Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities

In this paper we study manifolds $M_{\Sigma}$ with fibered singularities, more specifically, a relevant space $\Riem^{\psc}(X_{\Sigma})$ of Riemannian metrics with positive scalar curvature. Our main goal is to prove that the space $\Riem^{\psc}(X_{\Sigma})$ is homotopy invariant under certain surgeries on $M_{\Sigma}$.

It is convenient to denote βX := Y . Then we obtain a manifold with singularities of the type L (or just L-singularities) X Σ = X ∪ ∂X −βX × C(L), where C(L) is a cone over L. The metric g L on L easily extends to a scalar-flat metric g C(L) on the cone C(L) which is a product-metric near its base L ⊂ C(L). We say that a metric g on X Σ is a well-addapted Riemannain metric on X Σ if (i) the restriction g| X is a regular Riemannian metric which is a product-metric near ∂X ; (ii) the restriction g| βX×C(L) splits as a product-metric g| βX×C(L) = g βX + g C(L) .
We denote by R(X Σ ) the space of all well-adapted Riemannian metrics on X Σ , and by R psc (X Σ ) its subspace of psc-metrics. Then the above geometrical question is asking whether the space R psc (X Σ ) is non-empty. This existence question was addressed and even affirmatively resolved for some particular examples of the singularity types L (provided that all manifolds involved are spin and both X and βX are simply-connected, see [2]).
There is a particularly interesting example here. Let us consider spin manifolds, and choose L = S 1 with a non-trivial spin structure, so that L represents the generator η ∈ Ω spin 1 = Z 2 . We denote by Ω spin,η * (−) the bordism theory of spin manifolds with η -singularities, and by MSpin η the corresponding representing spectrum. It turns out, there exists a Dirac operator on spin-manifolds with η -singularities. Furthermore, there is a natural transformation α η : Ω spin,η * → KO η * which evaluates the index of that Dirac operator, where the "K -theory with η -singularities" KO η * (−) coincides with usual complex K -theory. Here is the result from [2]: [2]) Let X be a simply connected spin manifold with nonempty ηsingularity of dimension n ≥ 7. Assume βX = ∅. Then X admits a metric of positive scalar curvature if and only if α η ([X]) = 0 in the group KO η n ∼ = KU n .

1.3.
Existence of a psc-metric on a manifold with fibered singularities. There are more general objects, "manifolds with fibered singularities" (or pseudomanifolds with singularities of depth one). Here again, we start with a manifold X with boundary ∂X = ∅, which is a total space of the fiber bundle ∂X → βX with the fiber L. To get geometrically interesting objects, we assume that L is given a metric g L of non-negative constant scalar curvature and that the bundle ∂X → βX has a structure group G which is a subgroup of the isometry group Isom(g L ) of the metric g L . Then the bundle ∂X → βX is induced by a structure map f : βX → BG. Let C(L) be a cone over L with a "cone metric" g C(L) which restricts to g L on the base and is scalar-flat.
Furthemore, we assume the isometry action of the G extends to the one of the cone metric g C(L) .
Then there is a fiber bundle N (βX) → βX , which is given by "inserting" the cone C(L) as a fiber with the same structure group G. The actual manifold with fibered singularities is given as X Σ := X ∪ ∂X N (βX). Then a well-adapted Riemannain metric g on X Σ is a regular Riemannian metric restricted to X (which is also a product near the boundary), and g| N (βX) is determined by a requirement that the projection N (βX) → βX is a Riemannian submersion (which has a structure group G ⊂ Isom(g L )) and with the cone metric g C(L) on the fiber (we give a detailed definition in Section 2). We denote by R(X Σ ) the space of well-adapted Riemannain metrics on X Σ , and by R psc (X Σ ) its subspace of psc-metrics. Below we describe two interesting cases.
