isomorphism (for more details see [44]).Recall the following terminology for operators on a Γ-equivariant X -module H X : • An operator T is called locally compact if φT, T φ ∈ K ( H X ) for any φ ∈ C ( ˜ X ). • An operator T is called pseudo-local if [ φ, T ] ∈ K ( H X ) for any φ ∈ C ( ˜ X ). • An operator T is said to have finite propagation if for some R > φT φ ′ = 0 whenever φ, φ ′ ∈ C ( X ) satisfies d (supp( φ ) , supp( φ ′ )) > R .Here d denotes the metric on X .We note that the set of locally compact operators as well as the set of pseudo-localoperators form C ∗ -subalgebras of the bounded operators on H X . The C ∗ -algebra oflocally compact operators form an ideal in the C ∗ -algebra of pseudo-local operators.Associated with a Γ-presented space and the Hilbert space H X , there are anumber of C ∗ -algebras; we will make use of two: C ∗ ( ˜ X, H X ) Γ and D ∗ ( ˜ X, H X ) Γ .The C ∗ -algebra D ∗ ( ˜ X, H X ) Γ is defined to be the C ∗ -algebra closure of all Γ-invariant pseudo-local operators with finite propagation. The ideal C ∗ ( ˜ X, H X ) Γ ⊆ D ∗ ( ˜ X, H X ) Γ is the intersection of D ∗ ( ˜ X, H X ) Γ with the locally compact opera-tors on H X . We recall the following standard facts about the associated K -theorygroups. A mapping is coarse if for any R >
0, there is an
S > R is contained in a ball of radius S . Theorem 1.1.
The K -theory groups of C ∗ ( ˜ X, H X ) Γ and D ∗ ( ˜ X, H X ) Γ have thefollowing properties: (1) There are natural isomorphisms K ∗ ( C ∗ ( ˜ X, H X ) Γ ) ∼ = K ∗ ( C ∗ r (Γ)) and K ∗ ( D ∗ ( ˜ X, H X ) Γ /C ∗ ( ˜ X, H X ) Γ ) ∼ = KK ∗− ( C ( X ) , C ) = K an ∗− ( X ) . (2) Under these isomorphisms, the boundary mapping ∂ : K ∗ +1 ( D ∗ ( ˜ X, H X ) Γ /C ∗ ( ˜ X, H X ) Γ ) → K ∗ ( C ∗ ( ˜ X, H X ) Γ ) corresponds to the assembly map for free actions µ X : K an ∗ ( X ) → K ∗ ( C ∗ r (Γ)) . EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 5 (3)
Let ˜ Y be another Γ -represented space and H Y a Γ -equivariant Y -module.Whenever f : X → Y is a coarse Γ -equivariant mapping, there is a functo-rially associated mapping f ∗ : K ∗ ( C ∗ ( ˜ X, H X ) Γ ) → K ∗ ( C ∗ ( ˜ Y , H Y ) Γ ) . (4) Whenever f : X → Y is a continuous coarse Γ -equivariant mapping, thereis a functorially associated mapping f ∗ : K ∗ ( D ∗ ( ˜ X, H X ) Γ ) → K ∗ ( D ∗ ( ˜ Y , H Y ) Γ ) . The reader is directed to [18, 19, 44] for further details. Motivated by this result,when H X is clear from the context, we simply write C ∗ ( ˜ X ) Γ and D ∗ ( ˜ X ) Γ ; up tonatural isomorphism, a change of H X does not alter the relevant K -groups. Remark 1.2.
It is possible to construct full versions of C ∗ ( ˜ X, H X ) Γ and D ∗ ( ˜ X, H X ) Γ by using the maximal C ∗ -norm on the ∗ -algebra of Γ-invariant pseudo-local opera-tors with finite propagation. In this case, the boundary map in K -theory is the fullassembly map for free actions. The constructions of this paper also apply in thisframework.The next theorem follows from Theorem 1.1 and the existence of six term exactsequences in K -theory. Theorem 1.3 (The analytic surgery exact sequence, [20]) . There is a six termexact sequence K an ( X ) µ X −−−−→ K ( C ∗ r (Γ)) j X −−−−→ K ( D ∗ ( ˜ X ) Γ ) q X x y q X K ( D ∗ ( ˜ X ) Γ ) j X ←−−−− K ( C ∗ r (Γ)) µ X ←−−−− K an ( X ) , where j X : K ∗ ( C ∗ r (Γ)) → K ∗ ( D ∗ ( ˜ X ) Γ ) is the composition of the isomorphism ofTheorem 1.1 with the inclusion C ∗ ( ˜ X ) Γ → D ∗ ( ˜ X ) Γ and q X : K ∗ ( D ∗ ( ˜ X ) Γ ) → K an ∗ +1 ( X ) the composition of the quotient mapping D ∗ ( ˜ X ) Γ → D ∗ ( ˜ X ) Γ /C ∗ ( ˜ X ) Γ with the isomorphism of Theorem 1.1. Remark 1.4.
The motivation for the terminology “analytic surgery exact se-quence” is twofold. Firstly, it is the analytic analog of the surgery exact sequenceand secondly, there is a precise relationship between the two exact sequences (see[20, 21, 22, 41] for details); we also discuss this relationship in Section 5. We willreturn to the analytic surgery exact sequence in the context of the geometric modelfrom [11] in Section 2.1.2.
Higher Atiyah-Patodi-Singer index theory.
The higher index theory fora manifold with boundary can elegantly be described using the tools of the previoussubsection. Parts of the discussion on higher index theory resembles that occurringpreviously in this series of papers (e.g., [12, Section 1.2]). As such, we keep it shortand use the notation of [12]. In particular, our notation for Dirac operators will dif-fer from [40, 41]; we unfortunately need a notation allowing for more indices. Theyare used to indicate all the possible data associated with a geometric cycles. Wewill often (for simplicity) restrict the discussion to even-dimensional manifolds withodd-dimensional boundaries, the reader is referred to [32] for the general situation.
ROBIN J. DEELEY, MAGNUS GOFFENG
Let W be a compact spin c manifold with boundary (potentially with emptyboundary) equipped with some metric g W which is of product type near the bound-ary. We assume that D W is a choice of spin c -Dirac operator on W that is of producttype near ∂W ; the operator D W acts on the sections of the complex spinor bundle S W → W associated with the spin c -structure. Since D W is of product type, thereis an associated boundary operator D ∂W – a spin c -Dirac operator on ∂W .A smooth locally trivial bundle of finitely generated projective C ∗ r (Γ)-Hilbertmodules will be called a C ∗ r (Γ)-bundle. If E → W is a C ∗ r (Γ)-bundle equipped witha hermitean connection, there is an associated Dirac type operator D W E acting on S W ⊗ E that is of product type near ∂W . For simplicity, assume that the fibers of E are full Hilbert C ∗ r (Γ)-modules. As such, there is an associated boundary operator D ∂W E acting on S ∂W ⊗ E| ∂W , coinciding with the Dirac type operator constructedfrom E| ∂W and D ∂W . If W is even-dimensional, D W is odd in the grading on thegraded bundle S W . If W is odd-dimensional, ∂W is even-dimensional and D ∂W isodd in the grading on the graded bundle S ∂W .If W has a boundary ∂W , we let W ∞ denote the complete manifold obtainedfrom gluing on the cylinder [0 , ∞ ) × ∂W , equipped with the metric d t + g W | ∂W ,on to W . Any C ∗ r (Γ)-bundle E → W extends to W ∞ and the same holds for Diracoperators D W E . We denote these extensions by E ∞ → W ∞ and respectively D W ∞ E .If W = M is a closed manifold, the operator D M E is an elliptic element inthe Mishchenko-Fomenko calculus Ψ ∗ C ∗ r (Γ) ( M, E ). Hence it defines regular, self-adjoint, C ∗ r (Γ)-Fredholm operator on the C ∗ r (Γ)-Hilbert module L ( M, E ). In alarger generality, a Dirac operator on a C ∗ -bundle over a complete manifold definedfrom a hermitean connection is always self-adjoint and regular, see [16, Theorem2.3]. Returning to closed manifolds, the C ∗ r (Γ)-Fredholm operator D M E has a higherindex ind C ∗ r (Γ) ( D M E ) ∈ K dim( M ) ( C ∗ r (Γ)) . The parity of the K -theory group comes from the dimension of M : if dim( M ) iseven, D M E is self-adjoint and graded while if dim( M ) is odd, D M E is merely self-adjoint. Recall that by [32, Theorem 3] and [33, Proposition 2.10], there is asmoothing operator A ∈ Ψ −∞ C ∗ r (Γ) ( M, E ) such that D M E + A is invertible if and only ifind C ∗ r (Γ) ( D M E ) = 0. Such an operator is called a trivializing operator for D M E . Foreven-dimensional M , a trivializing operator A is by definition odd. The projection P A := 12 (cid:18) D M E + A | D M E + A | + 1 (cid:19) , is called a spectral section for D M E , see [32, Definition 3] and [33, Proposition 2.10].In the particular situation when M = ∂W and D M E = D ∂W E , it follows from [25,Theorem 6.2] that ind C ∗ r (Γ) ( D M E ) = 0. For any such choice of smoothing operator A ∈ Ψ −∞ C ∗ r (Γ) ( M, E ), we let A ∞ denote the odd operator on L ( W ∞ , S W ∞ ⊗ E ∞ )constructed from the operator χ ⊗ A : L ([0 , ∞ ) × ∂W, S +[0 , ∞ ) × ∂W ⊗ E ∞ | [0 , ∞ ) × ∂W ) (cid:9) , for some χ ∈ C ∞ [0 , ∞ ) vanishing near 0 and equaling 1 near ∞ . The proof of thenext theorem follows (more or less) by combining these facts with the techniquesof [33, Section 6]. EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 7
Theorem 1.5.
Let W be a compact even-dimensional spin c manifold with boundaryequipped with the following data: • a C ∗ r (Γ) -bundle E → W ; • a spin c -Dirac operator D W on W ; • a self-adjoint smoothing operator A ∈ Ψ −∞ C ∗ r (Γ) ( ∂W, E| ∂W ) such that D ∂W E + A is invertible.The associated C ∗ r (Γ) -linear operator D W ∞ E + A ∞ is a regular, self-adjoint, C ∗ r (Γ) -Fredholm operator on L ( W ∞ , S W ∞ ⊗ E ∞ ) . Definition 1.6.
Let ind
AP S ( D W E , A ) := ind C ∗ r (Γ) ( D W ∞ E + A ∞ ) ∈ K dim W ( C ∗ r (Γ)). Remark 1.7.
The constructions in [41, Section 2.3] of APS-indices are made in thecontext of coarse geometry and differs slightly from approach above. The geometricsetup considered in [41] is as follows. One has a continuous mapping g : W → B Γ,a vector bundle E → W and E = E ⊗ g ∗ L B Γ , where L B Γ := E Γ × Γ C ∗ r (Γ) → B Γ denotes the Mishchenko bundle. For this choice of C ∗ r (Γ)-bundle there is ageometrically constructed isomorphism L ( ˜ W , S ˜ W ⊗ ˜ E ) ∼ = L ( W, S W ⊗ E ) ⊗ π ℓ (Γ),where π : C ∗ r (Γ) → B ( ℓ (Γ)) denotes the regular representation.A Dirac operator D WE on E → W can be lifted to a Γ-equivariant operator ˜ D WE on the Galois covering ˜ W := E Γ × g W → W . Under the isomorphism above, D W E ⊗ π id ℓ (Γ) corresponds to ˜ D WE and A ⊗ id ℓ (Γ) to an operator ˜ A on L ( ∂ ˜ W , S ∂ ˜ W ⊗ ˜ E | ∂ ˜ W )such that ˜ D ∂WE + ˜ A is L -invertible. In [41, Section 2.3], there is a constructionof the index class ind C ∗ ( ˜ W ) Γ ( ˜ D WE + ˜ A ∞ ) ∈ K dim W ( C ∗ ( ˜ W ) Γ ); it is the image ofind AP S ( D W E , A ) under the isomorphism K ∗ ( C ∗ ( ˜ W ) Γ ) ∼ = K ∗ ( C ∗ r (Γ)). Theorem 1.8 (Gluing higher APS-indices) . Let Z be a compact even-dimensionalspin c -manifold and Y ⊆ Z ◦ a hyper surface separating Z into two compact mani-folds, W and W . That is, Z = W ∪ Y W with ∂W i = M i ˙ ∪ ( − i Y for i = 1 , and closed manifolds M and M . Furthermore, assume that the following data hasbeen fixed: • a spin c -Dirac operator D Z on Z of product type near ∂Z and Y , withassociated boundary operators D M , D M respectively D Y ; • a C ∗ r (Γ) -bundle E → Z with connection of product type near ∂Z and Y ; • trivializing operators A i ∈ Ψ −∞ C ∗ r (Γ) ( M i , S M i ⊗ E| M i ) of D M i E| Mi , for i = 1 , ; • a trivializing operator A Y ∈ Ψ −∞ C ∗ r (Γ) ( Y, S Y ⊗ E| Y ) of D Y E| Y .Then, ind AP S ( D Z E , A ˙ ∪ A ) = ind AP S ( D W E| W , A ˙ ∪ − A Y ) + ind AP S ( D W E| W , A ˙ ∪ A Y ) , where D W i is the restriction of D Z to W i , for i = 1 , . Theorem 1.8 follows from applying [32, Theorem 8] to the hyper surface ∂W ∪ ∂W in the closed manifold 2 Z . Remark 1.9.
The assumption on existence of the trivializing operators A , A and A Y in Theorem 1.8 is necessary and does not follow from [32, Theorem 3].In general, [32, Theorem 3] only implies the existence of a trivializing operator on ∂W and ∂W , but not that it decomposes into a direct sum acting on the disjointunion ∂W i = M i ˙ ∪ ( − i Y for i = 1 , ROBIN J. DEELEY, MAGNUS GOFFENG
Definition 1.10.
