Stratified Surgery and K-Theory Invariants of the Signature Operator

STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR

PIERRE ALBIN AND PAOLO PIAZZA

Abstract. In work of Higson-Roe the fundamental role of the signature as a homotopy and bordism invariant for oriented manifolds is made manifest in how it and related secondary invariants define a natural transformation between the (Browder-Novikov-Sullivan-Wall) surgery exact sequence and a long exact sequence of C∗-algebra K-theory groups. In recent years the (higher) signature invariants have been extended from closed oriented manifolds to a class of stratified spaces known as L-spaces or Cheeger spaces. In this paper we show that secondary invariants, such as the ρ-class, also extend from closed manifolds to Cheeger spaces. We revisit a surgery exact sequence for stratified spaces originally introduced by Browder-Quinn and obtain a natural transformation analogous to that of Higson-Roe. We also discuss geometric applications.

1. Introduction The discovery by Milnor of smooth manifolds that are homemorphic to S7 but not diffeomorphic to it, a milestone of modern mathematics, gave rise to the development of methods for classifying smooth manifolds within a given homotopy class. (The undecidability of the word problem makes an unrestricted classification impossible.) A convenient formulation of these methods is the surgery exact sequence of Browder, Novikov, Sullivan, and Wall which, roughly speaking, relates the ‘structure set’ S(X) of homotopy equivalences to a given smooth oriented manifold X (with a bordism equivalence relation) with the set N(X) of degree one maps preserving normal bundle information, known as ‘normal invariants’, (also with a bordism equivalence relation) and an algebraically defined L-group depending only on Γ = π1X, the fundamental group of X,

(1.1) ... / Lm+1(ZΓ) / S(X) / N(X) / Lm(ZΓ) . (See below and, e.g., [Wal99,Ran02a,L¨uc02,CLM] for more on the surgery exact sequence.) In a series of papers Higson and Roe [HR05a, HR05b, HR05c] established the remarkable arXiv:1710.00934v1 [math.KT] 2 Oct 2017 result that there are natural maps out of the surgery sequence (1.1), into a long exact sequence of K-theory groups of certain C∗-algebras and that these maps make the resulting ∗ ∗ Γ ∗ Γ diagram commute. The C -algebras in question are C (XΓ) and D (XΓ) , obtained as the closures of the Γ-equivariant operators on the universal cover XΓ of X that satisfy a ﬁnite propagation property and, in addition, are respectively ‘locally compact’ or ‘pseudolocal’. The former C∗-algebra is an ideal in the latter so we have a short exact sequence ∗ Γ ∗ Γ ∗ Γ ∗ Γ 0 → C (XΓ) → D (XΓ) → D (XΓ) /C (XΓ) → 0 which gives rise to a long exact sequence in K-theory known as the analytic surgery sequence of Higson and Roe. Making use of the canonical isomorphisms ∗ Γ ∗ Γ ∗ Γ ∗ K∗+1(D (XΓ) /C (XΓ) ) = K∗(X) and K∗(C (XΓ) ) = K∗(Cr Γ) 1 2 PIERRE ALBIN AND PAOLO PIAZZA

with the K-homology of X and the K-theory of the reduced C∗-algebra of Γ, the long exact sequence reads

∗ ∗ Γ ∗ (1.2) · · · → Km+1(Cr Γ) → Km+1(D (XΓ) ) → Km(X) → Km(Cr Γ) → · · · The result of Higson and Roe is thus a commutative diagram of long exact sequences

(1.3) Lm+1(ZΓ) / S(X) / N(X) / Lm(ZΓ)

γ ρ β γ ∗ 1 ∗ Γ 1 1 ∗ 1 Km+1(Cr Γ)[ 2 ] / Km+1(D (XΓ) )[ 2 ] / Km(X)[ 2 ] / Km(Cr Γ)[ 2 ]

1 1 where we use the short-hand A[ 2 ] to indicate A ⊗Z Z[ 2 ] whenever A is an Abelian group. These maps were recast by the second author and Schick [PS16] in a more index-theoretic light, using in a crucial way properties of the signature operator on Galois Γ-coverings. Geometric applications of the interplay between the two sequences have been given, for example, in [CW03], [Wah13], [Zen17], [WY15], [WXY16]. In this paper we generalize all of this to the setting of stratiﬁed spaces.

Theorem 1.1. Every m-dimensional, oriented, smoothly stratiﬁed Cheeger space, X,b with fundamental group Γ, gives rise to a commutative diagram

(1.4) LBQ(Xb × I) / SBQ(Xb) / NBQ(Xb) / LBQ(Xb)

ρ IndAPS β IndAPS ∗ 1 ∗ Γ 1 1 ∗ 1 Km+1(Cr Γ)[ 2 ] / Km+1(D (XbΓ) )[ 2 ] / Km(Xb)[ 2 ] / Km(Cr Γ)[ 2 ] between the Browder-Quinn surgery exact sequence and the Higson-Roe analytic surgery sequence. We prove a better version of this diagram in §6.6 involving the signature operator on all of the strata of X,b but refer the reader to the text so as not to introduce more notation.

The top row of (1.4) is the surgery sequence for stratified spaces of Browder-Quinn [BQ75]. One of our contributions in this paper is a detailed treatment in §2 of the Browder-Quinn surgery exact sequence in the setting of smoothly stratified spaces (a.k.a. Thom-Mather stratified spaces). The original treatment in [BQ75] is quite sparse and its generalization in [Wei94] uses algebraic tools applicable in its setting of homotopically stratified spaces while for our purposes it is necessary to have geometric proofs that stay within the category of smooth stratifications. Our treatment naturally draws heavily from these two sources. A feature of the Browder-Quinn surgery sequence is that if Xb is a Witt space [Sie83] or a Cheeger space [Ban06, ABL+15, ALMP16] then all of the spaces that arise in the surgery sequence are also Witt spaces, respectively Cheeger spaces. This allows us to bring to bear the analysis that we have developed in joint work with Eric Leichtnam and Rafe Mazzeo [ALMP12, ALMP16, ALMP17] to define the vertical maps in (1.4) and to show that the diagram commutes. Notice that while the vertical maps are defined in analogy with [PS16], there are substantial technical differences, especially in the Cheeger case, where ideal boundary conditions must be chosen. STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 3

In detail, the vertical maps out of the Browder-Quinn L-groups are Atiyah-Patodi-Singer index classes; the map out of the normal invariants NBQ(Xb) is given in terms of the fundamental class, in K-homology, associated to the signature operator on a Cheeger space; finally the rho-map is a true secondary invariant associated to a suitable perturbation of the signature operator. In the smooth case this rho-map is directly connected with well-known numeric rho invariants; we comment on the validity of this principle in the singular case at the end of the paper. As already remarked, all of these constructions depend upon the definition of ideal boundary conditions; these depend, in turn, on the choice of a mezzoperversity and a major theme in this article is the detailed analysis of the dependence of these classes on the choice of a mezzoperversity and the proof of the remarkable fact that our maps are in fact all independent of the choice of a mezzoperversity. We end this introduction with a remark on the spin-Dirac operator. One of the advan- tages of the approach to the Higson-Roe theorem given in [PS16] is that it applies also to the spin-Dirac operator, giving in particular the mapping of the Stolz surgery sequence for positive scalar metrics to the analytic surgery sequence of Higson-Roe, see [PS14]. In a future publication, building also on [AGR16, PV, AGR], we plan to discuss the extension of the results in [PS14] to the stratified setting.

The paper is organized as follows. In Section 2 we give a rigorous and detailed treatment of the relevant results stated in the paper by Browder and Quinn. In Section 3 we specialize to Cheeger spaces and give a coarse theoretic treatment of some of the results in [ALMP12, ALMP16,ALMP17]; in particular we define the fundamental K-homology class of a Cheeger space without boundary and the associated index class. Finally, in the invertible case, we introduce the rho class of an invertible perturbation of the signature operator. In Section 4 we pass to manifolds with boundary, with a particular emphasis on the notion of Cheeger space bordism. It is in this section that we explain the statement of the delocalized Atiyah- Patodi-Singer index theorem on Cheeger spaces, a key tool in our analysis, and we illustrate its proof, building on [PS14, PS16]. In Section 5 we recall and expand results around the Hilsum-Skandalis perturbation for the signature operator on the disjoint union of two Cheeger spaces that are stratified homotopy equivalent. In Section 6 we finally define the vertical maps in the diagram that maps the Browder-Quinn surgery sequence to the Higson-Roe surgery sequence; we prove the well-definedness of these maps and that they are independent of the choice of a mezzoperversity. We then prove the commutativity of the squares of the diagram. We end this section by observing that it is in fact possible to consider different diagrams, each one associated to an individual closed stratum. Section 7, the last section of the paper, presents some geometric applications of our main result, in the spirit of [CW03].

Acknowledgements. The first author was partially supported by Simons Foundation grant #317883 and NSF grant DMS-1711325, and is grateful to Sapienza Universitàdi Roma, to UniversitéParis 7 (Equipe´ Algèbresd’Opérateurs), and to M.I.T. for their hospitality during the completion of this work. Financial support for the visits to Sapienza was provided by Istituto Nazionale di Alta Matematica (INDAM). The second author thanks UniversitéParis 7 (Equipe´ Algèbresd’Opérateurs)for hospitality and financial support during the completion of this work; additional funding for these visits was provided by Istituto Nazionale di Alta Matematica and Ministero dell’Istruzione, dell’ Universitàe della Ricerca, MIUR, under the project Spazi di Moduli e Teoria di Lie. 4 PIERRE ALBIN AND PAOLO PIAZZA

We are grateful to Markus Banagl, Jim Davis, Wolfgang L¨uck, Thomas Schick, Georges Skandalis, Shmuel Weinberger and Vito Felice Zenobi for their interest in this work and for interesting and useful discussions.

Contents 1. Introduction 1 Notation 5 2. Browder-Quinn Surgery 5 2.1. Browder-Quinn stratified spaces and transverse maps 6 2.2. Smoothly stratified spaces and transverse maps 7 2.3. Surgery definitions 9 2.4. The Π-Π condition 11 2.5. Surgery theorem 16 2.6. Wall realization 18 3. K-theory classes associated to the signature operator on Witt and Cheeger spaces 21 3.1. Metric structures 21 3.2. Galois coverings 21 3.3. C∗ and D∗ algebras 22 3.4. K-homology classes 24 3.5. Higson-Roe sequences associated to a Thom-Mather space 24 3.6. Index classes 25 3.7. Rho classes associated to trivializing perturbations 25 4. Bordisms and associated K-theory classes 26 4.1. Bordisms of Cheeger spaces 26 4.2. The signature operator on Cheeger spaces with cylindrical ends 27 4.3. Perturbations and coarse APS-index classes 28 4.4. The delocalized APS index theorem 29 5. Stratified homotopy equivalences and associated perturbations 29 5.1. The Hilsum-Skandalis replacement 29 5.2. The compressed Hilsum-Skandalis replacement 30 5.3. The Hilsum-Skandalis perturbation 31 5.4. The compressed Hilsum-Skandalis perturbation 32 5.5. Passing to the Roe algebra 32 5.6. APS-index classes associated to degree one normal maps 33 6. Mapping the Browder-Quinn surgery sequence to analysis 33 6.1. The rho class of a stratified homotopy equivalence 33 6.2. The rho map from S (X) to K (D∗(X )Γ) 34 BQ b dim Xb+1 bΓ 6.3. The map from N (X) to K (X) 36 BQ b dim Xb b ∗ Γ 6.4. The index map from LBQ(Xb) to K∗(C (XbΓ) ) 38 6.5. Mapping stratified surgery to analysis 44 6.6. Mapping stratified surgery to analysis on all strata 45 7. Further considerations 46 7.1. A long exact sequence for the Browder-Quinn L-groups 46 7.2. Atiyah’s L2 signature theorem 49 STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 5

7.3. Torsion elements and the cardinality of the BQ-structure set. 51 References 54

Notation. Below we will occasionally use diagrams such as f (Mc; ∂1M,c ∂2Mc) −−→ (Yb; ∂1Yb , ∂2Yb).

This should be understood to imply that ∂Mc = ∂1Mc ∪ ∂2M,c ∂Yb = ∂1Yb ∪ ∂2Y,b also that ∂1Mc ∩ ∂2Mc = ∂(∂iMc) (and similarly for Yb), and that f restricts to maps

f| : ∂1Mc −→ ∂1Yb , f| : ∂2Mc −→ ∂2Y,b which we sometimes denote ∂1f, ∂2f, respectively. Occasionally it will be useful to decompose the boundary of a space into more than two pieces, in which case similar conventions are in effect. Our main object of study will be smoothly stratified spaces. As reviewed below, this will mean Thom-Mather straified pseudomanifolds. A bordism between two stratified spaces will be a stratified space with boundary and a bordism between two stratified spaces with boundary will be a stratified space with corners (as is well known, e.g., [Wal16, §8.3], this is only useful if restrictions are placed on part of the boundary). For a careful discussion of these concepts we refer the reader to [Ver84] (see also [Alb17, §6]). While we do not use the language of ‘n-ads’ as in [Wal99], it is clear that the constructions below extend to ‘n-ads of stratifed spaces’. Note that the boundary of a manifold with corners is not itself a manifold with corners, but rather a union of manifolds with corners with various identifications of boundary faces. Following Melrose (see, e.g., [Mel12]), an ‘articulated manifold (without boundary)’ is a finite union of connected components of the boundary of a compact manifold with corners (thus guaranteeing that the identifications of boundary faces are consistent). More generally, an ‘articulated manifold with corners’ is a finite union of boundary hypersurfaces of the boundary of a compact manifold with corners (not necessarily making up full connected components of the boundary). Similarly an ‘articulated stratified space (without boundary)’ refers to a finite union of connected components of the boundary of a compact stratified space with corners, and an ‘articulated stratified space with corners’ is a finite union of boundary hypersurfaces of a stratified space with corners. Working with ‘articulated stratified spaces’ is analogous to working with n-ads in the category of stratified spaces; for instance, if (Mc; ∂1M,c ∂2Mc) is as above and Mc is a stratified space with corners then ∂iMc are articulated stratified spaces with corners. An alternate approach, see, e.g., [Wal16, §2.6], is to ‘round the corners’ and work with spaces with boundary.

2. Browder-Quinn Surgery We will make use of the surgery theory for stratified spaces of Browder and Quinn [BQ75]. Some of the results we need for the purpose of defining maps into K-theory are implicit in their exposition so we have decided to include a more explicit description of this surgery theory. In carrying this out we have benefitted from Weinberger’s exposition [Wei94, §7.1] where some of the proofs below are sketched (e.g., the Π-Π theorem, Theorem 2.1), as well as 6 PIERRE ALBIN AND PAOLO PIAZZA

from [Wal99,Ran02b,DP,CLM,And77,Qui70a] and [DR88] (from which we adapted the proof of the Wall realization theorem, Theorem 2.5). We are also happy to acknowledge useful conversations and email exchanges with Shmuel Weinberger, Markus Banagl, Wolfgang Lück, and Jim Davis. 2.1. Browder-Quinn stratified spaces and transverse maps. Although there are many notions of stratified spaces, perhaps the most common is that of a Whitney stratified space. If L is a smooth manifold then a Whitney stratification of a subset Xb ⊆ L is a locally finite collection of pairwise disjoint smooth submanifolds covering X,b known as strata, satisfying the ‘frontier condition’ Y ∩ Y 0 6= ∅ =⇒ Y ⊆ Y 0 and ‘Whitney’s condition (B)’ concerning the relations of the tangent spaces of the strata. For example, Whitney showed [Whi65] that algebraic varieties admit Whitney stratifications. It was subsequently shown by Thom and Mather [Mat12] that in a Whitney stratified space neighborhoods of the strata have geometric structure and this was abstracted in the notion of Thom-Mather stratified space. A further abstraction was given by Browder and Quinn [BQ75] (cf. [HW01,DS90,Wei94]). They fix a category F of ‘manifolds with fibrations’ such as smooth manifolds and locally trivial smooth fiber bundles, PL manifolds and block bundles with manifold fibers, topological manifolds and locally trivial topological fiber bundles, or Poincaréspaces and maps whose homotopy fiber satisfies Poincaréduality. Although Browder-Quinn do not specify what properties are necessary in the category F , an important property is that there be a notion of pull-back in the category F . An F -stratified space is a topological space Xb filtered by subsets Xa indexed by a partially ordered set A satisfying the following. If for each a ∈ A we let [ X∂a = {Xb : b ∈ A, b < a} then each Xa is equipped with a closed neighborhood Na = N(Xa) of X∂a in Xa and a projection νa : ∂Na −→ X∂a such that i) Xa \ X∂a and ∂Na are manifolds in F , ii) Na is the mapping cylinder of νa (with ∂Na and X∂a corresponding to the top and bottom of the cylinder), iii) If a, b ∈ A, b < a, Wb = Xb \ int(Nb), then −1 νa : νa (Wb) −→ Wb is a fibration in F .

If Xb and Mc are two F -stratified spaces whose filtrations are indexed by the same partially ordered set A, then a filtration-preserving map f : Xb −→ Mc of F -stratified sets is said to be transverse if each fibration in Xb is the pull-back along f of the corresponding fibration in M.c Remark 1. When F is equal to the category of smooth manifolds and locally trivial smooth fiber bundles, Browder-Quinn F-stratified spaces are the same as Thom-Mather stratified spaces. One could show this by, for example, proceeding as in [ALMP12] and proving that any Browder-Quinn F -stratified space can be ‘resolved’ to an F -space ‘with corners’ and STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 7

iterated fibration structures, and any such can be collapsed to a Browder-Quinn F -stratified space. Instead of developing this, we will work directly with Thom-Mather stratified spaces in establishing the Browder-Quinn surgery sequence below. 2.2. Smoothly stratified spaces and transverse maps. From now on we will only work with F equal to smooth manifolds and locally trivial smooth fiber bundles, i.e., the setting of Thom-Mather stratified spaces. For this class of spaces there is a construction going back to Thom [Tho69] and carried out in [ALMP12] that replaces a stratified space, X,b with its ‘resolution’, X,e a manifold with corners and an iterated fibration structure. We now recall this construction.

Let Xb be a stratiﬁed space with singular strata

S(Xb) = {Y1,Y2,...,Y`}.

Each Yi is subset of Xb that inherits the structure of an open manifold (indeed, of the interior of a manifold with corners). We write Yi < Yj if the closure of Yj in Xb contains Yi. The closure of Yj in Xb is given by [ Ybj = {Yi : Yi ≤ Yj} and is itself a stratified space. Every point in Yi has a neighborhood in Xb homeomorphic to dim Yi a ball in R times the cone over a stratified space Zbi, known as the link of Yi in X.b The resolution of X,b denoted X,e is a smooth manifold with corners. Each stratum Yi of Xb corresponds to a collective boundary hypersurface BYi of X,b by which we mean a collection of boundary hypersurfaces no two of which intersect. Each collective boundary hypersurface participates in a fiber bundle,

φYi Zei − BYi −−−→ Yei, where the base is the resolution of Ybi and the typical ﬁber is the resolution of Zbi. If Yi and

Yj are strata of Xb with Yi < Yj then BYi ∩ BYj 6= ∅ and we have a commutative diagram of ﬁber bundle maps

φYj BYi ∩ BYj / BYiYj ⊆ BYj

φY φY Y i $ y i j Yei where BYiYj is a collective boundary hypersurface of BYj . We refer to a manifold with corners together with these collective boundary hypersurface fiber bundle maps as a manifold with an iterated fibration structure. There is a canonical ‘blow-down map’ between a manifold with corners and an iterated fibration structure Xe and a stratified space X,b β : Xe −→ Xb which collapses the fibers of the boundary fiber bundles to their base. Note that β is a diffeomorphism between the interior of Xe and the regular part of X.b A continuous map between stratified spaces is stratum preserving if the inverse image Fb of a stratum is a union of strata. A stratum preserving map Mc −−→ Xb is smooth if it lifts 8 PIERRE ALBIN AND PAOLO PIAZZA

Fe to a smooth map Mf −−→ X.e We denote the space of such maps by ∞ ∞ CΦ (M,f Xe) ⊆ C (M,f Xe) and the corresponding maps between Mc and Xb by C∞(M,c Xb). Note that we have a natural identiﬁcation ∞ ∞ β∗ : CΦ (M,f Xe) −→ C (M,c Xb) The smooth map Fe necessarily induces a ﬁber bundle map between the collective boundary hypersurfaces of Mf and those of X,e

Fe (2.1) Mf ⊇ BN / BY ⊆ Xe

φN φY Fg|N Ne b / Ye

Fb Fe We will say that Mc −−→ Xb and Mf −−→ Xe are transverse if the commutative diagram of fiber bundles (2.1) is a pull-back diagram. We denote the class of such maps by C−∞(M, X) t f e and C−∞(M, X); the identification β restricts to an identification t c b ∗ ∞ ∞ β : C− (M, X) −→ C− (M, X). ∗ t f e t c b Note that C−∞(M, X) are the transverse maps of Browder-Quinn. (Indeed, the fibrations ν t c b a of a Browder-Quinn stratified space correspond in the smooth category to the fiber bundle maps φY .) These maps are also used in [FM81, Part I, §4] and [GM83, §5.4] where they are called ‘normal non-singular’. A weaker notion called ‘homotopy transverse’ was used by Weinberger [Wei94, §5.2]. An example from [GM83] is the inclusion H ∩ X,b → Xb when Xb is a stratified subset of a smooth manifold and H is a smooth submanifold transverse to the strata of X.b Another example from the same source is the fiber bundle projection map for a fiber bundle over a stratified space with fiber a smooth manifold.