1.3.1. We assume that all manifolds are spin, and L = S 1 representing η ∈ Ω spin 1 , and G = S 1 . We obtain a corresponding bordism group Ω spin,η-fb * of such manifolds. Then there exists an appropriate Dirac operator on X Σ , and index map α η-fb : Ω spin,η-fb * → KO η-fb * evaluating the index of that Dirac operator. Here is the existence result for psc-metrics in that setting: [4]) Let X Σ = X ∪ N (βX) be a simply connected spin manifold with fibered η -singularity (i.e. X and βX are simply-connected and spin) of dimension 1.3.2. Again, we assume that all (L, G)-manifolds are spin: here L is equipped with a metric g L with constant scalar curvature s L = ( − 1), dim L = , and G is a subgroup of the isometry group of the metric g L . We assume L = ∂L and the G-action on L extends to a G-action on L. In this setting, an (L, G)-manifold X Σ could be given as a triple (X, βX, f ), where X is a manifold with boundary ∂X , which is a total space of a fiber bundle ∂X → βX (with a fiber L and a structure group G) given by a map f : βX → BG. In this setting and with n = dim X , we have indices α(βX) ∈ KO n− −1 and α cyl (X) ∈ KO n . Here are the existence results: spin manifold X Σ . Assume that X , βX , and G are all simply connected, that n − ≥ 6, and suppose that L is a spin boundary, say L = ∂L, with the standard metric g L on L extending to a psc-metric onL, and with the G-action on L extending to a G-action onL. Assume that the two obstructions α(βX) ∈ KO n− −1 and α cyl (X) ∈ KO n both vanish. Then X Σ admits a well-adapted psc-metric.
Then if X and βX are both simply connected and n − 8k ≥ 6, X Σ has an adapted psc-metric if and only if the α-invariants α(βX) ∈ KO n−8k−1 and α cyl (X) ∈ KO n both vanish.
There are several other interesting cases and also much more general results when the pseudo manifold X Σ has non-trivial fundamental group; see [6]. Now we are ready to address our main result concerning homotopy invariance of corresponding spaces of psc-metrics on X Σ .

Main result.
The homotopy-invariance of certain spaces of psc-metrics is a crucial property which has allowed detection of their non-trivial homotopy groups. Let M be a closed spin manifold.
An important consequence of the results due to Chernysh [7], Walsh [17,18,19] (see also recent work by Ebert and Frenk [8]) is that the homotopy type of the space R psc (M ) is an invariant of the bordism class [M ] ∈ Ω spin n (provided M is simply-connected and n ≥ 5). 1 1 There is also a similar result for non-simply connected manifolds.
Notice that if X Σ = X ∪ ∂X N (βX) is a pseudomanifold with (L, G)-singularities equipped with structure map f : βX → BG, then there are two types of surgery possible on X : (i) a surgery on its "resolution", i.e. the interior X ⊂ X Σ away from the boundary ∂X ; (ii) a surgery on the structure map f : βX → BG.
In case (i) all constructions are the same as in the case of closed manifolds, however, in case (ii), we have to be a bit more careful. Indeed, letB : βX βX 1 be the trace of a surgery on the map f : βX → BG, with ∂B = βX −βX 1 . Then the map f extends to a mapf :B → BG which gives a fiber bundlep : Z → B with the fiber L. This gives us a new manifold X 1 = X ∪ ∂X Z with boundary ∂X 1 , the total space over a new Bockstein βX 1 with the same fiber L. Also we obtain a new conical part N (βX 1 ) as above. All of this results in a new pseudomanifold with structure map f 1 =f | βX 1 : βX 1 → BG. Here is our main technical result: (i) Let i : S p ⊂ X be a sphere with trivial normal bundle, and X Σ,1 be the result of surgery on X Σ along S p . Then if n − p ≥ 3, the spaces R psc (X Σ ) and R psc (X Σ,1 ) are weakly homotopy equivalent.
(ii) Let i : S p ⊂ βX be a sphere with trivial normal bundle, and with f • ι : S p → BG homotopic to zero. LetB be a trace of the surgery along S p ⊂ βX with ∂B = βX −βX 1 and a structure mapf :B → BG. Then if n − − p ≥ 3, the spaces R psc (X Σ ) and R psc (X Σ,1 ) are homotopy equivalent, where X Σ,1 is given by (1).
Theorem A could be applied to a variety of interesting examples. Among these are: (1) Let L = k be the set of k points, and let G = Z k be its "isometry group". Then a ( k , Z k )-manifold X Σ = X ∪ ∂X N (βX) is assembled out of a manifold X with boundary ∂X equipped with free Z k -action, and a structure map βX → BZ k classifies the corre- (2) Let η ∈ Ω spin 1 be as above, i.e., [L] = η , and G = S 1 . Then, similarly, we arrive at the bordism groups Ω spin,η-fb * and the index map α η-fb : Ω spin,η-fb * → KO η-fb * , as in Theorem 5.1 as above.
