Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants

In this paper we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-Mather stratified space $M_\Sigma$ with singular stratum $\beta M$ (a closed manifold of positive codimension) and associated link equal to $L$, a smooth compact manifold. We briefly call such spaces manifolds with $L$-fibered singularities. Under suitable spin assumptions we give necessary index-theoretic conditions for the existence of wedge metrics of positive scalar curvature. Assuming in addition that $L$ is a simply connected homogeneous space of positive scalar curvature, $L=G/H$, with the semisimple compact Lie group $G$ acting transitively on $L$ by isometries, we investigate when these necessary conditions are also sufficient. Our main result is that our conditions are indeed sufficient for large classes of examples, even when $M_\Sigma$ and $\beta M$ are not simply connected. We also investigate the space of such psc metrics and show that it often splits into many cobordism classes.


Introduction
This paper continues a program begun in [10,15], and our previous paper [14] (hereafter called Part I ), to understand obstructions to positive scalar curvature on manifolds with fibered singularities, for metrics that are well adapted to the singularity structure.
As in [14], the stratified spaces (or singular manifolds) M Σ that we study are Thom-Mather pseudomanifolds of depth one, where we take the two strata to be spin. Topologically, M Σ is homeomorphic to a quotient space of a compact smooth manifold M with boundary ∂M . The manifold M is called the resolution of M Σ , and the quotient map M → M Σ is the identity on the interiorM of M , and on ∂M , collapses the fibers of a fiber bundle φ : ∂M → βM , with fibers all diffeomorphic to a fixed manifold L, called the link of the singularity, and with base βM sometimes called the Bockstein of M (by analogy with other cases in topology). While we do treat the general case for some of our results, we shall eventually adopt the main geometric assumptions from [14], namely that the bundle φ comes from a principal G-bundle p : P → βM , for some connected semisimple compact Lie group G that acts transitively on L by isometries for some fixed metric g L , and thus ∂M = P × G L. The transitivity of the action of G on L means that L = G/H is a homogeneous space which comes with a metric g L with constant positive scalar curvature, which we normalize to be the same as that of a sphere of the dimension = dim L.
Given a Lie group G and a link L = G/H as above, we say that a fiber bundle E → B with a fiber L is a geometric (L, G)-bundle if its structure group is G, where G acts on L by isometries of the metric g L .
An adapted metric g on M Σ will be defined by the following data: a Riemannian metric g M on M , which is a product metric dt 2 + g ∂M in a small collar neighborhood ∂M × [0, ε) of the boundary ∂M , a connection ∇ p on the principal G-bundle p, and a Riemannian metric g βM on βM . We require the metrics g L , g ∂M and g βM to be compatible in the sense that the bundle projection φ : (∂M, g ∂M ) → (βM, g βM ) is a Riemannian submersion with fibers (L, g L ). Furthermore, since the structure group G of the bundle φ : ∂M → βM acts by isometries of the metric g L , we can make the special metric g L on the fibers orthogonal to the metric g βM lifted up to the horizontal spaces for the connection ∇ p . Then M Σ is the result of gluing together the Riemannian manifold M and a tubular neighborhood N of βM along their common boundary ∂M . The complement of βM in N will look like a fiber bundle over βM whose fibers are open cones (0, R) × L, where R is the radius of the cones. We require these fibers to be actual metric cones with the metric dr 2 + r 2 g L (which is actually a warped product metric on the product of L with the interval (0, R)), transitioning smoothly near r = R to a product metric dr 2 + Cg L for a suitable positive constant C. In this paper, we allow for M and βM to have non-trivial fundamental groups; the simply-connected case was considered in Part I [14]. Throughout this paper we assume that the link L = G/H is a simply connected homogeneous space, where G is a compact semisimple Lie group acting on L by isometries of the metric g L , and where scal g L = scal S = ( − 1). (1.1) This normalization makes the cone on L scalar-flat with respect to the metric dr 2 + r 2 g L . This will be important for our eventual existence results since this will guarantee that if g βM has positive scalar curvature, then we can make the scalar curvature uniformly positive on the tubular neighborhood of βM . In this setting, it is easy to see (since L is simply connected) that π 1 (∂M ) = π 1 (βM ), and then Van Kampen implies that π 1 (M Σ ) = π 1 (M ).
Theorem 1.1 (obstruction theorem). Let M Σ be an n-dimensional compact pseudomanifold with resolution M , a spin manifold with boundary ∂M . Assume the following: (1) M is a spin manifold with boundary ∂M fibered over a connected spin manifold βM , (2) the fiber bundle φ : ∂M → βM is a geometric (L, G)-bundle.
The indices α Γ cyl (M, g M ) and α Γ β (βM ) do not depend on a choice of adapted metric g, and they both vanish if there exists an adapted psc metric on M Σ . Remark 1.2. In our case we fix the metric g L on L and the connection on the bundle φ : ∂M → βM coming from a connection ∇ p on the principal bundle p : P → βM . (This is harmless since the space of such connections is contractible.) Then an adapted wedge metric g on M Σ is determined (up to contractible choices) by a metric g βM (which then determines the metric g ∂M ) and by an extension g of g ∂M to M . Thus the space of adapted wedge metrics on M Σ is contractible, and from the analytic properties of Dirac operators we then obtain that the cylindrical α-class α cyl (M, g M ) is independent of g (once the metric g βM has been fixed). Hereafter, we will omit the metric g from the notation. Notice that there is also a wedge α-class α Γ w (M Σ , g) defined by considering the Dirac operator on the regular part of the pseudomanifold M Σ . This is also independent of g for (L, G)-fibered pseudomanifolds. The two classes α Γ cyl (M ) and α Γ w (M Σ ) are equal if the metric g βM is psc. However, in more general situations, the wedge α-class α Γ w (M Σ , g) and the cylindrical α-class α Γ cyl (M, g M ) both depend on the choice of metric g and they are in general different. We shall make all this very precise in the next section. Now we are ready for the existence result. (1) M is a spin manifold with boundary ∂M fibered over a connected spin manifold βM , (2) the fiber bundle φ : ∂M → βM is a geometric (L, G)-bundle.
Let Γ = π 1 (M ), Γ β = π 1 (βM ). Furthermore, assume n ≥ + 6, where = dim L, and that one of the following condition holds: (i) either L is the boundary of a spin G-manifoldL equipped with a G-invariant psc metric gL, which is a product near the boundary and satisfies gL| L = g L , (ii) or the embedding ∂M → M induces an isomorphism on π 1 , and moreover, ∂M = βM × L, where L is an even quaternionic projective space, and Ω spin * (BΓ) is free as an Ω spin * -module.
Remark 1. 4. We notice that the condition (i) holds when L is a sphere, an odd complex projective space, or when L = G.
In the last part of this paper, Section 6, we begin to analyze the homotopy type of the space R + w (M Σ ) of adapted metrics of positive scalar curvature on M Σ , in the case where this space is non-empty. One of the key results is the following.
Theorem 1.5. Let M Σ be an (L, G)-fibered spin pseudomanifold, and assume that M Σ admits an adapted psc metric. Fix a connection ∇ p on the associated principal G-bundle over βM . Let res Σ : R + w (M Σ ) → R + (βM ) be the forgetful map sending a psc metric g on M Σ , interpreted as a pair (g M , g βM ), to the metric g βM on βM . Then res Σ is surjective onto R + (βM ), and it has a (non-canonical ) section. In particular, there is a split injection of the homotopy groups of R + (βM ) into those of R + w (M Σ ).
We use this result to detect non-trivial homotopy groups of the space R + w (M Σ ). Namely, we let M Σ = M ∪ −N (βM ), as before. Once we fix a base point, a metric g 0 ∈ R + w (M Σ ), it determines corresponding metrics g βM,0 ∈ R + (βM ) and g M,0 ∈ R + (M ) g ∂M,0 (where g ∂M,0 is given by the metric g βM,0 and g L ). Then we have index-difference homomorphisms: inddiff g βM,0 : π q R + (βM ) → KO q+n− and inddiff g ∂M,0 : π q R + (M ) g ∂M,0 → KO q+n+1 , where KO k is the k-th coefficient group for real K-theory (equal to Z for k a multiple of 4 and Z/2 for k ≡ 1, 2 mod 8); see Section 6 and [11,24]. According to Theorem 1.5, we can choose a splitting (For q = 1 one might get a semidirect product instead of a direct sum, as there is no obvious reason why the fundamental group should be abelian.) In particular, we prove the following result (see Corollary 6.7 for more details): Theorem 1.6. Let M Σ be a spin (L, G)-fibered compact pseudomanifold with L a simply connected homogeneous space of a compact semisimple Lie group, and n − − 1 ≥ 5, where dim M = n, dim L = . Assume that M Σ admits an adapted psc metric. Then the composition is surjective rationally and onto the 2-torsion.
We address the more general case when M Σ has non-trivial fundamental group in Corollaries 6.8 and 6.9.
2 KO-obstructions on L-fibered pseudomanifolds 2.1 KO n -classes on L-fibered pseudomanifolds Let (M Σ , g) be a Thom-Mather space of depth one, endowed with an adapted wedge metric g. We denote as usual by βM the singular locus of M Σ and by L the link. The resolved manifold, a manifold with fibered boundary, will be denoted by M .
Remark 2.1. We emphasize that for now the fibration φ : ∂M → βM is assumed to be just a smooth fiber bundle with a fiber L, without any restriction on the structure group of that fibration. Then we say that (M Σ , g) is an L-fibered pseudomanifold, as opposed to an (L, G)fibered pseudomanifold, which is the special case when the structure group G of the fibration φ : ∂M → βM acts on L by isometries of a certain metric g L . However, the relevant analytical constructions and results concerning the Dirac operators we need are well-studied and understood even for the general case of L-fibered pseudomanifolds.
