Homotopy Invariance of the Space of Metrics with Positive Scalar Curvature

Homotopy Invariance of the Space of Metrics
with Positive Scalar Curvature on Manifolds
with Singularities††This paper is a contribution to the Special Issue on
Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday. The
full collection is available at https://www.emis.de/journals/SIGMA/Gromov.html

Boris BOTVINNIK a and Mark G. WALSH b

B. Botvinnik and M. Walsh

a) Department of Mathematics, University of Oregon, Eugene, OR, 97405, USA
<EMAIL_ADDRESS>http://darkwing.uoregon.edu/~botvinn/

b) Department of Mathematics and Statistics, Maynooth University, Maynooth,
Ireland<EMAIL_ADDRESS>
Received June 16, 2020, in final form March 24, 2021; Published online April
02, 2021

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

positive scalar curvature metrics; manifolds with singularities; surgery

53C27; 57R65; 58J05; 58J50

##  1 Introduction

###  1.1 Existence of a psc-metric

A classical result in this subject concerns the existence of metrics of
positive scalar curvature (psc-metrics) on a simply-connected smooth closed
manifold $M$. There are two cases here: either $M$ is a spin manifold or it is
not. Recall that in the case when $M$ is spin, there is an index $\alpha(M)\in
KO^{-n}$ of the Dirac operator valued in real $K$-theory. Here is the result:

######  Theorem 1.1 (Gromov–Lawson [12], Stolz [15]).

Let $M$ be a smooth closed simply connected manifold of dimension $n\geq 5$.

  1. $(i)$

If $M$ is spin, then $M$ admits a psc-metric if and only if the index
$\alpha(M)\in KO^{-n}$ of the Dirac operator on $M$ vanishes.

  2. $(ii)$

If $M$ is not spin, then $M$ always admits a psc-metric.

We denote by ${\mathcal{R}}^{\mathrm{psc}}(M)$ the space of psc-metrics on
$M$. Recall that one of the major tools used to prove Theorem 1.1 is the
surgery technique due to Gromov and Lawson (proved independently by Schoen and
Yau). In particular, Gromov–Lawson observed that a psc-metric survives
surgeries of codimension at least three (such surgeries are called
_admissible_). It turns out that the homotopy type of the space
${\mathcal{R}}^{\mathrm{psc}}(M)$ is invariant under such surgeries; see [8,
9, 18].

###  1.2 Existence of a psc-metric on a manifold with Baas–Sullivan
singularities

We start with the simplest case, where the geometrical picture is transparent.
Let $(L,g_{L})$ be a closed Riemannian manifold, in which the metric $g_{L}$
is assumed to have zero scalar curvature. Let $Y$ be a closed smooth manifold,
such that the product $Y\times L$ is a boundary of a smooth manifold $X$,
i.e., $\partial X=Y\times L$. Here is a natural geometrical question:

######  Question .

Does there exist a psc-metric $g_{Y}$ on $Y$, such that the product metric
$g_{Y}+g_{L}$ on $\partial X=Y\times L$ can be extended (being a product near
$\partial X$) to a psc-metric $g_{X}$ on $X$?

It is convenient to denote $\beta X:=Y$ (which is sometimes called the
Bockstein manifold of $X$). Thus, for now, the boundary of $X$ forms the total
space of a trivial bundle,

$\displaystyle\partial X=\beta X\times L\rightarrow\beta X.$

In this case, a manifold with Baas–Sullivan singularities of the type $L$
$($or just $L$-singularities$)$, and denoted $X_{\Sigma}$, is obtained as

$\displaystyle X_{\Sigma}:=X\cup_{\partial X}-\beta X\times C(L),$

where $C(L)$ is a cone over $L$ (and the minus sign represents a change of
orientation in the case when manifolds are oriented).

Assume for a moment that there is a psc-metric $g_{X}$ on $X$ such that
$g_{X}$ is a product metric of the form $g_{\partial X}+{\rm d}t^{2}$, near
$\partial X$, where the boundary metric $g_{\partial X}:=g_{X}|_{\partial X}$
satisfies $g_{\partial X}=g_{\beta X}+g_{L}$. Then we would like to extend the
metric $g_{X}$ to a psc-metric (with singularities) on $X_{\Sigma}$. (For
brevity we will use the term metric here with the understanding that on
$X_{\Sigma}$ (or on $C(L)$), metrics are necessarily singular.) To do this, we
begin by extending the scalar-flat metric, $g_{L}$, to a scalar-flat metric on
the cone $C(L)$, which we denote $g_{C(L)}$. There is an obvious way to do
this and details are provided in the appendix.

To obtain the desired metric on $X_{\Sigma}$, it will be necessary to attach
this conical metric, ${g_{C(L)}}$, to a cylindrical metric ${\rm
d}t^{2}+g_{L}$. While these metrics do not attach smoothly, it is possible to
bend one end of the cylinder inwards so that it matches the conical metric.
This process creates some positive but no negative scalar curvature. With this
in mind, we construct an “attaching” metric of non-negative scalar curvature
on the cylinder $X\times[0,1]$ which takes the form ${\rm d}t^{2}+g_{L}$ at
one end and smoothly attaches to ${g_{C(L)}}$ at the other. This metric is
denoted $g_{\mathrm{att}(L)}$ and details of its construction are given in the
appendix.

$g_{Y}+g_{L}+{\rm
d}t^{2}$$g_{Y}+g_{\mathrm{att}(L)_{\epsilon}}$$g_{Y}+g_{C(L)}$$(X,g_{X})$

Figure 1: Gluing together the components, $g_{X}$, $g_{Y}+g_{\mathrm{att}(L)}$
and $g_{Y}+g_{C(L)}$ of the metric $g_{X_{\Sigma}}$.

######  Remark 1.2.

We emphasize that, according to our construction, the metrics
$g_{\mathrm{att}(L)}$ and ${g_{C(L)}}$ are both invariant under the action of
the isometry group $\mathrm{Iso}(g_{L})$ of the metric $g_{L}$. In our work,
the role of $L$ will be played by either the circle $S^{1}$ (with $U(1)$ as
the isometry group), some other appropriate Lie group or a homogeneous space,
such as $S^{n}$ or $\mathbb{HP}^{n}$, equipped with the natural metric $g_{L}$
of non-negative constant scalar curvature.

Returning now to the manifold $X_{\Sigma}$, we define a metric
$g_{X_{\Sigma}}$ by

$\displaystyle
g_{X_{\Sigma}}:=g_{X}\cup(g_{Y}+g_{\mathrm{att}(L)})\cup(g_{Y}+g_{C(L)}),$

obtained by gluing in the obvious way; see Figure 1. The following fact is an
easy consequence of Proposition 6.2.

######  Proposition 1.3.

The construction of $g_{X_{\Sigma}}$ above yields a smooth $($singular$)$
metric of non-negative scalar curvature on $X_{\Sigma}$.