The above examples lead to the following two corollaries of Theorem A: Corollary B. Let X Σ be a spin ( k -fb)-manifold. Assume dim X ≥ 7 and that X and βX are simply-connected. Then the homotopy type of the space R psc (X Σ ) is a bordism invariant and depends only on the bordism class [X Σ ] ∈ Ω spin, k -fb n .
Corollary C. Let X Σ be a spin (η-fb)-manifold. Assume dim X ≥ 9 and that X and βX are simply-connected. Then the homotopy type of the space R psc (X Σ ) is a bordism invariant and depends only on the bordism class [X Σ ] ∈ Ω spin,η-fb n .
The cases addressed in Theorems 1.4 and 1.5 give interesting implications.
(3) Let L and G be as in Theorem 1.4, i.e. G is a simply connected Lie group, L is a spin boundary, say L = ∂L, with the standard metric g L on L extending to a psc-metric onL, and with the G-action on L extending to a G-action onL. Then an (L, G)-singular spin manifold X Σ determines an element in the relevant bordism group Ω spin,(L,G)−-fb n which fits to an exact triangle (see [5]): These examples lead to following corollary Corollary D. In both of the cases described in (3) and (4) (where = 2k in the case (4)).
In the last section we show that the cases (3) and (4) above lead to an interesting result concerning homotopy groups of R psc (X Σ ).

2.1.
Positive scalar survature on manifolds with boundary. Here we recall the main constructions and results from [18]. The set-up is as follows. Given a smooth compact n-dimensional manifold X (possibly with boundary ∂X = ∅), we denote by R(X), the space of all Riemannian metrics on X . The space R(X) is equipped with the standard C ∞ -topology, giving it the structure of a Fréchet manifold; see [15, Chapter 1] for details. For each metric g ∈ R(X), we denote by s g : X → R the scalar curvature on X of the metric g and by R + (X) ⊂ R(X) the subspace of psc-metrics on X .
In the case when ∂X = ∅, it is necessary to consider certain subspaces of R + (X) where metrics satisfy particular boundary constraints. With this in mind, we specify a collar embedding c : ∂X × [0, 2) → X around ∂X and define the space R + (X, ∂X) as: . Fixing a particular metric g ∈ R + (∂X), we define the subspace denotes the space of psc-metrics on Z which restrict as a product structure on the neighbourhood we fix a pair of psc-metrics g 0 ∈ R + (Y 0 ) and g 1 ∈ R + (Y 1 ) and consider the following subspace of R + (Z, ∂Z): We note that each metricḡ ∈ R + (Z, ∂Z) g 0 ,g 1 provides a psc-bordism (Z,ḡ) : We next assume X is a manifold whose boundary ∂X = Y 0 is equipped with the metric g 0 .
Furthemore, we assume that both spaces R + (X, ∂X) g 0 and R + (Z, ∂Z) g 0 ,g 1 are non-empty. Now, by making use of the relevant collars, we glue together X and Z to obtain a smooth manifold which we denote X ∪ Z and which has boundary ∂(X ∪ Z) = Y 1 ; see Fig. 1.
In particular, we obtain the space R + (X ∪ Z, Y 1 ) g 1 of psc-metrics which restrict as g 1 + dt 2 on Then for any metricḡ ∈ R + (Z, ∂Z) g 0 ,g 1 , we obtain a map: where h ∪ḡ is the metric obtained on X ∪ Z by the obvious gluing depicted in Fig. 1. Consider the case when the bordism Z : Y 0 Y 1 is an elementary bordism, i.e., when Z is the trace of a surgery on Y 0 with respect to an embedding φ : Then we have the following.
Then there exist metrics is a psc-bordism.
Such a bordism is usually called a Gromov-Lawson bordism (or GL-bordism for short). Here is a reformulation of the main technical result from [18]: be an elementary bordism as above with p, q ≥ 2. Then for any (2), is a weak homotopy equivalence.

2.2.
The case of manifolds with fibered singularities. Let L be a closed manifold with fixed metric g L of non-negative constant scalar curvature.
is simple if it satisfies either one of the following conditions: (a) the manifold (L, g L ) is such that s g L is a positive constant; For each case (a), (b), we fix a subgroup G of the isometry group of the metric g L . Before going forward with our constructions, we would like to clarify why the condition that the scalar curvature s g L is non-negative constant is important here.