Remark 2. Browder-Quinn considered these maps first in an equivariant situation. If G is a compact Lie group and L, L0 are spaces with G-actions then a map f : L −→ L0 is isovariant if, for any x ∈ L, g ∈ G, f(gx) = gf(x), and gf(x) = f(x) ⇐⇒ gx = x. An isovariant map is transverse linear if whenever H ⊆ G is a subgroup, and L{H} denotes the subset of L consisting of points whose isotropy group is conjugate to H, there are G-vector bundle tubular neighborhoods L{H} ⊆ U, (L0){H} ⊆ U 0 such that f restricts to a G-linear vector bundle map U −→ U 0. Transverse linear isovariant maps are examples of transverse maps for the stratification of a space into the orbit types of a group action. There are some properties of a stratified space Xb such that the existence of a transverse map fb : Mc −→ Xb implies that Mc also has this property. For example, if the dimensions of all of the links of Xb are odd, then the dimensions of all of the links of Mc must be odd as STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 9

well. A class of stratified spaces determined by such a property will be said to be preserved by transverse maps. As examples of such classes let us mention, in order of increasing generality: IP spaces, Witt spaces, and Cheeger spaces or L-spaces. Recall that a stratified space is a Witt space if, whenever the link of a stratum is even dimensional, its middle degree middle perversity intersection homology vanishes. A Witt space is an IP space (intersection Poincaréspace) if, whenever the link Zb of a stratum is odd dimensional, its middle perversity 1 homology in degree 2 (dim Zb − 1) is torsion-free. A stratified space is an L-space if there is a self-dual sheaf compatible with the intersection homology sheaves of upper and lower middle perversity [Ban06]. Smoothly stratified L-spaces are known as Cheeger spaces. 2.3. Surgery definitions. When carrying out surgery constructions we will need to use stratified spaces with corners. By a stratified space with corners we mean a stratified space with collared corners [Alb17, Definition 2] or an ‘abstract stratification with faces’ in the sense of Verona [Ver84, §5]. An ‘articulated stratified space with corners’ is a union of boundary hypersurfaces of a stratified space with corners; thus it is a union of stratified spaces with corners together with identifications of certain of their boundary hypersurfaces. Smoothness of maps to and from articulated spaces is defined in the natural way, i.e., continuity on the whole and smoothness on each stratified space with corners. Definition 1. Let Mc and Xb be oriented stratified spaces. i) A BQ-transverse map f : Mc −→ Xb is a transverse map that is orientation preserving and restricts to a diffeomorphism between strata of dimension less than five. ii) A BQ-normal map f : Mc −→ Xb is a BQ-transverse map such that, in the notation of §2.1, for each a ∈ A, f restricts to a degree one normal map

fa : Ma \ int(N(Ma)) −→ Xa \ int(N(Xa)) meaning that there is an smooth vector bundle τ −→ Xa \ int(N(Xa)) and a bundle isomor- ∗ phism b : νMa\int(N(Ma)) −→ fa τ covering fa. (We will not explicitly keep track of the bundle data as it will not affect our analytic maps.) iii) A BQ-equivalence f : Mc −→ Xb is a BQ-transverse map whose restriction to each stratum is a homotopy equivalence. (By Miller’s criterion [Mil13] (see [ALMP17, Corollary 1.11]), f is a BQ-equivalence if and only if there is a BQ-transverse map g : Xb −→ Mc and homotopies of f ◦ g and g ◦ f to the respective identities through BQ-transverse maps.) One could also work with simple homotopy equivalences and there is an s-cobordism theorem in this context [BQ75, pg. 34]. Given a stratified space, its dimension function will refer to the function on the poset of strata that assigns to each stratum its dimension. Definition 2. Let Xb be a smooth oriented stratified space, possibly with boundary. (Our convention is that the spaces below are allowed to be empty and a map between empty sets counts as an equivalence.) a) Let LBQ,d(Xb) denote the set of diagrams φ ω (2.2) (Mc; ∂Mc) −−→ (Yb; ∂Yb) −−→ Xb where Mc and Yb are oriented stratified spaces with corners, d is the dimension function of Mc (if omitted, Mc and Xb are assumed to have the same dimension function), φ is BQ-normal, 10 PIERRE ALBIN AND PAOLO PIAZZA

∂φ is a BQ-equivalence (between articulated stratiﬁed spaces with corners), and ω is BQ- transverse. We refer to these diagrams as (Browder-Quinn) L-cycles over X.b A null bordism of an L-cycle over Xb as above will mean a diagram

Φ Ω (Nb; ∂1N,b ∂2Nb) −−→ (Zb; ∂1Z,b ∂2Zb) −−→ Xb × I between stratiﬁed spaces with corners, with the same dimension function as Mc × I, where Φ is BQ-normal, Φ| : ∂2Nb −→ ∂2Zb is a BQ-equivalence, Ω is BQ-transverse, and

Φ| π◦Ω| φ ω (∂1N,b ∂12Nb) −−→ (∂1Z,b ∂12Zb) −−−−→ Xb = (Mc; ∂Mc) −−→ (Yb; ∂Yb) −−→ Xb

(with π : Xb × I −→ Xb the projection). In this case we say that the L-cycle is null bordant. The L-cycles over Xb naturally form an Abelian monoid with addition induced by disjoint union and zero given by the diagram with Mc = Yb = ∅. We say that two L-cycles over X,b α, β are equivalent if α + βop is null bordant, where βop denotes β with orientations reversed. This is an equivalence relation, known as L-bordism, and the set of equivalence classes, denoted LBQ,d(Xb), is known as the Browder-Quinn L-group of X.b b) The (Browder-Quinn) normal invariants, denoted NBQ(Xb) is the subset of LBQ(Xb) in which, with notation as above, Yb = Xb and ω = id . A normal bordism is an L-bordism between normal invariants which, in the notation above, has Zb = Xb × I and Ω = id . The set of normal invariants modulo normal bordisms is denoted NBQ(Xb).

If Xb has boundary we denote by NBQ(X,b ∂Xb) the subset of NBQ(Xb) in which, with notation as above, φ| is a (stratum preserving) diﬀeomorphism. A normal bordism, relative to ∂Mc ∂X, will be a normal bordism as above in which ∂ N = K × I and Φ| = φ| × id . The b 2 b b ∂2Nb ∂Mc corresponding set of normal invariants is denoted NBQ(X,b ∂Xb).

c) The (Browder-Quinn) Thom-Mather structures, denoted SBQ(Xb) is the subset of NBQ(Xb) in which, with notation as above, φ is a BQ-equivalence. Two such objects are equivalent if there is a normal bordism between them which, in the notation above, has Φ a BQ-equivalence. The set of equivalence classes is denoted SBQ(Xb) and is known as the (Browder-Quinn) structure set.

Similarly, if Xb has boundary then SBQ(X,b ∂Xb) is the subset of N (X,b ∂Xb) in which φ is a BQ-equivalence. Equivalence classes modulo normal bordism relative to ∂Xb in which Φ is a BQ-equivalence form the set SBQ(X,b ∂Xb).

Remark 3. A BQ-transverse map f : Xb −→ Wc induces a homomorphism f∗ : LBQ,d(Xb) −→ LBQ,d(Wc). It is easy to see that if f is a BQ-equivalence then it induces an isomorphism ∼ LBQ,d(Xb) = LBQ,d(Wc). Browder and Quinn point out [BQ75, Proposition 4.7] that this is true ∼ for any BQ-transverse f that satisﬁes the Π-Π condition below. For example, LBQ,d(Xb) = LBQ,d(Xb × I). Remark 4. The Pontrjagin-Thom constructions can be ﬁt together using transversality to identify NBQ(Xb) with homotopy classes of maps from Xb into G/O that are constant on strata STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 11

of dimension less than ﬁve. Since we do not use this we do not elaborate, see [Wei94, pg. 140].

Remark 5. If Xb is a smooth manifold, then LBQ,d(X) = Ld(π1(X)). Indeed, our deﬁnition 1 of LBQ,d(X) coincides with Wall’s Ld(X) from [Wal99, Chapter 9] save that we required Y in Deﬁnition 2 to be smooth. However X smooth and Wall’s realization theorem ( [Wal99, Theorem 10.4] and Theorem 2.5 below) shows that this does not change the group we obtain. The inclusion maps between the sets above descend to maps between the equivalence classes η θ SBQ(Xb) −−→ NBQ(Xb) −−→ LBQ(Xb), We will show that this sequence is exact and extends to the left

LBQ(Xb × I) / SBQ(Xb) in that there is an action of the L-group of Xb × I on the structure set of X.b Moreover the extended sequence is exact in that two elements of SBQ(Xb) have the same image under η if and only if they are in the same orbit of the L-group. For stratiﬁed spaces with boundary there are analogous sequences relative to the boundary. In fact the extension to the left of the sequence above, for a stratiﬁed space without boundary, is

η θ SBQ(X × I, X × ∂I) / NBQ(X × I, X × ∂I) / LBQ,d (X × I) b b b b Xb×I b

η θ / SBQ(X) / NBQ(X) / LBQ,d (X) b b Xb b where, e.g., d is the dimension function of X. One can iterate and extend the sequence Xb b further to the left. 2.4. The Π-Π condition. Definition 3. We say that a map h : Mc −→ Nb between stratified spaces with corners satisfies the Π-Π condition if: For every connected component of a stratum of N,Sb N , there is exactly one connected component of a stratum of M,Sc M , such that h(SM ) ∩ SN 6= ∅. Moreover, h(SM ) ⊆ SN and h∗ : π1(SM ) −→ π1(SN ) is an isomorphism. The π-π theorem (or surgery lemma) in our context is implicit in [BQ75] and presented by Weinberger in [Wei94, pg. 140], where a proof is also sketched. We formulate it as in Quinn’s thesis [Qui70b, Theorem 2.4.4] and prove it following [Wei94].

Theorem 2.1 (BQ Π-Π theorem). Let M,c Y,b Xb be stratiﬁed spaces with boundary, with the same dimension function, together with decompositions of their boundaries, e.g., ∂Xb = ∂0Xb ∪ ∂1X,b into two codimension zero stratiﬁed spaces with common boundary. Consider a diagram f ω (2.3) (Mc; ∂0M,c ∂1Mc) −−→ (Yb; ∂0Yb , ∂1Yb) −−→ (Xb; ∂0X,b ∂1Xb) in which f is BQ-normal, f| : ∂0Mc −→ ∂0Yb is a BQ-equivalence, ω is BQ-transverse and orientation preserving. 12 PIERRE ALBIN AND PAOLO PIAZZA

If the inclusion of ∂1Yb into Yb satisﬁes the Π-Π condition then there is a bordism between (2.3) and

0 0 0 f ω (2.4) (Mc ; ∂0M,c ∂1Mc ) −−→ (Yb; ∂0Yb , ∂1Yb) −−→ (Xb; ∂0X,b ∂1Xb) where f 0 satisﬁes the same properties as f but is moreover a BQ-equivalence. Explicitly this bordism is a diagram of oriented stratiﬁed spaces with corners

F ω×id Nb −−→ Yb × I −−−−→ Xb × I 0 in which ∂Nb = Mc ∪ Mc ∪ ∂0Mc × I ∪ P,b with F is a BQ-normal map satisfying

0 0 (F ;f,f ,f|×id,F |) (2.5) (Nb; M,c Mc , ∂0Mc× I, Pb) −−−−−−−−−−−→ (Yb × I; Yb × {0}, Yb × {1}, ∂0Yb × I, ∂1Yb × I). Proof. We proceed by induction on the depth of the stratified space. Our base case is when Y,b M,c and Xb are smooth manifolds with boundary and this is Theorem 3.3 in Wall’s book [Wal99], since in dimension less than five our maps are diffeomorphisms by definition. Suppose the theorem is established for all stratified spaces with boundary whose stratification has depth less than k and consider (2.3) where Yb (and hence M,c Xb) has a stratification of depth k. Denote the subsets of depth k by a † decoration and note that these are smooth manifolds and that transversality of the maps in (2.3) implies that these subsets are preserved by these maps. Thus we obtain

† † † † † f † † † ω † † † (2.6) (M ; ∂0M , ∂1M ) −−→ (Y ; ∂0Y , ∂1Y ) −−−→ (X ; ∂0X , ∂1X ) satisfying the same conditions as the diagram (2.3). Since the Π-Π condition holds, there is a bordism satisfying the same conditions as (2.5),

† † (2.7) N † −−−→F Y † × I −−−−−→ω ×id X† × I between (2.6) and

†0 † †0 †0 †0 f † † † ω † † † (M ; ∂0M , ∂1M ) −−−→ (Y ; ∂0Y , ∂1Y ) −−−→ (X ; ∂0X , ∂1X ) with f †0 a homotopy equivalence.

Transversality of f and ω guarantee that we can ﬁnd neighborhoods TM † ⊆ M,c TY † ⊆ Y,b † † † and TX† ⊆ Xb that ﬁber over M ,Y , and X respectively, such that each square in

f| ω| (2.8) TM † / TY † / TX†

φ† φ† † M Y φX † † † f † ω M / Y / X† where f| and ω| denote the restrictions of f and ω respectively, is a pull-back square. Note that for i = 0, 1 we have † −1 † ∂iMc ∩ TM † = (φM ) (∂iM ) and similarly for Yb and X,b so the top row of (2.8) is a diagram satisfying conditions analogous to (2.3). STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 13

We have an extension of this diagram to a similar diagram over the bordism (2.7)

† ω|×id †∗ F (2.9) F (TY † × I) / TY † × I / TX† × I

† † † F † ω ×id N / Y × I / X† × I

†∗ † with F (TY † )|M † = TM † and F the induced map on the pull-back. Restricting this diagram to the ‘ﬁxed’ ∂0 part of the boundary we get

f|×id ω|×id (∂0Mc ∩ TM † ) × I / (∂0Yb ∩ TY † ) × I / (∂0Xb ∩ TX† ) × I

f †|×id ω†|×id † † † ∂0M × I / ∂0Y × I / ∂0X × I and so the top row of (2.9) is an bordism, satisfying conditions analogous to (2.5), from the top row of (2.8) to

†0 †0 ∗ f ω| (2.10) (f ) TY † −−−→TY † −−→TX† †0 in which f is a BQ-equivalence. Now, as in [Bro68, Theorem 2.14] [Wei94, §4.3], we multiply each space in (2.3) by the unit interval and attach the top row of (2.9) to get a bordism

† [ †∗ (f×id)∪F [ Mc × I F (TY † ) −−−−−−−→ Yb × I TY † × I †∗ T ×{1}∼T ×{0} TM† ×{1}∼F (TY † )|M† Y † Y † ω×id ∪ω|×id [ −−−−−−−−−→ Xb × I TX† × I

TX† ×{1}∼TX† ×{0} from (2.3) to a similar diagram over Xb which we denote g ω (2.11) (Pb; ∂0Pb , ∂1Pb) −−→ (Yb; ∂0Yb , ∂1Yb) −−→ (Xb; ∂0X,b ∂1Xb) and which restricts to (2.10) in a neighborhood of the subsets of depth k. Now we remove these neighborhoods of the subsets of depth k to form + †0 ∗ + + Pb = Pb \ (f ) TY † , Yb = Yb \TY † , Xb = Xb \TX† . These are stratiﬁed spaces with corners (see, e.g., [Alb17, §6], [Ver84]) and we deﬁne

+ + + + ∂0Yb = (∂0Yb ∩ Yb ) ∪ ∂TY † , ∂1Yb = ∂1Yb ∩ Yb + + + + and similarly for Pb and Xb . Note that the restrictions of g and ω to ∂0Pb and ∂0Yb are BQ-equivalences so we have a diagram

+ + + g| + + + ω| + + + (Pb ; ∂0Pb , ∂1Pb ) −−→ (Yb ; ∂0Yb , ∂1Yb ) −−→ (Xb ; ∂0Xb , ∂1Xb ) satisfying conditions analogous to (2.3). (Note that though the stratiﬁed space now has corners of codimension two, one can ‘smooth out the corners’ as in [Wal16, §2.6], [HMM97, §3].) 14 PIERRE ALBIN AND PAOLO PIAZZA

Moreover, the compatibility between the stratifications and the boundary faces implies that each stratum of Yb + is homotopy equivalent to the corresponding stratum of Y,b and + the same is true for the strata of ∂1Yb . (Indeed iterating this process of removing tubular neighborhoods of deepest strata produces the resolution of Yb which does not change the + + homotopy type of the strata.) Significantly, the inclusion of ∂1Yb into Yb satisfies the Π-Π condition and since Yb + has depth less than k we can apply our inductive hypothesis to find a bordism, satisfying conditions analogous to (2.5),

G Ω Nb+ −−→ Yb + × I −−→ Xb+ × I between g| ω| + + + + + + + + + (Pc ; ∂0Pc , ∂1Pc ) −−→ (Yc ; ∂0Yc , ∂1Yc ) −−→ (Xd; ∂0Xd, ∂1Xd) 0 ω| +0 +0 +0 g + + + + + + and (Pd; ∂0Pd, ∂1Pd) −−→ (Yc ; ∂0Yc , ∂1Yc ) −−→ (Xd; ∂0Xd, ∂1Xd)

0 †0 ∗ with g a BQ-equivalence. Since the bordism does not change the spaces ∂((f ) TY † ), ∂TY † , ∂TX† or the maps between them, we can glue in the bordism (2.10) to ﬁnally obtain an bordism between (2.3) and (2.4) with

0 +0 †0 ∗ Mc = P ∪ (f ) TY † , 0 0 0 † and f = g ∪ f a BQ-equivalence as required. The key to applying the Π-Π theorem is a result of Wall that allows us to represent every 1 2 class in LBQ(Xb) by a ‘restricted’ representative. This is sometimes referred to as “L = L ” evoking the notation of [Wal99, Chapter 9].

Definition 4. Let Xb be a stratified space (possibly with boundary). An L-cycle over X,b φ ω (Mc; ∂Mc) −−→ (Yb; ∂Yb) −−→ X,b is a restricted L-cycle if ω : Yb −→ Xb satisfies the Π-Π condition. A null bordism of a restricted L-cycle over X,b Φ Ω (Nb; ∂1N,b ∂2Nb) −−→ (Zb; ∂1Z,b ∂2Zb) −−→ Xb × I, is a restricted null bordism if Ω: Zb −→ Xb × I satisfies the Π-Π condition. Remark 6. If Xb has depth zero then these are the restricted cycles of [Wal99, Chapter 9], see Remark 5. Theorem 2.2. Let Xb be a stratified space (possibly with boundary). Every element of LBQ(Xb) is L-bordant, relative to the boundary, to a restricted L-cycle over X.b If a restricted L-cycle over Xb is null bordant, then it participates in a restricted null bordism. Proof. Our proof is parallel to that of the Π-Π theorem. When possible we will simply refer back to the latter proof.