Let
E → M be a C ∗ r (Γ)-bundle on a closed odd-dimensionalmanifold M , D M E , • = ( D M E ,u ) u ∈ [0 , be a family of Dirac operators with vanishingindex, and P and P are spectral sections for D M E , and respectively D M E , . Then,the spectral flow of ( D M E , • , P , P ) is defined assf( D M E , • , P , P ) := [ P − Q ] − [ P − Q ] ∈ K ( C ∗ r (Γ)) , where Q • = ( Q t ) t ∈ [0 , is spectral section for D M E , • viewed as a C [0 , ⊗ C ∗ r (Γ)-linearoperator.For more details regarding Definition 1.10, see [32, Section 2.5]. We state [32,Proposition 8] for the convenience of the reader: Theorem 1.11 (Change of data in higher APS-indices) . Let W be a compacteven-dimensional spin c -manifold and E → W a C ∗ r (Γ) -bundle. Assume that D W E and D ′ W E are two Dirac type operators on S W ⊗ E as above. Take two trivializingoperators A, A ′ ∈ Ψ −∞ C ∗ r (Γ) ( ∂W, S ∂W ⊗ E| ∂W ) for D ∂W E respectively D ′ ∂W E . Let P := χ [0 , ∞ ) ( D ∂W E + A ) and P ′ := χ [0 , ∞ ) ( D ′ ∂W E + A ′ ) . Then, ind AP S ( D W E , A ) − ind AP S ( D ′ W E , A ′ ) = sf((1 − t ) D ∂W E + tD ′ ∂W E , P, P ′ ) . Remark 1.12.
In [32], a more convenient notation was introduced to treat the case D ∂W E = D ′ ∂W E . For a C ∗ r (Γ)-bundle E → M on a closed odd-dimensional manifold M and any two spectral sections P and P ′ of a Dirac operator D M E , there exists a C ∗ r (Γ)-linear projection Q ∈ L C ∗ r (Γ) ( L ( M, S M ⊗ E )) , such that P − Q and P ′ − Q are projections in K C ∗ r (Γ) ( L ( M, S M ⊗ E ) and one sets[ P − P ′ ] := [ P − Q ] − [ P ′ − Q ] ∈ K ( C ∗ r (Γ)) , using the isomorphism K ( K C ∗ r (Γ) ( L ( M, S M ⊗E ))) ∼ = K ( C ∗ r (Γ)) induced by Moritaequivalence (for further details see [32, Corollary 1]).1.3. Higher ρ -invariants. Assume that M is a closed odd-dimensional spin c -manifold equipped with a choice of spin c -Dirac operator D M defining the fun-damental class [ M ] := [ D M ] ∈ K an ( M ). Also, assume that f : M → B Γ is a con-tinuous map; in particular, ˜ M := E Γ × f M is a Γ-presented space with ˜ M /
Γ = M .Associated with this data, there is a Mishchenko bundle: L M := ˜ M × Γ C ∗ r (Γ) ∼ = f ∗ L B Γ . The Mishchenko bundle L M is a flat C ∗ r (Γ)-bundle. It is not difficult to see thatfor a vector bundle E → M and a spin c -Dirac operator D ME , µ M ([ M ] ∩ [ E ]) = ind C ∗ r (Γ) ( D ME ⊗L M ) . Here D ME ⊗L M denotes the Dirac operator on S M ⊗ E ⊗ L M obtained from twisting D ME by the flat connection on L M .If ind C ∗ r (Γ) ( D ME ⊗L M ) = 0, there is a particular choice of pre image in K ( D ∗ ( ˜ M ) Γ )associated with a choice of trivializing operator A ∈ Ψ −∞ C ∗ r (Γ) ( M, S M ⊗ E ⊗ L M ).Recall that for a smoothing operator A ∈ Ψ −∞ C ∗ r (Γ) ( M, S M ⊗ E ⊗ L M ), ˜ A denotes thesmoothing operator on L ( ˜ M , S ˜ M ⊗ ˜ E ) obtained from A ⊗ π id ℓ (Γ) . The followingtheorem is a consequence of the discussion in [41, Section 2.2]; the key technicalresult is [41, Proposition 2.8]. EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 9
Theorem 1.13.
Let M , f , E and D ME be as in the paragraphs just before thisstatement. Then, for any trivializing operator A ∈ Ψ −∞ C ∗ r (Γ) ( M, S M ⊗ E ⊗ L M ) , theelement χ [0 , ∞ ) ( ˜ D ME + ˜ A ) is a well defined projection in D ∗ ( ˜ M ) Γ giving the ρ -class : ρ ( D ME , A ) := h χ [0 , ∞ ) ( ˜ D ME + ˜ A ) i ∈ K ( D ∗ ( ˜ M ) Γ ) . Moreover, ρ ( D ME , A ) lifts [ M ] ∩ [ E ] , in the sense that q X (cid:0) ρ ( D ME , A ) (cid:1) = [ M ] ∩ [ E ] in K an ( M ) . Here the map q X is as in Theorem 1.3. The following lemma will come in handy when comparing higher ρ -invariantsconstructed from different geometric data. As in subsection 1.1, ˜ M denotes a Γ-presented space with M = ˜ M /
Γ. Moreover, we let j M : K ∗ ( C ∗ r (Γ)) → K ∗ ( D ∗ ( ˜ M ) Γ )denote the mapping induced from the isomorphism K ∗ ( C ∗ r (Γ)) ∼ = K ∗ ( C ∗ ( ˜ M ) Γ ) andthe inclusion C ∗ ( ˜ M ) Γ → D ∗ ( ˜ M ) Γ . Lemma 1.14.
Suppose that f : M → B Γ is a continuous mapping from a closedmanifold and that E → M is a vector bundle. Let ( D ME,u ) u ∈ [0 , be a family ofDirac operators thereon and A and let A be trivializing operators for the Diracoperators D ME ⊗L , respectively D ME ⊗L , . We let P := χ [0 , ∞ ) ( D ME ⊗L , + A ) and P := χ [0 , ∞ ) ( D ME ⊗L , + A ) denote the associated spectral sections. Then j M (cid:2) sf( D ME, • , P , P ) (cid:3) = ρ ( D ME, , A ) − ρ ( D ME, , A ) ∈ K ( D ∗ ( ˜ M ) Γ ) . Proof.
We set ˜ P i := χ [0 , ∞ ) ( ˜ D ME,i + ˜ A i ), for i = 0 ,
1, so ρ ( D ME,i , A i ) = [ ˜ P i ]. Itfollows from [41, Proposition 2.8] and [33, Proposition 2.10] that we can find a path( Q t ) t ∈ [0 , of spectral sections for ( D ME,u ) u ∈ [0 , with the following property: for i = 0 ,
1, the projections ˜ Q i ∈ D ∗ ( ˜ M ) Γ have the property that ˜ P − ˜ Q , ˜ P − ˜ Q ∈ C ∗ ( ˜ M ) Γ are projections and the class of sf( D ME, • , P , P ) is under K ( C ∗ r (Γ)) ∼ = K ( C ∗ ( ˜ M ) Γ ) equal to [ ˜ P − ˜ Q ] − [ ˜ P − ˜ Q ] (see Definition 1.10 and Remark 1.12),where (for i = 0 ,
1) the projections ˜ Q i are defined by taking the image of Q i ⊗ π id ℓ (Γ) under the isomorphism B ( L ( M, S M ⊗ E ⊗ L M ) ⊗ π ℓ (Γ)) ∼ = B ( L ( ˜ M , S ˜ M ⊗ ˜ E )) . The proposition now follows from the standard fact in K -theory that if p and q are projections in a ring R with pq = 0, then [ p ] + [ q ] = [ p + q ] in K ( R ) (see forexample [24, Lemma 3.4]). (cid:3) After suitable modifications to the definition of the ρ -invariant, both Theorem1.13 and Lemma 1.14 apply in the even-dimensional case (for more details see [40]for instance). Theorem 1.15 (Delocalized APS-theorem) . Let W be a compact manifold, F → W a vector bundle, D WF a Dirac operator on F and f : W → B Γ a continuousmapping with associated Mishchenko bundle L W . For any trivializing operator A of D ∂WF | ∂W ⊗L ∂W , j W (cid:0) ind AP S ( D WF ⊗L W , A ) (cid:1) = ι ∗ ρ ( D ∂WF | ∂W , A ) in K ∗ ( D ∗ ( ˜ W ) Γ ) , where ι : ∂W → W is the continuous inclusion and ι ∗ : K ∗ ( D ∗ ( ∂ ˜ W ) Γ ) → K ∗ ( D ∗ ( ˜ W ) Γ ) denotes the functorially associated mapping. The delocalized APS-theorem is stated as [41, Theorem 3.1] and proved therein the even-dimensional case. Other proofs of the delocalized APS-theorem (in theless general positive scalar curvature case) can be found in [40, 50].
Remark 1.16.
Suppose g X : X → B Γ is continuous and ˜ X ∼ = E Γ × g X X is a Γ-represented space with X compact. If f | ∂W equals the restriction to the boundaryof the composition of continuous maps ˆ f : W → X and g X : X → B Γ, then(2) j X (cid:0) ind AP S ( D WF ⊗L W , A ) (cid:1) = ( ˆ f | ∂W ) ∗ (cid:16) ρ ( D ∂WF | ∂W , A ) (cid:17) , as elements in K ∗ ( D ∗ ( ˜ X ) Γ ). The consequence (2) of the delocalized APS-theoremfollows because j X : K ∗ ( C ∗ r (Γ)) → K ∗ ( D ∗ ( ˜ X ) Γ ) factors as j X = ˆ f ∗ ◦ j W and( ˆ f | ∂W ) ∗ : K ∗ ( D ∗ ( ∂ ˜ W ) Γ ) → K ∗ ( D ∗ ( ˜ X ) Γ ) factors as ( ˆ f | ∂W ) ∗ = ˆ f ∗ ◦ ι ∗ . This factwill play a major role in proving that the isomorphism between the geometric modeland the analytic surgery group respects the bordism relation.2. The isomorphism λ S an : S geo ∗ ( X, L X ) ∼ −→ K ∗ ( D ∗ ( ˜ X ) Γ )Our goal is the construction of an isomorphism λ S an : S geo ∗ ( X, L X ) → K ∗ ( D ∗ ( ˜ X ) Γ ),which is compatible with the analytic surgery exact sequence of Theorem 1.3. Themap is defined using higher APS-index theory and higher ρ -invariants. The isomor-phism λ S an (in a sense) measures the failure of the delocalized APS-index formula tohold when the interior bundle on the manifold with boundary is a general C ∗ r (Γ)-bundle; this will be made more precise below in Remark 2.29.2.1. Decorated surgery cycles and associated higher invariants.
In orderto define the mapping λ S an , we need to recall some notions from the first two paperin this series [11, 12]. Similar to above, we assume that X is a compact metric spaceand g X : X → B Γ is a fixed continuous mapping. The space ˜ X := E Γ × g X X cannaturally be equipped with the structure of a Γ-presented space. The Mishchenkobundle L B Γ := E Γ × Γ C ∗ r (Γ) → B Γ is a locally trivial bundle of finitely generatedprojective C ∗ r (Γ)-modules and L X := ˜ X × Γ C ∗ r (Γ) = g ∗ X L B Γ . The Mishchenko bundle can be defined with respect to other completions as well.In fact, the geometric description of the analytic surgery group holds in an evenlarger generality, see [11].Whenever (
Z, Q, g ) is a triple consisting of a compact Hausdorff space Z , aclosed subset Q ⊆ Z and a continuous mapping f : Q → X , we can definerelative K -theory cocycles for assembly on ( Z, Q, f ) along L X to be quadruples ξ = ( E C ∗ r (Γ) , E ′ C ∗ r (Γ) , E C , E ′ C , α ) consisting of • a pair of locally trivial bundles of finitely generated projective C ∗ r (Γ)-modules E C ∗ r (Γ) , E ′ C ∗ r (Γ) → Z ; • a pair of vector bundles E C , E ′ C → Q ; • an isomorphism α : E C ∗ r (Γ) | Q ⊕ ( E ′ C ⊗ f ∗ L X ) → E ′ C ∗ r (Γ) | Q ⊕ ( E C ⊗ f ∗ L X )of C ∗ r (Γ)-bundles.This definition is [11, Definition 1.8]. Equipped with a suitable equivalence re-lation, the relative K -theory cocycles for assembly on ( Z, Q, f ) along L X can beused to build a K -theory group fitting together with assembly on the level of K -theory. For details, see [11, Section 1.3]. We will tacitly assume that the fibers of EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 11 E C ∗ r (Γ) , E ′ C ∗ r (Γ) → Z are full Hilbert C ∗ r (Γ)-modules. Full fibers can always be ob-tained without changing the K -theory class after adding a summand of the trivial C ∗ r (Γ)-bundle to E C ∗ r (Γ) and E ′ C ∗ r (Γ) .Recall from [11, Definition 2.1] that a cycle for S geo ( X, L X ) is a triple ( W, ξ, f )where: • W is an even-dimensional compact spin c -manifold with boundary; • f : ∂W → X is a continuous mapping; • ξ is a relative K -theory cocycle for assembly on ( W, ∂W, f ) along L X .From [12, Definition 3.1] we recall that a quintuple ( D E , D E ′ , D E , D E ′ ) is called achoice of Dirac operators on ξ if • D E and D E ′ are Dirac operators on S W ⊗ E C ∗ r (Γ) respectively S W ⊗ E ′ C ∗ r (Γ) defined from C ∗ r (Γ)-Clifford connections of product type near ∂W . • D E and D E ′ are Dirac operators on S ∂W ⊗ E C respectively S ∂W ⊗ E ′ C .Since the Dirac operators D E and D E ′ are defined from connections of product typenear the boundary, there are associated boundary Dirac operators D ∂W E and D ∂W E ′ acting on E ⊗ S ∂W respectively E ′ ⊗ S ∂W . Definition 2.1 (Definition 3.13 of [12]) . Let (
W, ξ, f ) be a cycle for S geo ∗ ( X, L X ).A decoration of ξ is a collection Ξ = ( ξ, ( D E , D E ′ , D E , D E ′ ) , ( A E , A E ′ , A )) where( D E , D E ′ , D E , D E ′ ) is a choice of Dirac operators and ( A E , A E ′ , A ) is a choice oftrivializing operators for ( W, ξ, f ), i.e., smoothing operators in the Mishchenko-Fomenko calculus of the following type: • A E ∈ Ψ −∞ C ∗ r (Γ) ( ∂W, E ⊗ S ∂W ) is a trivializing operator for D ∂W E ; • A E ′ ∈ Ψ −∞ C ∗ r (Γ) ( ∂W, E ′ ⊗ S ∂W ) is a trivializing operator for D ∂W E ′ ; • A ∈ Ψ −∞ C ∗ r (Γ) ( ∂W ˙ ∪ − ∂W, ( E ′ ⊗ S ∂W ⊗ f ∗ L ) ˙ ∪ − ( E ⊗ S ∂W ⊗ f ∗ L )) is atrivializing operator for (cid:16) D ∂WE ′ ⊗ f ∗ L ˙ ∪ − D ∂WE ⊗ f ∗ L (cid:17) .A decoration of a cycle ( W, ξ, f ) is a triple ( W, Ξ , f ) where Ξ is a decoration of ξ . It was shown in [12, Lemma 3.14] that any cycle admits a decoration; the proofis based on [32, Theorem 3] and [33, Proposition 10].In order to construct the higher APS-index class and relevant ρ -invariants,we need to introduce further notation. Let ( W, Ξ , f ) be a decorated cycle for S geo ∗ ( X, L X ). Associated with the decoration Ξ, we construct the Dirac operator¯ D ∂W × [0 , , acting on the bundle S ∂W × [0 , ⊗ ( E C ∗ r (Γ) | ∂W ⊕ E ′ C ⊗ f ∗ L X )over the cylinder ∂W × [0 , D ∂W E ⊕− D E ′ ⊗ f ∗ L X on ∂W × { } and the boundary operator α ∗ ( − D ∂W E ′ ⊕ D E ⊗ f ∗ L X ) on ∂W × { } .We also construct the smoothing operator A Ξ on the bundle S ∂W ×{ , } ⊗ ( E C ∗ r (Γ) | ∂W ⊕ E ′ C ⊗ f ∗ L X )by taking A Ξ := id ˙ ∪ α ∗ (cid:16)(cid:16) A E ˙ ∪ ( − A E ′ ) (cid:17) ⊕ A (cid:17) . By construction, the boundary operator ¯ D ∂W ×{ , } Ξ of D ∂W × [0 , has the propertythat ¯ D ∂W ×{ , } Ξ + A Ξ is invertible. Definition 2.2.