In the case (a), s g L is a positive constant. Denote by C(L) a cone over L and give the cone metric g C(L) = dr 2 + r 2 g L . This is a warped product metric on (0, R] × L away from the cone point (where r = 0). Then the scalar curvature of the metric g C(L) (away from the vertex) is given as where = dim L. Thus if s g L is a positive constant, we scale the metric g L to achive the idenity , and we will assume that the metric g L satisfies this condition. The case (b), when L = S 1 , has special features. First, according to (3) Euclidian metric, and we do not make any further assumptions.
Hence we assume that the cone C(L) is scalar-flat outside of its vertex in the above cases.
Let X Σ = X ∪ ∂X N (βX) be a pseudomanifold with (L, G)-singularities, with dim X = n, dim L = , and f : βX → BG is a corresponding structure map. Assuming that a link (L, g L ) is simple as above, we define the space of all well-adapted metrics R(X Σ ) as follows.
Definition 2.4. We say that a metric g on a pseudomanifold X Σ = X ∪ ∂X N (βX) is a Rimeannian well-adapted metric if it satisfies the following conditions: (i) the restriction g| X is a Riemannian metric such that g| X = g ∂ + dt 2 near the boundary (∂X, g ∂ ); (ii) the Riemannian manifold (∂X, g ∂ ) is a total space of a Riemannian submersion ∂X → βX .
As we mentioned above, there are two types of surgery that could be performed on X Σ : the first one on its resolution, the interior of X , and the second one on the structure map f : βX → BG.
We consider the latter.
Moreover, for now it is convenient to cut the singularity out and to work with a smooth manifold X whose boundary is fibered over βX with fiber L. We use the notation (X, βX, f ) for such manifold, where the boundary ∂X of X is a total space of the fiber bundle from the diagram: Here p L : E(L) → BG is the universal fiber bundle with the fiber L and the structure group G. Clearly these collars provide collars e i : E i × [0, 2) along ∂Ē .
In order to study the space R psc (X Σ ) of well-adapted psc-metrics on X Σ , we first study its closest relative, the space R psc (X, βX, f ) h β , defined as a subspace of R psc (X, ∂X) as follows.
Definition 2.5. Let h β ∈ R psc (βX) be a given metric. The space R psc (X, βX, f ) h β consists of all Riemannian metric g ∈ R psc (X, ∂X) subject to the conditions: • the restriction g| c(∂X×[0,1]) to the collar c(∂X × [0, 1]) splits as g ∂X + dt 2 , where g ∂X is a psc-metric on ∂X ; • the metric g ∂X on the total space ∂X of the fiber p : ∂X → βX is given by the psc-metric h β on the base βX and g ∂X restricts to g L (up to isometry from G) along every fiber L.
Consider again an elementary bordism (B,f ) : Let g βX = h β 0 . We assume that there is a psc-metrich β ∈ R psc (B) h β , where h β 1 is a psc-metric on the manifold B 1 . The metrich β provides a psc-bordism Furthemore, the structure mapf :B → BG determines a bordismĒ : E 0 E 1 , whereĒ is a pull-back of the universal (L, G)-fibration:Ē , where the metricsḡ ∂ , g ∂ 0 and g ∂ 1 are determined by the metricsh β , h β 0 and h β 1 respectively on the bases B , B 0 and B 1 and the metric g L on the fiber L. Now we glue the manifolds X andĒ (again, making use of collars near their boundaries) to obtain a manifold X 1 = X ∪ ∂XĒ with boundary ∂X 1 = E 1 , which is a total space of the (L, G)-fibration p 1 : Similarly to the case of manifolds with boundary we obtain a map: for any fixed metrich β ∈ R psc (B) h β 0 ,h β is a weak homotopy equivalence. Here, as above, the psc-metric g ∂ is given by the psc-bordism ) and the metric g L on the fiber L.

Proof of Theorem 2.6
3.1. Some standard metric constructions. Here we briefly a recall a couple of standard metric constructions. These constructions are discussed in detail in [19,Section 5].
We fix some constants δ > 0 and λ ≥ 0. Then a (δ -λ)-torpedo metric on the disk D n , denoted g n torp (δ) λ , is a psc-metric which roughly takes the form a round hemisphere of radius δ > 0 near the centre before transitioning into a cylinder of radius δ and length λ ≥ 0 near the boundary; see first image in Fig. 2 below.
We now consider product metrics g n−1 torp (δ) λ + dt 2 on the cylinder D n−1 × [0, L]. It is convenient to allow L to vary bearing in mind that there is an obvious family of rescaling maps (see the map ξ L at the end of section 2 in [19]) which allow us to compare such metrics, for any L > 0, on D n−1 × I .