We will prove by induction on the depth of the stratiﬁcations that, whenever we have

φ ω (2.12) Mc −−→ Yb −−→ Xb STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 15

with φ BQ-normal, ω BQ-transverse, and M,c Y,b Xb stratiﬁed spaces with corners with the same dimension functions, there is a bordism relative to the boundary to a similar diagram

φ0 ω0 Mc0 −−→ Yb 0 −−→ Xb in which ω0 satisﬁes the Π-Π condition. Speciﬁcally, there is a diagram

(Φ;φ,φ0,φ|×id) (Ω;ω,ω0,ω|×id) (Nb; M,c Mc0, ∂Mc × I) −−−−−−−−−→ (Zb; Y,b Yb 0, ∂Yb × I) −−−−−−−−−→ Xb × I in which Φ is BQ-normal and Ω is BQ-transverse and moreover

Mc ∩ Mc0 = ∅, Mc ∩ (∂Mc × I) = ∂M,c Mc0 ∩ (∂Mc × I) = ∂Mc0 and similarly for Z.b

We proceed by induction on the depth of the stratified space. Our base case is for smooth manifolds with corners. If dim Y < 5 then the theorem is automatic since f and ω are then both diffeomorphisms. Assuming dim Y ≥ 5, this case is handled by Wall in [Wal99, Theo- rems 9.4, 9.5], where he shows that this can be arranged by carrying out surgery on Y along 1-handles and 2-handles and then modifying M along the preimages of the corresponding embeddings. As pointed out in [Mil61, Proof of Lemma 3], a theorem of Whitney implies that for dim Y > 4 any homotopy class of maps from S1 or S2 into Y contains an embedding with image contained in the interior of Y. Thus all of the modifications in M and Y can be carried out in their interiors. Suppose the theorem is established for all stratified spaces with boundary whose stratification has depth less than k and consider (2.12) where Yb (and hence Mc) has a stratification of depth k. Denote the subsets of depth k by a † decoration and note that these are smooth manifolds and that transversality of the maps in (2.12) implies that these subsets are preserved by these maps. Thus we obtain

f † ω† (Mc†, ∂Mc†) −−→ (Yc†, ∂Yc†) −−−→ X,b

† an element of LBQ(Xc). Applying the base case to this situation we obtain an L-bordism, relative to the boundary, to an element satisfying the desired Π-Π condition. Proceeding as in the proof of Theorem 2.1 we can lift this L-bordism to neighborhoods of these subsets and then graft it onto the product of (2.12) with the unit interval to obtain an L-bordism, relative to the boundary, between (2.12) and an element of LBQ(Xb) analogous to (2.11), g α (Pb , ∂Pb) −−→ (Vb , ∂Vb) −−→ X,b where α : Vb −→ Xb satisﬁes the Π-Π condition for strata of depth k. Now we remove consistent tubular neighborhoods of the subsets of depth k to form Pb+, Vb +, and Xb+ as in the proof of Theorem 2.1. This gives a diagram g| α| + + + + + + + (Pc ; ∂0Pc , ∂1Pc ) −−→ (Vc ; ∂0Vc , ∂1Vc ) −−→ (Xb; ∂0X,b ∂1Xb) in which the stratiﬁcation on Vb + has depth less than k. By our inductive hypothesis there is a bordism, relative to the boundary, between this and another diagram

0 0 +0 +0 +0 g +0 +0 +0 α + (Pd; ∂0Pd, ∂1Pd) −−→ (Vd; ∂0Vd, ∂1Vd) −−→ (Xb; ∂0X,b ∂1Xb) 16 PIERRE ALBIN AND PAOLO PIAZZA

0 for which α0 : V + −→ Xb+ satisfies the Π-Π condition. Since the bordism is relative to the boundary we may glue in the previous bordism between neighborhoods of the strata of depth k to obtain a bordism, relative to the boundary, between our original diagram and g0∪(g|) α0∪(α|) +0 +0 Pd ∪ TP † −−−−−→ Vd ∪ TV † −−−−−→ X.b Finally, because of the compatibility between the stratifications and the boundary faces, the fact that α satisfies the Π-Π condition between the strata of depth k and α0 satisfies the Π-Π condition between strata of depth less than k means that α0 ∪ (α|) satisfies the Π-Π condition on all strata. Our final preliminary result is to point out that the sum in LBQ(Xb × I) which is induced by disjoint union can, when appropriate, be carried out by identifying boundary faces. Lemma 2.3. Let φ ω α :(Mc; ∂Mc) −−→ (Yb; ∂Yb) −−→ Xb × I ψ θ β :(Lb; ∂Lb) −−→ (Wc; ∂Wc) −−→ Xb × I be two L-cycles over Xb × I. Suppose that the pre-images in α lying above Xb × {0} coincide with the pre-images in β lying above Xb × {1}, ρ ξ γ :(Pb; ∂Pb) −−→ (Vb; ∂Vb) −−→ X.b

The class of α + β in LBQ(Xb × I) is represented by the union of the diagrams along their common boundary, α ∪γ β. Proof. Let N = (M ∪ L) × I, Z = (Y ∪ W ) × I, and consider a diagram b c Pb b b b Vb c Φ Ω (2.13) Nb −−→ Zb −−→ Xb × I where Φ has the form Φ(x, t) = (Φt(x), t) and similarly Ω(x, t) = (Ωt(x), t). By deﬁnition, Pb has a collar neighborhood in each of Mc and Lb and gluing these together we have a neighborhood of the form (−ε, ε) × P in M ∪ L and of the form (−ε, ε) × P × I in N. b c Pb b b b Similarly, we have a neighborhood of the form (−ε, ε) × V in Y ∪ W and of the form b b Vb c (−ε, ε) × Vb × I in Z.b Note that M ∪ L \ ((−ε, ε) × P ) = M t L, Y ∪ W \ ((−ε, ε) × V ) = Y t W. c Pb b c b b Vb c b c

We choose Φ so that Φ0 = φ ∪ ψ and for t > 0, Φt (−tε,tε)×P = id ×ρ, while oﬀ of this neighborhood Φ is essentially φ t ψ. We similarly choose Ωt. With these choices, since φ ∂Mc and ψ are BQ-equivalences, we recognize (2.13) as an L-bordism between the disjoint ∂Lb union of α and β and their union along γ. 2.5. Surgery theorem. With the preliminary results out of the way, we can establish the fundamental result of surgery: a normal map is normal bordant to an equivalence precisely when its surgery obstruction vanishes.

Theorem 2.4 (Exactness part 1). Let Xb be an oriented stratiﬁed space with boundary (possibly empty) and let h : Kb −→ ∂Xb STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 17

be a (stratum preserving) diﬀeomorphism. Given a stratiﬁed space Mc with boundary ∂Mc = K,b and a BQ-normal map φ : Mc −→ Xb extending h, there is a normal bordism relative to h between φ and a BQ-equivalence if and only if φ is L-null-bordant.

Equivalently, the sequence η θ SBQ(X,b ∂Xb) −−→ NBQ(X,b ∂Xb) −−→ LBQ(Xb) is exact. Proof. If there is a normal bordism, relative to h, between φ and a BQ-equivalence, φ0 : Mc0 −→ X,b say Φ id (Nb; M,c Mc0, Kb × I) −−→ (Xb × I; Xb × {0}, Xb × {1}, ∂Xb × I) −−−→ Xb × I, then this bordism witnesses the triviality of φ in LBQ(Xb).

On the other hand, if (φ;h) id (Mc; Kb) −−−−→ (Xb; ∂Xb) −−−→ Xb is a null bordant L-cycle over Xb then, since it is a restricted cycle, there is by Theorem 2.2 a restricted null bordism. That is to say, there are maps of stratified spaces with corners Φ Ω (Nb; ∂1N,b ∂2Nb) −−→ (Zb; ∂1Z,b ∂2Zb) −−→ Xb × I with Φ BQ-normal, Φ| : ∂2Nb −→ ∂2Zb a BQ-equivalence, Ω BQ-transverse, Φ| π◦Ω| φ id (∂1N,b ∂12Nb) −−→ (∂1Z,b ∂12Zb) −−−−→ Xb = (Mc; ∂Mc) −−→ (Xb; ∂Xb) −−−→ Xb , and Ω satisfies the Π-Π condition. Now ∂1Zb = Xb and Ω = id, so the inclusion of ∂1Zb ∂1Zb into Zb satisfies the Π-Π condition. Since Φ is a BQ-equivalence we can apply the Π-Π ∂2Nb Φ Ω theorem and find a bordism (not necessarily an L-bordism) between Nb −−→ Zb −−→ Xb × I and 0 0 0 0 Φ Ω (Nb ; ∂1Nb , ∂2Nb ) −−→ (Zb; ∂1Z,b ∂2Zb) −−→ Xb × I with Φ0 a BQ-equivalence, and

0 0 Φ | Φ| ∂2Nb −−−→ ∂2Zb = ∂2Nb −−→ ∂2Zb .

The bordism itself has the form Ψ Γ Lb −−→ Zb × I −−→ Xb × I2 or, more explicitly,

0 0 (Ψ;Φ,Φ ,Φ|×I,Ψ|) Ω×id 2 (Lb; N,b Nb , ∂2Nb×I, Pb) −−−−−−−−−−−→ (Zb×I; Zb×{0}, Zb×{1}, ∂2Zb×I, ∂1Zb×I) −−−−→ Xb×I , where this diagram serves to deﬁne P.b Note that 0 0 0 ∂Pb = Pb ∩ (Nb ∪ Nb ∪ ∂2Nb × I) = ∂1Nb ∪ ∂1Nb ∪ ∂12Nb × I = Mc ∪ ∂1Nb ∪ Kb × I 18 PIERRE ALBIN AND PAOLO PIAZZA

and recall that ∂1Zb = X,b so if we restrict this diagram to the boundary face P,b we ﬁnd 0 0 (Ψ|;φ,Φ |,h×id) id (Pb; M,c ∂1Nb , Kb × I) −−−−−−−−−−→ (Xb × I; Xb × {0}, Xb × {1}, ∂Xb × I) −−−→ Xb × I. We recognize this as a normal bordism, relative to h : Kb −→ ∂X,b (i.e., without changing h) between 0 (φ;h) id 0 (Φ |;h) id (Mc; Kb) −−−−→ (Xb; ∂Xb) −−−→ Xb and (∂1Nb ; Kb) −−−−−→ (Xb; ∂Xb) −−−→ Xb which, since Φ0| is a BQ-equivalence, proves the theorem. 2.6. Wall realization. In this section we follow Dovermann-Rothenberg [DR88, §8]. f Theorem 2.5 (Wall realization). Let Xb be a stratiﬁed space (without boundary) and Lb −−→ Xb a BQ-equivalence. Every element α ∈ LBQ(Xb × I) has a representative of the form F id (Wc; ∂−Wc , ∂+Wc) −−→ (Xb × I; Xb × {0}, Xb × {1}) −−−→ Xb × I with h F | i h f i ∂−Wc −−→ Xb × {0} = Lb −−→ Xb .

Note that this representative is an element of NBQ(Xb × I) and that the restriction F | ∂+Wc −−→ Xb × {1} gives another Thom-Mather structure on X.b Proof. Choose an L-cycle representing α, φ ω (Mc; ∂Mc) −−→ (Yb; ∂Yb) −−→ Xb × I and consider the null bordant L-cycle obtained from f, f×id id (Lb × I; Lb × ∂I) −−−−→ (Xb × I; Xb × ∂I) −−−→ Xb × I. Adding these L-cycles together produces another representative of α, φtf×id ωtid (Mc t Lb × I; ∂Mc t Lb × ∂I) −−−−−−→ (Yb t Xb × I; ∂Yb t Xb × ∂I) −−−−→ Xb × I, We can improve this representative using Theorem 2.2 to obtain φ0 ω0 (Mc0; ∂Mc t Lb × ∂I) −−→ (Yb 0; ∂Yb t Xb × ∂I) −−→ Xb × I, with φ0 and ω0 equal to φ and ω when restricted to the boundary, and with ω satisfying the Π-Π condition. Let us write 0 0 ∂1Mc = ∂Mc t Lb × {1}, ∂2Mc = Lb × {0} = Lb 0 0 ∂1Yb = ∂Yb t Xb × {1}, ∂2Yb = Xb × {0} = X.b By commutativity of Xb × {0} / Yb 0

ω0| ω0 Xb × {0} / Xb × I STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 19

0 0 we see that the inclusion ∂2Yb ,→ Yb satisﬁes the Π-Π condition. Since moreover the map 0 0 0 φ | : ∂1Mc −→ ∂1Yb is a BQ-equivalence, we can apply Theorem 2.1, the Π-Π theorem, relative to this part of the boundary to ﬁnd a BQ-equivalence φ00 Mc00 −−−→ Yb 0 00 00 00 00 0 00 0 0 where ∂Mc = ∂1Mc t ∂2Mc , with ∂1Mc = ∂1Mc and φ is equal to φ on ∂1Mc , and 00 00 00 00 0 φ : ∂2Mc −→ ∂2Yb a BQ-equivalence. The BQ-equivalence φ is related to φ by a bordism Φ Nb −−→ Yb 0 × I 0 00 0 where ∂Nb = Mc ∪ Mc ∪ (∂1Mc × I) ∪ P,b 0 00 0 0 0 00 0 0 00 Mc ∩ Mc = ∅, Mc ∩ (∂1Mc × I) = ∂1Mc = Mc ∩ (∂1Mc × I), ∂Pb = ∂2Mc t ∂2Mc .

The restriction of Φ to Pb yields an element γ of LBQ(Xb × I)

00 Φ| 0 id (Pb; L,b ∂2Mc ) −−→ (∂2Yb × I = Xb × I; Xb × {0}, Xb × {1}) −−−→ Xb × I of the kind required in the statement of the theorem. (Incidentally, note that the fact that Φ| is BQ-normal and not a BQ-equivalence is why the bordism N is not a null bordism for Pb b α.) The BQ-normal map Φ : Nb −→ Yb 0 × I is a null bordism of the L-cycle M 0 ∪ P −→ Y 0 ∪ X × I −→ X × I c Lb b b Xb b b which shows, by Lemma 2.3, that α and γ represent the same class in LBQ(Xb × I).

Corollary 2.6. Given [α] ∈ LBQ(Xb ×I) and [β] ∈ SBQ(Xb) we can use the theorem to choose representatives of the form f β : Mc −−→ Xb

(φ;id,φ2) id α :(Wc; M,c Mc0) −−−−−−→ (Mc × [0, 1]; Mc × {0}, Mc × {1}) −−−→ Mc × I,

0 and then the class of f ◦ φ2 : Mc −→ Xb in SBQ(Xb) is well-deﬁned and denoted ∂(α)(β). The map

LBQ(Xb × I) × SBQ(Xb) / SBQ(Xb) ([α], [β]) / ∂(α)(β) deﬁnes a group action of the Browder-Quinn L-group of Xb × I on the structure set of X.b

Proof. Since f : Mc −→ Xb is a BQ-equivalence, LBQ(Xb × I) = LBQ(Mc × I) and we can use id the Wall representation theorem starting with the BQ-equivalence Mc −−−→ Mc to represent α as above. If we ﬁx the representative β then any two representatives α, α0 of [α] as above can be glued together along their common boundary and the result γ ∈ LBQ(M × I) represents the zero element of LBQ(M ×I). It follows, from Theorem 2.4 applied to Mc×I, that γ is normal 0 bordant relative to the boundary to a BQ-equivalence. Thus f ◦ φ2 and f ◦ φ2 represent the 20 PIERRE ALBIN AND PAOLO PIAZZA

same element of SBQ(M).

f 0 If the BQ-equivalence β0 : Lb −−→ Xb represents the same class as β, then there is a bordism between them (F ;f,f 0) (Nb; M,c Lb) −−−−−−→ (Xb × I; Xb × {0}, Xb × {1}). Using the theorem we can ﬁnd a representative of [α] of the form

(ψ:id,ψ2) α0 :(Vb; L,b Lb0) −−−−−−→ (Lb × [0, 1]; Lb × {0}, Lb × {1}). Now let us glue these, and α, together in the following order by matching the ‘lower boundary’ of one row with the ‘upper boundary’ of the following row,

0 (ψ:id,ψ2) f ×id (Vb; L,b Lb0) / (Lb × [0, 1]; Lb × {0}, Lb × {1}) / Xb × I

0 id (F ;f,f ) (Nb; M,c Lb) / (Nb; M,c Lb) / Xb × I

(φ;φ2,id) f×id (Wcop; Mc0, Mc) / (Mc × [0, 1]; Mc × {0}, Mc × {1}) / Xb × I

0 0 We end up with an cycle in LBQ(X × I) with ∂(α )(β ) along the upper boundary and ∂(α)(β) along the lower boundary. Moreover, this cycle is null bordant since by Lemma 2.3, it represents the class [α] + 0 − [α] = 0. It follows as in the previous case that ∂(α0)(β0) and ∂(α)(β) represent the same element in SBQ(Xb).

Compatibility with the group operation on LBQ(Xb × I) is easy as the operation is given by stacking normal bordisms together as in Lemma 2.3. Corollary 2.7. [Exactness part 2] Let Xb be a stratiﬁed space (without boundary). The sequence

θ ∂ η NBQ(Xb × I, Xb × ∂I) / LBQ(Xb × I) / SBQ(Xb) / NBQ(Xb) is exact in that two elements of the L-group have the same action on the class of the identity map precisely when their diﬀerence is in the image of θ, and two elements in the structure set are in the same orbit precisely when they have the same image under η.

Proof. Given x1, x2 ∈ LBQ(Xb × I) such that ∂(x1)(id) = ∂(x2)(id) we can use the Wall representation theorem to ﬁnd

Fi id (Wci; X,b ∂+Wci) −−→ (Xb × I; Xb × {0}, Xb × {1}) −−−→ Xb × I

F1 F2 representing xi. Without loss of generality ∂+Wc1 −−−→ Xb and ∂+Wc2 −−−→ Xb can be taken to be the same representative of ∂(x1)(id), so that we may form

F1tF2 id (W1 t −W2; X, X) −−−−−→ (X × I; X × {0}, X × {1}) −−−→ X × I c ∂+Wci c b b b b b b and recognize this as a representative of a class in NBQ(Xb × I, Xb × ∂I) representing x1 − x2. The converse follows by similar reasoning. 0 If [β], [β ] ∈ SBQ(Xb) have the same image under η then there is a normal bordism α 0 between β and β . This normal bordism deﬁnes an element of LBQ(Xb × I) whose action on STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 21

0 SBQ(Xb) sends [β] to [β ]. For the same reason elements in the structure set that are in an orbit of the action of an element of the L-group have the same image under η.

3. K-theory classes associated to the signature operator on Witt and Cheeger spaces 3.1. Metric structures. In order to do analysis we endow a stratified space with a Riemannian metric. Let Xb be a smoothly stratified space and Xe its resolution to a manifold with corners and an iterated fibration structure. Recall from, e.g., [ALMP12, ALMP16, ALMP17], that an iterated incomplete edge metric (briefly, an iie-metric) is a Riemannian metric on the interior of Xe (or, better, a bundle metric on the iterated incomplete edge tangent bundle over all of Xe) that in a collar neighborhood of each collective boundary hypersurface BY takes the form 2 2 ∗ dx + x gZ + φY gY .

Here x is a boundary defining function for BY , i.e., a smooth non-negative function on Xe that is positive except at BY = {x = 0} where it vanishes to exactly first order, and gZ and gY are metrics with the same structure on the spaces Z and Y. (Thus this is really an inductive definition over the depth of a stratified space, with spaces of depth zero being assigned smooth Riemannian metrics, see loc. cit..) In particular we point out that an iie-metric on Xb includes a Riemannian metric on each stratum of Xb and that these metrics fit together continuously (even smoothly in that they lift to a smooth section over Xe). Thus endowing Xb with an iie-metric in particular gives Xb the structure of a ‘Riemannian Whitney (A) space’ in the sense of Pflaum [Pfl01, §2.4]. (Note that the latter concept is more general, e.g., if we were working with metrics that were 2 2` ∗ asymptotically of the form dx + x gZ + φY gY for any ` we would still get a ‘Riemannian Whitney (A) space’.) In particular, by Theorem 2.4.7 in [Pfl01], the topology on Xb is that of the metric space with distance between two points given by taking the infimum over rectificable curves joining them. As a metric space, Xb is complete and locally compact [Pfl01, Theorem 2.4.17] and hence a ‘length space’.