For a decorated cycle ( W, Ξ , f ) for S geo ∗ ( X, L X ), we define theclassesind AP S ( W, Ξ , f ) := ind AP S ( D W E , A E )+ ind AP S ( D − W E ′ , − A E ′ ) − ind AP S ( ¯ D Ξ , A Ξ ) ∈ K ∗ ( C ∗ r (Γ)); ρ ( W, Ξ , f ) := f ∗ [ ρ ( D E ˙ ∪ − D E ′ , A )] ∈ K ∗ ( D ∗ ( ˜ X ) Γ ); λ S an ( W, Ξ , f ) := j X ind AP S ( W, Ξ , f ) − ρ ( W, Ξ , f ) ∈ K ∗ ( D ∗ ( ˜ X ) Γ ) . The construction of the APS-index of a decorated cycle can be found in [12,Definition 3.15].
Remark 2.3.
Somewhat informally, the APS-index of a decorated cycle is to bethought of as the sum of the APS-index of ( W, E C ∗ r (Γ) ) ˙ ∪ ( − W, E ′ C ∗ r (Γ) ) “glued in the K -theory sense” along the cylinder ∂W × [0 ,
1] with the bundle data E C ∗ r (Γ) ⊕ E ′ C ⊗L X . Still informally, the higher ρ -class of a decorated cycle should be thought of asa difference of the higher ρ -class of E C with that of E ′ C . In general, however, thehigher ρ -class of E C and E ′ C are not separately well-defined!The informal discussion in the previous paragraph is motivated by the followingspecial case: suppose that E ′ C ∗ r (Γ) = E ′ C = 0 and we take the decorationΞ = (( E C ∗ r (Γ) , E C , α ) , ( D E , D E ) , ( A E , A )) . Then, Theorem 1.8 implies that ind
AP S ( W, Ξ , f ) = ind AP S ( ˆ D E , A ), where ˆ D E co-incides with D E outside a neighborhood of ∂W where ˆ D E is of product type withboundary operator α ∗ ( D E ⊗L ). In particular, λ S an ( W, Ξ , f ) = j X [ind AP S ( ˆ D E , A )] − f ∗ [ ρ ( D E , A )] ∈ K ∗ ( D ∗ ( ˜ X ) Γ ) . Lemma 2.4.
Given a cycle ( W, ξ, f ) for S ∗ ( X, L X ) , the class λ S an ( W, Ξ , f ) ∈ K ∗ ( D ∗ ( ˜ X ) Γ ) does not depend on the choice of decoration ( W, Ξ , f ) of ( W, ξ, f ) .Proof. For i = 0 and 1, let Ξ i = ( ξ, ( D E i , D E ′ ,i , D E,i , D E ′ ,i ) , ( A E i , A E ′ i , A i )) be twodecorations of ξ and P E ,i := χ [0 , ∞ ) ( D ∂W E ,i + A E i ) ,P E ′ ,i := χ [0 , ∞ ) ( − D ∂W E ′ ,i − A E ′ i ) ,P i := χ [0 , ∞ ) ( D ∂W ×{ , } Ξ i + A Ξ i ) and P i := χ [0 , ∞ ) ( D E ⊗ f ∗ L X ,i ˙ ∪ − D E ′ ⊗ f ∗ L X ,i + A i ) . We take a family ( D E ,u , D E ′ ,u , D E,u , D E ′ ,u ) u ∈ [0 , interpolating between the twochoices of Dirac operators. Lemma 1.14 implies that ρ ( W, Ξ , f ) − ρ ( W, Ξ , f ) = − j X sf( D E ⊗ f ∗ L X , • ˙ ∪ − D E ′ ⊗ f ∗ L X , • , P , P ) , while Theorem 1.11 implies thatind AP S ( W, Ξ , f ) − ind AP S ( W, Ξ , f )=sf( D E , • , P E , , P E , ) − sf( D E ′ , • , P E ′ , , P E ′ , ) + sf( ¯ D Ξ , • , ¯ P , ¯ P ) . EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 13
It follows from the construction of ¯ D Ξ , • thatsf( ¯ D Ξ , • , ¯ P , ¯ P ) + sf( D E ⊗ f ∗ L X , • ˙ ∪ − D E ′ ⊗ f ∗ L X , • , P , P )+ sf( D E , • , P E , , P E , ) − sf( D E ′ , • , P E ′ , , P E ′ , ) = 0 in K ∗ ( C ∗ r (Γ)) . We conclude that λ S an ( W, Ξ , f ) − λ S an ( W, Ξ , f ) = j X (0) = 0. (cid:3) Motivated by Lemma 2.4, we sometimes write λ S an ( W, ξ, f ) for λ S an ( W, Ξ , f ) forsome choice of decoration ( W, Ξ , f ). Theorem 2.5.
The mapping λ S an : S geo ∗ ( X ; L X ) → K ∗ ( D ∗ ( ˜ X ) Γ ) is a well definedisomorphism. Furthermore, it fits into the following commutative diagram: (3) K geo ∗ ( pt ; C ∗ r (Γ)) r −−−−→ S geo ∗ ( X ; L X ) δ −−−−→ K geo ∗ +1 ( X ) λ an y λ S an y λ an y K ∗ ( C ∗ r (Γ)) j X −−−−→ K ∗ ( D ∗ ( ˜ X ) Γ ) q X −−−−→ K ∗ ( D ∗ ( ˜ X ) Γ /C ∗ ( ˜ X ) Γ ) Remark 2.6.
The difficult part of the proof is showing that λ S an is well defined;the proof of that fact will occupy Subsection 2.2 and 2.4. Assuming that λ S an iswell defined, the left part of the diagram (3) commutes since λ S an ( r ( M, E C ∗ r (Γ) )) = j X ind AS ( M, E C ∗ r (Γ) ) = j X λ an ( M, E C ∗ r (Γ) ) . That the right hand side of the diagram commutes follows from Theorem 1.13,which implies that q X ( λ S an ( W, Ξ , f )) = q X ρ ( W, Ξ , f ) = f ∗ ([ M ] ∩ [ E C ]) − f ∗ ([ M ] ∩ [ E ′ C ])= λ an ( M, E C , f ) − λ an ( M, E ′ C , f ) = λ an ( δ ( W, ξ, f )) . The fact that λ S an is an isomorphism follows from the five lemma once λ S an is welldefined. In [40, 41], the delocalized APS-theorem is used to prove commutativity ofdiagrams of mappings into the analytic surgery exact sequence (see Theorem 1.3).We only use the delocalized APS-theorem to prove that λ S an respects bordism. Remark 2.7.
We will focus on the even-dimensional case, ∗ = 0. The odd-dimensional case is proved analogously. An alternative route to the isomorphism λ S an : S geo ( X ; L X ) → K ( D ∗ ( ˜ X ) Γ ) is as follows. View ˜ X × R as a Γ × Z -presentedspace with ˜ X × R / Γ × Z = X × S . One shows that there is a commutative diagramwith exact rows −−−−−→ S geo ( X, L X ) −−−−−→ S geo ( X × S , L X × S ) −−−−−→ S geo ( X, L X ) −−−−−→ λ S an y λ S an y −−−−−→ K ( D ∗ ( ˜ X ) Γ ) −−−−−→ K ( D ∗ ( ˜ X × R ) Γ × Z ) −−−−−→ K ( D ∗ ( ˜ X ) Γ ) −−−−−→ , and then defines λ S an : S geo ( X ; L X ) → K ( D ∗ ( ˜ X ) Γ ) from this diagram. We notethat the map S geo ( X, L X ) → S geo ( X × S , L X × S ) is given by product with [ S ] ∈ K ( S ) using S = B Z and [11, Section 4], while the map S geo ( X × S , L X × S ) →S geo ( X, L X ) is the forgetful one. The maps in the lower row are more intricate andthe reconciliation of this definition of λ S an : S geo ( X ; L X ) → K ( D ∗ ( ˜ X ) Γ ) with thedefinition using higher APS-theory would follow from an explicit product formulain analytic surgery theory. Proof that λ S an respects bordism. We must show that λ S an is well-defined;in this subsection the bordism relation is considered. As mentioned, the delocalizedAPS-theorem plays a key role. The precise relationship between the delocalizedAPS-theorem and bordisms in S geo ∗ ( X, L X ) is the content of the next result. Lemma 2.8.
Assume that ( W , ξ , f ) is a cycle for S geo ( X ; L X ) , with ξ = ( E C ∗ r (Γ) , E ′ C ∗ r (Γ) , E C , E ′ C , α ) . Assume that • f extends to a continuous mapping g : W → X ; • there exists vector bundles F C , F C → W extending E C , E ′ C → ∂W ; • α extends to an isomorphism over W : β : E C ∗ r (Γ) ⊕ F ′ C ⊗ g ∗ L X ∼ = E C ∗ r (Γ) ⊕ F C ⊗ g ∗ L X . Then λ S an ( W , ξ , f ) = 0 .Proof. Using Lemma 2.4, the general result follows upon proving the vanishing ina specific decoration. The decoration of ξ will be chosen in the following manner.We first choose Dirac operators ( D W E , D W E ′ , D ∂W E , D ∂W E ′ ). Since the vector bundles E C and E ′ C extend, we can find Dirac operators D W F and D W F ′ on S W ⊗ F C and S W ⊗ F ′ C of product type near ∂W with boundary operators D ∂W E and respectively D ∂W E ′ . As ( ∂W , E C ⊗ f ∗ L X ) = ∂ ( W , F C ⊗ g ∗ L X ) and ( ∂W , E ′ C ⊗ f ∗ L X ) = ∂ ( W , F ′ C ⊗ g ∗ L X ) it follows from the bordism invariance of the index thatind C ∗ r (Γ) ( D ∂W E ⊗ f ∗ L ) = ind C ∗ r (Γ) ( D ∂W E ′ ⊗ f ∗ L ) = 0 . We conclude that there exists smoothing operators A ∈ Ψ −∞ C ∗ r (Γ) ( ∂W , E C ⊗ f ∗ L X )and A ′ ∈ Ψ −∞ C ∗ r (Γ) ( ∂W , E ′ C ⊗ f ∗ L X ) with D ∂W E ⊗ f ∗ L + A and D ∂W E ′ ⊗ f ∗ L + A ′ invertible . We choose a decoration of the formΞ = ( ξ , ( D W E , D W E ′ , D ∂W E , D ∂W E ′ ) , ( A E , A E ′ , A ˙ ∪ − A ′ )) . Consider the even-dimensional spin c -manifold Z = W ∪ ∂W ∂W × [0 , ∪ ∂W − W . We define the C ∗ r (Γ)-bundle F C ∗ r (Γ) → Z by F C ∗ r (Γ) = ( E C ∗ r (Γ) ⊕ F ′ C ⊗ g ∗ L X ) ∪ α ( E ′ C ∗ r (Γ) ⊕ F C ⊗ g ∗ L X ) , where we glue over the cylinder using the isomorphism α . Since α extends to theisomorphism β on W ,( Z, F C ∗ r (Γ) ) ∼ = ∂ ( W × [0 , , ( E C ∗ r (Γ) ⊕ F ′ C ⊗ g ∗ L X ) × [0 , . Here W × [0 ,
1] forms a smooth manifold after “straightening the angle”, see forexample [43, Appendix A]. We consider the Dirac operator D Z F given by gluingtogether D W E ⊕ D W F ′ ⊗ g ∗ L on W with − D W E ′ ⊕ − D W F ⊗ g ∗ L on − W via linearlyinterpolating the boundary operators over the cylinder using the isomorphism α . EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 15
We can apply Theorem 1.8 to the closed manifold Z cut into several parts bythe hyper surface ∂W × { , } ; we obtain the identityind AS ( Z, F C ∗ r (Γ) ) = ind AP S ( W , Ξ , f )+ ind AP S ( D W F ′ ⊗ g ∗ L , A ′ ) − ind AP S ( D W F ⊗ g ∗ L , A ) . On the other hand, bordism invariance of the index implies that the left hand sidevanishes. Since ρ ( D ∂W E ˙ ∪ − D ∂W E ′ , A ˙ ∪ − A ′ ) = ρ ( D ∂W E , A ) − ρ ( D ∂W E ′ , A ′ ) , it follows that λ S an ( W , ξ , f ) = g ∗ j W ind AP S ( D W F ⊗ g ∗ L , A ) − f ∗ ρ ( D ∂W E , A ) − g ∗ j W ind AP S ( D W F ′ ⊗ g ∗ L , A ) + f ∗ ρ ( D ∂W E ′ , A ) . (4)The identity λ S an ( W , ξ , f ) = 0 follows from (4) and the delocalized APS-theorem(see Theorem 1.15, cf. Remark 1.16) which implies that g ∗ j W ind AP S ( D W F ⊗ g ∗ L , A ) − f ∗ ρ ( D ∂W E , A )= g ∗ j W ind AP S ( D W F ′ ⊗ g ∗ L , A ) − f ∗ ρ ( D ∂W E ′ , A ) = 0 . (cid:3) Lemma 2.9.