Recall that M Σ = M ∪ ∂M (−N (βM )), where N (βM ) is the tubular neighborhood of the singular locus, which is also the total space of a fiber bundle with fiber equal to the cone over the link L. This also determines a smooth fiber bundle φ : ∂M → βM which is a restriction of the bundle (2.1) to the link L. We denote by T (∂M/βM ) → βM the corresponding vertical tangent bundle. As in [14,Section 2.3] (to which we refer for further details) we fix a connection on the fiber bundle L → ∂M φ − → βM . The metric g on M Σ restricted to the regular part of N (βM ) is assumed to have the following structure: with r denoting the radial variable along the cone and where the isomorphism between the horizontal bundle H of the chosen connection and φ * T (βM ) has been used. (In this general setting we employ the notation g X/B for a metric on the vertical tangent bundle of a smooth fiber bundle X → B.) The resolved manifold M inherits two metrics: the restriction of g to M , a Riemannian metric of product type near the boundary, denoted g M , and the extension of the metric g onM ≡ M reg Σ to the wedge metric on the wedge tangent bundle w T M → M . (This was defined in [4,Section 4] and in [2] under the alternate name incomplete edge tangent bundle; see again Part I [14, Section 2.3] for a quick introduction to the wedge tangent bundle). We assume that M , or equivalently M reg Σ , is given a spin structure. This fixes a spin structure on (∂M, g ∂M ) also. Then we assume that βM is also spin and fix a spin structure for (βM, g βM ). This also fixes a spin structure for the vertical tangent bundle T (∂M/βM ) → βM endowed with the vertical metric g ∂M/βM . Let us recall some basic facts in spin geometry. We refer to [31, Chapter II, Section 7] for further details. Let C n denote the Clifford algebra, and : Spin n → Hom(C n , C n ) the representation given by left multiplication. Then we denote by S / g (M ) the bundle given by P spin × C n (here P spin is the principal Spin(n)-bundle defined by the spin structure). There is a fiberwise action of C n on S / g (M ) on the right which makes S / g (M ) a bundle of rank one C n -modules. The bundle S / g (M ) inherits a Levi-Civita connection ∇. Let D / g be the associated C n -linear Atiyah-Singer operator; thus, by definition, D / g = cl • ∇. The operator D / g has the usual local expression with {e j } a local orthonormal frame of vector fields and {e j } the dual basis defined by the metric. D / g is a Z/2-graded odd formally self-adjoint operator of Dirac type commuting with the right action of C n . For this operator the Schrödinger-Lichnerowicz formula holds: with κ g denoting the scalar curvature of g. See again [31, Chapter II, Section 7] and also [42] for more details on this crucial point.
Notation. Unless absolutely necessary we shall omit the reference to the wedge metric g in the bundle and the operator, thus denoting the C n -linear Atiyah-Singer operator simply by Moreover, we shall often use the shorter notation S / instead of S /(M ).
As in Part I, we can regard D / as a wedge differential operator of first order on L 2 (M, S /), initially with domain equal to C ∞ c M reg Σ , S / ⊂ L 2 (M, S /) (more on this below). We are looking for self-adjoint C n -linear extensions of this differential operator in L 2 (M, S /).
We are assuming that ∂M is a fiber bundle of spin manifolds, L → ∂M ϕ − → βM , and we fix a connection on this bundle so as to have a well-defined notion of horizontal and vertical subspaces. Consequently where⊗ denotes the graded tensor product. Extending work of Bismut and Cheeger on fibrations of spin manifolds, see [9, Section 4], Albin and Gell-Redman made a careful study of the Levi-Civita connection near the singular stratum of a depth-one spin pseudomanifold -see Sections 2.2 and 3.1 in [2]; this study implies the following structure of D / near the singular stratum: and exhibits D / as a wedge differential operator of order 1: D / = r −1 D / e , with D / e an edge differential operator. See Part I and of course [2] for an introduction to edge and wedge operators. With a small abuse of notation, widely used in family index theory, we denote by D / L the generic operator of the vertical family D / ∂M/βM and by spec L 2 (D / L ) its spectrum.
The following result has been discussed in Part I: Then the following holds: is essentially self-adjoint. (ii) Its unique self-adjoint extension, still denoted by D /, defines a C n -linear Fredholm operator and thus a class α w (M Σ , g) in KO n , with n = dim M Σ .
As explained in Part I, (i) and (ii) are direct consequences of the analysis developed in [2] (which builds in turn on the slightly more complicated case of the signature operator on Witt spaces, see [4]). The main step is the construction of a Cl n -linear parametrix for an operator D / satisfying (2.3); this is based crucially on the construction of a parametrix for the edge operator D / e associated to D /, using Mazzeo's edge pseudodifferential calculus [35].
The following result also follows directly from [2] and from the Schrödinger-Lichnerowicz formula, which is valid for D /: If M reg Σ , g has psc everywhere then (2.3) holds (this is clear from above, given that N (βM ), g| N (βM ) has psc) and the unique self-adjoint extension D / is L 2 -invertible; in particular Definition 2.4. We shall say that the stratified spin pseudomanifold (M Σ , g) is geometric-Witt if the metric g is such that spec L 2 (D / L ) ∩ (−1/2, 1/2) = ∅ for each fiber L.
This notion is taken from [2], where it is applied to any generalized wedge Dirac operator. If (M Σ , g) does not satisfy (2.3) but is such that the vertical metric g ∂M/βM is of psc along the fibers, then we can still define a wedge-alpha class α w (M Σ , g) ∈ KO n . Indeed, by the Schrödinger-Lichnerowicz formula applied to the vertical family we know that there exists > 0 such that for each fiber L. There is a different (and in fact preferable) realization of this class that makes use of a natural self-adjoint domain defined for any wedge Dirac operator with invertible vertical family along the fibers of the boundary fibration. This is the so called vertical APS domain, see [1,Definition 2.3]. By applying the definition to our case we thus obtain a closed self-adjoint extension of D /, denoted D VAPS (D /). Following [1] one proves that on this domain D / admits a parametrix, that is, an inverse modulo compacts, which can be used in order to see that D /, D VAPS (D /) is a C nlinear Fredholm operator on its domain endowed with the graph norm. We obtain in this way a class α w (M Σ , g) ∈ KO n ; one can prove that the class defined through the vertical APS domain and the class defined by the operator associated to the rescaled metric are in fact equal. See [1, Remark 4.10] for a sketch of the argument. Definition 2.5. We shall say that the stratified spin pseudomanifold (M Σ , g) is psc-Witt if the metric g is of psc along the links, i.e., if the vertical metric g ∂M/βV induces on each fiber L a metric of psc.
Given a psc-Witt stratified spin pseudomanifold (M Σ , g) we define its wedge alpha class in KO n by considering the C n -linear Fredholm operator D /, D VAPS (D /) . Remark 2.6. In this article, which concentrates on (L, G)-pseudomanifolds, with L = G/H a simply connected homogeneous space and G a compact semisimple Lie group acting on L by isometries, we do not need to consider the vertical APS domain D VAPS (D /) or, equivalently, the rescaled metric. Indeed, the proof of [2, Theorem 1.3] shows that (2.3) is automatic when scal g L = scal S = ( − 1), which is the normalization we have adopted, see (1.1). Put it differently, for the purposes of this article we can and we shall exclusively treat the geometric-Witt case. Remark 2.8. As usual, we have α w (M, g) = π * [D / g ], with π the mapping of the compact pseudomanifold M Σ to a point. Remark 2.9. We remark that the wedge α-class and the fundamental class [D / g ] are unchanged if g(t), t ∈ [0, 1], is a 1-parameter family of adapted wedge metrics that are geometric-Witt for any t ∈ [0, 1]. This is in fact a special case of Theorem 2.17 below.
Remark 2.10. Theorem 2.7, Remarks 2.8 and 2.9 also hold when (M Σ , g) is psc-Witt, with the condition in Remark 2.9 being that g(t) is psc-Witt for each t ∈ [0, 1]. As we shall not use this case, we do not discuss the details.
2.2 C * -algebras and KO classes associated to D / In this second part of our work we would like to bring in the fundamental group of the pseudomanifold M Σ . In fact, we will begin, more generally, with Galois Γ-covering spaces since this approach is more general and simplifies the techniques.
Let Γ be a discrete group. A Γ-cover of M Σ will be denoted by M Γ Σ π − → M Σ . We shall keep the notation M Σ → M Σ for the universal cover of M Σ . We notice that the Γ-cover M Γ Σ has a natural structure of depth-1 pseudomanifold, with strata equal to the inverse image of the strata of M Σ through the projection map π. See [40] and [39] for more on this background material. Hence, the singular stratum βM Γ of M Γ Σ 1 is a Γ-cover of βM , − → ∂M inducing a diffeomorphism between the fiber of p Γ over x ∈ βM Γ and the fiber of p over π(x). In the sequel we shall always endow M Γ Σ , or rather its regular part, with the lift of an adapted wedge metric on M ; this metric extends as a Γ-equivariant wedge metric on all of the resolution M Γ ; we denote this Γ-invariant metric by g Γ . Let us assume once again that M and βM are spin; then also M Γ and βM Γ will be spin. Consequently, M reg Σ and M Γ Σ reg are spin. Let D / Γ be the associated Γ-equivariant Cl n -linear Atiyah-Singer operator. Notice that since the link of M Γ Σ is again L the analysis to be developed for understanding the properties of D / Γ is obtained by combining the usual analysis of Γ-equivariant operators on Γ-covers of smooth compact manifolds and the wedge analysis on compact stratified spaces. This principle is explained in detail in [39].
For the next theorem, however, we shall rather use the Mishchenko-Fomenko operator associated to D /. Define the Mishchenko bundle V := M Γ Σ × Γ C * r,R (Γ), a bundle of finitely generated projective C * r,R (Γ)-modules of rank 1 over the whole M Σ . By definition D / MF is equal to D / twisted by V restricted to the regular part of M Σ . We shall also refer to this operator as the Atiyah-Singer-Mishchenko operator and if necessary we shall denote it more precisely by D / MF,g . We shall also consider the associated edge-operator, D / e MF , obtained by twisting D / e by the Mishchenko bundle V.
Theorem 2.11. Let M and βM be spin and let g be a wedge adapted metric. Consider a Galois Γ-cover M Γ Σ of M Σ and endow the regular part with the associated Γ-equivariant metric g Γ as above. Let us assume that (M Σ , g) is geometric-Witt. Consider the Atiyah-Singer-Mishchenko operator D / MF,g . Then there is a unique self-adjoint extension of D / MF,g , with domain denoted D(D / MF,g ) such that the following hold: (1) the pair D / MF,g , D(D / MF,g ) defines an unbounded Kasparov (R, C * r,R Γ ⊗ C n )-bimodule and thus an index class Ind w D / MF,g , M Γ Σ ∈ KKO n R, C * r,R Γ ; as usual we identify the group KKO n R, C * r,R Γ with the isomorphic group KO n C * r,R Γ , r,R Γ is the assembly map and if f : M Σ → BΓ is the classifying map of the Γ-cover, then Sketch of the proof . The proof is an easy adaptation of the corresponding result for the signature operator on Witt spaces, see [4, Proposition 6.4, Theorem 6.6], and for this reason we shall be brief. The operator D / MF acts on the sections of S / ⊗ V. Since the Mishchenko bundle on the tubular neighborhood of the singular locus βM is the pull-back of a bundle on βM , we see that the analysis to be developed in order to understand D / MF is no more difficult than the one already developed for D /. More precisely, following the proof of [4, Proposition 6.4], we see that N q (D / e MF ), the normal operator of D / e MF at q ∈ βM is equal, up to conjugation by a bundle isomorphism, to N q (D /) ⊗ Id C * r,R Γ and so its invertibility properties, that are crucial in the analysis developed in [2], are a consequence of those already established for N q (D /). The proof now proceeds parallel to the one given in [4, Section 6.3] for the signature operator on Witt spaces. Notice that in (2) we use again the fact that a metric which is psc everywhere is psc in N (βM ), g| N (βM ) and thus geometric-Witt.