The above construction motivates the notion of a well-adapted metric on
$X_{\Sigma}$. Before stating it, we fix a scalar-flat metric $g_{L}$ on $L$
and equip $X$ with a boundary collar $c\colon\partial X\times[-1,1]\rightarrow
X$, where $c(\partial X\times\\{-1\\})=\partial X$.

We say that a metric $g$ on $X_{\Sigma}$ is a _well-adapted Riemannian metric
on $X_{\Sigma}$_ if

  1. $(i)$

the restriction $g|_{X}$ satisfies the following conditions on the collar $c$:

$\displaystyle c^{*}g|_{\partial X\times[0,1]}=g_{\beta X}+g_{L}+{\rm
d}t^{2}\qquad\text{and}\qquad c^{*}g|_{\partial X\times[-1,0]}=g_{\beta
X}+g_{\mathrm{att}(L)},$

  2. $(ii)$

the restriction $g|_{\beta X\times C(L)}$ splits as a product-metric
$g|_{\beta X\times C(L)}=g_{\beta X}+g_{C(L)}$.

We denote by ${\mathcal{R}}(X_{\Sigma})$ the space of all well-adapted
Riemannian metrics on $X_{\Sigma}$, and by
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$ its subspace of psc-metrics. Thus,
Question 1 above is equivalently asking whether the space
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$ 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 $\beta X$ are simply-connected; see [3]).

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 $\eta\in\Omega^{\mathrm{spin}}_{1}=\mathbb{Z}_{2}$.
We denote by $\Omega^{\mathrm{spin},\eta}_{*}(-)$ the bordism theory of spin
manifolds with $\eta$-singularities, and by $\mathsf{MSpin}^{\eta}$ the
corresponding representing spectrum. We refer to [3] for details on cobordism
theory $\Omega^{\mathrm{spin},\eta}_{*}(-)$.

It turns out, there exists a Dirac operator on spin-manifolds with
$\eta$-singularities. Furthermore, there is a natural transformation
$\alpha^{\eta}\colon\Omega^{\mathrm{spin},\eta}_{*}\to KO_{*}^{\eta}$ which
evaluates the index of that Dirac operator, where the “$K$-theory with
$\eta$-singularities” $KO_{*}^{\eta}(-)$ coincides with the usual complex
$K$-theory. Here is the result from [3]:

######  Theorem 1.4 (Botvinnik [3]).

Let $X$ be a simply connected spin manifold with nonempty $\eta$-singularity
of dimension $n\geq 7$. Then $X$ admits a metric of positive scalar curvature
if and only if $\alpha^{\eta}([X])=0$ in the group $KO_{n}^{\eta}\cong
KU_{n}$.

###  1.3 Existence of a psc-metric on a manifold with fibred singularities

There are more general objects, namely, “manifolds with fibred singularities”
(or pseudo-manifolds with singularities of depth one). Here again, we start
with a manifold $X$ with boundary $\partial X\neq\varnothing$, which is the
total space of a fibre bundle $\partial X\to\beta X$ with the fibre $L$.
Usually the manifold $L$ is referred to as the _link_. To get _geometrically
interesting_ objects, we fix the following data on the manifold $L$:

  1. $(a)$

a metric $g_{L}$ on $L$ of non-negative constant scalar curvature,

  2. $(b)$

a subgroup $G$ of the isometry group $\mathrm{Isom}(g_{L})$ of the metric
$g_{L}$.

In particular, if $G$ is a nontrivial subroup, the metric $g_{L}$ is assumed
to be homogeneous. Henceforth, we assume that the bundle $\partial X\to\beta
X$ is a $G$-bundle; thus the bundle $\partial X\to\beta X$ is induced by a
structure map $f\colon\beta X\to BG$.

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. As we mentioned in Remark 1.2, the
attaching metric $g_{\mathrm{att}(L)}$ and the cone metric ${g_{C(L)}}$ are
both invariant under the action of the isometry group $\mathrm{Isom}(g_{L})$.
In particular, the isometry action of $G$ on $L$ extends automatically to
$C(L)$ and this gives rise to a fibre bundle $N(\beta X)\to\beta X$, obtained
by “inserting” the cone $C(L)$ as a fibre with the same structure group $G$.
Thus, we obtain a manifold with fibred singularities, $X_{\Sigma}$, given as
$X_{\Sigma}:=X\cup_{\partial X}-N(\beta X)$. Such a manifold is referred to as
an $(L,G)$-manifold.

Now we consider spin $(L,G)$-manifolds $X_{\Sigma}:=X\cup_{\partial X}-N(\beta
X)$; this means that we first fix a spin structure on $L$, and $X$ and $\beta
X$ are given spin structures which are respected by the bundle map $\partial
X\to\beta X$. We will assume that the isometry group $G$ preserves the fixed
spin structure on $L$. For short, we call such a bundle map $\partial
X\to\beta X$ a _geometric $L$-bundle_.

There is a natural concept of spin cobordism between $(L,G)$-manifolds.
Namely, we say that $W_{\Sigma}\colon X_{\Sigma}\rightsquigarrow X_{\Sigma,1}$
is a _spin $(L,G,\mbox{-}\mathrm{fb})$-cobordism between_

$\displaystyle X_{\Sigma}=X\cup_{\partial X}-N(\beta X)\qquad\mbox{and}\qquad
X_{\Sigma,1}=X_{1}\cup_{\partial X_{1}}-N(\beta X_{1}),$

if $W_{\Sigma}:=W\cup_{\partial^{s}W}-N(\beta W)$, where the boundary
$\partial W$ is decomposed as

$\displaystyle\partial W=X\cup\partial^{s}W\cup X_{1},$

and $\partial^{s}W\colon\partial X\rightsquigarrow\partial X_{1}$ is a spin
cobordism over a geometric $L$-bundle map $p\colon\partial^{s}W\to\beta W$
which restricts to given geometric $L$-bundles $\partial X\to\beta X$ and
$\partial X_{1}\to\beta X_{1}$. This gives a corresponding cobordism theory
$\Omega_{n}^{\mathrm{spin},(L,G)\mbox{-}\mathrm{fb}}(\mbox{-})$; see [5, 6, 7]
for more details.

It turns out there are several interesting cases when the Dirac operator is
well defined on such spin $(L,G)$-manifolds, and the index of such operator
gives a homomorphism

$\displaystyle\alpha^{(L,G)\mbox{-}\mathrm{fb}}\colon\
\Omega_{n}^{\mathrm{spin},(L,G)\mbox{-}\mathrm{fb}}\to
KO_{n}^{(L,G)\mbox{-}\mathrm{fb}},$

where $KO_{n}^{(L,G)\mbox{-}\mathrm{fb}}$ is an appropriate $K$-theory; see
[5, 6, 7] and the examples below in this section.