Let x i = ( W i , ξ i , f i ) , with ξ i = ( E C ∗ r (Γ) ,i , E ′ C ∗ r (Γ) ,i , E C ,i , E ′ C ,i , α i ) , becycles for i = 1 , . Assume that ∂W i = ( − i Y , for some spin c -manifold Y onwhich f = f , and ( E C ∗ r (Γ) , , E ′ C ∗ r (Γ) , , E C , , E ′ C , , α ) | Y = ( E C ∗ r (Γ) , , E ′ C ∗ r (Γ) , , E C , , E ′ C , , α ) | Y , as relative cycles for assembly on the pair of spaces ( Y, Y ) . Then λ S an ( x ) + λ S an ( x ) = j X (cid:16) ind AS ( Z, [ F C ∗ r (Γ) ] − [ F ′ C ∗ r (Γ) ]) (cid:17) , where Z := W ∪ Y W and F C ∗ r (Γ) , F ′ C ∗ r (Γ) → Z are the C ∗ r (Γ) -bundles obtained fromgluing together E ,C ∗ r (Γ) respectively E ′ ,C ∗ r (Γ) with E ,C ∗ r (Γ) respectively E ′ ,C ∗ r (Γ) .Proof. To simplify notation, set E := E C , = E C , , E ′ := E ′ C , = E ′ C , and f := f = f . We choose decorations of the following formΞ i = ( ξ i , ( D W i E i , D W i E ′ i , ( − i D YE , ( − i D YE ′ ) , ( A E i , A E ′ i , ( − i A i )) , where D W E = D W E near Y and D W E ′ = D W E ′ near Y . We assume that the decora-tions are chosen so that A E = A E , A E ′ = A E ′ and A = A .The bundles F C ∗ r (Γ) and F ′ C ∗ r (Γ) can be equipped with Dirac operators D Z F and D Z F ′ obtained by gluing D W E respectively D W E ′ with D W E respectively D W E ′ . Itfollows from Theorem 1.8 thatind AP S ( D W E , A E ) + ind AP S ( D W E , A E ) = ind( D Z F ) = ind AS ( Z, F ) , ind AP S ( D W E ′ , A E ′ ) + ind AP S ( D W E ′ , A E ′ ) = ind( D Z F ′ ) = ind AS ( Z, F ′ ) . In a similar fashion,ind
AP S ( D − Y × [0 , , A Ξ ) + ind AP S ( D Y × [0 , , A Ξ )= ind AS ( Y × S , ( E ⊕ E ′ ⊗ f ∗ L X ) × S ) = 0 , where the last identity follows from bordism invariance of the index. From all thesecomputations, the result follows upon noting that ρ ( W , Ξ , f )+ ρ ( W , Ξ , f )= f ∗ [ ρ ( D E ˙ ∪ − D E ′ , A )] + f ∗ [ ρ ( − D E ˙ ∪ D E ′ , − A )] = 0 . (cid:3) Proposition 2.10. If ( W, ξ, f ) ∼ bor then λ S an ( W, ξ, f ) = 0 in K ( D ∗ ( ˜ X ) Γ ) .Proof. By definition, x = ( W, ξ, f ) ∼ bor x =( W , ξ , f ) as in Lemma 2.8 such that ∂W = ∂W and W ∪ ∂W W being theboundary of a spin c -manifold Z , with the bundle data satisfying ξ | ∂W = ξ | ∂W and that ξ C ∗ r (Γ) ∪ ∂W ξ ,C ∗ r (Γ) belongs to the image of the restriction mapping K ( Z ; C ∗ r (Γ)) → K ( ∂Z ; C ∗ r (Γ)). Hence, the bordism invariance of the index,Lemmas 2.8 and 2.9 imply that 0 = λ S an ( x ) + λ S an ( x ) = λ S an ( x ). (cid:3) An intermezzo on vector bundle modification.
Recall that if W is amanifold, possibly with boundary, and V → W is a spin c -bundle of even rank, thevector bundle modification of W is the total space of the sphere bundle W V := S ( V ⊕ R ), where 1 R → W denotes the trivial real line bundle. In regard to thisprocess in general, we refer the reader to [2, 3, 5, 23]; the references [5, 23] are themost relevant in our context.To simplify the discussion, we assume that the rank of V is constant and let2 k = rk( V ). If P V → W denotes the principal Spin c (2 k )-bundle of spin c -frameson V , one can identify W V = P V × Spin c (2 k ) S k . On S k there is a distinguished Spin c (2 k )-equivariant vector bundle Q k → S k called the Bott bundle. The associ-ated vector bundle Q V := P V × Spin c (2 k ) Q k → W V is called the Bott bundle of V .For a C ∗ -algebra B and a B -bundle E B → W there is an associated vector bundlemodified B -bundle E VB := π ∗ V ( E B ) ⊗ Q V → W V , where π V : W V → W denotes the projection.Furthermore, there is a Spin c (2 k )-equivariant spin c -structure given by a Cliffordbundle S k → S k such that S W V = π ∗ V S W ˆ ⊗ S W V /W as Clifford bundles, where S W V /W := P V × Spin c (2 k ) S k . By construction, we can identify C ∞ ( W V , S W V ⊗ E VB ) = [ C ∞ ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) ⊗ ( π ∗ V E B ⊠ Q k ))] Spin c (2 k ) , where [ V ] Spin c (2 k ) denotes the Spin c (2 k )-invariant part of a Spin c (2 k )-representation V . In fact, after choosing a suitable Spin c (2 k )-invariant Laplacian, we can carryout the same identification for the Sobolev C ∗ r (Γ)-modules for any s ∈ N :(5) H s ( W V , S W V ⊗ E VB ) = (cid:2) H s ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) ⊗ ( π ∗ V E B ⊠ Q k )) (cid:3) Spin c (2 k ) . On the Clifford bundle S k ⊗ Q k there is a Spin c (2 k )-equivariant spin c -Diracoperator D Q such that the even part D + Q has a one-dimensional kernel, giving thetrivial Spin c (2 k )-representation, and ker D − Q = 0. We let e Q ∈ Ψ −∞ ( S k , S k ⊗ Q k )denote the projection onto this kernel, it is smoothing because of elliptic regularity.Since ker D + Q is Spin c (2 k )-invariant, so is e Q . For details, see [5, Proposition 3.11].The next definition and Propositions 2.12, 2.14, 2.16, and 2.17 should be com-pared with [5, Propositions 3.6 and 3.11]. EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 17
Definition 2.11 (Vector bundle modification of Dirac operators) . Let B be a C ∗ -algebra and E B → W a B -bundle over a spin c -manifold W equipped with a twistedspin c -Dirac operator D E . There is an associated vector bundle modified twistedspin c -Dirac operator on E VB → W V given by D V E := (cid:0) D E ˆ ⊗ ⊗ D Q (cid:1) | C ∞ ( W V ,S WV ⊗E VB ) . One verifies in local coordinates that the vector bundle modification of Diracoperators produces a well defined Dirac operator. Using the projection e Q , wewill decompose the Sobolev space H s ( W V , S W V ⊗ E VB ). The proof of the nextproposition follows from the construction of e Q . Proposition 2.12.
The operator ⊠ e Q acting on C ∞ ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) + ⊗ ( π ∗ V E B ⊠ Q k )) restricted to the Spin c (2 k ) -invariant part: e VQ := (1 ⊠ e Q ) | C ∞ ( W V ,S WV ⊗E VB ) , defines a projection e VQ in any of the Sobolev C ∗ r (Γ) -modules H s ( W V , S W V ⊗ E VB ) satisfying [ D V E , e VQ ] = 0 on H s ( W V , S W V ⊗ E VB ) ∀ s ∈ N > . Definition 2.13.
We define the complemented B -Hilbert module: E sB := (1 − e VQ ) H s ( W V , S W V ⊗ E VB ) ⊆ H s ( W V , S W V ⊗ E VB ) . For simplicity, we set E B := E B . Proposition 2.14.
Let W be a spin c -manifold, V → W a spin c -vector bundle ofeven rank and E B → W a B -bundle. There is a C ∞ ( W ) -linear isomorphism of B -Hilbert modules e VQ H s ( W V , S W V ⊗ E VB ) ∼ = H s ( W, S W ⊗ E B ) , which is graded if W is even-dimensional. In particular, for s ∈ N , there is anorthogonal direct sum decomposition of B -Hilbert modules (6) H s ( W V , S W V ⊗ E VB ) ∼ = H s ( W, S W ⊗ E B ) ⊕ E sB . The decomposition (6) respects the left action of C ∞ ( W ) and is graded if W iseven-dimensional.Proof. The isomorphism e VQ H s ( W V , S W V ⊗ E VB ) ∼ = H s ( W, S W ⊗ E B ) is constructedas the composition of the following chain of (graded) isomorphisms: e VQ H s ( W V , S W V ⊗ E VB ) ∼ = ker (cid:16) (1 ⊗ D Q ) | H s ( W V ,S WV ⊗E VB ) (cid:17) ∼ = H s ( W, S W ⊗ E B ) ⊗ ker D Q ∼ = H s ( W, S W ⊗ E B ) . Equation (6) follows from the existence of this isomorphism since e VQ preservesthe Sobolev scale by Proposition 2.12. The decomposition (6) respects the left C ∞ ( W )-action since e VQ commutes with the left C ∞ ( W )-action. (cid:3) Remark 2.15.
It is clear that (6) respects the C ( W )-action for s = 0 and the Lip ( W )-action for s ≤ Proposition 2.16.
For any f ∈ H ( W V , S W V ⊗ E VB ) , (1) h ( D E ˆ ⊗ f, (1 ˆ ⊗ D Q ) f i L ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) ⊗ ( π ∗ V E B ⊠ Q k )) + h (1 ˆ ⊗ D Q ) f, ( D E ˆ ⊗ f i L ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) ⊗ ( π ∗ V E B ⊠ Q k )) = 0(2) The inequality h (1 ˆ ⊗ D Q ) f, (1 ˆ ⊗ D Q ) f i L ( W V ,S WV ⊗E VB ) ≥ c h (1 − e VQ ) f, (1 − e VQ ) f i E B holds in B for a c > only depending on k . We omit the proof of Proposition 2.16 as it reduces to well known computationsby localizing to a coordinate chart. The number c > D Q on S k above zero. The main tool when dealingwith vector bundle modifications will be the following Lemma which decomposesoperators into a “nice” part and a “bad” part. Lemma 2.17.
Following the notation of Proposition 2.14, we let D E denote atwisted spin c -Dirac operator on E B . The direct sum decomposition (6) is compatiblewith D V E in the sense that there is a densely defined regular (odd) operator D E : E B → E B with domain E B such that (7) D V E = D E ⊕ D E . on the dense subspace H ( W, S W ⊗ E B ) ⊕ E B = H ( W V , S W V ⊗ E VB ) ⊆ L ( W V , S W V ⊗ E VB ) . Moreover, the decomposition (7) also has the following property: for any f ∈ E B ⊆ H ( W V , S W V ⊗ E VB ) there exists a c (depending only on k ) such that (8) h D V E f, D V E f i L ( W V ,S WV ⊗E VB ) ≥ c h f, f i E B in B. Proof.