Notation. We shall briefly denote the index class Ind
In case we want to be very precise about the Galois cover M Γ Σ involved in the definition of this class we shall also write Remark 2.12. Similarly to Remark 2.9, the class α Γ w (M Σ , g) ∈ KO * C * r,R Γ is unchanged if g(t), t ∈ [0, 1], is a 1-parameter family of adapted wedge metrics that are geometric-Witt for any t ∈ [0, 1]. This is in fact a special case of Theorem 2.17 below.
Remark 2.13. The above results can also be established for psc-Witt pseudomanifold, either by using the vertical APS domain or by rescaling in the fiber direction on the boundary. We shall not need this more general case.
Remark 2.14. The existence of a fundamental class [D /] ∈ K * (M Σ ) and of a class α Γ (M Σ , g) can be established more generally for spin pseudomanifolds of arbitrary depth, assuming of course suitable Witt conditions along the links. This is based heavily on the general edge pseudodifferential calculus developed by Albin and Gell-Redman in [1]. Details will appear in [3].

Cylindrical KO-theory classes and a gluing formula
We decompose and Let g be an adapted wedge metric on M Σ . Denote by g M the Riemannian metric equal to the restriction of g to M and by g N (βM ) the metric equal to the restriction of g to N (βM ), the collar neighborhood of the singular stratum. By assumption, Recall that an adapted wedge metric g is such that g M and g N (βM ) are of product type in a collar neighborhood of ∂M . We make the hypothesis that g is geometric-Witt and that the whole metric g ∂M is of psc. Now attach an infinite cylinder to M along the boundary ∂M and extend the metric to be constant on the cylinder; similarly, attach an infinite cylinder to N (βM ) and extend the metric. Because of the hypothesis that g ∂M is of psc we have well defined classes For the existence of the cylindrical class on M Γ we refer the reader to [18,33,47] and references therein. The existence of the index class Ind cyl,w D / MF , N (βM ) Γ follows from the above theorem and these references.
Notation. We shall briefly denote the above index classes as Proposition 2.15. Under the same assumptions as above, namely that g is geometric-Witt and that g| ∂M is of psc, the following gluing formula holds: Proof . The existence of these classes has been discussed above. The gluing formula has been discussed for the signature operator on Witt and Cheeger spaces by Albin-Piazza [5], based on a well known technique due to Bunke -see [18]; the same arguments, with minor modifications, apply here.