We have to modify the above definition of well-adapted Riemannian metric on
$X_{\Sigma}$ (we give a detailed definition in Section 2). Roughly, in this
setting, a _well-adapted Riemannian metric_ $g$ on $X_{\Sigma}=X\cup_{\partial
X}-N(\beta X)$ is a regular Riemannian metric restricted to $X$ (satisfying
certain product conditions near the boundary), and the restriction metric
$g|_{N(\beta X)}$ is determined by a requirement that the projection $N(\beta
X)\to\beta X$ is a Riemannian submersion (which has a structure group
$G\subset\mathrm{Isom}(g_{L})$) and with the cone metric $g_{C(L)}$ on the
fibre. We denote by ${\mathcal{R}}(X_{\Sigma})$ the space of well-adapted
Riemannian metrics on $X_{\Sigma}$, and by
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$ its subspace of psc-metrics; see
Section 2.4 for a precise definition of this subspace. Below we describe some
interesting cases.

1.3.1. We assume that all $(L,G)$-manifolds are spin, $L=S^{1}$ representing
$\eta\in\Omega_{1}^{\mathrm{spin}}$, and $G=S^{1}$. We obtain a corresponding
bordism group $\Omega^{\mathrm{spin},\eta\mbox{-}\mathrm{fb}}_{*}$ of such
manifolds. Then there exists an appropriate Dirac operator on $X_{\Sigma}$,
and index map
$\alpha^{\eta\mbox{-}\mathrm{fb}}\colon\Omega^{\mathrm{spin},\eta\mbox{-}\mathrm{fb}}_{*}\to
KO^{\eta\mbox{-}\mathrm{fb}}_{*}$ evaluating the index of that Dirac operator.
Here is the existence result for psc-metrics in that setting:

######  Theorem 1.5 (Botvinnik–Rosenberg [7]).

Let $X_{\Sigma}=X\cup N(\beta X)$ be a simply connected spin manifold with
fibred $\eta$-singularity $($i.e., $X$ and $\beta X$ are simply-connected and
spin$)$ of dimension $n\geq 7$. Then
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\neq\varnothing$ if and only if
$\alpha^{\eta\mbox{-}\mathrm{fb}}([X_{\Sigma}])=0$ in the group
$KO^{\eta\mbox{-}\mathrm{fb}}_{n}$.

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
$s_{L}=\ell(\ell-1)$, $\operatorname{{\text{\rm dim}}}L=\ell$, and $G$ is a
subgroup of the isometry group of the metric $g_{L}$. We assume
$L=\partial\bar{L}$, where $\bar{L}$ is a smooth compact manifold with
boundary, and the $G$-action on $L$ extends to a $G$-action on $\bar{L}$. In
this setting, an $(L,G)$-manifold $X_{\Sigma}$ could be described as a triple
$(X,\beta X,f)$, where $X$ is a manifold with boundary $\partial X$, which is
a total space of a fibre bundle $\partial X\to\beta X$ (with a fibre $L$ and a
structure group $G$) given by a map $f\colon\beta X\to BG$. In this setting we
have indices $\alpha(\beta X)\in KO_{n-\ell-1}$ and
$\alpha_{\mathrm{cyl}}(X)\in KO_{n}$, where $n=\operatorname{{\text{\rm
dim}}}X$.

Here are the existence results:

######  Theorem 1.6 (Botvinnik–Piazza–Rosenberg [5]).

Let $(X,\partial X,f)$ define a closed $(L,G)$-singular spin manifold
$X_{\Sigma}$. Assume that $X$, $\beta X$, and $G$ are all simply connected,
that $n-\ell\geq 6$, and suppose that $L$ is a spin boundary, say
$L=\partial\bar{L}$, with the standard metric $g_{L}$ on $L$ extending to a
psc-metric on $\bar{L}$, and with the $G$-action on $L$ extending to a
$G$-action on $\bar{L}$. Assume that the two obstructions $\alpha(\beta X)\in
KO_{n-\ell-1}$ and $\alpha_{{\rm cyl}}(X)\in KO_{n}$ both vanish. Then
$X_{\Sigma}$ admits a well-adapted psc-metric.

######  Theorem 1.7 (Botvinnik–Piazza–Rosenberg [5]).

Let $(X,\beta X,f)$ define a closed $(L,G)$-singular spin manifold
$X_{\Sigma}$, with $L=\mathbb{HP}^{2k}$ and $G={\rm Sp}(2k+1)$, $n\geq 1$.
Assume that $\partial X=\beta X\times L$, i.e., the $L$-bundle over $\beta X$
is trivial, or in other words that the singularities are of Baas–Sullivan
type. Then if $X$ and $\beta X$ are both simply connected and $n-8k\geq 6$,
$X_{\Sigma}$ has an adapted psc-metric if and only if the $\alpha$-invariants
$\alpha(\beta X)\in KO_{n-8k-1}$ and $\alpha_{{\rm cyl}}(X)\in KO_{n}$ both
vanish.

There are several other interesting cases and also more general results when
$X_{\Sigma}$ has non-trivial fundamental group; see [6]. Now we are ready to
address our main result concerning homotopy invariance of the corresponding
spaces of psc-metrics on $X_{\Sigma}$.

###  1.4 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
[8], Walsh [18, 19, 20] (see also recent work by Ebert and Frenck [9]) is that
the homotopy type of the space ${\mathcal{R}}^{\mathrm{psc}}(M)$ is an
invariant of the bordism class $[M]\in\Omega^{\mathrm{spin}}_{n}$ (provided
$M$ is simply-connected and $n\geq 5$).111There is also a similar result for
non-simply connected manifolds.

Notice that if $X_{\Sigma}=X\cup_{\partial X}-N(\beta X)$ is a pseudo-manifold
with $(L,G)$-singularities equipped with structure map $f\colon\beta X\to BG$,
then there are two types of surgery possible on $X$:

  1. $(i)$

a surgery on its “resolution”, i.e., the interior $X\subset X_{\Sigma}$ away
from the boundary $\partial X$,

  2. $(ii)$

a surgery on the structure map $f\colon\beta X\to BG$.

In case $(i)$ all constructions are the same as in the case of closed
manifolds; however, in case $(ii)$, we have to be more careful. Indeed, let
$\bar{B}\colon\beta X\rightsquigarrow\beta X_{1}$ be the trace of a surgery on
the map $f\colon\beta X\to BG$, with $\partial\bar{B}=\beta X\sqcup-\beta
X_{1}$ and $\beta X_{1}$, the manifold obtained from $\beta X$ by surgery on
$f$. Then the map $f$ extends to a map $\bar{f}\colon\bar{B}\to BG$ which
gives a fibre bundle $\bar{p}\colon Z\to\bar{B}$ with the fibre $L$. This
gives us a new manifold $X_{1}=X\cup_{\partial X}Z$ with boundary $\partial
X_{1}$, the total space over a new Bockstein $\beta X_{1}$ with the same fibre
$L$. Also we obtain a new conical part $N(\beta X_{1})$ as above. All of this
results in a new pseudo-manifold

$\displaystyle X_{\Sigma,1}=X_{1}\cup_{\partial X_{1}}-N(\beta X_{1}),\qquad
X_{1}=X\cup_{\partial X}Z,$ (1.1)

with structure map $f_{1}=\bar{f}|_{\beta X_{1}}\colon\beta X_{1}\to BG$. Here
is our main technical result:

######  Theorem 1.8.

Let $X_{\Sigma}=X\cup_{\partial X}-N(\beta X)$ be a pseudo-manifold with
$(L,G)$-singularities, with $\operatorname{{\text{\rm dim}}}X=n$,
$\operatorname{{\text{\rm dim}}}L=\ell$.