The decomposition (7) follows since e VQ respects the Sobolev scale and com-mutes with D V E ; a fact proved in Proposition 2.12. To prove (8), we note that wecan identify f ∈ E B ⊆ H ( W V , S + W V ⊗ E VB ) with a Spin c (2 k )-invariant section of(9) H ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) ⊗ ( π ∗ V E B ⊠ Q k ))on which D E ˆ ⊗ ⊗ D Q , by Proposition 2.16, anti commute in form sense.Again, by Proposition 2.16, we have that 1 ⊗ D Q ≥ c in the B -valued form senseon the space (1 − ⊗ e Q ) H ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) ⊗ ( π ∗ V E B ⊠ Q k )) . It follows from these observations that h D V E f, D V E f i L ( W V ,S − WV ⊗E VB ) = h ( D E ˆ ⊗ f, ( D E ˆ ⊗ f i L ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) ⊗ ( π ∗ V E B ⊠ Q k )) + h (1 ˆ ⊗ D Q ) f, (1 ˆ ⊗ D Q ) f i L ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) ⊗ ( π ∗ V E B ⊠ Q k )) ≥ h (1 ˆ ⊗ D Q ) f, (1 ˆ ⊗ D Q ) f i L ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) ⊗ ( π ∗ V E B ⊠ Q k )) ≥ c h f, f i L ( P V × S k , ( π ∗ V S W ˆ ⊠ S k ) ⊗ ( π ∗ V E B ⊠ Q k )) = c h f, f i E B (cid:3) EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 19
Definition 2.18 (Vector bundle modification of smoothing operators) . Let M be aclosed manifold, E B → M a B -bundle and A ∈ Ψ −∞ B ( M, E B ) a smoothing operator.The vector bundle modified smoothing operator is defined to be A V = ( A ˆ ⊗ e Q ) | L ( W V ,S WV /W ⊗E VB ) . It was shown in [12, Lemma 2.9] that the vector bundle modification of a trivi-alizing operator is again a trivializing operator. The proof of the next propositionfollows directly from the same method of proof as in Lemma 2.17.
Proposition 2.19.
Let W , E B → W and V → W be as in Lemma 2.17 with ∂W = ∅ . If A ∈ Ψ −∞ B ( W, E B ⊗ S W ) , then under the decomposition (6) , the smoothingoperator A V decomposes as A V = A ⊕ E B . Proposition 2.20.
Assume that W is an even-dimensional spin c -manifold andthat E B → W is a B -bundle. Let V → W be a spin c -vector bundle of even rank.For any a twisted spin c -Dirac operator D E on E B of product type near ∂W and anytrivializing operator A ∈ Ψ −∞ B ( ∂W, S ∂W ⊗ E B | ∂W ) , ind AP S ( D E , A ) = ind AP S ( D V E , A V ) in K ( C ∗ r (Γ)) . Proof.
Lemma 2.17 implies that D V, ∞E + A V ∞ can be realized as a direct sum of theoperators D V, ∞E + A V ∞ = ( D ∞E + A ∞ ) ⊕ D ∞ E , where D ∞ E := (1 − e V ∞ Q ) D W V ∞ E V . The operators D V, ∞E and D ∞E are regular and self-adjoint, therefore D ∞ E must also be regular and self-adjoint. It follows from Lemma2.17 (more precisely the inequality (8)) that D ∞ E is invertible; henceind AP S ( D V E , A V ) = ind C ∗ r (Γ) ( D ∞E + A ∞ ) + ind C ∗ r (Γ) D ∞ E = ind AP S ( D E , A ) . (cid:3) We now turn to closed manifolds and higher ρ -invariants. Our construction isbased on the [23, Section 8] (in particular, see the proof of [23, Theorem 8.18]). Wenote that in this case, we need an analog of Lemma 2.17 for the equivariant Diracoperators on L ( ˜ M ) for a covering Γ → ˜ M → M . On manifolds like ˜ M , we use theSobolev spaces coming from a suitable Γ-equivariant Laplacian type operator. Lemma 2.21.
Assume that M is a closed spin c -manifold, f : M → B Γ a continu-ous mapping, E → M a vector bundle, and D ME a Dirac operator on S M ⊗ E . Let V → M be a spin c -vector bundle of even rank. The projection e VQ ⊗ π id ℓ (Γ) respectsthe Sobolev scale H s ( ˜ M V , S ˜ M V ⊗ ˜ E V ) and decomposes (10) H s ( ˜ M V , S ˜ M V ⊗ ˜ E V ) = H s ( ˜ M , S ˜ M ⊗ ˜ E ) ⊕ E sL , in a (graded) C ∞ c ( ˜ M ) -linear way. Under the decomposition (10) , ˜ D M V E V = ˜ D ME ⊕ D E L , where D E L is a self-adjoint invertible operator with dom( D E L ) = E L . The proof follows the same lines as the one of Lemma 2.17. The fact that theprojection e VQ ⊗ π id ℓ (Γ) respects the Sobolev scale follows since it coincides withthe orthogonal projection onto the closed subspace H s ( ˜ M , S ˜ M ⊗ ˜ E ) ⊗ ker D Q ⊆ H s ( ˜ M V , S ˜ M V ⊗ ˜ E V ) . Let γ k denote the grading operator on S k ⊗ Q k and define an operator J Q bytaking J Q := iγ k D Q | D Q | − ∈ B ( L ( S k , S k ⊗ Q k )) on ker( D Q ) ⊥ and J Q | ker D Q = 0.Define J := (1 ˆ ⊗ J Q ) | E L ∈ B ( E L ) . Lemma 2.22.
The self-adjoint operator J ∈ B ( E L ) has the following properties (1) J = 1 (i.e., J is an involution); (2) J preserves the scale E sL ; (3) JD E L + D E L J = 0 on E L ; (4) J ∈ D ∗ ( ˜ M , E L ) Γ and commutes with the C ( ˜ M ) -action.Proof. It is clear that J = 1 because J Q = 1 − e Q . Moreover, J Q formally anticommutes with D Q , and 1 ˆ ⊗ J Q formally anti commutes with ˜ D EM ˆ ⊗
1; hence, J formally anitcommutes with D E L . It also follows that J preserves the scale E sL as it commutes with a suitable choice of Laplacian operator on ˜ M V coming from( ˜ D EM ) ⊗ ⊗ D Q . We can (by a density argument) conclude that JD E L + D E L J = 0 on E L .It remains to prove that J ∈ D ∗ ( ˜ M , E L ) Γ . Since e VQ ⊗ π id ℓ (Γ) commutes withthe C ∞ c ( ˜ M )-action on E L , it is clear that J commutes with the C ( ˜ M )-action andis pseudo-local with finite propagation; in fact, it has zero propagation. (cid:3) Lemma 2.23.
Suppose p ∈ D ∗ ( ˜ X ) Γ be a projection. Then, if there exists a self-adjoint involution J ∈ D ∗ ( ˜ X ) Γ such that • J commutes with the C ( ˜ X ) -action; • Jp + pJ = J ,then [ p ] = 0 in K ( D ∗ ( ˜ X ) Γ ) . Proof.
Recall that H X denotes the Γ-equivariant X -module used to define D ∗ ( ˜ X ) Γ .We set q = (1 + J ) /
2; it is a projection in D ∗ ( ˜ X, H X ) Γ . Consider the function Q ∈ C ([0 , , D ∗ ( ˜ X, H X ) Γ ) given by Q ( t ) = tJ + √ − t (2 p −
1) + 12 . Since J (2 p −
1) = − (2 p − J , the operator tJ + √ − t (2 p −
1) is a symmetryfor any t ; hence, Q ( t ) is projection valued. Furthermore, Q (0) = p and Q (1) = q .We conclude that [ p ] = [ q ] in K ( D ∗ ( ˜ X, H X ) Γ ). Since the conditions on J aresymmetric under J
7→ − J , the same argument applies to 1 − q = (1 − J ) / p ] = [1 − q ] in K ( D ∗ ( ˜ X, H X ) Γ ).After possibly tensoring with ℓ ( N ), we can assume that q H X or (1 − q ) H X is am-ple (i.e., that it is a Γ-equivariant X -module). For simplicity, we assume q H X satis-fies this assumption. Under the isomorphism K ( D ∗ ( ˜ X, H X ) Γ ) ∼ = K ( D ∗ ( ˜ X, q H X ) Γ )we can identify [ q ] with [1]. The result follows since [1] = 0 ∈ K ( D ∗ ( ˜ X, H X ) Γ ); afact that can be proved using a “swindle” argument. (cid:3) Lemma 2.24.
Let M be a closed spin c -manifold, f : M → X be a continuous map, V → M be a spin c -vector bundle of even rank, E → M be a complex vector bundle, D E be a twisted spin c -Dirac operator on E , and A ∈ Ψ −∞ C ∗ r (Γ) ( M, S M ⊗ E ⊗ f ∗ L X ) be a trivializing operator. Then, ρ ( D VE , A V ) = ρ ( D E , A ) in K ∗ ( D ∗ ( ˜ M ) Γ ) . EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 21
Proof.
It follows from Lemma 2.17 that χ [0 , ∞ ) ( ˜ D VE + ˜ A V ) = χ [0 , ∞ ) ( ˜ D E + ˜ A ) ⊕ χ [0 , ∞ ) ( D E L ) . Since χ [0 , ∞ ) ( ˜ D E + ˜ A ) ∈ D ∗ ( ˜ M , L ( ˜ M , S ˜ M ⊗ ˜ E )) Γ and χ [0 , ∞ ) ( ˜ D VE + ˜ A V ) ∈ D ∗ ( ˜ M , L ( ˜ M V , S ˜ M V ⊗ ˜ E V )) Γ , we can consider χ [0 , ∞ ) ( D E L ) ∈ D ∗ ( ˜ M , E L ) Γ . Furthermore, under suitable identi-fications, ρ ( D VE , A V ) = ρ ( D E , A ) + [ χ [0 , ∞ ) ( D E L )] . The result follows by noting that Lemmas 2.22 and 2.23 imply that [ χ [0 , ∞ ) ( D E L )] =0 in K ∗ ( D ∗ ( ˜ M ) Γ ). (cid:3) Proof that λ S an respects vector bundle modification. Finally, we turnto proving that λ S an respects vector bundle modification. In order to do so, we willmake use of a special choice of decoration for the vector bundle modified cycle. Definition 2.25 (Vector bundle modification of decorations) . Let (
W, ξ, f ) denotea cycle and Ξ = ( ξ, ( D E , D E ′ , D E , D E ′ ) , ( A E , A E ′ , A ))a decoration of ξ . For a spin c -vector bundle V → W of even rank, we define thefollowing decoration of ξ V :Ξ V := ( ξ, ( D V E , D V E ′ , D VE , D VE ′ ) , ( A E V , A E ′ V , A V )) , where (for simplicity) we have omitted the notation indicating restriction of thebundle V to ∂W .It follows from [12, Lemma 2.9] that Ξ V is a well defined decoration of ξ V . Proposition 2.26.
Let ( W, Ξ , f ) be a decorated cycle and V → W a spin c -vectorbundle of even rank. Then, ind AP S ( W, Ξ , f ) = ind AP S ( W V , Ξ V , f V ) .Proof. The proof follows by applying Proposition 2.20 to the terms involved in thedefinition of the APS-index of a decorated cycle (see Definition 2.2). (cid:3)
Proposition 2.27.
Let ( W, Ξ , f ) be a decorated cycle and V → W a spin c -vectorbundle of even rank. Then, ρ ( W, Ξ , f ) = ρ ( W V , Ξ V , f V ) .Proof. The proof follows from the identity ( ˜ D E ˙ ∪ ˜ D E ′ ) V = ˜ D VE ˙ ∪ ˜ D VE ′ and a directapplication of Lemma 2.24. (cid:3) Corollary 2.28.
The mapping λ S an : S geo ∗ ( X, L X ) → K ∗ ( D ∗ ( ˜ X ) Γ ) is well defined.Proof. The mapping λ S an trivially respects disjoint union/direct sum and vanisheson degenerate cycles. By Proposition 2.10, λ S an respects bordism. By combiningProposition 2.26 with Proposition 2.27, it follows that λ S an respects vector bundlemodification. (cid:3) Remark 2.29.
We remark that S geo ∗ ( X, L X ) = K ∗ ( D ∗ ( ˜ X ) Γ ) = 0 holds if and onlyif λ S an = 0. Such a statement is equivalent to an extended version of the delocalizedAPS-index theorem: λ S an ( W, Ξ , f ) = 0 ⇔ j X ind AP S ( W, Ξ , f ) = ρ ( W, Ξ , f ) . If Γ has torsion elements, the failure of the free assembly map to be an isomorphismis an obstruction to this extended version of the delocalized APS-index theorem.3.
Delocalized Chern characters
In this section, Chern characters on the geometric model for the analytic surgerygroup are considered. Following the modus operandi in this series of paper, a delo-calized Chern character should commute with a specified choice of Chern characteron K ∗ ( B Γ) and K ∗ ( C ∗ r (Γ)). As such, it will depend on the choice of a dense subal-gebra A ⊆ C ∗ r (Γ) as well as the choice of suitable homology theory of A . The targetgroup for the delocalized Chern character will depend heavily on these choices. In-deed, the delocalized Chern character (being a relative construction) depends onthe precise form of the Chern character on A at the level of cycles. As such, thechoice of homology theory plays an important role in its very definition. The choicewe make is a topological version of noncommutative de Rham homology. We follow[28]. This homology theory has previously been used in the context of higher indextheory in for instance [35, 36, 48]. Related constructions have been carried out in[1, 34, 37, 38].3.1. Noncommutative de Rham homology.
We begin with a rather long listof notation. Let A denote a unital algebra. The universal derivation on A isgiven by d A : Ω ( A ) = A : → Ω ( A ) := A ⊗ ( A / C A ), a d a = 1 ⊗ a . Fromthe universal derivation one constructs the universal Z -graded differential algebraof A as Ω ∗ ( A ) := L ∞ k =0 Ω k ( A ) where Ω k ( A ) := Ω ( A ) ⊗ A k = A ⊗ ( A / C A ) ⊗ C k for k >
0. The derivation on Ω ∗ ( A ) is defined from d A via the Leibniz rule,by an abuse of notation we denote the derivation on Ω ∗ ( A ) by d A . We defineΩ ab ∗ ( A ) := Ω ∗ ( A ) / [Ω ∗ ( A ) , Ω ∗ ( A )] where all commutators are graded. If A is aunital Fr´echet ∗ -algebra, after replacing all algebraic tensor products by projectivetensor products we arrive at the universal Z -graded differential Fr´echet algebraˆΩ ∗ ( A ) of A . We also define ˆΩ ab ∗ ( A ) := ˆΩ ∗ ( A ) / [ ˆΩ ∗ ( A ) , ˆΩ ∗ ( A )]. For details ontopological homology, see [17, 47].For an algebra A , we let ˜ A denote its unitalization. If A is a Fr´echet ∗ -algebra,then so is ˜ A . Definition 3.1.