3.2. Galois coverings. Let Xb be a smoothly stratified pseudomanifold of arbitrary depth. Consider a Galois covering π : XbΓ → Xb with Galois group Γ and fundamental domain FcΓ. There is a natural way to define a topological stratification on XbΓ. Decompose XbΓ into the preimages under π of the strata in Xb. Surjectivity of π ensures that each stratum in the covering is mapped surjectively onto the corresponding stratum in Xb. Since π is a local homeomorphism, it is straightforward to check that XbΓ and its fundamental domain are again topological stratified spaces. In fact, more is true: by using these local homeomorphisms we can induce a smooth stratification on XbΓ by simply pulling it up from the base, in either the Whitney as well as the Thom-Mather cases. It is important to point out that, by definition, the link of a point pe ∈ XbΓ is equal to the link of its image, p, in the base. This construction exhibits the 22 PIERRE ALBIN AND PAOLO PIAZZA

covering map π as a transverse map and thus if Xb belongs to a class C as above, then so does XbΓ. Needless to say, if XbΓ is the universal covering space of Xb, the individual strata in XbΓ need not be the universal covering of the corresponding strata in the base. We denote by XΓ the regular stratum of XbΓ and observe that it is a Galois covering of the regular stratum X of Xb with fundamental domain FΓ equal to the regular part of FcΓ. Let g be an admissible incomplete edge metric on X. We can lift g to the Galois covering XΓ where it becomes a Γ-invariant admissible incomplete edge metric ge. Moreover, there is an isometric embedding of FΓ into X with complement of measure zero. We denote by DΓ the signature operator on XΓ associated to such a metric.

3.3. C∗ and D∗ algebras. First of all, we need to fix an Hilbert space H with a unitary action of Γ and a C∗- representation from C0(XbΓ) to B(H) intertwining the two actions of Γ. Notice that the representation is associated to the stratified Galois covering XbΓ (and not to its regular part 2 ∗ XΓ). We take H = L (XΓ, Λ XΓ); the representation is given by the multiplication operator associated to the restriction of a function to the regular part XΓ. To these data we can ∗ ∗ Γ associate two C -algebras: the Roe algebra C (XbΓ,H) , obtained as the closure of the Γ- equivariant finite propagation locally compact bounded operators on H, and the Higson-Roe ∗ Γ algebra D (XbΓ,H) , obtained as the closure of the Γ-equivariant finite propagation pseudolocal bounded operators on H. Since we shall be eventually interested in the K-theory groups of these C∗-algebras and since the K-theory groups are independent of the choice of ∗ Γ the (adequate) Γ equivariant Cc(XbΓ)-module H, we shall adopt the notation C (XbΓ) and ∗ Γ ∗ D (XbΓ) for these two C -algebras. 1 Then we have the following fundamental

Proposition 3.1. Let (X,b g) and (XbΓ, ge) as above. Assume that Xb is a Cheeger-space and let W be a self-dual mezzoperversity for D. Then:

• there exists a closed Γ-equivariant self adjoint extension of DΓ associated to W, de- W noted DΓ . W ∗ Γ • if φ ∈ C0(R), then φ(DΓ ) ∈ C (XbΓ) . • if χ is a chopping function (i.e. χ : R → [−1, 1] is odd and limx→±∞ χ(x) = ±1) W ∗ Γ then χ(DΓ ) ∈ D (XbΓ) . Proof. (1) The pull-back of the mezzoperversity W along the covering map π is a mezzoperversity on XΓ, which we briefly denote WΓ. The definition of the domain associated to a mezzoperversity in [ALMP16] applies in the setting of XΓ, as the asymptotic expansions on which it relies are carried out in distinguished neighborhoods of points on the singular strata and these are the same on Xb or XbΓ. Similarly we can define DWΓ (d) and DWΓ (δ) as in [ALMP16] and see that they are mutually adjoint and that

W DWΓ (DΓ ) = DWΓ (d) ∩ DWΓ (δ) W so that DΓ with this domain is self-adjoint.

1 2 ∗ 2 For technical reasons having to do with functoriality one actually takes H = L (XΓ, Λ XΓ) ⊗ ` (N). STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 23

2 ∗ The analysis of [ALMP16] that establishes that DW (D) includes compactly into L (X;Λ X) implies that, for any compact subset K ⊆ XbΓ, W {u ∈ DWΓ (DΓ ) : supp(u) ⊆ K} includes compactly into H. (2) As far as the second item is concerned we initially tackle the local compactness of W W W φ(DΓ ). We have to prove that if g ∈ Cc(XbΓ) then gφ(DΓ ) and φ(DΓ )g are compact W operators. By taking adjoints it suffices to see that gφ(DΓ ) is compact. Using the Stone- Weistrass theorem it suffices to establish this property for the function φ(x) = (i + x)−1. W As this maps H into DWΓ (DΓ ), the local compactness of the inclusion of the latter into H W implies that of φ(DΓ ). Next we consider the finite propagation property: by a density argument it suffices to see such a property for smooth functions φ that are of rapid decay and have compactly supported Fourier transform. Thus, let φb be the Fourier transform of a smooth rapidly decaying φ and let us assume that the support of φb is contained in [−R/2,R/2]. We must show that there + W exists S ∈ R such that fφ(DΓ )g = 0 whenever the distance between the support of f and g is greater than S. Proceeding precisely as in [ALMP17, Theorem 5.3] we know that there W W exists a δ such that exp(isDΓ ) has propagation |s| if |s| < δ; thus f exp(isDΓ )g = 0 if the distance of the supports of f and g is greater than |s|, with s in the range (−δ, δ). Recall now that, by functional calculus, we can write 1 Z φ(DW ) = exp(isDW )φb(s)ds Γ 2π Γ where the integral converges weakly: 1 Z hφ(DW )u, vi = hexp(isDW )u, viφb(s)ds Γ 2π Γ

for each pair of compactly supported sections on XΓ. Assume initially that R < δ. Then, W from the above integral representation, we see that φ(DΓ ) has finite propagation (in fact, propagation R) which is what we wanted to prove. For the general case we use a trick P from [Roe87]. Write φ = j fj where the sum is finite and where fj has Fourier transform W supported in (Tj − δ/2,Tj + δ/2). Consider gj(x) = exp(−iTjx)fj(x). Then gj(DΓ ) has propagation δ by what we have just seen. Write now k Y exp(iTjx) = exp(iτ`x) with |τ`| < δ . `=1 Qk We have then fj(x) = `=1 exp(iτ`x)gj(x) and thus W W W W fj(DΓ ) = exp(iτ1DΓ ) ◦ · · · ◦ exp(iτkDΓ ) ◦ gj(DΓ ) . All the operators appearing on the right hand side have finite propagations and we know that the composition of two operators of finite propagation is again of finite propagation. W Thus fj(DΓ ) has finite propagation. The proof of item 2 is complete. (3) Let χ be a chopping function. Recall from [HR00, Section 10.6] that for every t > 0 there exists a chopping function χ with (distributional) Fourier transform supported in (−t, t). Moreover, if χ0 and χ are two arbitrary chopping function, then χ0 − χ = φ, with W W φ ∈ C0(R). This implies immediately that if χ0(DΓ ) is of finite propagation then χ1(DΓ ) is W W a limit of finite propagation operators; moreover χ0(DΓ )g − χ(DΓ )g is a compact operator 24 PIERRE ALBIN AND PAOLO PIAZZA for any g ∈ Cc(XbΓ). We choose a chopping function χ0 with Fourier transform supported in W (−δ/2, δ/2). Then we know, from the previous arguments, that χ0(DΓ ) is of propagation W δ. Hence χ(DΓ ) is a limit of finite propagation operators for each chopping function χ. It W W remains to see that χ(DΓ ) is pseudolocal, i.e. [f, χ(DΓ )] is compact for any f ∈ C0(XbΓ). By Kasparov’s Lemma, see [HR00, Lemma 5.4.7] and [HR00, Lemma 10.6.4], we know that W W χ(DΓ ) is pseudolocal if and only if fχ(DΓ )g is compact for any choice of f ∈ C(XbΓ) bounded and g ∈ Cc(XbΓ) with disjoint supports. Now, if η > 0 is the distance between the support of f and the support of g and if we choose a chopping function χ0 with Fourier W transform supported in (−η/2, η/2) then we know that fχ0(DΓ )g = 0. But then for an arbitrary chopping function χ we have W W W W W W fχ(DΓ )g = fχ0(DΓ )g + f(χ(DΓ ) − χ0(DΓ ))g = 0 + f(χ(DΓ ) − χ0(DΓ ))g W W and since f is bounded and (χ(DΓ ) − χ0(DΓ ))g is compact we see that on the right hand side we do have a compact operator as required. The proof of item 3 is now complete. 3.4. K-homology classes. Proposition 3.1 allows us to recover, in the bounded picture, the fundamental classes that were defined in [ALMP12] for Witt spaces and in [ALMP17] for Cheeger spaces. More precisely:

Proposition 3.2. If Xb is an n-dimensional Cheeger space endowed with a rigid iterated conic metric g and if W is a self-dual mezzoperversity adapted to g then there is a well W defined K-homology signature class [D ] ∈ Kn(Xb). 2 W Proof. Let χ be a chopping function; then χ − 1 is an element in C0(R) and thus χ(DΓ ) ∗ Γ ∗ Γ is an involution in the quotient D (XbΓ) /C (XbΓ) . Thus, using also the grading in even ∗ Γ ∗ Γ dimension, one defines an element in Kn+1(D (XbΓ) /C (XbΓ) ) which is precisely Kn(Xb) by Paschke duality. Remark 7. The class we have just defined does coincide with the one defined in [ALMP17]: this follows from the proof of [HR00, Theorem 10.6.5] and the correspondence between the unbounded and bounded picture for K-homology.

W Remark 8. The class [D ]Q ∈ Kn(Xb) ⊗ Q is independent of the choice of self-dual mez- W zoperversity W; indeed the homological Chern character of [D ]Q, in H∗(X,b Q) is equal to the L-class of the Cheeger space, see [ALMP17, Thorem 5.6], and we know that the L-class is independent of the choice of W, see [ALMP17, Section 5.1]. We shall come back to this point later on. 3.5. Higson-Roe sequences associated to a Thom-Mather space. If Xb is a Thom-Mather stratiﬁed space and XbΓ is a Galois covering with structure group Γ, then there is a short exact sequence of C∗-algebras ∗ Γ ∗ Γ ∗ Γ ∗ Γ 0 → C (XbΓ) → D (XbΓ) → D (XbΓ) /C (XbΓ) → 0 and thus a 6-term long exact sequence in K-theory: (3.1) ∗ Γ ∗ Γ ∗ Γ ∗ Γ ∗ Γ · · · → Km+1(C (XbΓ) ) → Km+1(D (XbΓ) ) → Km+1(D (XbΓ) /C (XbΓ) ) → Km(C (XbΓ) ) → · · · STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 25

This is the analytic surgery sequence of Higson and Roe associated to the Γ-compact Γ-space ∗ Γ ∗ Γ XbΓ. Since we have the canonical isomorphism K∗+1(D (XbΓ) /C (XbΓ) ) = K∗(Xb) we can also rewrite (3.1) as

∗ Γ ∗ Γ ∗ Γ (3.2) · · · → Km+1(C (XbΓ) ) → Km+1(D (XbΓ) ) → Km(Xb) → Km(C (XbΓ) ) → · · ·

∗ Γ ∗ Moreover, recall that we have a canonical isomorphism K∗(C (XbΓ) ) = K∗(Cr Γ). Now, in particular, all of the above is true with XbΓ equal to the universal covering of Xb and Γ = π1(Xb). Consider now a closed stratum Yb of Xb: thus if Y is a stratum of Xb then [ Yb := Y = {Yi : Yi ≤ Y }

Consider Γ(Yb) := π1(Yb); then we have a 6-term exact sequence similar to (3.2) but associated to the universal covering, Γ(Y ) − Y → Y of Y . b bΓ(Yb ) b b

3.6. Index classes. Let now Xb be an n-dimensional Cheeger space and let us choose a self-dual mezzoper- W versity W. Then, by Proposition (3.2), we have a K-homology class [D ] ∈ Kn(Xb) = ∗ Γ ∗ Γ Kn+1(D (XbΓ) /C (XbΓ) ) and thus an index class

W W ∗ Γ (3.3) Ind(DΓ ) := ∂[D ] ∈ Kn(C (XbΓ) ), with ∂ the connecting homomorphism in the Higson-Roe surgery sequence. Following the proof given in [PS14, Proposition 2.1] this class corresponds to the one considered in [ALMP16] ∗ Γ ∗ through the canonical isomorphism K∗(C (XbΓ) ) = K∗(Cr Γ); notice that the class deﬁned in [ALMP16] is a Mishchenko class, obtained by twisting the signature operator by the ∗ Mishchenko bundle XbΓ ×Γ Cr Γ. Both in the Higson-Roe formalism, see [HR10], and in the Mishchenko formalism, we can also consider the index class with values in the maximal version of our C∗-algebras. We denote the maximal group C∗-algebra associated to Γ as C∗Γ. Finally, if Yb is a closed m-dimensional stratum with fundamental group Γ(Yb) then W induces a self-dual mezzoperversity W(Yb) for Yb and we obtain a K-homology class [DW(Yb )] ∈ W(Yb ) ∗ Km(Yb) and thus an Index class Ind(D ) ∈ Km(C (Γ(Yb)). Γ(Yb ) r

3.7. Rho classes associated to trivializing perturbations. Let Xb be a Cheeger space endowed with an iie metric g. We initially assume that Xb is odd dimensional. Let W be a self-dual mezzoperversity for Xb and let DW be the corresponding signature operator, an unbounded self-adjoint operator on L2(X, Λ∗X). (Recall that DW is a short notation for the pair (D, DW (D)), the (extension of the) signature operator on (X, g), the regular part of Xb endowed with the Riemannian metric g, with domain deﬁned by the self-dual mezzoperversity W.) Given a Galois Γ-covering XbΓ of Xb, we also have the W 2 ∗ Γ-equivariant signature operator DΓ , a self-adjoint unbounded operator on L (XΓ, Λ XΓ). 2 ∗ W Let now A be a bounded Γ-equivariant self-adjoint operator on L (XΓ, Λ XΓ). Then DΓ +A, W with domain equal to the domain of DΓ , is also self-adjoint. Following [PS16, Section 2B] we W 2 ∗ Γ make the assumption that DΓ +A is L -invertible and that A ∈ M(C (XbΓ) ), the multiplier ∗ Γ algebra of C (XbΓ) . We refer to A as a trivializing perturbation. Then, using Proposition 26 PIERRE ALBIN AND PAOLO PIAZZA

3.1 and [PS16, Proposition 2.8], we see that W DΓ + A ∗ Γ (3.4) W is an element in D (XbΓ) . |DΓ + A| W W Moreover, DΓ + A/|DΓ + A| is clearly an involution and thus W 1 DΓ + A W + 1 2 |DΓ + A| ∗ Γ W is a projection in D (XbΓ) . We define the rho class associated to DΓ + A as W W 1 DΓ + A ∗ Γ (3.5) ρ(DΓ + A) := W + 1 in K0(D (XbΓ) ) . 2 |DΓ + A| In the even dimensional case we consider the grading associated to the Hodge ? operator ∗ Γ and we demand that the trivializing perturbation A ∈ M(C (XbΓ) ) be odd with respect to W this grading; thus DΓ + A can be written as W,− − 0 DΓ + A W,+ + DΓ + A 0 We now fix a chopping function χ equal to the sign function on the spectrum of the invertible W iie ∗ iie ∗ operator DΓ + A; we also fix a Γ-equivariant isometry u :Λ−( T XΓ) → Λ+( T XΓ) and consider the induced bounded Γ-equivariant operator on the space of L2 sections of these W bundles, call it U. Observing that χ(DΓ + A) is also odd (see [HR00, Lemma 10.6.2]) we W ∗ Γ consider Uχ(DΓ + A)+ which is a unitary in D (XbΓ) . We then define the rho class in the even dimensional case an W W ∗ Γ (3.6) ρ(DΓ + A) := [Uχ(DΓ + A)+] in K1(D (XbΓ) ) As explained in [PS16, page 118] this is well defined, independent of the choice of u.

4. Bordisms and associated K-theory classes 4.1. Bordisms of Cheeger spaces. We recall here some fundamental facts established in [ALMP17]. Assume that M is a topological space and denote by Sign(M) the bordism group of four-tuples (X,b g, W, r : Xb −→ M) where Xb is an oriented Cheeger space of dimension n, g is an adapted iterated incomplete conic metric metric (briefly an iie metric), W is a self-dual Hodge mezzoperversity adapted to g and r : Xb −→ M is a continuous map. An admissible bordism between (X,b g, W, r : Xb −→ M) and (Xb0, g0, W0, r0 : Xb0 −→ M) is a four-tuple (Xc, G, W ,R : Xc−→ M) consisting of: i) a smoothly stratified, oriented, compact pseudomanifold with boundary Xc, whose boundary is Xb t (−Xb0), and whose strata near the boundary are collars of the strata of Xb or Xb0, ii) an iie metric G on Xc that near the boundary is of the collared form dx2 + g or dx2 + g0, iii) an adapted self-dual mezzoperversity W that extends, in a collared way, that of Xb and Xb0, iv) a map R : Xc−→ M that extends r and r0. We shall briefly say that (X,b g, W, r : Xb −→ M) and (Xb0, g0, W0, r0 : Xb0 −→ M) are Cheeger- bordant through (Xc, G, W ,R : Xc−→ M). We are mainly interested in the case M = BΓ, STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 27

so that a map r : Xb → BΓ deﬁnes a Galois Γ-covering XbΓ. We have the following important results, see [ALMP17] for proofs: Theorem 4.1. If (X,b g, W, r) and (Xb0, g0, W0, r0) are n-dimensional and Cheeger-bordant through (Xc, G, W ,R : Xc−→ M) then: 1] the numeric Fredholm indices associated to DW and (D0)W0 are equal; W 2] there exists a well deﬁned relative K-homology class [D ] ∈ Kn+1(Xc, ∂Xc); 0 3] if ∂ : Kn+1(Xc, ∂Xc) → Kn(∂Xc) ≡ Kn(Xb ∪ (−Xb )) is the connecting homomorphism associated to the long exact sequence of the pair (Xc, ∂Xc) then

W W 0 W0 1 ∂[D ] = [D ] − [(D ) ] in Kn(∂Xc) ⊗ [ ]; Z Z 2 4] the signature index classes associated to (X,b g, W, r : Xb −→ BΓ) and (Xb0, g0, W0, r0 : 0 ∗ 1 Xb −→ BΓ) are equal in K∗(Cr Γ) ⊗Z Z[ 2 ]. 5] If W and W0 are adapted to g and g0 on the same Cheeger space Xb and r : X → M is a continuous map then (X,b g, W, r : Xb → M) is Cheeger-bordant to (X,b g0, W0, r : Xb → M). In particular, the numeric Fredholm index, in Z, and the signature index class, in ∗ 1 K∗(Cr Γ) ⊗Z Z[ 2 ], are independent of the choice of self-dual mezzoperversity. ∗ Remark 9. The statements in [ALMP17] are given with values in K0(C Γ) ⊗Z Q but it is ∗ 1 easy to see that the arguments given there establish the same results in K0(C Γ) ⊗Z Z[ 2 ]. The main idea behind the formulation and the proof of item 5] is due to Markus Banagl, see [Ban06]. W ∗ 1 Notation. Since the signature index class Ind(DΓ ) ∈ K∗(Cr Γ) ⊗Z Z[ 2 ] associated to a Galois covering r : Xb → BΓ is in fact independent of the cloice of W, we shall often denote it simply by Ind(DΓ) or even Ind(XbΓ). 4.2. The signature operator on Cheeger spaces with cylindrical ends. Let Γ−XcΓ → Xcbe a Galois Γ-covering of an even dimensional Cheeger space with boundary. We consider a rigid iie metric g on the regular part X which is collared near ∂X and we lift it to a Γ-equivariant rigid iie metric on XΓ. We also consider the Cheeger spaces with cylindrical ends, Xc∞, XcΓ,∞, obtained by attaching (−∞, 0] × ∂Xc and (−∞, 0] × ∂XcΓ to Xc, XcΓ respectively. We endow (−∞, 0]×∂X and (−∞, 0]×∂XΓ with product metrics and we obtain in this way global metrics on X∞ and X∞,Γ. If W is a self-dual mezzoperversity on X then we obtain in a natural way a self-dual mezzoperversity W∞ on X∞ and thus, by lifting, a Γ-equivariant self-dual mezzoperversity WΓ,∞ on X∞,Γ. We denote by ∂W the self-dual mezzoperversity induced on the boundary, see Theorem 4.1. Finally, we denote by P0 the multiplication operator by the characteristic function of the attached semi-cylinder.