  1. $(i)$

Let $i\colon S^{p}\subset X$ be a sphere with trivial normal bundle, and
$X_{\Sigma}^{\prime}$ be the result of surgery on $X_{\Sigma}$ along $S^{p}$.
Then if $2\leq p\leq n-3$, the spaces
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$ and
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma}^{\prime})$ are weakly homotopy
equivalent.

  2. $(ii)$

Let $i\colon S^{p}\subset\beta X$ be a sphere with trivial normal bundle, and
with $f\circ\iota\colon S^{p}\to BG$ homotopic to zero. Let $\bar{B}$ be a
trace of the surgery along $S^{p}\subset\beta X$ with $\partial\bar{B}=\beta
X\sqcup-\beta X_{1}$ and a structure map $\bar{f}\colon\bar{B}\to BG$. Then if
$2\leq p\leq n-\ell-3$, the spaces ${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$
and ${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma,1})$ are homotopy equivalent,
where $X_{\Sigma,1}$ is given by (1.1).

######  Remark 1.9.

As it deals with surgery on the interior, part $(i)$ of Theorem 1.8 above
follows directly from Chernysh’s theorem [8]. Our contribution in this paper
is in proving part $(ii)$.

Theorem 1.8 could be applied to a variety of interesting examples. Among these
are:

  1. (1)

Let $L=\left<k\right>$ be the set of $k$ points, and let $G=\mathbb{Z}_{k}$ be
its “isometry group”. Then a $(\left<k\right>\\!,\mathbb{Z}_{k})$-manifold
$X_{\Sigma}=X\cup_{\partial X}-N(\beta X)$ is assembled out of a manifold $X$
with boundary $\partial X$ equipped with a free $\mathbb{Z}_{k}$-action, and a
structure map $\beta X\to B\mathbb{Z}_{k}$ classifies the corresponding
$k$-folded covering $\partial X\to\beta X=\partial X/\mathbb{Z}_{k}$. Here
$N(\beta X)$ is given by inserting the cone $C\\!\left<k\right>$ instead of
$\left<k\right>$ in the fibre bundle $\partial X\to\beta X$. Assuming that all
manifolds are spin, we obtain corresponding bordism groups
$\Omega^{\mathrm{spin},(\left<k\right>\\!,\mathbb{Z}_{k})\mbox{-}\mathrm{fb}}_{*}$
and the corresponding transformation
$\alpha^{(\left<k\right>\\!,\mathbb{Z}_{k})\mbox{-}\mathrm{fb}}\colon\Omega^{\mathrm{spin},(\left<k\right>\\!,\mathbb{Z}_{k})\mbox{-}\mathrm{fb}}_{*}\to
KO_{*}(B\mathbb{Z}_{k})$ which evaluates the index of the corresponding Dirac
operator; see [7] for details.

  2. (2)

Let $\eta\in\Omega_{1}^{\mathrm{spin}}$ be as above, i.e., $[L]=\eta$ and
$G=S^{1}$. Then, similarly, we arrive at the bordism groups
$\Omega^{\mathrm{spin},\eta\mbox{-}\mathrm{fb}}_{*}$ and the index map
$\alpha^{\eta\mbox{-}\mathrm{fb}}\colon\Omega^{\mathrm{spin},\eta\mbox{-}\mathrm{fb}}_{*}\to
KO^{\eta\mbox{-}\mathrm{fb}}_{*}$, as in Theorem 1.4 as above; see [5, 7] for
details.

The above examples lead to the following two corollaries of Theorem 1.8:

######  Corollary 1.10.

Let $X_{\Sigma}$ be a spin $(\left<k\right>\\!\mbox{-}\mathrm{fb})$-manifold.
Assume $\operatorname{{\text{\rm dim}}}X\geq 7$ and that $X$ and $\beta X$ are
simply-connected. Then the homotopy type of the space
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$ is a bordism invariant and depends
only on the bordism class
$[X_{\Sigma}]\in\Omega^{\mathrm{spin},\left<k\right>\mbox{-}\mathrm{fb}}_{n}$.

######  Corollary 1.11.

Let $X_{\Sigma}$ be a spin $(\eta\mbox{-}\mathrm{fb})$-manifold. Assume
$\operatorname{{\text{\rm dim}}}X\geq 9$ and that $X$ and $\beta X$ are
simply-connected. Then the homotopy type of the space
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$ is a bordism invariant and depends
only on the bordism class
$[X_{\Sigma}]\in\Omega^{\mathrm{spin},\eta\mbox{-}\mathrm{fb}}_{n}$.

######  Remark 1.12.

To simplify the presentation, we consider only the spin case here. However,
there are also non-spin cases where similar results hold. We leave this
intentionally outside of this paper.

The cases addressed in Theorems 1.6 and 1.7 give interesting implications.

  1. (3)

Let $L$ and $G$ be as in Theorem 1.6, i.e., $G$ is a simply connected Lie
group, $L$ is a spin boundary, say $L=\partial\bar{L}$, with the standard
metric $g_{L}$ on $L$ extending to a psc-metric on $\bar{L}$, and with the
$G$-action on $L$ extending to a $G$-action on $\bar{L}$. Then an
$(L,G)$-singular spin manifold $X_{\Sigma}$ determines an element in the
relevant bordism group $\Omega_{n}^{\mathrm{spin},(L,G)-\mbox{-}\mathrm{fb}}$
which fits to an exact triangle (see [5]): Here the indices $\alpha(\beta
X)\in KO_{n-\ell-1}$ and $\alpha_{{\rm cyl}}(X)\in KO_{n}$ can be thought of
as homomorphisms from $\Omega^{\mathrm{spin},(L,G)\mbox{-}\mathrm{fb}}_{*}$ to
a relevant $K$-theory.

  2. (4)

Let $L=\mathbb{HP}^{2k}$ and $G={\rm Sp}(2k+1)$, $n\geq 1$. Assume that
$\partial X=\beta X\times L$, i.e., the $L$-bundle over $\beta X$ is trivial,
or in other words that the singularities are of Baas–Sullivan type. Then a
closed $(L,G)$-singular spin manifold $X_{\Sigma}$ determines an element in
the corresponding bordism group
$\Omega^{\mathrm{spin},(L,G)\mbox{-}\mathrm{fb}}_{n}$, and, as above, the
index homomorphism from $\Omega^{\mathrm{spin},(L,G)\mbox{-}\mathrm{fb}}_{*}$
to a relevant $K$-theory.