The de Rham homology H dR ∗ ( A ) of an algebra A is defined as thehomology H ∗ (Ω ab ∗ ( ˜ A )). The topological de Rham homology ˆ H dR ∗ ( A ) of a Fr´echetalgebra A is defined as the topological homology of ˆΩ ab ∗ ( ˜ A ) (i.e., closed cyclesmodulo the closure of the exact cycles).We use the notation C ∗ ( A ) for the Hochschild complex, C λ ∗ ( A ) for the cycliccomplex and b for the boundary operator on both of these complexes. We use HH ∗ ( A ) and HC ∗ ( A ) to denote the Hochschild homology groups respectively thecyclic homology groups of A . For details, see [10, 28]. The construction of the S -operator on cyclic homology is found in for instance [10, Page 11]. Proposition 3.2 (Theorem 2.15 of [28]) . The map (11) Ω abk ( ˜ A ) ∋ a d a · · · d a k a ⊗ a ⊗ · · · a k ∈ C λk ( A ) / Im( b ) induces an isomorphism H dR ∗ ( A ) ∼ = Im( S : HC ∗ +2 ( A ) → HC ∗ ( A )) = ker( B : HC ∗ ( A ) → HH ∗ +1 ( A )) . EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 23
Proof.
Excision (of split short exact sequences) implies that the reduced cyclichomology HC ( ˜ A ) of ˜ A coincides with the cyclic homology of A . It follows from[28, Theorem 2.15] that the mapping (11) induces an isomorphism H dR ∗ ( A ) = H ∗ (Ω ab ∗ ( ˜ A )) ∼ = Im( ¯ S : HC ∗ +2 ( ˜ A ) → HC ∗ ( ˜ A )) . By naturality of the S -operator, the quotient mapping induces an isomorphismIm( ¯ S : HC ∗ +2 ( ˜ A ) → HC ∗ ( ˜ A )) ∼ = Im( S : HC ∗ +2 ( A ) → HC ∗ ( A )) proving theLemma. (cid:3) We will restrict our attention to Fr´echet ∗ -algebra completions A (Γ) of C [Γ]. Tosimplify notations, we write ˜ A (Γ) and ˜ C [Γ] for their respective unitalizations. Byfunctoriality, there is a differential graded map j Ω , A : Ω ab ∗ ( ˜ C [Γ]) → ˆΩ ab ∗ ( ˜ A (Γ)) . For an element g ∈ Γ, we let
There is an isomorphism b Γ : H
The relative complex of A (Γ) delocalized in < g > is defined tobe the homology of the mapping cone complexˆΩ rel ∗ ( A (Γ); < g > ) := ˆΩ ab ∗ ( ˜ A (Γ)) ⊕ ˆΩ
In [49], delocalized homology is used in the context of higher Atiyah-Patodi-Singer index theory. Beware that another definition is used; it is based ona quotient construction rather than a relative construction.
Proposition 3.6.
The delocalized homology groups fits into a long exact sequenceon homology: . . . → ˆ H dR ∗ ( A (Γ)) r dR −−→ ˆ H del,
If the induced mapping H
The main usage of delocalized homology will be to detect invariantson the analytic and geometric surgery groups. This is done by means of pairingthe Chern character, defined in the next section, with a cyclic cocycle τ on A (Γ)satisfying τ ( g , g , . . . , g k ) = 0 whenever g g · · · g k = e . Assume for a moment thatΓ has polynomial growth or is hyperbolic and let τ be a group cocycle which haspolynomial growth. As Proposition 3.8 shows, such a group cocycle τ will inducea cyclic cocycle of the type described above, and a mapping ˆ H del (Γ) → C . Thechoice of dense subalgebra and that of extending cocycles is in general a subtle one,addressed in for instance [9, 14, 35, 36, 42, 48].3.2. The delocalized Chern character of a surgery cycle with connection.
We now turn to defining the delocalized Chern character for a geometric surgerycycle. As in the case of the isomorphism to the analytic surgery group, the con-struction is carried out by a map that cycles to cycles, which depends on a choiceof geometric data. We then prove that it gives a well defined mapping from classesto classes. As before, X denotes a finite CW-complex. Definition 3.10 (Definitions 3 . .
13 of [12]) . Let x = ( W, ξ, f ) be a cycle for S geo ∗ ( X, A (Γ)) with ξ = ( E A (Γ) , E ′A (Γ) , E C , E ′ C , α ) . A connection for x is a collection ∇ ξ = ( ∇ W , ∇ E , ∇ E ′ , ∇ E , ∇ E ′ ) consisting of: EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 25 • a spin c -connection ∇ W on W being of product type near ∂W ; • hermitean A (Γ)-linear connections ∇ E and ∇ E ′ on E A (Γ) respectively E ′A (Γ) ; • hermitean connections ∇ E and ∇ E ′ on E C respectively E ′ C .We write a surgery cycle with connection as ( W, ξ, ∇ ξ , f ).We recall from [48, Section 1.2] that for any manifold W , we obtain bicomplexes C ∞ ( W, ∧ ∗ T ∗ W ⊗ ˆΩ ∗ ( ˜ A (Γ))) and C ∞ ( W, ∧ ∗ T ∗ W ⊗ ˆΩ ab ∗ ( ˜ A (Γ))) with total differentialbeing d W + d A . If E A (Γ) → W is a smooth A (Γ)-bundle, it extends via tensorproduct to a smooth ˜ A (Γ)-bundle ˜ E ˜ A (Γ) := E A (Γ) ⊗ A (Γ) ˜ A (Γ) → W .A connection ∇ E on E A (Γ) can be extended to a connection ∇ ˜ E ⊗ d A on the space C ∞ ( W, ∧ ∗ T ∗ W ⊗ ˜ E ˜ A (Γ) ⊗ ˜ A (Γ) ˆΩ ∗ ( ˜ A (Γ))). We recall from [48, Lemma 1.1] that theelementCh A (Γ) ( ∇ E ) := tr E (cid:2) exp( − ( ∇ ˜ E ⊗ d A ) )) (cid:3) ∈ C ∞ ( W, ∧ ∗ T ∗ W ⊗ ˆΩ ab ∗ ( ˜ A (Γ)))is a well defined form; it is closed with respect to the total differential d W + d A .Given two connections ∇ E and ∇ ′E on E → W , we let ¯ ∇ denote the connection on E × [0 , → W × [0 ,
1] obtained from linearly interpolating between ∇ E and ∇ ′E .Furthermore, we set Ch A (Γ) ( ∇ E , ∇ ′E ) := Ch A (Γ) ( ¯ ∇ ) . Definition 3.11.
Let (
W, ξ, ∇ ξ , f ) be a surgery cycle with connection. We canassociate the following forms:(1) The interior Chern form of ( W, ξ, ∇ ξ , f ) is defined to beCh int ( W, ξ, ∇ ξ , f ) := Ch A (Γ) ( ∇ E ) − Ch A (Γ) ( ∇ E ′ ) ∈ C ∞ ( W, ∧ ∗ T ∗ W ⊗ ˆΩ ab ∗ ( ˜ A (Γ))) . (2) The boundary Chern form of ( W, ξ, ∇ ξ , f ) is defined to beCh ∂ ( W, ξ, ∇ ξ , f ) := Ch A (Γ) ( ∇ E ⊗ ∇ f ∗ L A ) − Ch A (Γ) ( ∇ E ′ ⊗ ∇ f ∗ L A ) , as an element of C ∞ ( ∂W, ∧ ∗ T ∗ ∂W ⊗ ˆΩ ab ∗ ( ˜ A (Γ))), where ∇ f ∗ L A denotes theflat connection on the Mishchenko bundle f ∗ L A → ∂W .(3) The Chern-Simons term is defined viaCS( W, ξ, ∇ ξ , f ) := Z Ch A (Γ) ( ∇ E ⊕ ∇ E ′ ⊗ ∇ f ∗ L , α ∗ ( ∇ E ′ ⊕ ∇ E ⊗ ∇ f ∗ L )) , as an element of C ∞ ( ∂W, ∧ ∗ T ∗ ∂W ⊗ ˆΩ ab ∗ ( ˜ A (Γ))).The proof of the next result follows from [35, Section IV] and [36, Section 4.3]. Proposition 3.12. If E C → M is a vector bundle with hermitean connections ∇ E and ∇ ′ E , then the Chern form Ch A (Γ) ( ∇ E ⊗ ∇ f ∗ L ) is in the image of the injection C ∞ ( ∂W, ∧ ∗ T ∗ ∂W ⊗ ˆΩ
Using the notation of the previous paragraph and Definition 3.11,we have that d A (cid:18) Z W Ch int ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ W ) − Z ∂W CS(
W, ξ, ∇ ξ , f ) ∧ T d ( ∇ ∂W ) (cid:19) = j
Since Ch int ( W, ξ, ∇ ξ , f ) is d W + d A -closed, it follows from Stokes’s theoremthatd A Z W Ch int ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ W ) = Z W d W Ch int ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ W )= Z ∂W Ch int ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ ∂W ) . Let ¯ ∇ ξ denote the connection on ( E A | ∂W ⊕ E ′ C ⊗ f ∗ L ) × [0 , → ∂W × [0 , ∇ E ⊕ ∇ E ′ ⊗ ∇ f ∗ L and α ∗ ( ∇ E ′ ⊕∇ E ⊗ ∇ f ∗ L ). Partially integrating over [0 , Z ∂W Ch int ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ W ) = Z ∂W [Ch A (Γ) ( ∇ E ) − Ch A (Γ) ( ∇ E ′ )] ∧ T d ( ∇ ∂W )= Z ∂W [Ch A (Γ) ( ¯ ∇ ξ ) | t =1 − Ch A (Γ) ( ¯ ∇ ξ ) | t =0 ] ∧ T d ( ∇ ∂W )+ j
W, ξ, ∇ ξ , f )] ∧ T d ( ∇ ∂W ) . (cid:3) Definition 3.14.
The delocalized Chern cycle of a geometric surgery cycle withconnection is the closed chain of ˆΩ rel ∗ ( A (Γ); < e > ) given byCh del ( W, ξ, ∇ ξ , f ):= R W Ch int ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ W ) − R ∂W CS(
W, ξ, ∇ ξ , f ) ∧ T d ( ∇ ∂W ) R ∂W Ch ∂
The delocalized Chern character ch del ( W, ξ, ∇ ξ , f ) ∈ ˆ H del,
Proof.
Let ∇ ξ and ∇ ′ ξ be two choices of connections on the geometric surgery cycle( W, ξ, f ). Then,Ch del ( W, ξ, ∇ ξ , f ) − Ch del ( W, ξ, ∇ ′ ξ , f )= R W Ch int ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ W ) − R ∂W CS(
W, ξ, ∇ ξ , f ) ∧ T d ( ∇ ∂W ) R ∂W Ch ∂
W, ξ, ∇ ′ ξ , f ) ∧ T d ( ∇ ∂W ) R ∂W Ch ∂
Our next goalis to study the delocalized Chern character as a mapping from classes to classes;that is, prove that it respects the relations in the group and is compatible with theother Chern characters in the geometric analogue of the surgery exact sequence.We define the localized Chern characterch
The delocalized Chern character from Definition 3.14 induces awell defined mapping ch del : S geo ∗ ( X, A (Γ)) → ˆ H del,
Lemma 3.17.
Assuming that ch del : S geo ∗ ( X, A (Γ)) → ˆ H del,
The left part of the diagram commutes by construction, using Proposition3.12. The middle part of the diagram commutes because for a geometric cycle withconnection ( M, E A , ∇ E , ∇ M ) for K geo ∗ ( pt ; A (Γ)), it holds on the level of cycles that r dR (Ch( M, E A , ∇ E , ∇ M )) = (cid:18)R M Ch( ∇ E ) ∧ T d ( ∇ M )0 (cid:19) = Ch del ( r ( M, E A , ∇ E , ∇ M )) . Finally, the right part of the diagram also commutes on the level of cycles, becausefor a geometric surgery cycle with connection (
W, ξ, ∇ ξ , f ) it holds that δ dR (ch delX ( W, ξ, f )) = Z ∂W Ch ∂
Assume that ( W , ξ , f ) is a cycle as in Lemma 2.8. Let ∇ ξ =( ∇ W , ∇ E , ∇ E ′ , ∇ E , ∇ E ′ ) be a connection for ξ , ∇ F a connection on F C extending ∇ E and ∇ F ′ a connection on F ′ C extending ∇ E ′ . In the notation of Lemma 2.8,we have that Ch del ( W, ξ, ∇ ξ , f )= d rel
W, ξ, ∇ ξ , f ) ∧ T d ( ∇ ∂W ) R ∂W Ch ∂
Lemma 3.19.