It is shown in [19,Section 5], provided n ≥ 4, that any such product metric g n−1 torp (δ) λ + dt 2 can Figure 2. The metrics g n torp (δ) λ , g n torp+ (δ) λ andĝ n torp (δ) λ 1 ,λ 2 (bottom) on the manifolds D n , D n + and D n stretch (top) followed by the boot metric g n boot (δ) Λ,l be moved by isotopy through psc-metrics to a particular psc-metric called a δ -boot metric. We do not provide a precise definition of such a metric here as full details can be found in [19]. However, we would like to explain the basic idea. The metric is denoted g n boot (δ) Λ,l . Here Λ > 0 is some possibly large constant andl = (l 1 , l 2 , l 3 , l 4 ) ∈ (0, ∞) 4 . This metric should be thought of as defined on D n−1 × [0, l 3 ] and, roughly, takes the form: g n−1 torp (δ) l 4 + dt 2 when t is near l 3 and g n−1 torp (δ) l 2 + dt 2 when t is near 0.
Remark 3.2. The constant Λ > 0 in the definition of the boot metric controls the bending arc used in pushing out the "toe" and will in part depend on δ . This bending arc may need to be quite large to maintain positive scalar curvature. In turn, this puts constraints on the components l 2 and l 3 of the vectorl . We will not concern ourselves with this now, except to say that sufficiently large Λ, l 2 and l 3 can always be found.

3.2.
Back to the proof of Theorem 2.6. The proof follows from that of [18,Theorem A]. We will provide a brief review of the main steps of that proof and show that it goes through perfectly well in our case. The strategy is to decompose the map into a composition of three maps as shown in the commutative diagram below.
Here the right vertical map denotes inclusion. We will define the spaces R psc boot (X, βX, f ) h β std and R psc Estd (X, βX, f ) h β 1 and the maps µ boot and µ Estd in due course. The point is to show that each of these maps is a weak homotopy equivalence.
We denote by k = n − − 1 = dim βX = B 0 . We consider carefully the elementary bordism . The manifoldB is given by attaching a handle D p+1 × D q+1 to B 0 along the embeddings φ : We would like to have some flexibility for the embedding φ. We introduce the following family of rescaling maps: where ρ ∈ (0, 1]. We set and N ρ := φ ρ (S p × D q+1 ), abbreviating N := N 1 and φ := φ 1 . Let T φ be the trace of the surgery on B 0 with respect to φ. We denote by R psc std (B 0 ), the space defined as follows: According to Chernysh's theorem [7,8], the inclusion is a weak homotopy equivalence. A major step in the proof of this theorem is the fact (which follows easily enough from the original Gromov-Lawson construction in [11]) that for any pscmetric h β ∈ R psc (B 0 ), there is an isotopy h β t , t ∈ I of metrics in R psc (B 0 ) connecting h β 0 = h β to a psc-metric h β std ∈ R + std (Y ). By a well known argument, see [18,Lemma 2.3.2], this isotopy gives rise to a concordance:h β con on B 0 × [0, λ + 2] for some λ > 0 which takes the form of product metrics: Note that on the slice N 1 2 × [0, 1], the metrich β con pulls back to a metric of the form Making use of [18, Lemma 5.2.5] we can perform an isotopy of the metrich β con , adjusting only on N 1 2 ×[0, 1], to replace the g q+1 torp +dt 2 factor in (7) with g q+2 boot (1) Λ,l for some appropriately large Λ > 0 and withl satisfying l 1 = l 4 = 1. We denote the resulting psc-metrich β pre on B 0 × [0, λ + 2]. We consider B 0 × [0, λ + 2] as a long collar ofB and assume that the mapf restricted to B 0 × [0, λ + 2] is given byf (x, t) = f 0 (x). LetĒ 0 be a manifold given by pulling back the fiber bundlē wheref 0 is a restriction off . We now use the metrich β pre on B 0 × [0, λ + 2] to extend the metric g ∂ from the boundary ∂M to a metricḡ ∂ pre onĒ 0 by "inserting" the metric g L to the fibers L via the mapf 0 : B 0 × [0, λ + 2] → BG.