Spin bordism of geometric-Witt pseudomanifolds
It is well-known that in the smooth context the Fredholm index of the spin-Dirac operator and the index class of the spin-Dirac operator twisted by the Mishchenko bundle define group homomorphisms where for the numeric index we limit ourselves to n = 4k. More generally, the α invariant of the Atiyah-Singer operator D / defines a group homomorphism α : Ω spin n → KO n while the α Γ -invariant of the Atiyah-Singer-Mishchenko operator D / MF defines a group homomorphism The results for the spin-Dirac operator, see (2.6), is well known and treated in several references. See for example the detailed treatment in [7] and its extension to the C * r Γ-index class in [32]. For the passage to the Atiyah-Singer operator, see [31, Chapter IV, Section 4, equation (4.3)] and [42] or, for a purely analytic argument, the appendix to this paper, Section A. Definition 2.16. Let (M Σ , g) and (M Σ , g ) be two spin-stratified pseudomanifolds of dimension n and let us assume that they are both geometric-Witt. Let (W Σ ,ḡ) be a spin-stratified pseudomanifold of dimension (n + 1) with collared boundary and adapted wedge metric, also of product-type near the boundary, such that We shall say that (W Σ ,ḡ) provides a geometric-Witt bordism between (M Σ , g) and (M Σ , g ) if the adapted wedge metricḡ is globally psc-Witt.
If Γ is a discrete group as above and f : M Σ → BΓ and f : M Σ → BΓ are classifying maps, then we shall say that (M Σ , g, f ) is geometric-Witt bordant to (M Σ , g , f ) if there exists (W Σ ,ḡ) as above and a continuous map F : W Σ → BΓ restricting to f and f on the boundary. We use the notation (W Σ ,ḡ, F ) : If there exists a geometric-Witt bordism (W Σ ,ḡ, F ) : where we recall that the above classes are defined in terms of Sketch of proof . Once an analytic proof of the spin-bordism invariance is given in the smooth closed case, the argument for geometric-Witt pseudomanifolds is the same. For this reason, we just give a sketch of the argument. Indeed, by general principles, the proof is reduced to a collar neighborhood of the boundary where the two directions, the one of the boundary and the one normal to the boundary, are completely decoupled. Now, for smooth spin manifolds we have provided a purely analytic proof of the spin-bordism invariance of the α-class and of the α Γ -class in the appendix; 2 it is very easy to extend this proof to the geometric-Witt case, exactly as it is done for the signature operator on Witt pseudomanifolds in order to establish the Witt-bordism invariance of the signature index and of the C * r Γ signature-index class. See the proof of [ 3 KO-obstructions on (L, G)-fibered pseudomanifolds In this section we shall finally meet the obstructions to the existence of a wedge metric of psc on an (L, G)-fibered pseudomanifold. We treat first the general case of L-fibered pseudomanifolds and then specialize to the case of (L, G)-fibered pseudomanifolds; the latter are the pseudomanifolds for which we shall prove an existence theorem.
15, and in particular (2.5). Summarizing, if g is an adapted wedge metric of psc on an L-fibered pseudomanifold of dimension n, then the following necessary condition is fulfilled: Consider now the singular locus of M Γ Σ , denoted βM Γ . We already observed that the covering map M Γ Σ → M Σ induces a Γ-cover βM Γ → βM . Consider the Atiyah-Singer operator on (βM, g βM ) and let us denote it by D / β g βM . We also have an Atiyah-Singer-Mishchenko operator D / β MF,g βM , obtained by twisting D / β g βM with the Mishchenko bundle associated to the Γ-cover Since βM is a closed smooth spin manifold we can in fact adopt the notation α Γ (βM ) ∈ KO n− −1 C * r,R (Γ) , given that this class does not depend on the particular choice of the metric g βM . See [42]. Unless further assumptions are made on the fibration L → ∂M → βM we cannot infer that also α Γ (βM ) = 0 in KO n− −1 C * r,R (Γ) , where = dim L. In the next section, on the other hand, we shall specialize this discussion to the case of (L, G)fibered pseudomanifolds and get consequently more precise information.

KO-classes on (L, G)-fibered pseudomanifolds
We follow the notation of the previous section but we now assume that M Σ is an (L, G)-fibered pseudomanifold endowed with an adapted wedge metric g. We know that (M Σ , g) is geometric-Witt -see Remark 2.6. By Theorem 2.11 there exists a well defined wedge-alpha class α Γ w (M Σ , g) in KO n C * r,R (Γ) . Moreover, from Theorem 3.5 of Part 1 (item (1)) we know that g| ∂M is of psc and so we also have a cylindrical class α Γ cyl (M, g M ) ∈ KO n C * r,R (Γ) . As already remarked in the Introduction, the space of adapted wedge metrics on an (L, G)-fibered pseudomanifold is contractible. Thus α Γ w (M Σ , g) ∈ KO n C * r,R (Γ) does not depend on the choice of the adapted wedge metric g. See Remark 2.12. Similarly, the cylindrical α class α Γ cyl (M, g M ) does not depend on the choice of g; indeed, if g and g are two adapted wedge metrics joined by a 1-parameter family of adapted wedge metric, then g M and g M are joined by a path of metrics that are uniformly of psc on the boundary ∂M . The result then follows from well-known properties of index classes on manifolds with cylindrical ends. See for example [26] or [34]. Thus we can adopt the notation for the two classes.
Finally, we remark that under the assumption that (βM, g βM ) is of psc, the equality holds generally; indeed, in this case the wedge metric g restricted to N (βM ) has psc -see again Theorem 3.5 of Part I (item (2)) -and thus the result follows from Proposition 2. 15. Assume now that our (L, G)-fibered pseudomanifolds is endowed with an adapted wedge metric of psc. We then certainly have that since this is true even in the general L-fibered case. However, in this particular case, because of Theorem 3.5 of Part I (item (3)), we have additionally that This discussion proves the obstruction theorem, Theorem 1.1 in the introduction. Our task in the next two sections will be to show that under suitable additional assumptions these necessary conditions for the existence of an adapted wedge metric of psc are also sufficient.
Recall now that for these special pseudomanifolds we have defined a bordism theory Ω spin,(L,G)-fb * (−); see Part I, Section 4. We end this section by observing that in this special case of (L, G)-fibered pseudomanifolds we can frame the alpha classes of this section in the following elegant way.
. We know that (L, G)-fibered pseudomanifolds are geometric-Witt. Fix an adapted wedge metric g on M Σ and set