Proposition 4.2. Let D∞ and D∞,Γ be the the signature operators on X∞ and X∞,Γ re- W W spectively. By employing W∞ and WΓ,∞ we can deﬁne self-adjoint extensions D∞ and D∞,Γ

Proof. Extend the iterated fibration structure from X to X∞ by including the cylindrical direction in the base of each fiber bundle. Define

W D∞ = (D∞, DW∞ (D∞)), where DW ,∞(D) = {u ∈ Dmax(D∞) : at each singular stratum,

u satisﬁes the ideal boundary condition corresponding to W∞}. 28 PIERRE ALBIN AND PAOLO PIAZZA

It is easy to see that this is a self-adjoint domain. Indeed, this domain is localizable (an element is in the domain if and only if it is in the domain after multiplying by any function ∞ in CΦ (X), see [ALMP16, §2] and, e.g., the discussion after assumption 3.8 in [ALMP16].) and so it suffices to show that the corresponding domain on the full cylinder ∂X × R is self-adjoint; here we could either use Fourier transform in the R-factor to reduce to the self-adjointness of D∂W , or alternately recognise ∂X × R as a cover of ∂X × S1, consider the pull-back of ∂W to this product, and then appeal to Proposition 3.1 above. 4.3. Perturbations and coarse APS-index classes. In this subsection we shall define APS-index classes associated to the self-adjoint operator W ∗ ∗ Γ D∞,Γ. We consider the C -algebra C (XcΓ ⊂ XcΓ,∞) obtained by closing the operators ∗ Γ T in Cc (XcΓ,∞) with the additional property that ∃ R > 0 such that φT = 0 = T φ whenever φ ∈ Cc(XcΓ,∞) and d(supp(φ), XcΓ) > R. See for example [PS14, Definition 1.7]. ∗ Γ A similar definition can be given for D (XcΓ ⊂ XcΓ,∞) . One can prove that the inclusion c : XcΓ ,→ XcΓ,∞ induces K-theory isomorphisms: ∗ Γ ∗ Γ ∗ Γ ∗ Γ (4.1) K∗(C (XcΓ) ) ' K∗(C (XcΓ ⊂ XcΓ,∞) ); K∗(D (XcΓ) ) ' K∗(D (XcΓ ⊂ XcΓ,∞) ) . ∗ Γ ∗ Γ ∗ Γ Notice that C (XcΓ ⊂ XcΓ,∞) ) and D (XcΓ ⊂ XcΓ,∞) are ideals in D (XcΓ,∞) . ∂W We assume the existence of a trivializing perturbation C∂ for the signature operator DΓ ∂W on ∂XΓ: this means, as before, that C∂ is bounded, that DΓ + C∂ (with domain equal to ∂W 2 ∗ Γ the domain of DΓ ) is L -invertible and that C∂ ∈ M(C (∂XcΓ) ). C∂ ⊗ IdR then defines 2 a bounded operator on L (∂XcΓ × R). We can then graft this perturbation on XΓ,∞ and W obtain a bounded perturbation C∞ for D∞,Γ. In the case of interest to us it will be the case that C∞ is a limit of finite propagation operators and so we assume this property in what follows. In fact, we might more generally consider a perturbation B∞ which is a limit of finite propagation operators and such that ∗ Γ (4.2) P0B∞P0 − P0C∞P0 ∈ C (XcΓ ⊂ XcΓ,∞) .

Proposition 4.3. Let C∂, C∞ and B∞ be as above. If φ ∈ C0(R) and if χ is a chopping ∂W function equal to the sign function on the spectrum of DΓ + C∂, then: W ∗ Γ 1] φ(DΓ,∞ + B∞) ∈ C (XcΓ,∞) ; W ∗ Γ 2] χ(DΓ,∞ + B∞) ∈ D (XcΓ,∞) ; W ∗ Γ 3] χ(DΓ,∞ + B∞) is an involution modulo C (XcΓ ⊂ XcΓ,∞) . Proof. For 1] and 2] we use Proposition 3.1 and the purely functional analytic arguments given in [PS16, Lemma 2.25]. For the third item we use the proof of Proposition 2.26 in [PS16], which is once again purely functional analytic.

Given C∂ as above, choosing B∞ = C∞ and using Proposition 4.3 we can deﬁne a coarse relative index class rel W W ∗ Γ Ind (DΓ,∞ + C∞) := ∂[χ(DΓ,∞ + C∞)] ∈ K∗(C (XcΓ ⊂ XcΓ,∞) ) and thus, using (4.1), a coarse APS-index class

W −1 rel W ∗ Γ ∗ Ind(DΓ ,C) := c∗ (Ind (DΓ,∞ + C∞)) ∈ K∗(C (XcΓ) ) ' K∗(Cr Γ) STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 29

Notice that the left hand side is just notation; we have not really deﬁned a perturbation C on XcΓ. One can prove, following the arguments in the proof of [PS16, Proposition 2.33], that for C∂, C∞ and B∞ as above:

rel W rel W ∗ Γ Ind (DΓ,∞ + C∞) = Ind (DΓ,∞ + B∞) ∈ K∗(C (XcΓ ⊂ XcΓ,∞) ) where the right hand side is well deﬁned because of item 3] of Proposition 4.3.

4.4. The delocalized APS index theorem. W W W W ∂W ∂W Let X , XΓ, X∞, XΓ,∞, D , DΓ , D∞, DΓ,∞, D , DΓ and C∂ as in the previous sub- sections. We assume X to be even dimensional. By assumption C∂ is a trivializing pertur- ∂W ∗ Γ bation for DΓ ; assume that C∂ ∈ C (∂W ) ), so that C∂ is a norm limit of ﬁnite propagation operators. Consequently C∞ is also a norm limit of ﬁnite propagation operators. We ∂W ∗ Γ can consider the rho class ρ(DΓ + C∂) ∈ K0(D (∂XΓ) ) and the coarse-APS index class W ∗ Γ ∗ Γ ∗ Γ Ind(DΓ ,C) ∈ K0(C (XΓ) ). Let ι : C (XΓ) ,→ D (XΓ) be the natural inclusion and ∗ Γ ∗ Γ consider j∗ : K0(D (∂XΓ) )) → K0(D (XΓ) ) induced by the inclusion of ∂XΓ into XΓ. Our main tool in the next section will be the delocalized APS index theorem for perturbed signature operators on Cheeger spaces:

Theorem 4.4. If the trivializing perturbation C∂ is a norm limit of ﬁnite propagation operators, then the following equality holds

W ∂W ∗ Γ (4.3) ι∗(Ind(DΓ ,C)) = j∗(ρ(DΓ + C∂)) in K0(D (XΓ) ). Proof. All the arguments given in [PS14] and then [PS16] are functional analytic with the exception of the proof of Proposition 5.33 in [PS16]. However, the alternative proof of this particular Proposition given by Zenobi, see Proposition 3.20 in [Zen17] , applies verbatim to the present context. Let now X be odd dimensional. After inverting 2 we can reduce the delocalized APS index theorem on X to the one on X × S1 by a suspension argument. This is discussed ∗ Γ carefully in [Zen17, §5] where a diﬀerent description of the group K∗(D (XbΓ) ) is given for metric spaces with Γ-actions. These arguments apply in our situation largely unchanged.

5. Stratified homotopy equivalences and associated perturbations 5.1. The Hilsum-Skandalis replacement. Let Xb be a Cheeger space, r : Xb −→ BΓ the classifying map for the universal cover of X,b G (r) the Mishchenko bundle associated to r, and WX a self-dual mezzoperversity on X.b If Mc is another Cheeger spaces and f : Mc −→ Xb a stratiﬁed homotopy equivalence then (see [ALMP17, Theorem 4.6]), there is a ‘Hilsum-Skandalis replacement’ for the pull-back of diﬀerential forms by f, HS(f): L2(X;Λ∗iieT ∗X ⊗ G (r)) −→ L2(M;Λ∗iieT ∗M ⊗ G (f ◦ r)),

] that we can use to deﬁne a self-dual mezzoperversity WM = f (WX ) on M.c These data satisfy G (r) G (f◦r) G (r) G (f◦r) • HS(f)d = d HS(f) and HS(f)(DWX (d )) ⊆ DWM (d ), 30 PIERRE ALBIN AND PAOLO PIAZZA

2 G (r) • There is an L -bounded operator Υ acting on DWX (d ), such that 0 Id −HS(f) HS(f) = dG (r)Υ + ΥdG (r), where HS(f)0 denotes the adjoint with respect to the quadratic form deﬁned by the Hodge operator. We point out that the boundedness of HS(f) on L2(X;Λ∗iieT ∗X ⊗ G (r)), together with the ﬁrst of these properties, implies that HS(f) is bounded as a map

G (r) G (f◦r) HS(f): DWX (d ) −→ DWM (d ), when these spaces are endowed with the respective d-graph norm. Similarly Υ is bounded G (r) 2 as an operator on the Hilbert space DWX (d ). Note however that HS(f) does not map L diﬀerential forms into the maximal domain of d; indeed, if a diﬀerential form extends to be smooth on the closure of Xe and its exterior derivative fails to be in L2, then the same will be true of its image under HS(f).

5.2. The compressed Hilsum-Skandalis replacement. Following [PS07a], we will also make use of a compressed version of the Hilsum-Skandalis replacement. In this case the replacement will make use of a ﬁxed mezzoperversity and will have the property that it maps all of the L2 diﬀerential forms into the domain of d. Recall W f ]W that one of the main results in [ALMP17] is that the resolvents of DG(r) and DG(r◦f) are ∗ Cr Γ-compacts.

Definition 5. Let Xb be a Cheeger space, WX a self-dual mezzoperversity, r : Xb −→ BΓ the classifying map for the universal cover of Xb and f : Mc −→ Xb be a smooth stratified map. For each µ : R −→ R an even, rapidly decreasing function, we define the compressed Hilsum-Skandalis replacement of f to be the operator

2 ∗iie ∗ 2 ∗iie ∗ HSµ(f): L (X;Λ T X ⊗ G (r)) −→ L (M;Λ T M ⊗ G (f ◦ r)), ] f WX WX HSµ(f) = µ(D ) ◦ HS(f) ◦ µ(D ). As elements of the functional calculus we know that, e.g., µ(DWX ): L2(X;Λ∗iieT ∗X ⊗ G (r)) −→ L2(X;Λ∗iieT ∗X ⊗ G (r)) commutes with DWX and is a bounded operator with range contained in the domain of DWX . In fact the range is contained in the domain

∞ \ ` DWX (D ) = {ω ∈ DWX (D): Dω, . . . , D ω ∈ DWX (D)} `∈N as x`µ(x) is a rapidly decreasing function for any ` ∈ N. Since this domain is compactly included in L2(X;Λ∗iieT ∗X ⊗G (r)), it follows that µ(DWX ) is a compact operator. Moreover since µ is even and d commutes with (d + δ)2, d commutes with µ(D). Thus,

] WX f WX HSµ(f) is a compact operator and HSµ(f)d = d HSµ(f). The compressed Hilsum-Skandalis replacement satisﬁes properties similar to those of HS(f), see Lemma 9.7 in [PS07a]. STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 31

5.3. The Hilsum-Skandalis perturbation. On X t −M consider the operators dX 0 τX 0 dXt−M = , τXt−M = , 0 dM 0 −τM and, for t ∈ [0, 1], the operator

] ] Lt : DWX tf WX (dXt−M ) −→ DWX tf WX (dXt−M ) (5.1) Id −HS(f)0HS(f) (1 − itγΥ) ◦ HS(f)0 L = . t HS(f) ◦ (1 + itγΥ) Id

] We point out that Lt is bounded as an operator on the space DWX tf WX (dXt−M ) endowed √ ∗ with its dXt−M -graph norm, and let |Lt| = Lt Lt denote the operator deﬁned by the functional calculus on this Hilbert space (or equivalently as a bounded operator on L2- diﬀerential forms). As in [HS92], the Hilsum-Skandalis replacement can be used to construct a perturbation of the signature operator

W ] D X 0 DWX tf WX = on X t (−M) 0 −DWM b c that results in an invertible operator. Indeed, for suﬃciently small t, the operator

] WX tf WX 1 −1 D + Ct(f) = i Ut ◦ Dt ◦ Ut

] WX tf WX is invertible, where Dt is the operator obtained from D by making two replacements: 0 dX tHS(f) dXt−M 7→ , τXt−M 7→ sign (τXt−M ◦ Lt) = τXt−M ◦ sign(Lt) 0 dM and 1/2 1/2 Ut = |τXt−M ◦ Lt| = |Lt| .

] WX tf WX Lemma 5.1. The operator Ct(f) is a bounded operator relative to D ; that is, Ct(f) is bounded as a map

2 ] Ct(f): DWX tf WX (D) −→ L (X t −M;Λ∗(X t −M)).

] WX tf WX The operator D + Ct(f) is invertible for small enough t > 0.

] WX tf WX Proof. The boundedness of Ct(f) relative to D follows from the fact that Lt is a bounded operator on DWX (d). With notation similar to [Zen17, Proof of Proposition 3.4], we can write 0 tHS(f)0 E = , sign(L ) = Id +G ,U = Id +H0,U −1 = Id +F 0 t 0 0 t t t t t t

0 0 ] with Et,Gt,Ht,Ft bounded operators on DWX tf WX (d), with its d-graph norm as well as 2 on L (X t −M;Λ∗(X t −M)). Then, e.g., in the even dimensional case we can write

1 0 0 Dt = i (1+Ft )◦((d+Et)+τXt−M ◦(1+Gt)◦(d+Et)◦τXt−M ◦(1+Gt))◦(1+Ht) = D+Ct(f) 32 PIERRE ALBIN AND PAOLO PIAZZA

2 ] and it follows that Ct(f) is bounded as a map from DWX tf WX (D) to L (X t −M;Λ∗(X t −M)). Note that L0 satisﬁes Id 0 L = R0R,R = 0 HS(f) Id

and, since R is invertible, this shows that L0 is invertible and hence Lt is invertible for small ] WX tf WX enough t. The invertibility of D + Ct(f) as an unbounded operator with domain ] DWX tf WX (DXt−M ) now follows as in [HS92, Lemme 2.1], [Wah13, §3]. A similar result holds for the signature operator on X t(−M), with mezzoperversity given ] by WX and f WX and twisted by the Mishchenko bundle G (r) on X and G (f ◦ r) on M. In G (r) G (f◦r) this case we use the Hilsum-Skandalis replacement HS(f): DWX (d ) −→ DWM (d ).

5.4. The compressed Hilsum-Skandalis perturbation. We can repeat the argument from the previous subsection replacing HS(f) by HSµ(f). The resulting perturbation, which we denote Ct,µ(f) and refer to as the compressed Hilsum- Skandalis perturbation, satisﬁes an improved version of Lemma 5.1.

] Lemma 5.2. The operator Ct,µ(f) extends from DWX tf WX (DXt−M ) to a compact operator

2 2 Ct,µ(f): L (X t −M;Λ∗(X t −M)) −→ L (X t −M;Λ∗(X t −M)).

] WX tf WX The operator D + Ct,µ(f) is invertible.

Proof. If Lt,µ is the operator obtained as in (5.1) but using HSµ(f), then it is an invertible operator of the form Id +Ht,µ with Ht,µ a compact operator such that both Ht,µ and its 2 ∞ ] adjoint map L (X t −M;Λ∗(X t −M)) into DWX tf WX (DXt−M ). It follows from, e.g., the 0 0 argument used in Lemma A.12 of [PS07a], that each of the operators Et,µ,Gt,µ,Ht,µ,Ft,µ deﬁned as in the proof of Lemma 5.1 will also have this property. Hence Ct,µ(f) will be a compact operator. The invertibility of the perturbed signature operator follows from [HS92, Lemme 2.1]. Also in this case we can extend the whole analysis to the signature operators twisted by the appropriate Mishchenko bundles; we state and use this result in Proposition 5.3 below.

5.5. Passing to the Roe algebra. ∗ 2 ∗ Let B(E) denote the operators acting on the Hilbert Cr Γ-module E := L (X, Λ X ⊗ G(r)) ⊕ L2(M, Λ∗M ⊗ G(r ◦ f)). Recall that there is a C∗-homomorphism

2 2 Lπ : B(E) → B(L (XΓ, Λ∗XΓ) ⊕ L (MΓ, Λ∗MΓ))

∗ Γ and that Lπ induces an isomorphism between K(E) and the Roe algebra C (XbΓ t (−McΓ)) ∗ Γ and between B(E) and the multiplier algebra M(C (XbΓ t (−McΓ)) ) of the Roe algebra.

Proposition 5.3. The compressed Hilsum-Skandalis perturbation Cµ,t(f) is an element in K(E). Consequently, if Cµ,t(f) := Lπ(Cµ,t(f)), then

∗ Γ (5.2) Cµ,t(f) ∈ C (XbΓ t (−McΓ)) )) . STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 33

5.6. APS-index classes associated to degree one normal maps. Let M and X Cheeger spaces with boundary. We denote Mc := ∂M and Xb := ∂X . We let Z := X t (−M ). We assume the existence of a smooth transverse stratified map F : M → X ; we further assume that F sends ∂M into ∂X and that F∂ := F |∂Y is a stratified homotopy equivalence from ∂M ≡ Mc into ∂X ≡ Xb, as in the previous subsection. Finally we assume the existence of a classifying map X → BΓ; the latter, together with F , defines a Γ-covering Γ → ZΓ → Z . We fix a self-dual mezzoperversity W on X and ] consider the induced mezzo-perversity F W on M . This gives Z , and thus ZΓ, a self-dual ] mezzo-perversity W t F W . We consider now Zb := ∂Z ≡ Xb t (−Mc) and ZbΓ := ∂ZΓ; this gives a Galois Γ-covering of Cheeger-spaces without boundary

Γ − ZbΓ → Zb ≡ Γ − ∂ZΓ → ∂Z . By our discussion above there is a well defined (compressed) Hilsum-Skandalis perturbation ∗ C(F∂) ∈ C (ZbΓ) (for simplicity, we will no longer include the t, µ sub-indices in the notation for the perturbation); this is a trivializing perturbation for the signature operator on ∂ZΓ with domain fixed by ∂W t f ]∂W . We thus obtain a well defined APS coarse index class W tF ]W ∗ that we shall denote as Ind(DΓ ,C(F∂)) in K∗(C (ZΓ)). Proposition 5.4. If F is a global stratified transverse homotopy equivalence, then

W tF ]W Ind(DΓ ,C(F∂)) = 0 Proof. There is a well-defined Hilsum-Skandalis perturbation, C(F ), on the manifold with cylindrical ends, ZΓ,∞. (Notice that this is an ‘un-compressed’ perturbation.) Proceeding as in [HS92] and [Wah13], one can show that the associated perturbed signature operator is invertible. One can then proceed as in [Wah13] and show that the index class in W tF ]W the statement of the proposition, viz. Ind(DΓ ,C(F∂)), is equal to the index class of the invertible operator and hence vanishes. This is done in two steps. The first is a spectral flow argument, which is purely functional analytic and hence applies in our setting. The second is a relative index theorem following Bunke, of the type we will discuss below in Proposition 6.6, and also applies in our setting.

6. Mapping the Browder-Quinn surgery sequence to analysis 6.1. The rho class of a stratiﬁed homotopy equivalence. Let f : Mc → Xb be a transverse stratiﬁed homotopy equivalence. Let Γ be π1(Xb). Let Zb := Xb t (−Mc). The Cheeger space Zb comes equipped with two maps induced respectively by f and the identity and by f and the classifying map for Xb: φ : Zb → Xb , u : Zb → BΓ .