These examples lead to following corollary

######  Corollary 1.13.

In both of the cases described in (3) and (4), the homotopy type of the space
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$ is a bordism invariant and depends
on the bordism class
$[X_{\Sigma}]\in\Omega^{\mathrm{spin},(L,G)\mbox{-}\mathrm{fb}}_{n}$, provided
$n-\ell\geq 6$ $($where $\ell=2k$ in the case $(4))$.

######  Remark 1.14.

To see how Theorem 1.8 implies the corollaries, it is enough to notice that in
all those cases, the dimensional assumptions imply that a cobordism
$W_{\Sigma}\colon X_{\Sigma}\rightsquigarrow X_{\Sigma,1}$ could be modified
so that the embedding $X_{\Sigma,1}\subset W_{\Sigma}$ is two-connected in an
appropriate sense, i.e., the induced embeddings $\beta X_{1}\subset\beta W$
and $(X_{1},\partial X_{1})\hookrightarrow(W,\partial W)$ are both
2-connected. Then the argument goes exactly as in the case of closed
manifolds.

In the last section we show that the cases (3) and (4) above lead to an
interesting result concerning homotopy groups of
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$.

##  2 Preliminaries

###  2.1 Positive scalar curvature on manifolds with boundary

Here we recall the main constructions and results from [20]. The set-up is as
follows. Given a smooth compact $n$-dimensional manifold $X$ (possibly with
boundary $\partial X\neq\varnothing$), we denote by ${\mathcal{R}}(X)$, the
space of all Riemannian metrics on $X$. The space ${\mathcal{R}}(X)$ is
equipped with the standard $C^{\infty}$-topology, giving it the structure of a
Fréchet manifold; see [16, Chapter 1] for details. For each metric
$g\in{\mathcal{R}}(X)$, we denote by $s_{g}\colon X\rightarrow\mathbb{R}$ the
scalar curvature on $X$ of the metric $g$ and by
${\mathcal{R}}^{\mathrm{psc}}(X)\subset{\mathcal{R}}(X)$ the subspace of psc-
metrics on $X$.

In the case when $\partial X\neq\varnothing$, it is necessary to consider
certain subspaces of ${\mathcal{R}}^{\mathrm{psc}}(X)$, where metrics satisfy
particular boundary constraints. With this in mind, we specify a collar
embedding $c\colon\partial X\times[0,2)\hookrightarrow X$ around $\partial X$
and define the space ${\mathcal{R}}^{\mathrm{psc}}(X,\partial X)$ as:

$\displaystyle{\mathcal{R}}^{\mathrm{psc}}(X,\partial
X):=\big{\\{}h\in{\mathcal{R}}^{\mathrm{psc}}(X)\colon c^{*}h|_{\partial
X\times I}=h|_{\partial X}+{\rm d}t^{2}\big{\\}},$

where $I:=[0,1]\subset[0,2)$. Fixing a particular metric
$g\in{\mathcal{R}}^{\mathrm{psc}}(\partial X)$, we define the subspace
${\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)_{g}\subset{\mathcal{R}}^{\mathrm{psc}}(X,\partial X)$ of all psc-metrics
$h\in{\mathcal{R}}^{\mathrm{psc}}(X,\partial X)$, where $(c^{*}h)|_{\partial
X\times\\{0\\}}=g$.

Let $Z\colon Y_{0}\rightsquigarrow Y_{1}$ be a bordism between
$(n-1)$-dimensional manifolds $Y_{0}$ and $Y_{1}$ given together with collars
$c_{i}\colon Y_{i}\times[0,2)\hookrightarrow Z$, $i=0,1$ near the boundary
$\partial Z=Y_{0}\sqcup Y_{1}$. Then ${\mathcal{R}}^{\mathrm{psc}}(Z,\partial
Z)$ denotes the space of psc-metrics on $Z$ which restrict as a product
structure on the neighbourhood $c_{i}(Y_{i}\times I)\subset Z$, $i=0,1$; i.e.,
$c_{i}^{*}\bar{g}=g_{i}+{\rm d}t^{2}$ on $Y_{i}\times I$ for some pair of
metrics $g_{i}\in{\mathcal{R}}^{\mathrm{psc}}(Y_{i})$, $i=0,1$. Now we fix a
pair of psc-metrics $g_{0}\in{\mathcal{R}}^{\mathrm{psc}}(Y_{0})$ and
$g_{1}\in{\mathcal{R}}^{\mathrm{psc}}(Y_{1})$ and consider the following
subspace of ${\mathcal{R}}^{\mathrm{psc}}(Z,\partial Z)$:

$\displaystyle{\mathcal{R}}^{\mathrm{psc}}(Z,\partial
Z)_{g_{0},g_{1}}:=\big{\\{}\bar{g}\in{\mathcal{R}}^{\mathrm{psc}}(Z,\partial
Z)\mid c_{i}^{*}\bar{g}=g_{i}+{\rm d}t^{2}\text{ on
}Y_{i}\times[0,1],\,i=0,1\big{\\}}.$

We note that each metric $\bar{g}\in{\mathcal{R}}^{\mathrm{psc}}(Z,\partial
Z)_{g_{0},g_{1}}$ provides a psc-bordism
$(Z,\bar{g})\colon(Y_{0},g_{0})\rightsquigarrow(Y_{1},g_{1})$. We next assume
$X$ is a manifold whose boundary $\partial X=Y_{0}$ is equipped with the
metric $g_{0}$. Furthermore, we assume that both spaces
${\mathcal{R}}^{\mathrm{psc}}(X,\partial X)_{g_{0}}$ and
${\mathcal{R}}^{\mathrm{psc}}(Z,\partial Z)_{g_{0},g_{1}}$ 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 $X\cup Z$ and which has boundary
$\partial(X\cup Z)=Y_{1}$; see Figure 2.

In particular, we obtain the space ${\mathcal{R}}^{\mathrm{psc}}(X\cup
Z,Y_{1})_{g_{1}}$ of psc-metrics which restrict as ${g_{1}}+{\rm d}t^{2}$ on
$c_{1}(Y_{1}\times[0,1])\subset Z\subset X\cup Z$. Then for any metric
$\bar{g}\in{\mathcal{R}}^{\mathrm{psc}}(Z,\partial Z)_{g_{0},g_{1}}$, we
obtain a map:

$\displaystyle\mu_{Z,\bar{g}}\colon\ {\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)_{g_{0}}$ $\displaystyle\longrightarrow{\mathcal{R}}^{\mathrm{psc}}(X\cup
Z,Y_{1})_{g_{1}},$ $\displaystyle h$ $\displaystyle\longmapsto h\cup\bar{g},$
(2.1)

where $h\cup\bar{g}$ is the metric obtained on $X\cup Z$ by the obvious gluing
depicted in Figure 2.