Fundamental groups and KO-obstructions for (L, G)-fibered pseudomanifolds
We assume now that Γ = π 1 (M Σ ). We also consider π 1 (βM ). In this section we want to be particularly precise about the discrete groups involved in the various Galois coverings.
• In all of this section we assume that M Σ has an adapted wedge metric g of psc.
Under this assumption we observe that (M, g M ) is of psc and we certainly have that where the first class is relative to M Σ , the universal cover of M Σ , and the second class is relative to M π 1 (M Σ ) , the restriction of M Σ to the inverse image of M . Also, since βM admits a metric of psc, we also have, as already remarked, where this class is relative to the π 1 (M Σ )-cover of βM obtained by taking the restriction of M Σ to the inverse image of βM (this is in fact the singular locus of M Σ ). In addition we also have where this class is relative to βM , the universal cover of βM , π 1 (βM ) → βM → βM , and it is defined in terms of D / MF,π 1 (βM ) , the operator D / β g βM twisted by the Mishchenko bundle We remark that if L is simply connected (for example a sphere, complex or quaternionic projective space, or complex Grassmannian), then the fibration L → ∂M ϕ − → βM gives that ϕ * : π 1 (∂M ) → π 1 (βM ) is an isomorphism; moreover, the tubular neighborhood N of βM has a deformation retraction down to βM . An easy application of Van Kampen's theorem then proves that Thus in this case there are exactly two fundamental groups to keep track of, π 1 (M Σ ) ≡ π 1 (M ) and π 1 (βM ).
We summarize the index obstructions when L is 1-connected and we take into account the fundamental groups, in the following proposition.
Let g be an adapted wedge metric of psc on M Σ . We endow M with the metric g M := g| M . Under the above hypothesis we have the following vanishing results: where the first two classes are relative to M Σ , the universal cover of M Σ , and to its restriction to the inverse image of M respectively; the third class is relative to the restriction of M Σ to the inverse image of βM , and the fourth class is defined in terms of βM , the universal cover of βM . (BΓ). In more detail, this means that there are maps ξ : M Σ → BΓ and ξ β : βM → BΓ such that the following diagram of isomorphisms is commutative:   Proof . We start with some (L, G)-bordism W Σ : M Σ M Σ over BΓ. In particular, we have mapsξ : and the (L, G)-fiber bundle ∂ (1) W is given by the mapf : βW → BG, wheref | βM = f andf | βM = f . Then we denote by By assumption, the maps ξ and ξ β induce isomorphisms of fundamental groups. As already remarked, the restrictionsξ| M Σ = ξ andξ β | βM Σ = ξ β may not be isomorphisms on π 1 . However, the commutativity of the diagrams (with the vertical arrows induced by the inclusions) implies that the homomorphismsξ * : π 1 (W ) → Γ and (ξ β ) * : π 1 (βW ) → Γ are surjective.
Then the first step is to do surgery on βW to make the pair (βW, βM ) 2-connected. We begin by killing the kernel of π 1 (βW ) Γ. This proceeds as in the proof of [14,Theorem 4.7], the point being that since this kernel maps trivially to π 1 (BΓ), the surgery can be done over BΓ without any extra effort. After taking care of the fundamental group, the situation is the same as in [14,Theorem 4.7]. The process for making the pair (W, M ) 2-connected is completely analogous.
Once we have established Theorem 4.1, we get a corresponding bordism theorem; see Theorem 4.6 from Part I.
between such objects is an (L, G)-singular spin pseudomanifold W Σ = W ∪ ∂W N (βW ) with the boundary δW Σ = M Σ −M Σ given together with mapsξ : W → BΓ andξ β : βW → BΓ β such thatξ   To generalize the arguments in [14] for proving existence of well-adapted positive scalar curvature metrics in some cases, we now need an exact sequence into which the bordism group of Definition 4.5 will fit. In general this is quite complicated, so we only do an easy case as follows.
Proof . First of all, note that the assumption on L gives the surjectivity of β. For if the map βM → BΓ β × BG represents an element of Ω spin n− −1 (BΓ β × BG) and we form the associated (L, G)-bundle where the composite π 1 (βM ) → π 1 (βW ) → Γ β (along the top triangle) is an isomorphism, and thus the homomorphism π 1 (βW ) → Γ β is surjective. The first step is to do surgery on βW to make the pair (βW, βM ) 2-connected. We begin by killing the kernel of the surjection π 1 (βW ) Γ β , just as in the proof of Theorem 4.1, so that the inclusion of βM → βW induces an isomorphism on π 1 . Once this is done, we have an exact sequence π 2 (βM ) → π 2 (βW ) → π 2 (βW, βM ) Then we need to do surgery on certain embedded 2-spheres in βW , generating π 2 (βW, βM ). These embedded 2-spheres have trivial normal bundle because of the spin assumption, and map null-homotopically to the classifying space BG (since the Lie group G is simply connected, hence automatically 2-connected -π 2 of any Lie group vanishes -and thus BG is 3-connected). So the necessary surgery is possible. The process for making the pair (W, M ) 2-connected is completely analogous.
The following result is a straightforward generalization of known bordism results for psc metrics. Here Theorem 4.10 provides all the necessary tools. Proof . This proceeds like the proof Theorem 4.6 from Part I, using Theorem 4.10.