Let ∂X × [−1, 0] ⊂ X be a collar, such that ∂X × {0} = ∂X , where, by assumption, every slice ∂X ×{t} is a total space of the (L, G)-fibration over B 0 ×{t}. We consider a manifold X ∪Ē 0 which we identify with the original manifold X by deforming linearly the manifold (∂X × [−1, 0]) ∪ ∂XĒ0 to the collar ∂X × [−1, 0]. For any element g ∈ R psc (X, βX, f ) h β 0 , the metric g ∪ ∂Xḡ ∂ pre on X ∪Ē 0 (obtained by obvious gluing) is denoted by g std , an element of the space R psc (X, βX, f ) h β std . This gives a map We denote R psc boot (X, βX, f ) h β std := Im(µḡ∂ pre ). This new metric is depicted in the bottom left of Fig.  3, with the original metric g depicted in the top left. For clarity, this figure depicts only the case when L is a point. Lemma 6.5.5 of [18], consolidating work from previous sections, shows that in the case when L is a point, the map µḡ pre is a weak homotopy equivalence.
Note that any element of the space R psc boot (X, βX, f ) h β std := Im(µḡ∂ pre ) has, near the boundary, a standard pieceḡ ∂ pre which is determined by the metrich β pre = ds 2 p + g q+2 boot (1) Λ,l . Replacing this Figure 3. Representative elements of the spaces from the commutative diagram (6) above in the case when L is a point with g p+1 torp + g q+1 torp near the boundary, we obtain a map the image ofμ, we obtain the lower horizontal map µ Estd in diagram (6). A typical element in the image of this map is depicted in the lower right of Fig. 3. This lower horizontal map is demonstrably a homeomorphism.
It remains to show that the inclusion R psc is a weak homotopy equivalence. Note that the notation "Estd" used in describing the former space (originating in [18]) is intended to convey the fact that these metrics take a standard form on a much larger region than typical metrics in R psc (X 1 , βX 1 , f 1 ) h β 1 and are thus "Extra-standard".

Proof of Theorem A
Let X Σ = X ∪ ∂X N (βX) be a pseudomanifold as above, where X is a manifold with boundary ∂X . We consider the spaces of psc-metrics R psc (X, ∂X) and R psc (∂X) which are connected by the restriction map res : R psc (X, ∂X) → R psc (∂X), res : g → g| ∂X .
This map is very important for us because of the following fact: Theorem 4.1. [7,8] The restriction map res : R psc (X, ∂X) → R psc (∂X) is a Serre fiber bundle.
Now we consider two pseudomanifolds X Σ = X ∪ ∂X N (βX) and X Σ,1 = X 1 ∪ ∂X N (βX 1 ), where X 1 = X ∪ ∂X Z , and the manifold Z is given by an elementary psc-bordismB : βX βX 1 and a structure mapf :B → BG, so that f =f | βX and f 1 =f | βX 1 . Namely, the manifold Z is a total space of the following smooth bundle: , h β 1 ∈ R psc (βX 1 ) be metrics as in Theorem 2.6, and g ∂ 0 ∈ R psc (∂X), g ∂ 1 ∈ R psc (∂X 1 ) be corresponding Riemannian submersion metrics which restrict to g L on each fiber L over βX (respectively, over βX 1 ). It is important to keep in mind that the metrics g ∂ 0 and g ∂ 1 are determined by the corresponding metrics h β 0 and h β 1 and by the maps f : βX → BG and f 1 : of the corresponding restriction maps: res 0 : R psc (X, ∂X) → R psc (∂X), res 1 : R psc (X 1 , ∂X 1 ) → R psc (∂X 1 ).
We denote by R psc (∂X, g L ; f ) the space of psc-metrics g ∂ which are submersion metrics on the total space ∂X given by some psc-metric h β on βX and by the metric g L on the fiber (which given by the map f : βX → BG). We have the inclusion map ι : R psc (∂X, g L ; f ) → R psc (∂X). Now, by definition, we obtain the space R psc (X Σ ) as a pull-back in the following diagram Since we fixed the map f : βX → BG, it follows that the space R psc (∂X, g L ; f ) is homeomorphic to the space R psc (βX). Let g β ∈ R psc (βX) and g ∂ ∈ R psc (∂X, g L ; f ) be a corresponding submersion metric. Clearly, we can identify the fiber res −1 Σ (g ∂ ) ⊂ R psc (X Σ ) with the space R psc (X, βX, f ) h β . We obtain the following diagram of fiber bundles: (βX 1 , h β 1 ) be an elementary psc-bordism (with p, q ≥ 2), which is given together with a mapf :B → BG such that f =f | βX and f 1 =f | βX 1 . Let Z be a manifold given by (8) equipped with corresponding Riemannian submersion metrics g ∂ 0 ∈ R psc (∂X), g ∂ 1 ∈ R psc (∂X 1 ) determined by the given data. Then the psc-bordism (B,h β ) determines a pscbordism (Z,ḡ ∂ ) : (∂X, g ∂ 0 ) (∂X 1 , g ∂ 1 ). Theorem 2.2 and Theorem 2.6 give us the following homotopy equivalences: We obtain the following commutative diagram: where all horizontal rows are Serre fiber bundles. Thus, it is evident that the circled maps above are weak homotopy equivalences. This proves Theorem A.