The delocalized Chern character respects the bordism relation in S geo ∗ ( X ; A (Γ)) .Proof. Suppose that (
W, ξ, f ) ∼ bor S geo ∗ ( X ; A (Γ)). Hence, there is a cycle x = ( W , ξ , f ) as in Lemma 2.8 and Lemma 3.18 such that ∂W = ∂W and W ∪ ∂W W being the boundary of a spin c -manifold Z , with bundle data such that E ∪ ∂W E → ∂Z extends to a bundle F → Z and E ′ ∪ ∂W E ′ → ∂Z extends to abundle F ′ → Z . Choose connections ∇ F , ∇ F ′ , ∇ F , ∇ F ′ and ∇ Z of product typenear the relevant boundaries and use these connections to define a connection ∇ ξ on ( W, ξ, f ).It follows thatCh del ( W, ξ, ∇ ξ , f )= R ∂Z (Ch A (Γ) ( ∇ F ) − Ch A (Γ) ( ∇ F ′ )) ∧ T d ( ∇ ∂Z )+ R W (cid:0) Ch A (Γ) ( ∇ E ) − Ch A (Γ) ( ∇ E ′ ) (cid:1) ∧ T d ( ∇ W ) − R ∂W CS ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ ∂W ) R ∂W (cid:0) Ch
W, ξ, ∇ ξ , f ) bea cycle with connection ∇ ξ = ( ∇ W , ∇ E , ∇ E ′ , ∇ E , ∇ E ′ ) and V → W a spin c -vectorbundle of even rank. We define the vector bundle modified connection ∇ Vξ by ∇ Vξ := ( π ∗ ∇ W ⊗ ∇ W V /W , π ∗ ∇ E ⊗ ∇ Q , π ∗ ∇ E ′ ⊗ ∇ Q , π ∗ ∇ E ⊗ ∇ Q , π ∗ ∇ E ′ ⊗ ∇ Q ) . The vector bundle modification of the cycle with connection (
W, ξ, ∇ ξ , f ) is definedby ( W, ξ, ∇ ξ , f ) V := ( W V , ξ V , ∇ Vξ , f V ) . The following proposition follows directly from the proof of [28, Theorem 1.26].
Proposition 3.21.
Let ( W, ξ, ∇ ξ , f ) be a geometric cycle for surgery with connec-tion and V → W a spin c -vector bundle of even rank. Then Z W V Ch int ( W, ξ, ∇ ξ , f ) V ∧ T d ( ∇ VW ) = Z W Ch int ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ W ); Z ∂W V CS
Smooth sub-algebras and the delocalized Chern character.
Supposethat A (Γ) ⊆ C ∗ r (Γ) is a Fr´echet ∗ -sub algebra. Following [11], the geometric surgerygroup S geo ∗ ( B Γ; A (Γ)) can be defined and it fits into a geometric surgery exactsequence with K geo ∗ ( B Γ) and K geo ∗ ( pt ; A (Γ)). By naturality, there is a commutingdiagram with exact rows: . . . δ −−−−−−→ K geo ∗ ( B Γ) µ −−−−−−→ K geo ∗ ( pt ; A (Γ)) r −−−−−−→ S geo ∗ ( B Γ; A (Γ)) δ −−−−−−→ K geo ∗− ( B Γ) −−−−−−→ . . . = y y y = y . . . δ −−−−−−→ K geo ∗ ( B Γ) µ −−−−−−→ K geo ∗ ( pt ; C ∗ r (Γ)) r −−−−−−→ S geo ∗ ( B Γ; C ∗ r (Γ)) δ −−−−−−→ K geo ∗− ( B Γ) −−−−−−→ . . ., where the vertical arrows are induced by the inclusion A (Γ) ⊆ C ∗ r (Γ). If the in-duced mapping K geo ∗ ( pt ; A (Γ)) → K geo ∗ ( pt ; C ∗ r (Γ)) is an isomorphism; we call sucha subalgebra a K -smooth algebra. Examples of K -smooth algebra include densesmooth subalgebras and Banach algebra completions A (Γ) such that K ∗ ( A (Γ)) → K ∗ ( C ∗ r (Γ)) is an isomorphism, see [11, Appendix]. It follows from the five Lemmathat for a K -smooth subalgebra A (Γ) ⊆ C ∗ r (Γ), the inclusion mapping induces anisomorphism S geo ∗ ( B Γ; A (Γ)) ∼ = S geo ∗ ( B Γ; C ∗ r (Γ)); which after composing with λ − an produces a delocalized Chern character on the analytic surgery group.Assume that Γ is a group of polynomial growth or hyperbolic. We considerthe Schwartz algebra S (Γ) which since Γ has property ( RD ) (see [15] for more onProperty ( RD )) is a dense smooth subalgebra of C ∗ r (Γ). By the argument above,there is a delocalized Chern character e ch del : S geo ∗ ( B Γ; C ∗ r (Γ)) → ˆ H del ∗ (Γ) . The reader is directed to Proposition 3.8 for notation.Recalling the notations of Subsection 2.1, we let ( W, Ξ , f ) be a decorated cyclefor S geo ∗ ( X ; S (Γ)). We can assume that the choice of Dirac operators contained inΞ is defined from a choice of connections ∇ ξ on ξ and also that all the trivializ-ing operators are induced from trivializing operators with coefficients in S (Γ), see[33, Proposition 2.10]. Following [36, 39] we define the delocalized ρ -invariant of( W, Ξ , f ) byˆ ρ ( W, Ξ , f ) : = ˆ ρ ( D E ⊗ f ∗ L ˙ ∪ D E ′ ⊗ f ∗ L + A )= ˆ ρ ( D ∂W E + A E ) + ˆ ρ ( − D ∂W E ′ − A E ′ ) + ˆ ρ ( ¯ D ∂ Ξ + A Ξ ) ∈ ˆ H del ∗ (Γ) . EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 31
Here ˆ ρ denotes the delocalized ρ -invariant of a Dirac operator perturbed by asmoothing operator, see more in [36, Section 4.6] and [48]. It is the part of thehigher η -invariant belonging to the range of the projection mapping π del : ˆ H dR ∗ ( S (Γ)) ∼ = H ∗ ( B Γ) ⊗ HC ∗ +2 ( C ) ⊕ ˆ H del ∗ (Γ) → ˆ H del ∗ (Γ) . Even though the higher η -invariant is not closed, its delocalized part is, givinga well defined delocalized ρ -invariant. It follows from [33], that ind AP S ( W, Ξ , f )can be constructed as an element of K ∗ ( S (Γ)) and as such has a Chern characterch S (Γ) (ind AP S ( W, Ξ , f )) ∈ ˆ H dR ∗ ( S (Γ)). The following Proposition is an immediateconsequence of Wahl’s higher Atiyah-Patodi-Singer index theorem [48, Theorem9.4]. Proposition 3.22.
Let Γ be a discrete group of polynomial growth or hyperbolic.Then ˆ ρ ( W, Ξ , f ) satisfies that π del (cid:0) ch S (Γ) (ind AP S ( W, Ξ , f )) (cid:1) − ˆ ρ ( W, Ξ , f ) = e ch del ( W, ξ, f ) . We return to the case of a general group Γ and consider two unitary representa-tions σ , σ : Γ → U ( k ). Assume that σ and σ extend continuously to a Fr´echetalgebra completion A (Γ) of C [Γ]. Two such representations give rise to a relative η -invariant ρ σ ,σ : S geo ( B Γ; A (Γ)) → R , see more in [12]. This relative η -invariantfactorizes over the construction of Higson-Roe [23], see [12, Section 3]. On theother hand, σ and σ define a continuous cyclic 0-cocycle τ σ ,σ on A (Γ) suchthat τ σ ,σ (1) = 0. By Remark 3.9, we obtain a pairing with delocalized de Rhamhomology ˆ H del,
7→ h τ σ ,σ , x i ∈ C .The mapping h τ σ ,σ , ·i : ˆ H del,
Let A (Γ) be a Fr´echet algebra completion of C [Γ] and σ , σ :Γ → U ( k ) two unitary representations continuous in the topology of A (Γ) . Thefollowing identity holds for any w ∈ S geo ( B Γ; A (Γ)) : ρ σ ,σ ( w ) = h τ σ ,σ , ch del ( w ) i . Proof.
Let (
W, ξ, ∇ ξ , f ) be a cycle for S geo ( B Γ; A (Γ)) representing w . By thediscussion above, h τ σ ,σ , ch del ( w ) i = ω ◦ ( σ − σ ) ∗ (cid:18) Z W Ch int ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ W ) − Z ∂W CS(
W, ξ, ∇ ξ , f ) ∧ T d ( ∇ ∂W ) (cid:19) . Following the notation in [12, Section 3], we define the vector bundles E i := L A (Γ) ⊗ σ i C k , G i := E A (Γ) ⊗ σ i C k and G ′ i := E ′A (Γ) ⊗ σ i C k . We let ∇ G i denote the connection constructed from ∇ E and ∇ G ′ i the connectionconstructed from ∇ E ′ . Similarly, ∇ E,i and ∇ E ′ ,i are constructed from twisting ∇ E and ∇ E ′ , respectively, by the flat connection on f ∗ E i . We define ¯ ∇ i as theconnection on ( G i ⊕ E ′ C ⊗ f ∗ E i ) × [0 , → ∂W × [0 ,
1] obtained from interpo-lating between ∇ G i ⊕ ∇ E ′ ,i and α ∗ ( ∇ G ′ i ⊕ ∇ E,i ). In terms of the connections( ∇ G , ∇ G , ∇ G ′ , ∇ G ′ , ¯ ∇ , ¯ ∇ ), we can write h τ σ ,σ , ch del ( w ) i as ω ◦ ( σ − σ ) ∗ (cid:18) Z W Ch int ( W, ξ, ∇ ξ , f ) ∧ T d ( ∇ W ) − Z ∂W CS(
W, ξ, ∇ ξ , f ) ∧ T d ( ∇ ∂W ) (cid:19) = Z W (cid:18) ch( ∇ G ) − ch( ∇ G ) (cid:19) ∧ T d ( ∇ W ) − Z W (cid:18) ch( ∇ G ′ ) − ch( ∇ G ′ ) (cid:19) ∧ T d ( ∇ W )+ Z ∂W × [0 , (cid:0) ch( ¯ ∇ ) − ch( ¯ ∇ ) (cid:1) T d ( ∇ ∂W ) = ρ σ ,σ ( w ) , where we in the last equality use the definition of ρ σ ,σ (see [12, Definitions 3.1and 3.2], and also [12, Remark 3.3]). (cid:3) Remark 3.24.
It would be interesting to extend the content of Proposition 3.23to higher cocycles. The mere formulation for higher cocycles is analytically moresubtle, see Remark 3.9. In fact, already the problem of extending unitary rep-resentations is analytically subtle. If a finite dimensional unitary representation σ extends to a dense holomorphically closed ∗ -subalgebra A (Γ) ⊆ C ∗ r (Γ), then σ automatically extends to C ∗ r (Γ). In particular, if Γ has property (T), this causessubstantial difficulties for proving rigidity results for relative η -invariants, see [12,Section 5].4. Mappings from Stolz’ positive scalar curvature exact sequence
The positive scalar curvature exact sequence of Stolz describes the relationshipbetween positive scalar curvature metrics on spin manifolds with fundamental groupΓ and the spin bordism class of the manifolds. For details regarding the backgroundto this section, we refer to [40, 45, 46].Let Γ be a finitely generated discrete group and X be a finite CW-complex whichis also a closed subspace of B Γ; also let n ∈ N . We write Ω Spinn ( X ) for the spinbordism group of X . That is, Ω Spinn ( X ) consists of equivalence classes of pairs( M, f : M → X ), where M is a closed n -dimensional spin-manifold (where theequivalence relation is spin-bordism); addition is defined using disjoint union.The bordism group Pos Spinn ( X ) of spin manifolds over X with a positive scalarcurvature metric is defined as follows: a cycle for Pos Spinn ( X ) is a triple ( M, f : M → X, g ) consisting of a cycle (
M, f ) for Ω
Spinn ( X ) and a metric g on M ofpositive scalar curvature. If ( W, f : W → X, g ) is as in a cycle for Pos
Spinn ( X ),but W has boundary, then we say that ( ∂W, f | ∂W , g | ∂W ) ∼ bor
0. The equivalenceclasses of cycles under the associated bordism relation leads to the abelian group,Pos
Spinn ( X ); again, addition is defined via disjoint union.The forgetful mapping ( M, f, g ) ( M, f ) gives a well defined morphism ofabelian groups Pos
Spinn ( X ) → Ω Spinn ( X ) . EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 33
This mapping is of relevance for positive scalar curvature metrics because of resultsof Gromov-Lawson-Rosenberg implying that a closed spin manifold M of dimension n ≥
5, with classifying mapping f : M → Bπ ( M ), admits a positive scalarcurvature metric if and only if [ M, f ] ∈ im(Pos Spinn ( Bπ ( M )) → Ω Spinn ( Bπ ( M ))),see more in for instance [46, Theorem 3.5].The final ingredient in Stolz’ positive scalar curvature exact sequence is a relativegroup R Spinn ( X ) constructed from the mapping Pos Spinn ( X ) → Ω Spinn ( X ). A cyclefor R Spinn ( X ) is a triple ( W, f : W → X, g ) where W is an n -dimensional spinmanifold with boundary and g is a metric on ∂W with positive scalar curvature.The bordism relation in R Spinn ( X ) is constructed similarly to any relative group:a cycle for R Spinn ( X ) with boundary is a triple (( Z, W ) , f : Z → X, g ) where Z isan n + 1-dimensional spin manifold with boundary, W ⊆ ∂Z is a regular compactdomain and g is a metric on ∂Z \ W ◦ with positive scalar curvature. The bordismrelation is defined in the usual manner; in particular, the reader should note that ∂ (( Z, W ) , f : Z → X, g ) := (
W, f | W , g ∂W )There are by construction well defined morphismsΩ Spinn ( X ) → R Spinn ( X ) , ( M, f ) ( M, f, ∅ ) , and R Spinn +1 ( X ) → Pos
Spinn ( X ) , ( W, f, g ) ( ∂W, f | ∂W , g ) . The positive scalar curvature exact sequence of Stolz is the long exact sequence . . . → Pos
Spinn ( X ) → Ω Spinn ( X ) → R Spinn ( X ) → Pos
Spinn − ( X ) → . . . We now turn to mapping the positive scalar curvature sequence into the geomet-ric analog of the analytic surgery exact sequence. Using the notations of [40], wedefine β geo : Ω Spinn ( X ) → K geon ( X ) , ( M, f ) ( M, M × C , f ) . The mapping β geo is clearly well defined, because it maps spin bordisms to bor-disms in K geon ( X ). In order to define the mappings from R Spinn ( X ) respectivelyPos Spinn ( X ), we need the following Proposition. Proposition 4.1.