$(X,h)$$g_{0}+{\rm d}t^{2}$$g_{0}+{\rm d}t^{2}$$g_{1}+{\rm
d}t^{2}$$(Z,\bar{g})$

Figure 2: Attaching $(X,h)$ to $(Z,\bar{g})$ along a common boundary $\partial
X=Y_{0}$.

Consider the case when the bordism $Z\colon Y_{0}\rightsquigarrow Y_{1}$ is an
_elementary bordism_ , i.e., when $Z$ is the trace of a surgery on $Y_{0}$
with respect to an embedding $\phi\colon S^{p}\times D^{q+1}\to Y_{0}$ with
$p+q+1=n-1=\operatorname{{\text{\rm dim}}}Y_{0}$. Then we have the following.

######  Lemma 2.1 (surgery lemma; see [8, 9, 18]).

Let $g_{0}\in{\mathcal{R}}^{+}(Y_{0})$ be any metric. Assume $q\geq 2$. Then
there exist metrics $g_{1}\in{\mathcal{R}}^{+}(Y_{1})$ and
$\bar{g}\in{\mathcal{R}}^{+}(Z,\partial Z)_{g_{0},g_{1}}$ such that
$(Z,\bar{g})\colon(Y_{0},g_{0})\rightsquigarrow(Y_{1},g_{1})$ 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 [20]:

######  Theorem 2.2.

Let $Z\colon Y_{0}\rightsquigarrow Y_{1}$ be an elementary bordism as above
with $p,q\geq 2$. Then for any metric $g_{0}\in{\mathcal{R}}^{+}(Y_{0})$ there
exist metrics $g_{1}\in{\mathcal{R}}^{+}(Y_{1})$ and
$\bar{g}\in{\mathcal{R}}^{+}(Z,\partial Z)_{g_{0},g_{1}}$ such that the map

$\displaystyle\mu_{Z,\bar{g}}\colon\ {\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)_{g_{0}}\xrightarrow{\simeq}{\mathcal{R}}^{\mathrm{psc}}(X\cup
Z,Y_{1})_{g_{1}}$

defined by (2.1), is a weak homotopy equivalence.

###  2.2 Lifting to a Riemannian submersion

An important aspect of our work involves lifting psc-metrics from the base of
a smooth fibre bundle to a psc-submersion metric on its total space. In this
section we briefly recall some pertinent facts and establish some notation.

Let $p\colon M\rightarrow B$, be the associated smooth fibre bundle to a
principal $G$-bundle with respect to a Lie group, $G$, a compact smooth base
manifold $B$ and compact smooth fibre manifold $F$, on which $G$ acts. Thus,
the bundle has a structure map $f\colon B\rightarrow BG$. We denote by
$\mathcal{HD}(p)$ the space of all horizontal distributions. This space is
topologised in the obvious way as a subspace of smooth sections of the
Grassmann bundle obtained by replacing each tangent space to $M$ with the
Grassmannian of $\operatorname{{\text{\rm dim}}}B$-dimensional subspaces. The
following fact is elementary.

######  Proposition 2.3.

The space $\mathcal{HD}(p)$ is convex.

Suppose we have a $G$-invariant metric on $F$, denoted $g_{F}$, a metric on
$B$ denoted $g_{B}$ and a horizontal distribution
$\mathcal{H}\in\mathcal{HD}(p)$. By a well-known construction [2, Proposition
9.59], the triple, $(g_{B},g_{F},\mathcal{H})$, determines a unique Riemannian
submersion metric (with totally geodesic fibre metrics isometric to $g_{F}$)
on the total space $M$. For simplicity, we fix the $G$-invariant metric on
$F$, $g_{F}$, and consider constant multiples of this metric, $\tau g_{F}$,
for some $\tau\in(0,\infty)$. This ability to scale up or down the fibre
metric will give us important flexibility later on. We consider only the case
when the fibre metric $g_{F}$ has non-negative scalar curvature and the base
metric on $B$ has positive scalar curvature. Thus, we assume
${\mathcal{R}}^{\mathrm{psc}}(B)\neq\varnothing$.

We notice that the construction of a submersion metric varies continuously
with respect to the triple $(g_{B},\tau g_{F},\mathcal{H})$ and so we obtain a
map

$\displaystyle\mathcal{S}\colon\
{\mathcal{R}}^{\mathrm{psc}}(B)\times\mathcal{HD}(p)\times(0,\infty)\longrightarrow{\mathcal{R}}(M),$

where $\mathcal{S}(h,\mathcal{H},\tau)$ is the unique submersion metric with
base $(B,h)$, fibre $(F,\tau g_{F})$ and horizontal distribution
$\mathcal{H}$.

We denote by ${\mathcal{R}}_{\mathcal{S}}(M)$, the image of the map,
$\mathcal{S}$, in ${\mathcal{R}}(M)$ and by
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(M)$ its subspace of psc-metrics.
Thus, each element of ${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(M)$ is a
psc-submersion metric with respect to some base metric
$g_{B}\in{\mathcal{R}}^{\mathrm{psc}}(B)$, some fibre metric $\tau g_{F}$ on
$F$ (for some scaling constant $\tau>0$) and some horizontal distribution
$\mathcal{H}\in\mathcal{HD}(p)$. There are obvious maps from the space
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(M)$ to the spaces
$\mathcal{HD}(p)$, $(0,\infty)$ and ${\mathcal{R}}^{\mathrm{psc}}(B)$, sending
a submersion metric to its respective horizontal distribution, fibre scaling
constant and base metric. We consider the latter, denoting it

$\displaystyle\mathcal{B}\colon\
{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(M)\rightarrow{\mathcal{R}}^{\mathrm{psc}}(B).$

Thus, for any psc-submersion
$\bar{g}\in{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}$, $\mathcal{B}(\bar{g})$
is the base metric on $B$. Finally, for any base metric
$h\in{\mathcal{R}}^{\mathrm{psc}}(B)$, we denote by
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(M)_{h}$, the pre-image
$\mathcal{B}^{-1}(h)$.

We will be interested in constructing a “lifting” map

$\displaystyle{\mathcal{R}}^{\mathrm{psc}}(B)\to{\mathcal{R}}^{\mathrm{psc}}(M)$

which sends each psc-metric on $B$, to a submersion metric on the the total
space of the bundle $p\colon M\to B$. This will require some preliminary work.

From [2, Proposition 9.70] we know that the scalar curvature of such a metric,
$\mathcal{S}(h,\mathcal{H},\tau)$, denoted $s_{M}$, is given by the following
formula of O’Neill:

$\displaystyle s_{M}=s_{h}\circ p+\frac{1}{\tau}s_{F}-\tau|A|^{2},$

where $s_{h}$ and $s_{F}$ are the scalar curvatures of the metrics
$h\in{\mathcal{R}}^{\mathrm{psc}}(B)$ and $g_{F}$, and $A$ is the well-known
tensor which is the obstruction to the integrability of the distribution. Note
that the other well-known terms in this formula, in particular the tensor,
$T$, vanish since the fibres are totally geodesic [2, Theorem 9.59]. By
assumption, the sum of the first two terms in this formula is positive and so
in order to ensure that $s>0$ we need to minimise the effect of $A$.