Some further developments
In this section we would like to emphasize that recent results concerning homotopy groups of the spaces R psc (M ) (of psc-metrics (see [3,9,10,12,13]) could be applied directly and indirectly to the case of manifolds with (L, G)-fibered singularities. In particular, we would like to attract the attention of topologically-minded experts to relevant conjectures and results from the recent work [5,6].

5.1.
Index-difference map. We mentioned earlier that the homotopy-invariance of various spaces of psc-metrics is a crucial property in helping detect their non-trivial homotopy groups. With this in mind, there is a secondary index invariant, the index-difference map which is defined as follows. Let g 0 ∈ R psc (M ) be a base point. Then for any psc-metric g on M , there is an interval g t = (1 − t)g 0 + tg of metrics such that a corresponding curve of the Dirac operators D gt starts and ends at the subspace (Fred n ) × ⊂ Fred n of invertible Dirac operators.
Since the subspace (Fred n ) × is contractible, the curve D gt is a loop in the space Fred n of all Dirac operators. This space, in turn, is homotopy equivalent to the loop space Ω ∞+n KO representing a shifted KO -theory, i.e. π q (Ω ∞+n KO) = KO n+q . Thus the curve D gt gives an element in Ω ∞+n+1 KO, well-defined up to homotopy, to determine the map (11).

5.2.
Results and conjectures. The reader should note that much is also known about the spaces of psc-metrics for non-simply connected manifolds; see [9,10]. We will however return to the same examples we considered above. We have the following conjectures concerning examples (1) and (2): Conjecture 5.2. Let X Σ be a spin ( k -fb)-manifold. Assume dim X ≥ 7 and X and βX = ∅ are simply-connected and R psc (X Σ ) = ∅ with a base point g 0 ∈ R psc (X Σ ). Then there is an index-difference map (13) inddiff k g 0 : R psc (X Σ ) → Ω ∞+n+1 KO k which induces a non-trivial homomorphism in the homotopy groups (14) (inddiff k g 0 ) * : π q (R psc (X Σ )) → KO k n+q+1 when the target group KO k n+q+1 (KO with Z k -coefficients) is non-trivial.
It turns out that the above examples (3) and (4) (and many others, see [5]) lead to particular results concerning the homotopy groups of the spaces R psc (X Σ ). Let X Σ = X ∪ ∂X −N (βX) be a spin manifold with (L, G)-singularities. Let g ∈ R psc (X Σ ) be a well-adapted metric. Then g determines the metrics g ∂X ∈ R psc (∂X) and g βX ∈ R psc (βX) such that the bundle ∂X → βX is a Riemannian submersion. We fix the metric g βX,0 . This gives rise to a Serre fiber bundle res Σ : R psc (X Σ ) → R psc (βX) with fiber R psc (X Σ ) g βX ,0 , where R psc (X Σ ) g βX is the space of all metrics g ∈ R psc (X Σ ) which restrict to g βX,0 on R psc (βX). Since the metric g ∂X,0 on ∂X is determined by the metric g βX,0 , the fiber R psc (X Σ ) g βX ,0 coincides with the space R psc (X) g ∂X ,0 . Here is the result we need: Then there exists a section s : R psc (βM ) → R psc (M Σ ) to res Σ . In particular, there is a split short exact sequence: Here is one of the conclusions we would like to emphasize: where dim M = n, dim L = . Let g 0 ∈ R psc (M Σ ) = ∅ be a base point giving corresponding base points, the metrics g βM,0 ∈ R psc (βM ), g ∂M,0 ∈ R psc (∂M ) and g M,0 ∈ R psc (M ) g ∂M,0 .