In particular, there is a well defined Γ covering Γ − ZbΓ → Zb. We let uΓ : ZbΓ → EΓ be the Γ-equivariant lift of u. We fix a self-dual mezzoperversity W on Xb and consider the associated self-dual mezzoperversity f ]W on −Mc. We call W t f ]W the resulting self- Wtf ]W Wtf ]W dual mezzoperversity on Zb. We then have self-adjoint extensions D on Z, DΓ on ZΓ and, by Proposition 5.3, a well defined (compressed) Hilsum-Skandalis perturbation ∗ Γ ∗ Cf ∈ C (XbΓ t (−McΓ)) ≡ C (ZbΓ) 34 PIERRE ALBIN AND PAOLO PIAZZA

f Deﬁnition 6. The rho-class ρ(Mc −→ X,b W) associated to f : Mc → Xb and the self-dual mezzoperversity W is given by

f Wtf ]W ∗ Γ (6.1) ρ(Mc −→ X,b W) := φ∗(ρ(DΓ + Cf )) ∈ Kdim X+1(D (XbΓ) ) The universal rho class is, by deﬁnition,

f Wtf ]W ∗ (6.2) ρΓ(Mc −→ X,b W) := (uΓ)∗(ρ(DΓ + Cf )) ∈ Kdim X+1(DΓ) We shall see in the next subsection that the rho class of a stratified homotopy equivalence is independent of W and descends to SBQ(Xb). 6.2. The rho map from S (X) to K (D∗(X )Γ). BQ b dim Xb+1 bΓ Proposition 6.1. The rho class associated to a transverse stratified homotopy equivalence f : Mc → Xb and a self-dual mezzoperversity W on Xb satisfies the following properties: 1] it is independent of the choice of W; 2] it gives a well-defined map (6.3) ρ :S (X) −→ K (D∗(X )Γ) BQ b dim Xb+1 bΓ f f We denote by ρ[Mc −→ Xb] the image of [Mc −→ Xb] through the rho map. Proof. Let g and g0 two iie-metrics on Xb and let W and W0 be two self-dual mezzoperversities adapted respectively to g and g0. Let r : Xb → BΓ be a classifying map. Recall, following Banagl, how it is proved that (X,b g, W, r) is Cheeger-bordant to (X,b g0, W0, r); we refer the reader to [ALMP17, Section 4.4] for the details. We consider the pseudomanifold with boundary X = Xb × [0, 1]t. Instead of the product stratification, we stratify X using the strata of Xb as follows: i) The regular stratum X of Xb contributes X × [0, 1] ii) Every singular stratum of X,Yb k, contributes three strata to X , Y k × [0, 1/2),Y k × (1/2, 1],Y k × {1/2}. The link of X at Y k × [0, 1/2) and Y k × (1/2, 1] is equal to Zk, while the link of X at Y k × {1/2} is seen to be the (unreduced) suspension of Zk,SZk. Since the lower middle perversity intersection homology of SZk, when dim Zk = 2j − 1, is given by  m k I Hi−1(Z ) i > j m k  I Hi(SZ ) = 0 i = j  m k I Hi(Z ) i < k we see that X satisfies the Witt condition at the strata Y k × {1/2}. Put it differently, we do not need to fix a self-dual mezzoperversity at this stratum. Let us endow X with any iie metric G such that, for some t0 > 0, G = g + dt2,G = g0 + dt2. X×[0,t0) X×(1−t0,1] Next we endow X with a self-dual mezzoperversity W as follows: let Y 1,...,Y T be an ordering of the strata of Xb with non-decreasing depth. Denote W = {W 1 −→ Y 1,...,W T −→ Y T }, W0 = {(W 1)0 −→ Y 1,..., (W T )0 −→ Y T } STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 35

j j j j and denote the fiber of, e.g., W −→ Y at the point q ∈ Y , by Wq . Let us define 1 1 W− −→ Y × [0, 1/2) by requiring that the Hodge-de Rham isomorphism identifies all of the fibers. Once this is 2 2 3 done, we can define W− −→ Y × [0, 1/2) in the same way, and inductively define W− −→ 3 T T Y × [0, 1/2),...,W− −→ Y × [0, 1/2). j j We define W+ −→ Y × (1/2 × 1] in the same way to obtain 1 1 1 1 W = {W− −→ Y × [0, 1/2),W+ −→ Y × (1/2, 1],..., T T T T W− −→ Y × [0, 1/2),W+ −→ Y × (1/2, 1]}, a self-dual mezzoperversity over X . So, in words, we extend the metrics g and g0 arbitrarily to an iie metric G without changing them in collar neighborhoods of the boundary, and then we choose a Hodge mezzoperversity by extending the de Rham mezzoperversities trivially from Y i to Y i ×[0, 1/2) on the left and from Y i to Y i ×(1/2, 1] on the right. Since the strata induced by Y k × [0, 1/2) are disjoint from the strata induced by Y k × (1/2, 1], there is no compatibility required between the corresponding mezzoperversities. Finally, define R : X −→ BΓ by R(ζ, t) = r(ζ). The result is a Cheeger-bordism (X , G, W ,R : X −→ BΓ), between (X,b g, W, r : Xb −→ BΓ) and (Xb0, g0, W0, r : Xb −→ BΓ). Let us go back to the proof of our Proposition. Let f : Mc → Xb be a transverse stratified homotopy equivalence. We want to show that the rho class is independent of the choice of the self-dual mezzoperversity W on Xb. Let g, W and g0, W0 as above and consider f ∗g, f ]W and f ∗(g0), f ]W0 on Mc. We can consider M := Mc× [0, 1], stratified as above. Remark now that, by definition, the map F : M → X , F (ζ, t) = f(ζ) is such that F ]W , adapted to F ∗G, is precisely equal to the self-dual mezzoperversity producing the Cheeger bordism between (M,c f ∗g, f ]W, (r ◦ f): Mc → BΓ) and (M,c f ∗(g0), f ]W0, (r ◦ f): Mc → BΓ). Moreover, F is a (transverse) stratified homotopy equivalence between M and X . We thus have a stratified Cheeger-space with boundary, Z := (−M ) t X which is the disjoint union of two stratified Cheeger spaces with boundary, endowed with a stratified homotopy equivalence F : M → X , with self-dual mezzoperversities F ]W on M and W on X and with a classifying map into BΓ, the latter producing a Galois Γ-covering Γ−ZΓ → Z ; moreover, by construction, the self-dual mezzoperversity on the manifold with boundary (−M ) t X restricts to give f ]W t W on one boundary, the one corresponding to t = 0, and f ]W0 t W0 on the other boundary, the one corresponding to t = 1. For later use we denote by j0 and j1 the obvious inclusions of (−McΓ) t XbΓ into ZΓ as the t = 0 and t = 1 boundary respectively. We now apply Proposition 5.4 and obtain that W tF ]W ∗ Γ Ind(D ,C(F∂)) = 0 in K∗(C (ZΓ) ) . By applying the delocalized APS-index theorem we then obtain that Wtf ]W W0tf ]W0 0 ∗ Γ (6.4) 0 = (j0)∗(ρ(D + Cf )) − (j1)∗(ρ(D + Cf )) in K∗(C (ZΓ) ) .

Observe now that there is an obvious Γ-equivariant map ZΓ → XΓ = XbΓ × [0, 1], induced by F and the identity, and thus, by projecting onto the ﬁrst factor, a Γ-equivariant map 36 PIERRE ALBIN AND PAOLO PIAZZA

ZΓ → XbΓ. We can push-forward the equality (6.4) through this map and use functoriality in order to obtain f f 0 ∗ Γ 0 = ρ(Mc −→ X,b W) − ρ(Mc −→ X,b W ) ∈ K∗+1(D (XbΓ) ) with ∗ = dim Xb ; this shows indeed that the rho class is independent of the choice of self-dual mezzoperversity. The proof of item 2] is very similar. 6.3. The map from N (X) to K (X). BQ b dim Xb b We have defined NBQ(Xb) as equivalence classes of transverse degree one normal maps into Xb which are diffeomorphisms when restricted to strata of dimension less than five. Our task is to f map an element [M −→ X] ∈ N (X) to K (X), or, more precisely, to K (X)⊗ [1/2]. c b BQ b dim Xb b dim Xb b Z Following the original treatment of Higson and Roe in the smoth setting, we shall in fact f forget about the normal data encoded in [Mc −→ Xb] ∈ NBQ(Xb).

f ] (6.5) β[M −→ X] := f [Df W ] − [DW ] ∈ K (X) ⊗ [1/2] c b ∗ dim Xb b Z We then have the following Proposition 6.2. 1] The right hand side of (6.5) is independent of the choice of self-dual mezzoperversity W. f0 f1 2] The map β is well defined: if [Mc0 −→ Xb] = [Mc1 −→ Xb] in NBQ(Xb), then f ]W W f ]W W (6.6) (f ) [D 0 ] − [D ] = (f ) [D 1 ] − [D ] ∈ K (X) ⊗ [1/2] 0 ∗ 1 ∗ dim Xb b Z Proof. We establish both statements by adapting an argument due to Higson and Roe and by making use of Theorem 4.1 above, item 3]. Thus let W and W0 be two self-dual mezzoperversities, adapted to iie metrics g and g0 respectively. We must show that f [Df ]W ] − [DW ] − (f [Df ]W0 ] − [DW0 ]) = 0 ∈ K (X) ⊗ [1/2] ∗ ∗ dim Xb b Z We initially follow the construction exploited in the previous subsection. Thus we consider X := Xb ×[0, 1] and M := Mc×[0, 1], both stratified àla Banagl. We consider the transverse map F : M → X , F (ζ, t) = f(ζ) and consider G, F ∗G, W and F ]W as in the previous subsection. We thus have a stratified Cheeger-space with boundary, Z := (−M ) t X ≡ (−(Mc × [0, 1])) t (Xb × [0, 1]) which is the disjoint union of two stratified Cheeger spaces with boundary, endowed with a stratified transverse map F : M → X , with self-dual mezzoperversities F ]W on M and W on X ; moreover, by construction, the self-dual mezzoperversity F ]W t W on the manifold with boundary (−M )tX restricts to give f ]W tW on one boundary, the one corresponding to t = 0, and f ]W0 t W0 on the other boundary, the one corresponding to t = 1. Remark first of all that the K-homology groups of a disjoint union of two spaces A t B is equal to the direct sum of the individual K-homology groups. We define two group homomorphisms

Φ: K∗(Z , ∂Z ) → K∗(X , ∂X ) = K∗(Xb × [0, 1], Xb × {0, 1}) ,

φ : K∗(∂Z ) → K∗(∂X ) = K∗(Xb × {0, 1}) STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 37

as follows:

Φ(αM , βX ) = F∗αM − βX , φ(α0, α1, β0, β1) = (f∗α0 − β0, f∗α1 − β1) It is easy to check, using the functoriality properties of the connecting homomorphism in the long exact sequence of a pair, that the following diagram is commutative:

∂t K∗+1(Z , ∂Z ) / K∗(∂Z )

Φ φ ∂ K∗+1(X , ∂X ) / K∗(∂X )

The bottom horizontal homomorphism is part of the long exact sequence

∂ ι K∗+1(X , ∂X ) −→ K∗(∂X ) −→ K∗(X ) which can be rewritten as

∂ ι (6.7) K∗+1(Xb × [0, 1], Xb × {0, 1}) −→ K∗(Xb × {0, 1}) −→ K∗(Xb × [0, 1]) . Notice that there is a natural group homomorphism

ψ : K∗(Xb × {0, 1}) → K∗(Xb) , ψ(γ0, γ1) = γ0 − γ1 and that ψ factors as follows:

K (X × {0, 1}) ∗ b ψ ι , 2 K∗(Xb) π K∗(Xb × [0, 1]) with π induced by the projection onto the ﬁrst factor. Using these remarks and Theorem 4.1, which in the present context states that

F ]W f ]W f ]W0 W W0 ∂t([D t W ]) = ([D ], [D ], [D ], [D ]) we then have f ]W W f ]W0 W0 f∗[D ] − [D ] − (f∗[D ] − [D ]) f ]W W f ]W0 W0 = ψ(f∗[D ] − [D ], f∗[D ] − [D ]) f ]W W f ]W0 W0 = π ◦ ι(f∗[D ] − [D ], f∗[D ] − [D ]) = π ◦ ι ◦ φ([Df ]W ], [Df ]W0 ], [DW ], [DW0 ])

F ]W tW = π ◦ ι ◦ φ ◦ ∂t([D ]) = π ◦ ι ◦ ∂ ◦ Φ[DF ]W tW ] = 0 where in the last step we have used the exactness of (6.7). This establishes item 1]. Item 2] is similar, but easier. 38 PIERRE ALBIN AND PAOLO PIAZZA

∗ Γ 6.4. The index map from LBQ(Xb) to K∗(C (XbΓ) ). We ﬁnally consider the (APS) index homomorphisms L (X × [0, 1]) −−−−→IndAPS K (C∗(X )Γ) , L (X) −−−−→IndAPS K (C∗(X )Γ) . BQ b dim Xb+1 bΓ BQ b dim Xb bΓ First, we restate some of the constructions above in a way that will be useful to the present task.

Lemma 6.3. Every [α] ∈ LBQ(Xb × [0, 1]) can be represented by a diagram of the form (φ;id,ψ) id α :(Mc; X,b Xb0) −−−−−−→ (Xb × [0, 1]; Xb × {0}, Xb × {1}) −−−→ Xb × I.

Let α1 and α2 be two such diagrams representing the same class and β the diagram obtained by gluing α1 and −α2,

0 0 (Φ;ψ2,ψ1) id β :(Wc; Xb2, Xb1) −−−−−−−→ (Xb × [0, 1]; Xb × {0}, Xb × {1}) −−−→ Xb × I, then there are stratiﬁed spaces with corners P,b Qb together with a BQ-normal map Γ,

(Γ;γ0,γ1) (Pb; ∂0Pb , ∂1Pb) −−−−−−→ (Qb; ∂0Q,b ∂1Qb) such that γ0 Φ ∂0Pb −−→ ∂0Qb = Wc −−→ Xb × I and γ1 is a BQ-equivalence. Proof. We can represent [α] in this way directly from the Wall representation theorem (as in Corollary 2.6). Taking G Wc = Mc1 −Mc2. Xb and gluing α1 and −α2 along their common boundary yields β, an element of NBQ(Xb ×[0, 1]). Since Lemma 2.3 implies that β represents [α1] − [α2] = 0 in LBQ(Xb × [0, 1]), Theorem 2.4, implies that the class of β in NBQ(Xb × [0, 1], ∂(Xb × [0, 1])) is in the image of SBQ(Xb × [0, 1], ∂(Xb × [0, 1])). It follows that there is a normal bordism between β and an element in SBQ(Xb × [0, 1], ∂(Xb × [0, 1])), which yields Γ : Pb −→ Qb as above. ∗ Γ We can now deﬁne the homomorphism Ind : LBQ(Xb × [0, 1]) → K∗(C (XbΓ) ). Consider an element [α] ∈ LBQ(Xb × [0, 1]) and assume, thanks to the previous Lemma, that (φ;id,ψ) id α :(Mc; X,b Xb0) −−−−−−→ (Xb × [0, 1]; Xb × {0}, Xb × {1}) −−−→ Xb × I.

We have a classifying map Xb → BΓ, with Γ = π1(Xb). This induces, through φ : Mc → Xb × [0, 1] a Γ-Galois covering ZbΓ on Zb := (−Mc) ∪ (Xb × [0, 1]) We now fix an incomplete iterated conic metric on Xb and we choose an adapted mezzoperversity W for the resulting signature operator. We take the associated product mezzoperversity W × [0, 1] on Xb × [0, 1]. We endow −Mc with the induced metric and the induced ] mezzoperversity φ (W × [0, 1]). We lift all these data to ZbΓ. Recall at this time that φ is not an homotopy equivalence. On the other hand, since ψ (and, of course, the identity id Xb −→ Xb) is a stratified homotopy equivalence we see that there exists a well defined (APS) index class, obtained by using the compressed Hilsum-Skandalis perturbation associated to ] ∗ Γ ψ t id and to (ψ W t W) t (W t W) on ∂Zb: this index class belongs to K∗(C (ZbΓ) ). We STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 39

∗ Γ deﬁne IndAPS(α, W) as the push-forward of the above index class to K∗(C (XbΓ) ). In what ∗ Γ ∗ follows we shall use the canonical isomorphism K∗(C (XbΓ) ) ' K∗(Cr Γ). Lemma 6.4. If W and W0 are two mezzoperversities on Xb then 0 ∗ IndAPS(α, W) = IndAPS(α, W ) in K∗(Cr Γ). Proof. Consider the stratiﬁed manifolds with corners

M := Mc × [0, 1]t and X := (Xb × [0, 1]) × [0, 1]t.

We stratify (Xb ×[0, 1])×[0, 1]s as we did in the proof of Proposition 6.1; the mezzoperversity W on Xb induces a product mezzoperversity, denoted W × [0, 1], on (Xb × [0, 1]) × {0}; similarly, the mezzoperversity W0 on Xb induces a mezzoperversity, denoted W0 × [0, 1] on (Xb × [0, 1]) × {1}. We know that there is a mezzoperversity W on X ≡ (Xb × [0, 1]) × [0, 1]t 0 interpolating between W × [0, 1] and W × [0, 1]. Similarly, we stratify −(Mc × [0, 1]t) as in Proposition 6.1; let Φ : M → X the map Φ(m, t) = φ(m); the mezzoperversity Φ]W interpolates between φ](W × [0, 1]) and φ](W0 × [0, 1]). Consider now the stratified manifold with corners Z := −M t X ≡ −(Mc × [0, 1]t) t (Xb × [0, 1]) × [0, 1]t . This is a Cheeger space with corners. Consider the following boundary hypersurfaces of Z : F = −Mc × {s = 0} t ((Xb × [0, 1]) × {s = 0}) t −Mc × {s = 1} t ((Xb × [0, 1]) × {s = 1}) and G = −(∂Mc × [0, 1]s) t (∂(Xb × [0, 1]) × [0, 1]s). Consider the signature operator on Z := −M t X ; using appropriate Hilsum-Skandalis perturbations we can perturb this operator and make it invertible at G. To fix notation, let us assume that Xb is odd dimensional, so that Mc and Xb × [0, 1] are even dimensional. We ∗ can define a bivariant class B ∈ KK1(CF (Z ),Cr Γ), with CF (Z ) denoting the continuous functions on Z which vanish on F 2. Consider πF : F → point and πZ : Z → point. Denote by ι the natural inclusion F,→ Z := −M t X and by q : Z → F the restriction map to F . Obviously πF = πZ ◦ ι. Consider the semi-split short exact sequence j q 0 → CF (Z ) −→ C(Z ) −→ C(F ) → 0 Part of the associated long exact sequence in K-theory is

∗ δ ∗ ι∗ ∗ KK1(CF (Z ),CΓ) −→ KK0(C(F ),Cr Γ) −→ KK0(C(Z ),CΓ) .

In particular, by exactness, ι∗ ◦ δ = 0. Then, on the one hand a classic argument shows that F 0 ∗ ∗ π∗ (δB) = IndAPS(α, W) − IndAPS(α, W ) in KK0(C,Cr Γ) = K0(Cr Γ) F Z Z and, on the other hand, π∗ (δB) = π∗ ◦ ι∗(δB) = π∗ ◦ ι∗ ◦ δ(B) = 0 by exactness. Thus 0 IndAPS(α, W) − IndAPS(α, W ) = 0 as required.

2 ∗ Without further hypothesis we could only deﬁne a bivariant class in KK1(C∂Z (Z ),Cr Γ) 40 PIERRE ALBIN AND PAOLO PIAZZA

We shall need a generalization of the previous Lemma. Fix an incomplete iterated conic metric on X × [0, 1] (product-type near the boundary) and assume that W is a mez- b Xb×[0,1] zoperversity for the resulting signature operator on X ×[0, 1]; thus W is not necessarily b Xb×[0,1] the product mezzoperverity associated to a mezzoperversity on Xb. We can still define an index class Ind (α, W ) by considering φ](W ) on M and proceeding as above. APS Xb×[0,1] Xb×[0,1] c Lemma 6.5. If W and W0 are two mezzoperversities on X × [0, 1] then Xb×[0,1] Xb×[0,1] b 0 ∗ IndAPS(α, W ) = IndAPS(α, W ) in K∗(C Γ). Xb×[0,1] Xb×[0,1] r Proof. The proof given for Lemma 6.4 can be easily adapted to this more general case. We now describe a gluing theorem following Bunke [Bun95]. Let Zb be a stratified space with boundary and Hb a compact hypersurface transverse to the stratification that does not meet the boundary of Z.b We can view Zb as two stratified spaces with boundary glued along H,b [ Zb = Zb1 Zb2 Hb We decompose a Γ-cover of Zb accordingly: 1 [ 2 ZbΓ = ZbΓ ZbΓ.