This formula varies continuously with respect to the choices of $h$ and
$\mathcal{H}$; recall $g_{F}$ is fixed. In particular, we write
$A:=A_{h,\mathcal{H}}$. Thus, by the compactness of $M$, there are continuous
real valued parameters

$\displaystyle\mathrm{m}(h)=\operatorname*{{\text{\rm min}}}\\{s_{h}\circ
p(x)\colon x\in
M\\}\qquad\text{and}\qquad\mathrm{M}_{A}(h,\mathcal{H})=\operatorname*{{\text{\rm
max}}}\\{|A_{h,\mathcal{H}}(x)|\colon x\in M\\}.$

It follows that the scalar curvature, $s$, of the metric
$\mathcal{S}(h,\mathcal{H},\tau)$ satisfies

$\displaystyle
s\geq\mathrm{m}(h)+\frac{1}{\tau}s_{L}-\tau\mathrm{M}_{A}(h,\mathcal{H})^{2}.$

To ensure positivity of $s$, we define $\bar{\tau}(h,\mathcal{H})$ by

$\displaystyle\bar{\tau}(h,\mathcal{H}):=\frac{\mathrm{m}(h)}{2\mathrm{M}_{A}(h,\mathcal{H})^{2}}.$
(2.2)

Thus, we obtain a map

$\displaystyle\mathcal{S}^{\mathrm{psc}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(B)\times\mathcal{HD}(p)\to$ $\displaystyle\
{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(M),$
$\displaystyle(h,\mathcal{H})\mapsto$ $\displaystyle\
\mathcal{S}(h,\mathcal{H},\bar{\tau}(h,\mathcal{H})).$

By fixing some horizontal distribution, $\mathcal{H}$, we define the map

$\displaystyle\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(B)\rightarrow{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(M),$

by
$\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}(h)=\mathcal{S}^{\mathrm{psc}}(h,\mathcal{H})$.

We now deduce some facts about this construction concerning homotopy. As an
immediate consequence of the convexity of $\mathcal{HD}(p)$, we obtain the
following.

######  Proposition 2.4.

For any horizontal distributions,
$\mathcal{H},\mathcal{H}^{\prime}\in\mathcal{HD}(p)$, the maps
$\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}$ and
$\mathcal{S}_{\mathcal{H}^{\prime}}^{\mathrm{psc}}$ are homotopy equivalent.

The lemma below will play an important role later on.

######  Lemma 2.5.

The maps $\mathcal{B}$ and $\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}$ are
homotopy inverse, thus determining a homotopy equivalence:

$\displaystyle{\mathcal{R}}^{\mathrm{psc}}(B)\simeq{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(M).$

###### Proof.

The composition $\mathcal{B}\circ\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}$ is
precisely the identity map on ${\mathcal{R}}^{\mathrm{psc}}(B)$. Consider now
an arbitrary element
$\bar{g}\in{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(M)$. This element is
uniquely represented as a triple
$\bar{g}=(\bar{g}_{B},\mathcal{H}_{\bar{g}},\tau_{\bar{g}})$, consisting of a
base metric $\bar{g}_{B}=\mathcal{B}(\bar{g})$, a horizontal distribution and
a scaling constant. Now,

$\displaystyle\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}\circ\mathcal{B}(\bar{g})=\mathcal{S}(\bar{g}_{B},\mathcal{H},\bar{\tau}(\bar{g}_{B},\mathcal{H})).$

Thus, the desired homotopy is given by

$\displaystyle
H_{t}(\bar{g}):=\mathcal{S}(\bar{g}_{B},(1-t)\mathcal{H}+t\mathcal{H}_{\bar{g}},\bar{\tau}(\bar{g}_{B},(1-t)\mathcal{H}+t\mathcal{H}_{\bar{g}})),$

where $t\in[0,1]$. That the resulting submersion metric $H_{t}(\bar{g})$ has
positive scalar curvature for all $t$ is guaranteed by the construction of the
function $\bar{\tau}$. ∎

We now consider a variation of the scenario above which will be of great
importance for us. We replace the bundle $p\colon M\rightarrow B$, with a
smooth fibre bundle $\bar{p}\colon\bar{E}\to\bar{B}$. In this case, the base
$\bar{B}$ and total space are smooth compact bordisms of manifolds, denoted
$\bar{B}\colon B_{0}\rightsquigarrow B_{1}$ and $\bar{E}\colon
E_{0}\rightsquigarrow E_{1}$, respectively. Here $E_{i}=E|_{B_{i}}$ for
$i\in\\{0,1\\}$. This bundle is also a $G$-bundle with fibre $L$, as above,
and with structure map $\bar{f}\colon\bar{B}\to BG$ restricting to structure
maps $f_{i}\colon B_{i}\to BG$ of the corresponding boundary bundles
$p_{i}=\bar{p}|_{E_{i}}\colon E_{i}\to B_{i}$. These boundary bundles are of
exactly the type we discussed earlier with the role of $M$ played by $E_{i}$.
We equip the bordism $\bar{B}$ with collars $c_{i}\colon
B_{i}\times[0,2)\to\bar{B}$, $i=0,1$, along the boundary $\partial\bar{B}$. We
will now introduce the following simplifying notation:

$\displaystyle\bar{B}_{c_{i}}:=c_{i}(B_{0}\times[0,1]),\qquad\bar{B}_{c_{i}}^{\prime}:=\bar{B}\setminus\bar{B}_{c_{i}},$
$\displaystyle\bar{E}_{c_{i}}:=\bar{E}|_{\bar{B}_{c_{i}}}\qquad\text{and}\qquad\bar{E}_{c_{i}}^{\prime}:=\bar{E}\setminus\bar{E}_{c_{i}},\qquad\text{where}\quad
i\in\\{0,1\\}.$

As usual, we assume that $L$ is equipped with a fixed metric $g_{L}$ of non-
negative scalar curvature. Suppose that $h_{0}$ and $h_{1}$ are a pair of psc-
metrics on $B_{0}$ and $B_{1}$ and that $\bar{h}$ is a psc-metric in the
(assumed to be non-empty) space of psc-bordism metrics
${\mathcal{R}}^{\mathrm{psc}}\big{(}\bar{B},\partial\bar{B}\big{)}_{h_{0},h_{1}}$.
Given some horizontal distribution $\bar{\mathcal{H}}$ on the bundle
$\bar{p}$, the construction above allows us to “lift” the psc-metric $\bar{h}$
to psc-submersion metrics on the total space $\bar{E}$. In particular, to
ensure our lifted metric has a product structure near the boundary of
$\bar{E}$, we can impose the following admissibility condition on horizontal
distributions.