Let ( W, ξ, h ) be a cycle for S geo ∗ ( X ; L X ) . Assume that • The boundary takes the form ∂W = M V for a spin manifold M and somespin c -vector bundle V → M of rank k ; • the mapping h is of the form h = f V for a continuous h : M → X ; • the relative K -theory cocycle has the form ξ = ( E C ∗ r (Γ) , E ′ C ∗ r (Γ) , Q V , C k , α ) where Q V → M V denotes the Bott bundle and E C ∗ r (Γ) , E ′ C ∗ r (Γ) → W are C ∗ r (Γ) -bundles.Then given a positive scalar curvature metric g on M with associated spin Diracoperator D g , there is a decoration of ξ that takes the form Ξ g = ( ξ, ( D E , D E ′ , D Vg , D ) , ( A E , A E ′ , ∪ A )) . After constructing a Dirac operator D B on the ball bundle M B := B ( V ⊕ R ) → M with boundary operator D , the class ν ( W, ξ, f, g ) := j X (ind AP S ( W, Ξ g , f )) − j X (cid:16) ind AP S ( M B , (( L k M B , C k , id) , ( D B , D ) , A ) , f ) (cid:17) ∈ K ∗ ( D ∗ ( ˜ X ) Γ ) , does not depend on the choice of Ξ g but only on the class [ W, ξ, f ] and the metric g on M .Proof. Existence of decorations of the specified form follows directly from the in-vertibility of spin Dirac operators defined from positive scalar curvature metrics andthe techniques of Subsection 2.3. By the delocalized APS-theorem (see Theorem1.15)(14) j X ind AP S ( M B , (( L k M B , C k , id) , ( D B , D ) , A ) , f ) = ρ ( D , A ) . Equation (14) and the construction of λ S an imply that λ S an ( W, ξ, f ) − ρ ( D Vg ,
0) = j X (ind AP S ( W, Ξ g , f )) − j X (cid:16) ind AP S ( M B , (( L k M B , C k , id) , ( D B , D ) , A ) , f ) (cid:17) . By Lemma 2.24, ρ ( D Vg ,
0) = ρ ( D g ,
0) (which clearly only depend on g ). The resultnow follows since λ S an ( W, ξ, f ) only depends on the class [
W, ξ, f ] (see Theorem2.5). (cid:3)
Our next goal is the definition of a mapind geo Γ : R Spinn ( X ) → K geon ( pt ; C ∗ r (Γ)) . To do so, further notation is required. Let (
W, f, g ) be a cycle for R Spinn ( B Γ).We define the relative K -theory cocycle for assembly ξ f := ( f ∗ L X , C , id f ∗ L X ) on( W, ∂W ). Since g is a positive scalar curvature metric on ∂W , we can find adecoration Ξ f,g of ξ f as in Proposition 4.1. We defineind geo Γ : ( W, f, g ) ( pt, ind AP S ( W, Ξ f,g , f )) . Proposition 4.2.
The map ind geo Γ : R Spinn ( X ) → K geon ( pt ; C ∗ r (Γ)) is well definedand fits into a commuting diagram Ω Spinn ( X ) −−−−→ R Spinn ( X ) β geo y ind geo Γ y K geo ∗ ( X ) µ −−−−→ K geo ∗ ( pt ; C ∗ r (Γ)) , Proof.
Recall the notation λ an : K geo ∗ ( X, B ) → KK ∗ ( C ( X ) , B ) for the analyticassembly mappings, these are isomorphisms for a finite CW-complex X . It followsfrom [40, Theorem 1.31] that the diagramsΩ Spinn ( X ) −−−−→ R Spinn ( X ) λ an ◦ β geo y λ an ◦ ind geo Γ y K ∗ ( X ) µ −−−−→ K ∗ ( C ∗ r (Γ)) , commutes. The proof is complete (since λ an is an isomorphism). (cid:3) EALIZING THE ANALYTIC SURGERY GROUP GEOMETRICALLY: PART III 35
Definition 4.3.
For (
W, ξ, f, g ) as in Proposition 4.1, we set ν geo ( W, ξ,f, g ) := r ∗ ( pt, ind AP S ( W, Ξ g , f )) − r ∗ ( pt, ind AP S ( M B , (( L k M B , C k , id) , ( D B , D ) , A ) , f )) ∈ S geo ∗ ( X, L X )where r ∗ denotes the map from the K -theory of C ∗ r (Γ) to S geo ∗ ( X, L X ) (see [11,Theorem 3.8]). Remark 4.4.
Despite the fact that ν geo ( W, ξ, f, g ) is defined from a class in K ∗ ( C ∗ r (Γ)) that depends on a choice of Ξ g , it follows from Proposition 4.1 andcommutativity of the left half of the diagram in Theorem 2.5 that ν geo ( W, ξ, f, g ) iswell defined. In fact, λ S an ν geo ( W, ξ, f, g ) = ν ( W, ξ, f, g ) (in the notation of Proposi-tion 4.1).
Definition 4.5.
Let ρ P,geo Γ : Pos Spinn ( X ) → S geon +1 ( X, L X ) denote the map definedvia ρ P,geo Γ : ( M, f, g ) ν geo ( W, ξ, h, g ) − [ W, ξ, h ] , where ( W, ξ, h ) is any cycle for S geon +1 ( X, L X ) such that on the level of cycles δ ( W, ξ, h ) = ( M V , Q V , f V ) − ( M V , k C , f V ) , for some spin c -vector bundle V → M of some even rank 2 k .It is not clear that ρ P,geo Γ is well-defined. First, we show that there exists a cyclein S geon +1 ( X, L X ) with the property required in the definition of this map. Lemma 4.6.
Let ( M, f, g ) be a cycle in Pos
Spinn ( X ) . Then, there exists ( W, ξ, h ) ∈S geon +1 ( X, L X ) such that (at the level of cycles in geometric K -homology) δ ( W, ξ, h ) = ( M V , Q V , f V ) − ( M V , k C , f V ) , for some spin c -vector bundle V → M of some even rank k .Proof. Given a cycle (
M, f, g ) for Pos
Spinn ( X ), Lichnerowicz formula implies that λ an ◦ µ geo [ M, C , f ] = ind C ∗ r (Γ) ( D g ) = 0, where µ geo : K geo ∗ ( X ) → K geo ∗ ( pt, C ∗ r (Γ))denotes the free assembly and λ an : K ∗ ( pt ; C ∗ r (Γ)) → K ∗ ( C ∗ r (Γ)) the isomorphismbetween realizations of the K -theory of C ∗ r (Γ); hence µ geo [ M, C , f ] = 0 as a classin K geo ∗ ( pt, C ∗ r (Γ)). Using the notion of “normal bordism” (see [11, 43]), thereexists a spin c -vector bundle V → M of even rank, say 2 k , a spin c -manifold W withboundary M V and a K -theory class x ∈ K ( W ; C ∗ r (Γ)) such that( ∂W, x | ∂W ) = ( M V , [ f ∗ L X ] V ) = ( M V , [ C ] V ⊗ ( f V ) ∗ L X ) , as cycles with K -theory coefficients. For simplicity, we set h := f V . Since [ C ] V =[ Q V ] − [2 k C ], by definition, there exists a relative K -theory cocycle for assemblyon ( W, ∂W, h ) of the form ( E C ∗ r (Γ) , E ′ C ∗ r (Γ) , Q V , C k , α ) with x = [ E C ∗ r (Γ) ] − [ E ′ C ∗ r (Γ) ].This completes the proof as ( W, ( E C ∗ r (Γ) , E ′ C ∗ r (Γ) , Q V , C k , α ) , h ) has the requiredproperty. (cid:3) Theorem 4.7.
The map ρ P,geo Γ is well defined. Furthermore, ρ P,geo Γ fits into acommuting diagram with exact rows: . . . −−−−−−→ Ω Spinn ( X ) −−−−−−→ R Spinn ( X ) −−−−−−→ Pos
Spinn − ( X ) −−−−−−→ Ω Spinn − ( X ) −−−−−−→ . . . βgeo y ind geo Γ y ρP,geo Γ y βgeo y . . . δ −−−−−−→ K geon ( X ) µ −−−−−−→ K geon ( pt ; C ∗ r (Γ)) r −−−−−−→ S geon ( X ; L X ) δ −−−−−−→ K geon − ( X ) −−−−−−→ . . ., where the top row is Stolz’ positive scalar curvature sequence and the bottom row isthe geometric model for the analytic surgery exact sequence.Proof. To show that ρ P,geo Γ is well defined, we consider λ S an ◦ ρ P,geo Γ ( M, f, g ) ∈ K n +1 ( D ∗ ( ˜ X ) Γ ). By the construction of λ S an , (14) and Lemma 2.24, λ S an ◦ ρ P,geo Γ ( M, f, g )= (cid:18) j X ind AP S ( W, Ξ g , f ) − ρ ( D , A ) | {z } ν ( W,ξ,f,g ) (cid:19) − (cid:18) j X ind AP S ( W, Ξ g , h ) − ρ ( W, Ξ g , h ) | {z } λ S an ( W,ξ,h ) (cid:19) = ρ ( D g , . It follows that, on the level of cycles, the mapping λ S an ◦ ρ P,geo Γ coincides withthe mapping ρ Γ : Pos Spinn ( X ) → K n +1 ( D ∗ ( ˜ X ) Γ ) constructed in [40, Section 1.2].Since λ S an is an isomorphism, ρ P,geo Γ is well defined because ρ Γ is well defined.Commutativity of the diagram follows from [40, Theorem 1.31] (which states that ρ Γ makes the analogous diagram involving the analytic surgery exact sequencecommutative). (cid:3) Theorem 4.8. If Γ has property ( RD ) , the following diagram commutes: Pos spinn ( X ) ˆ ρ Γ & & ▲▲▲▲▲▲▲▲▲▲ ρ P,geo Γ / / S geon ( X ; S (Γ)) ch del (RD) x x ♣♣♣♣♣♣♣♣♣♣♣ ˆ H deln (Γ) where the map ˆ ρ Γ is defined in [40, Section 2.3] (also see [31] ) and S (Γ) denotesthe Schwartz algebra. Outlook and mappings from the surgery exact sequence
The terminology “analytic surgery group” is due to Higson and Roe [20, 21, 22]who mapped the surgery exact sequence into the analytic surgery exact sequenceusing Poincar´e complexes. In recent work by Piazza and Schick [41] the sameprogram was carried out using techniques of higher index theory. The combinedoutcome of the results of [41], Theorem 2.5 and Theorem 3.16 can be summarizedin a diagram with exact rows: . . . −−−−−−→ N ∗ ( X ) −−−−−−→ L ∗ ( Z [Γ]) −−−−−−→ S ∗ ( X ) −−−−−−→ N ∗− ( X ) −−−−−−→ . . . β y indΓ y ρ y β y . . . −−−−−−→ K ∗ ( X ) µ −−−−−−→ K ∗ ( C ∗ r (Γ)) r −−−−−−→ K ∗ ( D ∗ ( ˜ X ) Γ ) δ −−−−−−→ K ∗− ( X ) −−−−−−→ . . . λan x ind AS x λ S an x λan x . . . −−−−−−→ K geo ∗ ( X ) µ −−−−−−→ K geo ∗ ( pt ; A (Γ)) r −−−−−−→ S geo ∗ ( X ; A ) δ −−−−−−→ K geo ∗− ( X ) −−−−−−→ . . . ch
The upper part of the diagram commutes up to factors of two. The upper partof the diagram are the maps from “surgery to analysis” in [41]. This summary of“mapping surgery to analysis” and “mapping geometry to analysis” indicates thatone can define explicit define analytic invariants from the surgery exact sequenceby “inverting” the isomorphism from “geometry to analysis”. This process shouldbe viewed in the context of the “Baum approach” to index theory discussed inthe introduction. In the Baum’s approach, one “inverts” the isomorphism fromgeometric to analytic K -homology.It is also possible to map the surgery exact sequence directly into the geometricmodel for the analytic surgery exact sequence along the lines of Section 4, but wewill not discuss it in detail. The reader shoud note that there are two issues thatmake it more difficult than the case of the positive scalar curvature exact sequence.Firstly, in the surgery exact sequence case, it is more natural to work with orientedmanifolds. As such, one should use a geometric model for the analytic surgerygroup (built on oriented manifolds) following [13, 23, 29]. Secondly, there is a tech-nical difference: on a positive scalar curvature manifold the boundary operator isalways invertible, while for the signature operator one would need to make use ofthe trivializing operator constructed by Hilsum-Skandalis (see [41]). The procedureof mapping “surgery to geometry” directly circumvents inverting the mapping from“geometry to analysis” by using higher Atiyah-Patodi-Singer index theory. At thecost of computing a higher index, this mapping provides a geometric methodologyto define invariants on the surgery exact sequence – avoiding the analytic difficultiesof delocalized ρ -invariants and the Higson-Roe’s analytic structure group. Acknowledgements . The authors wish to express their gratitude towards KarstenBohlen, Heath Emerson, Nigel Higson, Paolo Piazza, Thomas Schick and CharlotteWahl for discussions. They also thank the Courant Centre of G¨ottingen, the LeibnizUniversit¨at Hannover, the Graduiertenkolleg 1463 (
Analysis, Geometry and StringTheory ) and Universit´e Blaise Pascal Clermont-Ferrand for facilitating this collab-oration.
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