HbΓ We assume that Zb is Cheeger, we fix an iterated incomplete edge metric g which is of product type near Hb; finally, we fix a selfdual mezzoperversity W adapted to g. This restricts to a selfdual mezzoperversity on the hypersurface Hb which we denote WH . Similarly, we obtain selfdual mezzoperversities W 1 on Zb1 and W 2 on Zb2. We lift all these structures to the Γ-covers with minimal change of notation. Let DΓ be the signature operator on ZΓ, the regular part of ZbΓ. We assume that a trivializing perturbation Q∂ of the boundary operator has been fixed; the latter gives a grafted perturbation Q∞ on the associated manifold with cylindrical ends and thus an index class W ∗ ∗ Γ ∗ IndAPS(DΓ ,Q∂) ∈ K∗(Cr Γ), where the canonical isomorphism K∗(C (ZbΓ) ) ' K∗(Cr Γ) has been used. We can assume, without loss of generality, that the perturbation Q∞ is localized away from HbΓ. The signature operator near HΓ, the regular part of HbΓ, will decompose in the usual way, WH given that the metric is of product type near HΓ. Let CH be a perturbation of DΓ such WH 3 that DΓ + CH is invertible and let W 1 1 W 2 2 DΓ,∞ + CH,∞,DΓ,∞ + CH,∞ 1 2 be perturbed differential operators on the spaces obtained from ZbΓ, ZbΓ by attaching an infinite half-cylinder along HbΓ. We obtain in this way well defined index classes W 1 1 W 2 2 ∗ IndAPS(DΓ ,Q∂ t CH ) , IndAPS(DΓ ,Q∂ t CH ) in K∗(Cr Γ) j j where Q∂ is Q∂ restricted to ∂ZbΓ ∩ ZbΓ.

3a simple argument using the cobordism invariance of the signature index class with Cheeger boundary conditions shows that such a perturbation always exists. STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 41

Proposition 6.6. (Gluing) With notation as above, the index classes satisfy

W W 1 1 W 2 2 ∗ IndAPS(DΓ ,Q∂) = IndAPS(DΓ ,Q∂ t CH ) + IndAPS(DΓ ,Q∂ t CH ) in K∗(Cr Γ) If Zb is without boundary, then W W 1 W 2 ∗ Ind(DΓ ) = IndAPS(DΓ ,CH ) + IndAPS(DΓ ,CH ) in K∗(Cr Γ) Proof. A happy byproduct of the functional analytic nature of Bunke’s proof is that it applies W j j almost unchanged to our setting. Let D denote either of DΓ,∞ +CH,∞, j ∈ {1, 2}, with D(D) its self-adjoint domain. Replace the definition of the spaces H`, ` ≥ 0, in [Bun95, (2)] by ` ` ` 2 X k 2 H = D(D ), kφk` = kD φkL2 k=0 where the norm on the right hand side is the pointwise norm coming from the Hilbert C∗- module structure. Note that since D2 + Id is invertible we can take a compact exhaustion of the regular part ∞ j of our space, approximate the constant function one, and find a non-negative f ∈ Cc (ZΓ,∞) such that D2 + f is invertible, thereby satisfying Bunke’s assumption 1. With these conventions, the analytical results in §1.2 of Bunke now hold verbatim save that the expressions R(λ)grad(f)R(λ) should be replaced by R(λ)(grad(f) + [C, f])R(λ) where C is the perturbation at H. This replacement is still a compact operator and hence the argument in Proposition 1.13 of Bunke yields well-defined index classes. The argument in Theorem 1.15 of Bunke then yields the equality of the index classes we seek, once we take into account that the index of a translation invariant operator on the infinite cylinder vanishes. Recall that an articulated stratified space without boundary Lb is the (entirety of the) boundary of a stratified space with corners. Thus Lb is a finite union of stratified spaces with corners together with identifications of their boundary faces and the absence of boundary says that there are no unmatched faces. If Lb is the boundary of a Cheeger space, Lb = ∂X,b so in particular each of its constituent stratified spaces with corners is a Cheeger space, then a choice of mezzoperversity on Xb induces compatible mezzoperversities on the constituents of L,b and the boundary identifications (which are stratified diffeomorphisms) give rise to Hilsum-Skandalis perturbations, and so we have an index class, ∗ Ind((∂Xb)Γ) ∈ K∗(Cr Γ), where Γ is the fundamental group of X.b Lemma 6.7. If Xb is a Cheeger space with corners then ∗ 1 Ind((∂Xb)Γ) = 0 in K∗(Cr Γ)[ 2 ]. Proof. Our convention is that every boundary hypersurface Mc of Xb is collared, i.e., has a neighborhood of the form [0, 1)ρ ×M in X, consistent with the stratification of X. We refer Mc c b b to ρ as a boundary defining function for M. By a ‘total boundary defining function’ for Mc c X, we mean a function ρ obtained by taking the product of boundary defining functions, b ∂Xb 42 PIERRE ALBIN AND PAOLO PIAZZA

one per boundary hypersurface of X.b Since the boundary hypersurfaces are collared, for all ε > 0 suﬃciently small the set {ρ ≥ ε} is a stratiﬁed space with boundary and ∂X can be ∂Xb b obtained from ∂{ρ ≥ ε} by ‘introducing corners’ (i.e., partitioning it and considering as an ∂Xb articulated manifold). Cobordism invariance of the signature on Cheeger spaces [ALMP17, Theorem 4.8] implies that the signature of ∂{ρ ≥ ε} vanishes in K (C∗Γ)[ 1 ] (see Remark ∂Xb ∗ r 2 9) and then Proposition 6.6 implies that the signature of ∂Xb vanishes. The following Proposition is the main result of this Subsection:

Proposition 6.8. Let W be a mezzoperversity for Xb. Let

0 (φ1;id,ψ1) id α1 = (Mc1; X,b Xb1) −−−−−−−→ (Xb × [0, 1]; Xb × {0}, Xb × {1}) −−−→ Xb × I

0 (φ2;id,ψ2) id α2 = (Mc2; X,b Xb2) −−−−−−−→ (Xb × [0, 1]; Xb × {0}, Xb × {1}) −−−→ Xb × I be two elements in LBQ(Xb × [0, 1]). Assume that [α1] = [α2] in LBQ(Xb × [0, 1]). Then

IndAPS(α1, W) = IndAPS(α2, W). Consequently, using also the Lemma 6.4, we can deﬁne a map L (X × [0, 1]) 3 ζ −→ Ind (ζ) ∈ K (C∗Γ) BQ b APS dim Xb+1 r by setting IndAPS(ζ) := IndAPS(α, W) for any representative α of ζ, [α] = ζ, and any choice of mezzoperversity W. This map is a homomorphism of abelian groups.

Proof. By Lemma 6.3 and the gluing formula, Proposition 6.6, it suﬃces to show that if α1 is null bordant then IndAP S(α1, W) = 0. Let Φ Ω (Nb; ∂1N,b ∂2Nb) −−→ (Zb; ∂1Z,b ∂2Zb) −−→ b(X × I) × I be a null bordism of α. Thus, Φ is BQ-normal, Φ| : ∂2Nb −→ ∂2Zb is a BQ-equivalence, Ω is BQ-transverse, and

Φ| π◦Ω| (∂1N,b ∂12Nb) −−→ (∂1Z,b ∂12Zb) −−−−→ Xb

0 (φ1;id,ψ1) id = (Mc1; X,b Xb1) −−−−−−−→ (Xb × [0, 1]; Xb × {0}, Xb × {1}) −−−→ Xb × I . By Lemma 6.7, we know that

∂1Φ ∂2Φ ] ∂1Zb ∂1Zb ] ∂2Zb ∂2Zb Ind((∂1Nb −−→ ∂1Zb) ∪ (∂2Nb −−→ ∂2Zb), (∂1Φ (W ) t W ) t (∂2Φ (W ) t W )) = 0 for any mezzoperversity WZb on Z.b By Proposition 6.6, we can write this as the sum of two APS indices,

∂1Φ ] ∂1Zb ∂1Zb ∂2Φ ] ∂2Zb ∂2Zb IndAP S(∂1Nb −−→ ∂1Z,b ∂1Φ (W ) t W ) + IndAP S(∂2Nb −−→ ∂2Z,b ∂2Φ (W ) t W )

However, the second summand is equal to zero since Φ| : ∂2Nb −→ ∂2Zb is a BQ-equivalence, and hence so is the ﬁrst summand. The fact that IndAPS is a homomorphism of abelian groups follows from Lemma 2.3 and the gluing formula. The Proposition is proved. STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 43

We can use the gluing result in Proposition 6.6 to simplify the APS index map from an L-cycle α ∈ LBQ(Xb × I) if it restricts to be a diﬀeomorphism on the boundary.

Proposition 6.9. Let α ∈ LBQ(Xb × I) be given by (φ;id,ψ) id α = (Mc; X,b Xb0) −−−−−−→ (Xb × [0, 1]; Xb × {0}, Xb × {1}) −−−→ Xb × I

0 1 where ψ is a diﬀeomorphism Xb −→ X.b Let G(α) ∈ LBQ(Xb × S ) be the LBQ-cycle given by 0 G(φ) id 1 G(α) = G(Mc) = M/c (Xb ∼ψ Xb ) −−−−→ Xb × [0, 1]/(Xb × {0} ∼ Xb × {1}) −−−→ Xb × S .

Given a mezzoperversity W on Xb we have ∗ IndAPS(α, W) = Ind(G(α), W) in K∗(Cr Γ) where on the left W is lifted to Xb × [0, 1] and pulled-back to M,c while on the right W is lifted to Xb × S1 and pulled-back to G(Mc). Note that the index map on the right does not require boundary conditions; put differently, this is the index class of a cycle involving Cheeger spaces without boundary. Remark 10. The index of (G(α), W) is, by definition, the index of the signature operator on 1 G(Mc) ∪ (Xb × S ) (twisted using a reference map to BΓ, with Γ = π1Xb). Another application of Proposition 6.6 shows that this index coincides with that of the signature operator on G Mc Xb × I, Xb∼Xb×{0} 0 Xb ∼ψXb×{1} as in the original definition of Higson-Roe [HR05c].

Proof. Let Nb ⊆ G(Mc) and Yb ⊆ Xb × S1 be the image of the boundary of M,c respectively of Xb × [0, 1], under the identification maps Mc −→ G(Mc), Xb × [0, 1] −→ Xb × S1. Without loss of generality we assume that G(φ) is collared near these subsets, i.e., that there are neighborhoods on which G(φ)| : Nb × (−1, 1) −→ Yb × (−1, 1) is the identity on the second factor. We assume that the stratifications respect the product structure of these neighborhoods. We will apply our gluing result to this situation. Let Zb = G(Mc) t (Xb × S1) endowed with the natural map to BΓ, Γ = π1X,b and let 1 1 1 1 Hb = (Nb × {− 2 , 2 }) t (Yb × {− 2 , 2 }). Given a mezzoperversity W on Xb adapted to a wedge metric on Zb that respects the product decomposition of the neighborhood (Nb t Yb) × (−1, 1), we can apply Proposition 6.6 to see that W W1 W1 1 1 Ind(DΓ ; Zb) = IndAPS(DΓ ,CH ; Mc t (Xb × [0, 1])) + IndAPS(DΓ ,CH ;(Nb t Yb) × (− 2 , 2 )) where CH is the Hilsum-Skandalis perturbation and we have used more explicit notation than is our wont. Finally note that the first summand on the right hand side is Ind(α, W), 44 PIERRE ALBIN AND PAOLO PIAZZA

and the second term vanishes, as it is the index of an invertible translation-invariant operator on an infinite cylinder (indeed, as we have already remarked, even though we denote our classes as APS classes, they are really classes on manifolds with cylindrical ends). 6.5. Mapping stratified surgery to analysis. At this point we have shown that all of the maps in the following diagram are well defined and independent of the choice of mezzoperversity: (6.8) η θ LBQ(Xb × I) / SBQ(Xb) / NBQ(Xb) / LBQ(Xb)

ρ IndAPS β IndAPS K (C∗Γ)[ 1 ] / K (D∗(X )Γ)[ 1 ] / K (X)[ 1 ] / K (C∗Γ)[ 1 ] dim Xb+1 r 2 dim Xb+1 bΓ 2 dim Xb b 2 dim Xb r 2 We now establish the commutativity of this diagram. The key fact for establishing commutativity of the ﬁrst square is the behaviour of the rho class under composition ( [Wah13, Theorem 9.1], [PS16, (4.14)]):

Proposition 6.10. Let L,b M,c V,b be Cheeger spaces and f g Mc −−→ V,b Lb −−→ Mc transverse stratified homotopy equivalences. We fix a self-dual mezzoperversity W on Vb and ] ] ] we consider the induced mezzoperversities f W on Mc and g (f W) on Lb. If VbΓ is a Γ- ∗ covering of Vb then we lift these mezzoperversities to the induced coverings f VbΓ on M and ∗ ∗ ∗ Γ g (f (VbΓ)) on L. The following identity holds in K∗(D (VbΓ) ): f◦g id ] g ] f ρ(Lb −−−→ V,b W)] + fe∗(ρ(Mc −−−→ M,c f W)) = fe∗(ρ(Lb −−→ M,c f W)) + ρ(Mc −−→ V,b W]. where in the first summand on the right-hand side it is the rho class of the perturbed signature ∗ ∗ ∗ operator on the covering g (f (VbΓ)) t f (VbΓ) that appears. Consequently, with a small abuse of notation, we have f◦g id g f (6.9) ρ[Lb −−−→ Vb] + fe∗(ρ[Mc −−−→ Mc]) = fe∗(ρ[Lb −−→ Mc]) + ρ[Mc −−→ Vb]. Remark 11. It can be shown that the ρ-invariant of the identity map vanishes, but we will not need this here. Proof. The proof given in [PS16], based in turn on the proof of [Wah13, Proposition 7.1] and on the delocalized APS index theorem, applies to the present situation.

Recall now how LBQ(Xb × I) acts on SBQ(Xb). If [α] ∈ LBQ(Xb × I) and [β] ∈ SBQ(Xb) then we can choose representatives of the form f β : Mc −−→ Xb

(φ;id,φ2) id α :(Wc; M,c Mc0) −−−−−−→ (Mc × [0, 1]; Mc × {0}, Mc × {1}) −−−→ Mc × I, 0 and then the class of f ◦ φ2 : Mc −→ Xb in SBQ(Xb) is well-deﬁned and denoted ∂(α)(β). The map

LBQ(Xb × I) × SBQ(Xb) / SBQ(Xb) ([α], [β]) / ∂(α)(β) STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 45

deﬁnes the group action of the Browder-Quinn L-group of Xb × I on the structure set of X.b In order to show that the ﬁrst diagram in (6.8) commutes we need to show that

(6.10) ρ(∂(α)(β)) − ρ(β) = ι∗(IndAPS(α)) with Ind (α) ∈ K (C∗(X )Γ)[ 1 ] = K (C∗Γ)[ 1 ] and ι : C∗(X )Γ → D∗(X )Γ APS dim Xb+1 bΓ 2 dim Xb+1 r 2 bΓ bΓ the natural inclusion. The left hand side of (6.10) is, by deﬁnition,

f◦φ2 f ρ[Mc0 −−→ Xb] − ρ[Mc −−→ Xb] . We now apply Proposition 6.10 and obtain that this diﬀerence equals:

0 φ2 id fe∗ρ[Mc −→ Mc] − fe∗ρ[Mc −→ Mc] and a direct application of the delocalized APS index theorem shows that this diﬀerence is precisely equal to ι∗(IndΓ,APS(α)). This establishes the commutativity of the ﬁrst square in the diagram. The second square in the diagram is proved to commute exactly as in [PS16]. Finally, for the third square, we observe that the image of a class in NBQ(Xb) is a union of two closed Cheeger spaces and that for such an element in LBQ(Xb) the APS-index class is just the index class of Subsection 3.6; the commutativity of the third square then follows by the functoriality of the boundary map in the Higson-Roe surgery sequence.

6.6. Mapping stratified surgery to analysis on all strata. The use of transverse maps in the definition of the Browder-Quinn surgery sequence implies that there are well-defined restriction maps from the long exact sequence of a stratified space to the corresponding sequence of a singular stratum. Recall from §2.2 that if Y ∈ S(Xb) then the closure of Y in Xb is a stratified space denoted Y.b An L-cycle over Xb restricts to an L-cycle over Y,b and a null bordism over Xb restricts to an L-cycle over Y.b The restriction of a normal invariant or a Thom-Mather structure from Xb to Yb is an L-cycle of the same type. Thus we have commutative diagrams

η θ LBQ(Xb × I) / SBQ(Xb) / NBQ(Xb) / LBQ(Xb)

η θ LBQ(Yb × I) / SBQ(Yb) / NBQ(Yb) / LBQ(Yb) which we can extend arbitrarily to the left. Note that the vertical arrows are generally neither injective nor surjective. Let us introduce the abbreviations, M K (C∗; S(X))[ 1 ] = K (C∗Γ)[ 1 ] ⊕ K (C∗Γ(Y ))[ 1 ], [j] b 2 dim Xb+j r 2 dim Yb +j r b 2 Y ∈S(X) M K (D∗; S(X))[ 1 ] = K (D∗(X )Γ)[ 1 ] ⊕ K (D∗(Y )Γ(Yb ))[ 1 ] [j] b 2 dim Xb+j bΓ 2 dim Yb +j bΓ(Yb ) 2 Y ∈S(X) M K (D∗/C∗; S(X))[ 1 ] = K (X)[ 1 ] ⊕ K (Y )[ 1 ]. [j] b 2 dim Xb+j−1 b 2 dim Yb +j−1 b 2 Y ∈S(X) 46 PIERRE ALBIN AND PAOLO PIAZZA

By restricting to each singular stratum an making use of their respective commutative diagram (6.8) we end up with a combined diagram (6.11) η θ LBQ(Xb × I) / SBQ(Xb) / NBQ(Xb) / LBQ(Xb)

⊕ Ind ⊕ρ ⊕β ⊕ Ind ∗ 1 ∗ 1 ∗ ∗ 1 ∗ 1 K[1](C ; S(Xb))[ 2 ] / K[1](D ; S(Xb))[ 2 ] / K[1](D /C ; S(Xb))[ 2 ] / K[0](C ; S(Xb))[ 2 ]

Remark 12. In [Wei94, §12.4], for Witt spaces with simply connected links (also known as ‘supernormal spaces’, see [CW91]) we ﬁnd that

1 1 M 1 LBQ(Xb)[ 2 ] = L(Zπ1(Xb))[ 2 ] ⊕ L(Zπ1(Yb))[ 2 ] Y ∈S(X)

(where Y,b the ‘closed stratum’, is the closure of Y in Xb with the induced stratiﬁcation) so that in this case the vertical arrows in (6.11) map from the algebraic L-groups.

7. Further considerations In [CW03], Chang and Weinberger use the surgery exact sequence of a manifold to show that torsion in its fundamental group implies the existence of infinitely many homotopy equivalent manifolds that are homeomorphically distinct. In this section we use the argument in loc. cit. as a launching pad to discuss related topics. First, as an answer to ‘how to map in to/out of a BQ L-group?’ we establish a couple of long exact sequences. Secondly we establish Atiyah’s L2-signature theorem for Cheeger spaces. Finally we combine these to discuss a version of the result of [CW03] for Cheeger spaces. 7.1. A long exact sequence for the Browder-Quinn L-groups. It is interesting to connect the Browder-Quinn L-groups with the usual (Wall) L-groups of a smooth manifold. If Xb is a smoothly stratified space then, on the one hand, recall that we can identify the regular part of Xb with the (interior of) the resolution of X,b X,e and so the inclusion, i, of the regular part induces i∗ Lk(Xe) −−→ LBQ,dXe (k)(Xb), where dXe (k) is the dimension function that is equal to k on the inverse image of the regular part of X.b On the other hand, if X† is a minimal stratum (a stratum of greatest depth), and hence a closed manifold, then the fact that the Browder-Quinn L-groups are defined using transverse maps means that we have a natural restriction map L (X) −−→R L (X†), BQ,d b dX† † where dX† is the restriction of the dimension function to X .