The existence theorem in the non-simply connected case
In this section we will use the results from Section 4 to obtain existence theorems for welladapted positive scalar curvature metrics on singular pseudomanifolds in suitable cases. These results are parallel to those in [14, Section 6], but without the assumption that M and ∂M are simply connected. However, we will have to assume something about the fundamental groups. We denote by Bott 8 a simply connected spin 8-manifold with A-genus 1, which one can take to be Ricci-flat [27]. This is often called a "Bott manifold", because taking the product with it in spin bordism corresponds, after applying the α-transformation to KO * , to the Bott periodicity map KO * ∼ = − → KO * +8 . Let Γ be a discrete group. We need the following definitions. if for any closed connected spin manifold M with fundamental group Γ, the vanishing of the generalized index α Γ (M ) ∈ KO n (C * r,R (Γ)) implies that M × Bott 8 k admits a psc metric for some sufficiently large k.
We recall that Stolz ([45] and [46, Section 3]) sketched a proof that the s GLR conjecture holds whenever the Baum-Connes assembly map is injective, which is true for a very large class of groups (conjecturally, all groups!). The GLR conjecture is satisfied for a more restricted class of groups, but including free groups, fundamental groups of oriented surfaces, and free abelian groups (as long as one takes n bigger than the rank of the group). It also holds [13] for finite groups with periodic cohomology, which includes finite cyclic groups and quaternion groups. But it fails for some groups [20,43]. Our first main result is a generalization of [14, Theorem 6.3]. Recall the setting: We fix a simply connected homogeneous space L = G/H, where G is a compact semisimple Lie group acting on L by isometries of the metric g L , where scal g L = scal S = ( − 1), where = dim L.
Theorem 5.1. Let M Σ be an n-dimensional compact pseudomanifold with resolution M , a spin manifold with boundary ∂M . Assume the following: (1) M is a spin manifold with boundary ∂M fibered over a connected spin manifold βM , (2) the fiber bundle φ : ∂M → βM is a geometric (L, G)-bundle.
Let Γ = π 1 (M ), Γ β = π 1 (βM ). Furthermore, assume n ≥ + 6 and that the following condition holds: • the link L is a spin G-boundary of a G-manifoldL equipped with a psc metric gL which is a product near the boundary and satisfies gL| L = g L .
Proof . Before we start with the proof we emphasize that under the assumptions above the index α Γ cyl (M ) is well defined, independent of the chosen adapted wedge metric g; see Remark 2.9 and Proposition 3.1. According to Theorems 4.11 and 4.9, it suffices to show that the bordism class 4 is a sum of classes of pseudomanifolds with well-adapted positive scalar curvature metrics. We use the exact sequence (4.2). Since α Γ β (βM ) = 0 and Γ β satisfies the GLR conjecture, and since dim βM = n − − 1 ≥ 5, the manifold βM admits a psc metric. Then, since the fibration φ : ∂M → βM is a geometric (L, G)-bundle; we can identify ∂M with P × G L, where P → βM is the corresponding principal G-bundle. We use a bundle connection on P to construct a well-adapted psc metric on the tubular neighborhood N of βM in M Σ . In particular, we obtain a psc metric on the boundary ∂N , where the (L, G)-fiber bundle ∂N → βM is also a Riemannian submersion. We use the G-manifoldL which bounds L to construct anL-bundle M = P × GL over βM associated to the above principal G-bundle P → βM . By assumptions,L is given a psc metric gL which is a product metric near ∂L = L. Then M has a bundle metric of positive scalar curvature, and joining M to N , we get an (L, G)-singular spin manifold Theorem 5.2. Let a link L and pseudomanifold M Σ be as in Theorem 5.1, but this time only assume that the groups Γ and Γ β satisfy the s GLR conjecture. Then M Σ stably admits a well-adapted metric of positive scalar curvature, i.e., M Σ × Bott 8 k admits such a metric for some k ≥ 0.
Proof . The proof of this is exactly the same as for Theorem 5.1, the only difference being that first we need to cross with some copies of the Bott manifold to get positive scalar curvature on βM , and then we might need to cross with additional copies of the Bott manifold to get positive scalar curvature on M . Remark 5.3. It is possible without great effort to adapt the arguments above to the case where βM is disconnected. We leave details to the reader. Now we present a generalization of [14,Theorem 6.7]. It was shown in [14,Lemma 6.6] that the class of HP 2k , k ≥ 1, is not a zero-divisor in the spin bordism ring Ω spin * . It then follows that if H * (BΓ; Z) is torsion free and the Atiyah-Hirzebruch spectral sequence converging to Ω spin * (BΓ) collapses, so that Ω spin * (BΓ) is a free Ω spin * -module Ω spin * ⊗ Z H * (BΓ; Z), then the class of HP 2k does not annihilate any non-zero class in Ω spin * (BΓ). This condition on Γ is satisfied if BΓ is stably homotopy-equivalent to a wedge of spheres -for example, if Γ is a free group or free abelian group. Then if n − 8k ≥ 6, the pseudomanifold M Σ has an adapted psc metric if and only if the α-invariant α Γ cyl (M ) ∈ KO n C * r,R (Γ) vanishes. Remark 5.5. Note that in this case we have isomorphisms π 1 M ∼ = π 1 ∂M ∼ = π 1 βM . Also recall that if the bundle ∂M → βM is trivial, then the singularities are of Baas-Sullivan type.
Proof . Necessity was proved in Section 3, so we need to prove existence of an adapted metric of positive scalar curvature assuming the vanishing condition. Since ∂M = βM × L is a spin boundary over BΓ (namely, it is the boundary of M ), its class is trivial in Ω spin * (BΓ). But the class of L cannot annihilate any non-zero class in Ω spin * (BΓ), by [14,Lemma 6.6], so in fact βM must be a spin boundary over BΓ.  In this section we shall study the space R + w (M Σ ) of well-adapted wedge psc metrics on M Σ . Our first goal is to establish a relationship between the space R + w (M Σ ) and corresponding spaces R + (βM ), R + (∂M ), R + (M ) of psc metrics. We consider the map res Σ : R + w (M Σ ) → R + (βM ) which takes an adapted wedge metric g on M Σ to its restriction g βM on βM , and we will observe that this map is a Serre fiber bundle. We show (this was stated before as Theorem 1.5) that there exists a non-canonical section s : R + (βM ) → R + w (M Σ ); this fact, together with recent results on the homotopy type of the space of psc metrics on spin manifolds, [11,24], implies that the homotopy groups of the space R + w (M Σ ) are rationally at least as complicated as the groups KO * C * r (π 1 (βM )) ⊗ Q. Next, we elaborate on a direct definition of the index-difference homomorphism which enables us to detect some nontrivial classes in π q R + w (M Σ ) . Finally, if the fundamental group of M Σ has an element of finite order, we show that the Cheeger-Gromov rho invariant on a depth-1 pseudomanifold, introduced and studied in [39], can be used to show that π 0 R + w (M Σ ) = ∞.
6.1 Controlling the homotopy type of R + w (M Σ ) We fix an (L, G)-fibered compact pseudomanifold M Σ = M ∪ ∂M N (βM ). Here M is a compact manifold with boundary ∂M which is a geometric (L, G)-bundle over βM . Recall that the G acts on L by isometries of a given metric g L , and the (L, G)-bundle φ : ∂M → βM is given by a principal G-bundle p : P → βM so that ∂M = P × G L. Then we identify the space R + w (M Σ ) of well-adapted wedge psc metrics on M Σ with the space of associated triples (g M , g βM , ∇ p ), where g M and g βM are psc metrics on M and βM respectively, and ∇ p is a connection on the principal bundle P . By definition of well-adapted wedge metrics, the triples (g M , g βM , ∇ p ) are subject to the following conditions: (i) g M = dt 2 + g ∂ near the boundary ∂M ⊂ M , for some Riemannian metric g ∂ on ∂M ; (ii) the bundle map φ gives a Riemannian submersion φ : (∂M, g ∂ ) → (βM, g βM ), and the connection ∇ p on P defines a G-connection ∇ φ for the submersion.
We denote by R + (M, ∂M ) the space of Riemannian psc metrics g on M such that g = dt 2 + g ∂ near the boundary ∂M , for some metric g ∂ on ∂M . There is an obvious restriction map res : R + (M, ∂M ) → R + (∂M ), (6.1) which is known to be a Serre fiber bundle, see [23, Theorem 1.1]. For a given metric h ∈ R + (∂M ), we denote by R + (M ) h = res −1 (h) the corresponding fiber of (6.1). We notice that there is a map ι L : R + (βM ) → R + (∂M ) which is given by lifting a metric g βM on βM to a Riemannian submersion metric g ∂M on the total space ∂M of the (L, G)-bundle φ : ∂M → βM , using the connection ∇ φ of (ii) above, by putting the metric g L on each fiber L. This map is injective since the metrics g βM and ι L (g βM ) determine one another. Then, by definition, a wedge psc metric g on M Σ determines a unique psc metric g βM ∈ R + (βM ). This gives a well-defined map res Σ : R + w (M Σ ) → R + (βM ). Thus we see that the space R + w (M Σ ) is a pull-back in the following diagram: As we mentioned above, the restriction map res : R + (M, ∂M ) → R + (∂M ) is a Serre fiber bundle, and it is easy to see that the map res Σ : R + w (M Σ ) → R + (βM ) from (6.2) is also a Serre fiber bundle. 5 Let g βM ∈ R + (βM ) and g ∂M ∈ R + (∂M ) be such that ι L (g βM ) = g ∂M . We have the following commutative diagram of fiber bundles: where R + w (M Σ ) g βM and R + (M ) g ∂M are the corresponding fibers. Note that the downward map on the left is a homotopy equivalence, in fact a homeomorphism, since any element of R + (M ) g ∂M defines a unique psc wedge metric.
Theorem 6.1. Let M Σ be an (L, G)-fibered compact pseudomanifold with L a simply connected homogeneous space of a compact semisimple Lie group. Also assume that M Σ admits an adapted wedge metric of positive scalar curvature. Then there exists a section s : R + (βM ) → R + w (M Σ ) to res Σ . In particular, there is an injection of homotopy groups Proof . We want to construct a (non-canonical) section to the "forgetful map" sending a psc metric g (interpreted as a triple (g M , g βM , ∇ p ), as above) to the metric g βM . As before, we will fix the connection ∇ p once and for all, and we fix a metric g βM ∈ R + (βM ). Given another metric g βM in R + (βM ), we consider the linear homotopy g βM (t) = (1 − t)g βM + tg βM within the space R(βM ) of all Riemannian metrics. This homotopy may go out of the subspace R + (βM ), but the scalar curvature function along the homotopy is bounded below by some constant −c, c > 0. If we scale the metrics g βM (t) by λ 2 , then the scalar curvature scales by scal λ 2 g βM (t) = λ −2 scal g βM (t) . We consider the family of metrics g ∂M (t) = ι L λ 2 g βM (t) on ∂M . Recall that scal L = ( − 1) > 0, so we obtain Clearly, there exists λ > 0 such that scal g ∂M (t) > 0 for all t > 0. Then we lift the curve of metrics g ∂M (t) to a curve of metrics g(t) ∈ R + (M, ∂M ) using homotopy lifting in the fiber bundle res : R + (M, ∂M ) → R + (∂M ).
In particular, we obtain an adapted wedge metric (g M , g βM , ∇ p ) ∈ R + w (M Σ ). It is important to notice the following: 1. The curve of the adapted wedge metrics (g M (t), g βM (t), ∇ p ) is not, in general, a curve in the space R + w (M Σ ). Indeed, the scalar curvature on the tubular neighborhood N (βM ) is dominated by the sum of zero scalar curvature on the cone over L and by the scalar curvature λ −2 scal g βM (t) . 6 6 An important comment: to apply the argument of [14,Theorem 3.5(2)] in order to rescale a metric g βM in R + (βM ) so that ιL(g βM ) extends to a psc metric on the whole tubular neighborhood N of βM , one needs to make use of a positive lower bound on the scalar curvature. So while g βM lifts to R + w (MΣ) (after rescaling by a positive constant), the same is not true for the other metrics in the path in R(βM ) from g βM to g βM .
2. The correspondence g βM → (g M , g βM , ∇ p ) can be made continuous (in the C 2 topology) as long as the metrics g βM stay inside a compact set K ⊂ R + (βM ).
This is enough to get a (non-canonical) section s g βM : R + (βM ) → R + w (M Σ ) of the fiber bundle res Σ in (6.2).
Here is an interesting example of obtaining information about π 0 R + (M Σ ) . Let L = G = SU(2) = S 3 . Since π 7 (BS 3 ) = π 6 S 3 = Z/12, there are 12 distinct principal S 3bundles over βM = S 7 , all of them of the rational homotopy type of S 7 ×S 3 . For any one of these S 3 -bundles over S 7 , the total space is ∂M = ∂N , where M = N is the disk bundle of our S 3bundle over S 7 . Define M Σ = N ∪ ∂M −N , the double of N . Note that N is actually the unit disk bundle of a quaternionic line bundle, so our M Σ in this case is in fact a smooth manifold, though not every Riemannian metric on M Σ is adapted for the (L, G)-fibered structure, and we are only interested in this special subclass of metrics. We know that π 0 R + w (M Σ ) contains π 0 R + (βM ) = π 0 R + S 7 as a direct summand. By [26,Theorem 4.47], π 0 R + S 7 contains a copy of Z, and the same argument shows π 0 R + S 11 contains a copy of Z. By taking a connected sum of (M Σ , g Σ ) with S 11 , g k for suitable metrics g k on S 11 (the connected sum takes place on the interior of M , so it doesn't change the adapted wedge structure of the metric near βM = S 7 ), we see that π 0 R + w (M Σ ) contains Z ⊕ Z, with one summand coming from βM and one coming from the interior of M . Remark 6.4. We recall that for a closed manifold X, the homotopy type of the space R + (X) does not change under admissible surgeries; in particular, this implies that the homotopy type of R + (X) depends only on the bordism class of X in a relevant bordism group [23, Theorem 1.5]. As we have seen, on an (L, G)-pseudomanifold M Σ , we can do two types of surgeries: surgeries on the interior of the resolution M , and surgeries on βM . It turns out that the homotopy type of the space R + w (M Σ ) is also invariant with respect to corresponding admissible surgeries, [16].
There are several methods for detecting non-trivial elements in π q R + (βM ) : the indexdifference homomorphisms and (higher) rho-invariants are certainly two very efficient tools in this direction. In the Sections 6.2 we shall elaborate further on these two tools. Notice that some of the results proven by rho-invariants can also be recovered using the index-difference homomorphisms (6.5) explained below; see [24, Remark 1.1.2].

The index-difference homomorphism
First we recall some results from [11,24]. LetX be a compact closed spin manifold, dimX = k ≥ 5. Leth 0 ∈ R + X be a base point. Assume there is a map f :X → BΓ, where Γ is a discrete group such that f * : π 1X → Γ is split surjective. (If π 1X is trivial, we assume Γ = 0.) We also need the case when the boundary ∂X = X is non-empty, with a given metric h ∈ R + (X). As before, we denote by R + X h the subspace of metricsh in R + (X, X) which restrict to h on the boundary X. Leth 0 ∈ R + X h be a base point. Again, we assume that there is a map f :X → BΓ, such that f * : π 1X → Γ is split surjective. In both these settings (without using split surjectivity of f * yet) we have the index-difference homomorphisms Since it will be useful later, we remind the reader about where these index difference maps come from. Say for simplicity thatX is a closed manifold. A class in π q R + X is represented by a family g t , t ∈ S q , of psc metrics onX, with g t 0 , t 0 the basepoint in S q , equal to our basepoint psc metric. Let us denote here by g 0 this base-point metric. From this family we get a warped product metric g on S q ×X, that restricts on the copy ofX over t ∈ S q to g t , and that is the usual flat (if q ≤ 1) or round (if q ≥ 2) metric on the copy of S q over each point inX. By the usual argument that "isotopy implies concordance" [25, Lemma 3], we can, without changing the homotopy class of our map t → g t in π q R + X , assume that g has positive scalar curvature on S q ×X. Our class in π q R + X is trivial if and only if it can be extended over the disk D q+1 . So extend g to a metric on D q+1 ×X which restricts to a product metric dr 2 + g on a neighborhood of the boundary. Extend it to a complete metric on R q+1 ×X by allowing r, the distance to the center of the disk, to go to ∞, and taking the metric to be dr 2 + g for all r ≥ 1.
Since the metric has psc outside a compact set, the Dirac operator on R q+1 ×X for this metric is Fredholm, and we get an index in KO q+1+k C * r (Γ) as usual. One way to see this property is to use b-calculus techniques; this point of view will be useful later on in this section. This index is an obstruction to triviality of our class in π q R + X , since it would be 0 if we could extend our family of metrics to a psc family over the disk. Summarizing, we have defined a homomorphism An alternative way to construct the index difference is the following. Once we choose a base point g 0 = g t 0 ∈ R + X , then for any metric g ∈ R + X , there is a linear path g t = (1 − t)g 0 + tg of metrics in the space R X of all Riemannian metrics. Then we have a curve of the corresponding Dirac operators D gt with the ends of the curve, D g 0 and D g being in the contractible space of C k -linear invertible operators. Such a curve determines a loop in the space KO k which classifies C k -linear Fredholm operators (where the space of invertible operators is collapsed to a point). One can easily replace this by the classifying space KO k C * r (Γ) for the K-theory KO * +k C * r (Γ) in the non-simply connected case. Thus in homotopy we get the index-difference homomorphism which depends on a choice of the base point g 0 ∈ R + X . The equivalence of these two approaches to the index difference (with a discussion of how to trace them back to the work of Gromov-Lawson and Hitchin, respectively) may be found in [21,22] (see also [17]). Next we recall the following relevant results: Theorem 6.5 (see [11,24]).
Then the images of both index-difference homomorphisms inddiff Γ h 0 from (6.5) generate the target group KO q+k+1 C * r (Γ) ⊗ Q as a Q-vector space.
Recall also the following stable result. As in the beginning of Section 5, we denote by Bott 8 a Bott manifold, which is a simply connected spin 8-manifold withÂ(Bott) = 1. By the work of Joyce [27], we may choose a Ricci flat metric g on Bott. Then for a closed manifoldX there are induced maps and write R + X Bott −1 for the homotopy colimit. Similarly, if ∂X = X = ∅, and h ∈ R + (X), we define the space R + X h Bott −1 . Assume we have a reference map f :X → BΓ which is surjective on the fundamental groups. Then the index-difference homomorphisms inddiff Γ h 0 extend to the corresponding index-difference homomorphisms Here is one more relevant result: Theorem 6.6 (see [24,Theorem B] and [17]). If Γ is a torsion-free group satisfying the Baum-Connes conjecture, and f :X → BΓ is split surjective on the fundamental groups, then the homomorphisms (6.6) are surjective for all q ≥ 0. Now we are ready to apply those results to the case of (L, G)-pseudomanifolds. Let M Σ be an (L, G)-fibered compact pseudomanifold. We notice that if the space R + w (M Σ ) is not empty and g 0 ∈ R + w (M Σ ) is a base point, then it determines base points in the corresponding spaces: the metrics g βM,0 ∈ R + (βM ), g ∂M,0 ∈ R + (∂M ) and g M,0 ∈ R + (M ) g ∂M,0 .
If M Σ is spin and simply connected, then we have the following commutative diagram: where the homomorphisms inddiff g M,0 and inddiff g βM,0 are both nontrivial whenever the target groups are. In particular, the homomorphism inddiff g 0 : π q R + w (M Σ ) → KO q+n+1 ⊕ KO q+n− is surjective rationally and surjective onto the torsion of KO q+n+1 ⊕ KO q+n− .
Proof . We use Corollary 6.2 to choose a splitting . Then we can construct the homomorphism inddiff g 0 as a direct sum . By Theorem 6.5(a), the individual index difference maps in this decomposition are non-trivial whenever the target KO k is non-zero. Since KO k = Z/2 whenever it has torsion, that implies that the map surjects onto the torsion. Similarly, the part of Theorem 6.5(a) about the rational groups implies rational surjectivity. Now we address the case when M Σ is not simply-connected. Let θ : Γ β → Γ be a group homomorphism as in Section 4. We consider a triple (M Σ , ξ, ξ β ) as an object representing an element in the bordism group Ω Thus we obtain the homomorphism . The same argument as above and Theorem 6.5(b) prove the following: Corollary 6.8. Let M Σ be a (L, G)-fibered compact pseudomanifold with L a simply connected homogeneous space of a compact semisimple Lie group, and n − − 1 ≥ 5, where dim M = n, dim L = . Let g 0 ∈ R + w (M Σ ) = ∅ be a base point giving corresponding base points, the metrics g βM,0 ∈ R + (βM ), g ∂M,0 ∈ R + (∂M ) and g M,0 ∈ R + (M ) g ∂M,0 . Let (M Σ , ξ, ξ β ) represent an element in the bordism group Ω spin,(L,G)-fb n BΓ β θ − → BΓ , then we have the following commutative diagram: If, in addition, the pair (Γ, Γ β ) is such that • Γ and Γ β both satisfy the rational Baum-Connes conjecture, • Γ and Γ β are both torsion free and have finite rational homological dimension d, and • q > d − 2n + 2 + 1, Let R + (M Σ ) Bott −1 be the homotopy colimit We have the following conclusion from Theorems 6.6 and 6.1: is surjective.
Example 6.10. In [24, Section 1.1.3], the authors provided an example of a group π 0 such that KO 7+k (C * r (π 0 )) ⊗ Q has countably infinite dimension for each k ≥ 0, 7 as well as a 4dimensional closed spin manifold X with π 1 X = π 0 . Then, according to [24,Theorem C], the manifold Y = X × S 2 has the property that the group π 7+k R + (Y ) ⊗ Q has countably infinite dimension for each k ≥ 0. Then it is easy to construct a pseudomanifold M Σ with L = G = SU(2) and βM = Y . Indeed, we let M = X × D 3 × L, ∂M = X × S 2 × L, βM = X × S 2 and M Σ = M ∪ ∂M (Y × c(L)). Then, clearly, Theorem 6.1 and Corollary 6.8 imply that π 7+k R + w (M Σ ) ⊗ Q has countably infinite dimension for each k ≥ 0.
6.3 A direct approach to the index-difference homomorphism for (L, G)-fibered pseudomanifolds Let M Σ be as above, thus a (L, G)-fibered pseudomanifold, and let g 0 , g 1 ∈ R + w (M Σ ) be two well adapted wedge psc metrics. Then the "difference" between g 0 and g 1 could be detected directly by a wedge relative index as follows. We consider M Σ ×[0, 1] as a pseudomanifold with boundary, and equip it with a well adapted wedge metricḡ of product-type near the boundary and equal to g 0 on M Σ × {0} and to g 1 on M Σ × {1}. Now we attach infinite cylinders to M Σ × [0, 1] to are equivalent if there exists a manifold W with corners such that ∂W = ∂ (0) W ∪ ∂ (1) W , where ∂ (1) W fibers over a manifold with boundary βW , i.e., ∂ (1) W : ∂M ∂M and βW : βM βM are usual spin bordisms between closed spin manifolds, and with Γ = π 1 (M Σ ). We shall mainly be interested in ρ w,CG and we shall denote it simply by ρ w . Summarizing, we have a well-defined homomorphism: We shall prove that under the present assumptions • ker f Γ β ,Γ has infinite cardinality, • there is a free and transitive action of ker f Γ β ,Γ on C(M Σ , ξ, ξ β ).
Indeed, as x is null-bordant in Ω Consider now Γ = Z n , the cyclic group of order n. It is explained in [37], building on specific examples provided in [12], that the Cheeger-Gromov rho invariant defines a map ρ: Pos spin m (BZ n ) → R which has an image of infinite cardinality. To prove this, one only needs to show that ρ is a homomorphism and that it is non-trivial; the property that it is a homomorphism is a consequence of the definition of rho invariant whereas the fact that it is non-trivial follows from the specific examples in [12] (lens spaces). Remark that we have a well defined rho-homomorphism ρ rel w : Pos spin,(L,G)-fb m BΓ β Bθ − − → BΓ → R, obtained by composing the homomorphism (6.7) with the homomorphism (6.10): Recall now that we are under the assumption that there is an injection j : Z n → Γ. Such an injection induces homomorphisms Consider the following diagram, where for typographic reasons we omit the superscripts spin and fb, By naturality this is a commutative diagram. Consider K := ker f Zn . We know from [37] that ρ| K has an image of infinite cardinality. For κ ∈ K consider We also have with g the wedge metric defined by g M , g βM and ∇ p . Indeed: whereas, thanks to Lemma 2.2 in [37] and naturality, we have • Do we have the analogue of Stolz sequence (7.1) in the singular setting?
• If so, can one map the analogue of sequence (7.1) to a Higson-Roe type analytic exact sequence as in [38]? This would involve the definition of a higher rho class associated to a wedge psc metric. For the signature operator on stratified spaces that are Witt or Cheeger, a similar result has been proved in [5], mapping the Browder-Quinn surgery sequence to the Higson-Roe analytic surgery sequence, and we expect the analysis there to play a role here.
• Is there an R-group R • Can we use the Stolz (L, G)-sequences above, the mapping to Higson-Roe, and the techniques of Xie-Yu-Zeidler [49] in order to give a lower bound on the rank of the group Pos spin,(L,G)-fb n (BΓ)?