Both of these maps ﬁt into a long exact sequence of Browder-Quinn L-groups. An example of the former is found in [BQ75] and of the latter in [Wei94, §6]. We treat these as special cases of long exact sequences in L-groups associated with inclusions. STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 47

We recall from, e.g., [Wal16, Lemma 8.3.1], a standard construction of long exact sequences in cobordism. Suppose that α and β denote two possible types of structure a manifold can have, and that a β structure implies an α structure (for example, if Y ⊆ X and α structure could be ‘is endowed with a map to X’ and an β structure ‘is endowed with a map to Y ). α β Denote by Ωn, Ωn the cobordism groups of n-dimensional manifolds with the corresponding α,β structure, and denote by Ωn the cobordism group of manifolds with boundary with an α structure and a compatible β structure on the boundary then, with the obvious maps, there is a long exact sequence

β α α,β ∂ β ... Ωn −→ Ωn −→ Ωn −−→ Ωn−1 −→ .... The proof of exactness in loc. cit. does not depend on the speciﬁc structures α and β and adapts easily to the situations we consider below.

One type of relative L-group. In [Wal99], Wall explains how to associate to a map h : V −→ W between two manifolds an L-group that moreover ﬁts into a long exact sequence with the L-groups of V and W. We now observe that the same is true if Vb and Wc are smoothly stratiﬁed spaces and h : Vb −→ Wc is h a transverse map between them. Let us denote by LBQ,d(Vb −−→ Wc) the set of commutative diagrams of the form

φ η (7.1) (Sb; ∂1S,b ∂2Sb) / (Tb; ∂1Tb , ∂2Tb) / Wc O O O h ? φ| ? η| ∂2Sb / ∂2Tb / Vb

where Sb is a smoothly stratiﬁed space with dimension function d, and boundary ∂1Sb ∪ ∂2S,b and similarly Tb , φ is a BQ-normal map and restricts to a BQ-equivalence between ∂1Sb and h ∂1Tb and η is BQ-transverse. As usual LBQ,d(Vb −−→ Wc) then denotes cobordism classes of such cycles. As above, these groups ﬁts into a long exact sequence

∂ h∗ j h ∂ h∗ ... −−→ LBQ,d(Vb) −−−→ LBQ,d(Wc) −−→ LBQ,d(Vb −−→ Wc) −−→ LBQ,d−1(Vb) −−−→ ... where the maps are given by

0 0 0 0 h 0 0 φ 0 0 η i h 0 0 φ 0 0 h◦η i h∗ (Sb ; ∂Sb ) −−→ (Tb , ∂Tb ) −−→ Vb = (Sb ; ∂Sb ) −−→ (Tb , ∂Tb ) −−−−→ Wc ,

 φ00 η00  (Sb00; ∂Sb00, ∅) / (Tb00; ∂Tb00, ∅) / Wc h φ00 η00 i  O O O  j (S00; ∂S00) −−−→ (T 00, ∂T 00) −−−→ W =   , b b b b c  h   ? ?  ∅ / ∅ / Vb  φ η  (S; ∂ S, ∂ S) / (T ; ∂ T , ∂ T ) / W b 1 b 2 b b 1 b 2 b cO  O O  h φ| η| i   ∂  h  = (∂2Sb; ∂(∂2Sb)) −−→ (∂2Tb; ∂(∂2Tb)) −−→ Vb .  ? φ| ? η|  ∂2Sb / ∂2Tb / Vb 48 PIERRE ALBIN AND PAOLO PIAZZA

If we apply this to the inclusion of the regular part, i : Xe −→ X,b we obtain

∂ i∗ j i ∂ i∗ (7.2) ... −−→ Lk(Xe) −−→ LBQ,dXe (k)(Xb) −−→ LBQ,dXe (k)(Xe −−→ Xb) −−→ Lk−1(Xe) −−→ ... (cf. [BQ75, Proposition 4.8]).

Another type of relative L-group. (For this sequence cf. [DS90, Theorem 5.4].) Let Xb be a stratiﬁed space, Σb ⊆ Xb a closed subset of Xb made up of a union of strata. Let LBQ,d(Xb; Σ)b denote the L-cycles over X with dimension function d whose restriction to Σb is a BQ-equivalence, and LBQ,d(Xb; Σ)b the corresponding bordism classes. We allow Σb = ∅ for which

LBQ,d(Xb; ∅) = LBQ,d(Xb). Analogously to the above, if Σb0 ⊆ Xb is another closed subset of Xb made up of a union of 0 0 strata, with Σb ⊆ Σb then there is a relative group LBQ,d(Xb; Σb ⊆ Σb ) with classes represented by diagrams of the form

φ ω (7.3) (Mc; ∂0M,c ∂1Mc) −−→ (Nb; ∂0N,b ∂1Nb) −−→ Xb such that φ is BQ-normal, ω is BQ-transverse, ∂0φ is a BQ-equivalence, ∂1φ restricted to the preimage of Σb0 is a BQ-equivalence, and φ restricted to the preimage of Σb is a BQ-equivalence. There are natural inclusion maps

0 0 LBQ,d(Xb; Σb ) −→ LBQ,d(Xb; Σ)b , LBQ,d(Xb; Σ)b −→ LBQ,d(Xb; Σb ⊆ Σb ) which ﬁt into a long exact sequence

0 0 ∂1 0 ... −→ LBQ,d(Xb; Σb ) −→ LBQ,d(Xb; Σ)b −→ LBQ,d(Xb; Σb ⊆ Σb ) −−→ LBQ,d−1(Xb; Σb ) −→ .... Exactness of this sequence follows from the the usual construction of relative sequences in bordism, see [Wal16]. Let us consider the case where Σb = ∅ and Σb0 = X†, a minimal stratum of X.b We point out that there are compatible restriction maps

† † † LBQ,d(Xb; ∅) −→ Ld(Y )(X ), LBQ,d(Xb; ∅ ⊆ X ) −→ Ld(Y )(X ), both of which are onto, since any L-cycle over X† has a lift to an L-cycle over X,b

φ ω † ∗ ∗ (M; ∂M) −−→ (N, ∂N) −−→ X 7→ (ω ◦ φ) TX† −→ ω TX† −→ (TX† ,→) Xb

† (where TX† is a tubular neighborhood of X in Xb). The restriction map LBQ,d(Xb; ∅ ⊆ † † † X ) −→ Ld(Y )(X ) is also injective. Indeed, assume that (7.3) is such that φ restricts to X to be a BQ-equivalence and consider

φ×id (7.4) M × [0, 1] −−−−→ N × [0, 1] −−−−→ω×id X × [0, 1]. Since φ × id restricted to ∂M × [0, 1] ∪ M × {1} is a BQ-equivalence over X†, we recognize (7.4) as a null bordism for (7.3). Thus we have established the long exact sequence

† † † (7.5) ... −→ LBQ,d(Xb; X ) −→ LBQ,d(Xb) −→ Ld(X†)(X ) −→ LBQ,d−1(Xb; X ) −→ .... STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 49

7.2. Atiyah’s L2 signature theorem. If M is a closed even-dimensional manifold and L −→ M is a regular cover with transformation group Γ then given any elliptic diﬀerential operator on M, Atiyah’s L2-index theorem asserts the equality of its index with the Γ-equivariant L2-index of its lift to L. Let C∗Γ denote the maximal group C∗-algebra associated to Γ. Both of these numeric indices can be ∗ ∗ obtained from the C Γ index of the elliptic operator, an element in K0(C Γ), by applying ∗ two traces: τΓ : K0(C Γ) → C is obtained by extending the trace τΓ : CΓ → C given by X τΓ( αγ γ) := αe γ ∗ whereas τ : K0(C Γ) → C is obtained by extending the trace τ : CΓ → C, X X τ( αγ γ) := αγ . γ γ Atiyah’s theorem can be summarized as

(τΓ − τ) ◦ IndΓ = 0. In [ALMP16, Theorem 6.5], it is shown that the signature operator admits a parametrix which is ε-local; once we have this key information, Atiyah’s original proof carries over to the setting of Cheeger spaces essentially unchanged. However, as some of the arguments will be useful below, we present instead a proof that follows [CW03, Appendix].

Recall that whenever Xb is an even-dimensional Cheeger space and r : Xb −→ BΓ is the classifying map of a regular Γ-cover we have a signature class

G (r) ∗ 1 ∗ 1 Ind(XbΓ) := Ind(D ) ∈ K0(C Γ) ⊗Z Z[ 2 ] ≡ K0(C Γ)[ 2 ] where G (r) denotes the flat bundle of C∗Γ-modules corresponding to r, and DG (r) is the twisted signature operator. (As already remarked around Theorem 4.1, the index class in ∗ K0(C Γ) is defined using a choice of a mezzoperversity but it is in fact independent of this ∗ 1 choice in K0(C Γ)[ 2 ], which is why we omit it from the notation.) Also in loc. cit., for any Che topological space L there are bordism groups Ωn (L) of pairs (M,c f : Mc −→ L) where Mc is a Cheeger space of dimension n and f is continuous. The signature class Ind(DG (r)) only depends on the bordism class [(X, r : X −→ BΓ)] ∈ ΩChe (BΓ), b b dim Xb and defines a group homomorphism [ALMP17, Corollary 5.11]

Che ∗ 1 σΓ :Ω (BΓ) −→ K0(C Γ)[ ] dim Xb 2

Proposition 7.1. Any homomorphism f :Γ1 −→ Γ2 between discrete groups induces a commutative diagram σ Che Γ1 ∗ 1 Ωn (BΓ1) / Kn(C Γ1)[ 2 ]

Bf∗ f∗ σ Che Γ2 ∗ 1 Ωn (BΓ2) / Kn(C Γ2)[ 2 ] 50 PIERRE ALBIN AND PAOLO PIAZZA

Proof. Given r : Xb −→ BΓ1 we want to show 1 Ind(DG ((Bf)◦r)) = f (Ind(DG (r))) ∈ K (C∗Γ )[ ]. ∗ n 2 2

For each Γi, let ∗ C Γi − UΓi −→ BΓi be the universal Mishchenko bundle and note that ∗ ∗ ∗ (Bf) UΓ2 = UΓ1 ⊗C Γ1 C Γ2,

where the tensor product makes use of f∗, and correspondingly ∗ ∗ G ((Bf) ◦ r) = G (r) ⊗C Γ1 C Γ2. Thus we have G ((Bf)◦r) G(r) ∗ ∗ Ind(D ) = Ind(D ) ⊗C Γ1 C Γ2. On the other hand, the map

∗ 1 ∗ 1 f∗ ∗ 1 ∗ 1 Kn(Cr Γ)[ 2 ] = KKn(C,C Γ1)[ 2 ] −−→ KKn(C,C Γ2)[ 2 ] = Kn(Cr Γ2)[ 2 ] ∗ ∗ is precisely obtained by tensoring KK-cycles · 7→ · ⊗C Γ1 C Γ2, and so the proof is complete. Theorem 7.2 (Atiyah’s L2-signature theorem for Cheeger spaces). If Xb is an even-dimensional Cheeger space and r : Xb −→ BG is the classifying space of a regular cover, then G (r) G (r) τΓ(Ind(D )) = τ(Ind(D )). Proof. (We follow [CW03, Appendix].) Given any injective group homomorphism f :Γ1 −→ Γ2 we have a commutative diagram

∗ (7.6) K(C Γ1) τ τΓ

z f $ ZRd ∗ :

τ τΓ ∗ K(C Γ2) (see [PS07b, Lemma 2.22]) which combined with Proposition 7.1 yields a commutative diagram

Che (7.7) Ω (BΓ1) σ σ(2)

z Bf $ ZRd ∗ :

σ σ(2) Che Ω (BΓ2) Now let {e} −−→i G −−→h A STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 51

be the inclusion of the identity into G and an injection of G into an acyclic group A. (Recall that a group A is called acyclic if BA is an acyclic space and that any group has an injective homomorphism into an acyclic group, discrete if G is discrete.) As BA is acyclic its suspension is contractible and since, by the Eilenberg-Steenrod axioms, generalized homology theories are stably invariant, they must vanish on BA. From [ALMP17] Che we know that Ω∗ is a generalized homology theory and so Che Che (B(hi))∗ :Ω∗ (pt) −→ Ω∗ (BA) is an isomorphism. Thus from the commutative diagram

ΩChe(pt)

σ σ(2) Bi∗

| σ σ(2) " o ΩChe(BG) / Z b < R

Bh∗ σ σ(2)

ΩChe(BA)

Che Che the equality σ = σ(2) on Ω (BG) reduces to the equality on Ω (pt) where it is immediate.

7.3. Torsion elements and the cardinality of the BQ-structure set. In this subsection we adapt an argument of [CW03] to the setting of Cheeger spaces. For any Cheeger space Xb of odd dimension with π1Xb = Γ we have a group homomorphism

α (7.8) L (X × I) / BQ b ; C

Ind τΓ−τ APS ' ∗ K0(C Γ)

where τ and τΓ are as in §7.2, which is actually valued in R, because the index class is “self- adjoint”. Combining Proposition 6.9 with Atiyah’s L2-signature theorem (Theorem 7.2) we have the following result. Proposition 7.3. The homomorphism α vanishes on the image of the map

θ :NBQ(Xb × I, Xb × ∂I) −→ LBQ(Xb × I) from the surgery exact sequence of X.b By exactness of the surgery sequence this proposition says that α vanishes on those elements of LBQ(Xb × I) that act trivially on SBQ(Xb). Conversely, if x ∈ LBQ(Xb × I) is such that α(x) 6= 0, then we can show that x acts non-trivially on SBQ(Xb). 52 PIERRE ALBIN AND PAOLO PIAZZA

Indeed, let ρ :S (X) −→ K (D∗(X )Γ) be the ρ-map from §6.2 and let ρ be the BQ b dim Xb+1 bΓ Γ composition with the natural map induced by the classifying map ∗ ρΓ :SBQ(Xb) −→ K0(DΓ).

In [PZ16] it is shown that ρΓ(ι) = 0, where ι denotes the class in SBQ(Xb) represented by the identity map (the context in [PZ16] is that of diﬀerentiable or topological manifold, but it is easy to see that the same arguments establish the more general statement given here). On the other hand, Benameur-Roy in [BR15], have deﬁned a homomorphism ∗ βCG : K0(DΓ) −→ R for which our main result, together with Corollary 3.20 in [BR15], implies the following.

Proposition 7.4. The map ρCG = βCG ◦ ρΓ :SBQ(Xb) → R satisﬁes

ρCG(ι) = 0 and ρCG(∂(x)(ι)) = α(x) where ∂(x)(ι) denotes the action of LBQ(Xb × I) on SBQ(Xb) defined in Corollary 2.6. In particular, if α(x) 6= 0, then x acts non-trivially on ι. It is pointed out in [CW03] that, since α is a homomorphism into R, the existence of a non-zero element in its range implies that its range has infinite cardinality. They also point out that, if Ck is the cyclic group of order k then, for any ` ∈ N, the homomorphism αk : L4`(ZCk) −→ R defined as in (7.8) has range of infinite cardinality. Thus the idea is to use these elements to find elements in the range of α. Assume that the dimension of Xb is 4`−1, for some ` > 1. Let i : Xe −→ Xb be the inclusion of the regular part, which we recall is a BQ-transverse map, and let dXe (4`) be the dimension function for Xb that is equal to 4` on the inverse image of the regular part of X,b so that as in §7.1 we have a homomorphism

i∗ : L4`(Xe) = L4`(Zπ1(Xe)) −→ LBQ,dXe (4`)(Xb) = LBQ,dXe (4`)(Xb × I).

Note that we also have a homomorphism αe :L4`(Zπ1(Xe)) −→ R deﬁned as above.

Proposition 7.5. If the map i∗ : π1(Xe) −→ π1(Xb) is injective and there is a monomorphism p : Ck −→ π1(Xe) then the following diagram commutes

p∗ i∗ L4`(ZCk) / L4`(Zπ1(Xe)) / LBQ,dXe (4`)(Xb × I)

α α e k α ' R v Proof. The commutativity of the left triangle is a classical result, already used by Chang- Weinberger. To see that the second triangle commutes, i.e., that αe = α ◦ i∗, it suﬃces in view of (7.6) to show that whenever F : Xb1 −→ Xb2 is a BQ-transverse map we have a commutative diagram

F∗ (7.9) LBQ,d(Xb1) / LBQ,d(Xb2)

IndAPS IndAPS ∗ 1 f∗ ∗ 1 K∗(C Γ1)[ 2 ] / K∗(C Γ2)[ 2 ] STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR 53

where Γj := π1(Xbj) and f :Γ1 −→ Γ2 is the map induced by F. (For our application F is i given by the inclusion Xe −−→ X.b )

If γ ∈ LBQ,d(Xb1) is represented by

0 (φ;id,ψ) id (Mc; Xb1, Xb1) −−−−→ (Xb1 × [0, 1]; Xb1 × {0}, Xb1 × {1}) −−−→ Xb1 × I G(r) then IndAPS([γ]) is equal to Ind(ðsign ) where ðsign is the signature operator on Zb = Mc t (Xb1 × I) and, if R : Xb1 × [0, 1] −→ BΓ1 is the classifying map of the universal cover of Xb1 × [0, 1], then

r : Zb −→ BΓ1 is given by R ◦ φ t R Notice that it is implicit the choice of a mezzoperversity and the statement that the index class is in fact independent of this choice. It follows that

G((Bf)◦r) IndAPS(F∗[γ]) = Ind(D )

G (r) ∗ 1 and by Proposition 7.1 this is equal to f∗(Ind(D )) ∈ K(Cr Γ2)[ 2 ], as required. An immediate consequence of the discussion above is the inﬁnite cardinality of the structure set.

Corollary 7.6. Let Xb be a Cheeger space of dimension 4` − 1, ` > 1, such that π1(Xe) has an element of ﬁnite order and i∗ : π1(Xe) −→ π1(Xb) is injective. There exist elements xj ∈ LBQ,d(Xb × I), j ∈ Z, such that α(xi) 6= α(xj) for i 6= j. Consequently, the elements ∂(xj)(ι) are all distinct in SBQ(Xb). In particular

| SBQ(Xb)| = ∞

fi Remark 13. Write ∂(xi)(ι) = [Mcj −→ Xb] ∈ SBQ(Xb) with fi a transverse stratified homotopy equivalence. If Xb is a Witt space of depth one, then we claim that Mci is not stratified diffeomorphic to Mck for i 6= k. Indeed, let

(φ;id,fi) id xi = [(Wci; X,b Mci) −−−−→ (Xb × [0, 1]; Xb × {0}, Xb × {1}) −→ Xb × [0, 1]] and consider α(xi). This is the difference of two numbers: one is the Von Neumann-index on the total space of a Galois Γ-covering with boundary and the other is the usual index on the base of such Galois covering. We point out that the operators we are considering are invertible on the boundary because they have been perturbed by the Hilsum-Skandalis perturbation. We now write the APS-index formula, upstairs and downstairs, following [PV], and take the difference; we find ourselves with the Cheeger-Gromov rho invariant of the signature operator of Mci t(−Xb) pertubed by the Hilsum-Skandalis perturbation associated to fi. Now we proceed as in [PS07a, Section 10], taking an ε-concentrated Hilsum-Skandalis perturbation and letting ε ↓ 0. We then obtain, finally, that α(xi) is equal to the difference of the Cheeger-Gromov rho-invariants of Mci and Xb. Since α(xi) 6= α(xk) for i 6= k we can conclude that the Cheeger- Gromov rho-invariants of Mci and Mck are indeed different, and the statement follows from the stratified diffeomorphism invariance of the Cheeger-Gromov rho invariant on Witt spaces of depth 1, established in [PV]. 54 PIERRE ALBIN AND PAOLO PIAZZA

Remark 14. If Xb has depth one we can use van Kampen’s theorem to see that the map π1(Xe) −→ π1(Xb) induced by inclusion is an isomorphism if and only if the link of the singular stratum of Xb is simply connected (i.e., Xb is supernormal). For Xb of arbitrary depth, using Remark 12 in the setting of simply connected links lets us argue as above to see that torsion in the fundamental group of any ‘closed stratum’ Yb forces | SBQ(Xb)| = ∞. Indeed note that the condition on fundamental groups is superseded by the injective map

1 1 L(Zπ1(Yb))[ 2 ] −→ LBQ(Xb)[ 2 ].

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