Torsion-free groups
In the smooth case one can prove that if π 1 (M ) is torsion-free and the Baum-Connes map µ max : K * (BΓ) → K * (C * max Γ) is an isomorphism, then the Cheeger-Gromov rho invariant of a positive scalar curvature metric on a spin manifold M of odd dimension must vanish. See [36] and also [8] for a new proof based on the Higson-Roe analytic surgery sequence. Notice that this is a no-go result: the Cheeger-Gromov rho invariant for manifolds with fundamental group satisfying these conditions cannot be used in order to distinguish metrics of psc (indeed, it is equal to 0).
Can one extend this vanishing result to (L, G)-pseudomanifolds and the Cheeger-Gromov rho invariant of a wedge metric?
A direct approach would build on the proof of Piazza-Schick in [36]. A different route would use the results of the previous Section 6.4, the Benameur-Roy map K 0 (D * Γ ) → R of [8] and the exactness of the Higson-Roe analytic surgery sequence.

Groups with torsion
On the other hand, one would like to define and use higher rho invariants in order to distinguish wedge metrics of positive scalar curvature, especially for fundamental groups with torsion. See for example [6,33,48] for the case of smooth closed manifolds.

Stratifications with higher depth
For applications to algebraic varieties, moduli spaces, and other natural examples of stratified spaces, it would be nice to generalize our theory to pseudomanifolds with higher depth, in other words, with singular strata of multiple dimensions. While the analytic part concerning Dirac operators is largely under control, thanks to [1] (see also [40]), on the geometric side we expect several complications: • The geometry of the tubular neighborhoods of the singular strata gets to be rather complicated, as one would need to consider iterated Riemannian submersions and careful curvature estimates.
• Proving the appropriate bordism theorem would be quite complicated.
• Perhaps one could say more in the case of Baas-Sullivan singularities, as in [10], which did consider some pseudomanifolds with higher depth.

Topological questions
• The "mixed fundamental group" bordism group Ω (L,G) n BΓ β Bθ − − → BΓ seems to be very hard to compute; it doesn't just fit into an exact sequence like the one we developed in [14].
But is there a spectral sequence computing Ω

A On the spin-bordism invariance of the α-invariant
In this appendix we shall discuss briefly an analytic proof of the spin-bordism invariance of the α class and of the α Γ class. We have already observed, building on the case of the signature operator on Witt spaces treated in great detail in [4,Section 7], that given an analytic proof in the smooth case, it extends mutatis mutandis to the pseudomanifold case.

A.1 A generator of KO 1 (R)
Consider R with its standard spin structure. Then we have D / R and its class in KO 1 (R) := KKO(C 0 (R), C 1 ).
But δ is the inverse of the isomorphism KO 0 (point) → KO 1 ((0, 1)) obtained by taking the Kasparov product with the generator D / (0,1) of KO 1 ((0, 1)). This again follows from Kasparov's results on Poincaré duality in KO-theory, [  Proof . We write part of the long exact sequence in KO associated to the short exact sequence 0 → C ∂X (X) → C(X) → C(∂X) → 0.

A.3 Spin bordism invariance
We have the induced exact sequence with ι the inclusion of ∂X into X. Notice, in particular, that Consider π Y : Y → point; obviously π Y = π X • ι. Then we have Note that for the α class the spin bordism invariance is usually proved by identifying it with the Atiyah-Milnor-Singer invariant. See [31]. The generalization of this approach to the α Γ class is treated in [41] and (more extensively) in [42].