######  Definition 2.6.

A horizontal distribution, $\bar{\mathcal{H}}$ on the bundle
$\bar{p}\colon\bar{E}\to\bar{B}$ is said to be admissible provided it takes
the form of a product $\mathcal{H}_{i}\times\mathbb{R}$ on the restrictions
$\bar{E}_{c_{i}}\to\bar{B}_{c_{i}}$ for each $i\in\\{0,1\\}$.

We denote by $\mathcal{HD}_{\mathrm{adm}}(\bar{p})$, the subspace of
admissible horizontal distributions on $\bar{p}$. That this space is convex
follows from an elementary exercise.

For our purpose, we need to do something slightly more delicate. Let us assume
that the boundary component $E_{0}$ of the total space is already equipped
with a psc-metric $g_{0}$, one determining a Riemannian submersion of positive
scalar curvature, $p_{0}\colon(E_{0},g_{0})\to(B_{0},h_{0})$, with respect to
the fibre $(L,\tau_{0}g_{L})$ for some constant $\tau_{0}>0$, and horizontal
distribution $\mathcal{H}_{0}\in\mathcal{HD}(p_{0})$. Thus,
$g_{0}\in{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(E_{0})$ and more
specifically lies in the image of the map

$\displaystyle S_{\mathcal{H}_{0}}^{\mathrm{psc}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(B_{0})\to{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(E_{0}).$

We wish to extend the psc-metric $g_{0}$ to a certain psc-metric on the total
space $\bar{E}$. This is done in the following lemma.

######  Lemma 2.7.

Let $\bar{p}\colon\bar{E}\to\bar{B}$ denote the bundle described above with
structure map $\bar{f}\colon\bar{B}\to BG$ and fibre $L$, equipped with a
fixed metric $g_{L}$ of non-negative scalar curvature. Let

$\displaystyle p_{0}=\bar{p}|_{E_{0}}\colon\ (E_{0},g_{0})\to(B_{0},h_{0})$

be a psc-Riemannian submersion with respect to the fibre $(L,\tau_{0}g_{L})$
for some constant $\tau_{0}>0$, and horizontal distribution
$\mathcal{H}_{0}\in\mathcal{HD}(p_{0})$. For compact spaces $K_{1}$ and
$K_{2}$, let
$\bar{h}(x)\in{\mathcal{R}}^{\mathrm{psc}}\big{(}\bar{B},\partial\bar{B}\big{)}$,
be a continuous family of psc-bordism metrics on $\bar{B}$, indexed by $x\in
K_{1}$, and let $\bar{\mathcal{H}}(y)\in\mathcal{HD}_{\mathrm{adm}}(\bar{p})$
be a continuous family of admissible horizontal distributions indexed by $y\in
K_{2}$. We further assume that, for all $x\in K_{1}$ and $y\in K_{2}$,
$\bar{h}(x)|_{B_{0}}=h_{0}$ and
$\bar{\mathcal{H}}(y)|_{E_{0}}=\mathcal{H}_{0}$.

Then there is a constant $\bar{\tau}_{\operatorname*{{\text{\rm min}}}}>0$ and
a corresponding family of psc-metrics $\bar{g}(x,y)$, $(x,y)\in K_{1}\times
K_{2}$ on $\bar{E}$ which satisfies the following conditions.

  1. $(i)$

$\bar{p}\colon\big{(}\bar{E},\bar{g}(x,y)\big{)}\to\big{(}\bar{B},\bar{h}(x)\big{)}$
is a Riemannian submersion with respect to the horizontal distribution
$\bar{\mathcal{H}}(y)$.

  2. $(ii)$

Each psc-metric $\bar{g}(x,y)$ takes the form of a product ${g}_{i}(x,y)+{\rm
d}t^{2}$ near the boundary components $E_{i}$, and satisfies
${g}_{0}(x,y)=g_{0}$ on $E_{0}$.

  3. $(iii)$

The Riemannian submersion obtained by restriction to $E_{1}$,

$\displaystyle p_{1}=\bar{p}|_{E_{1}}\colon\
\big{(}E_{1},g_{1}(x,y):=\bar{g}(x,y)|_{E_{1}}\big{)}\to\big{(}B_{1},h_{1}(x):=\bar{h}(x)|_{B_{1}}\big{)},$

has fibre $(L,\bar{\tau}_{\operatorname*{{\text{\rm min}}}}g_{L})$.

###### Proof.

We begin by specifying $\bar{g}=\bar{g}(x,y)$ as, for all $(x,y)$, the product
$g_{0}+{\rm d}t^{2}$ near the boundary component $E_{0}$. More precisely,
$\bar{g}:=g_{0}+{\rm d}t^{2}$ on $\bar{E}_{c_{0}}\cong E_{0}\times[0,1]$.
Thus, near $E_{0}$, the metric $\bar{g}$ forms part of a submersion
$p_{0}\times\mathrm{Id}\colon\big{(}E_{0}\times I,g_{0}+{\rm
d}t^{2}\big{)}\rightarrow\big{(}B_{0}\times I,h_{0}+{\rm d}t^{2}\big{)}$ with
fibre $(L,\tau_{0}g_{L})$ and horizontal distribution
$\bar{\mathcal{H}}|_{\bar{E}_{c_{0}}}=\mathcal{H}_{0}\times\mathbb{R}$.

Our next task is to extend this metric as a psc-submersion over the rest of
$\bar{E}$. To maintain positive scalar curvature it may be necessary to
continuously rescale the fibre metric, but in such a way as to preserve the
original fibre metric near $E_{0}$. To simplify things, we make adjustments to
the fibre metric only in $\bar{E}_{c_{0}}$ (and away from $E_{0}$), where the
metric, $\bar{g}$, takes the form of a product $\bar{g}=g_{0}+{\rm d}t^{2}$.
Consider, for constants $\tau>0$ and $b_{\tau}\geq 2$, smooth monotonic
functions $\gamma_{\tau}\colon[0,b_{\tau}]\rightarrow[0,\tau]$ satisfying

  1. $(i)$

$\gamma_{\tau}(t)=\tau_{0}$ when $t\in[0,1]$,

  2. $(ii)$

$\gamma_{\tau}(t)=\tau$ when $t\in[b_{\tau}-1,b_{\tau}]$.

We define $g_{0}(t)$ to be the submersion metric obtained on
$E_{0}\times\\{t\\}\to B_{0}\times\\{t\\}$ with base metric $h_{0}$ and
horizontal distribution $\mathcal{H}_{0}$ as before but replacing the fibre
metric $\tau_{0}g_{L}$ with $\gamma_{\tau}(t)g_{L}$. Now replace the
submersion $p_{0}\times\mathrm{Id}\colon\big{(}\bar{E}_{c_{0}}=E_{0}\times
I,g_{0}+{\rm d}t^{2}\big{)}\rightarrow\big{(}B_{0}\times I,h_{0}+{\rm
d}t^{2}\big{)}$, with
$p_{0}\times\mathrm{Id}\colon\big{(}E_{0}\times[0,b_{\tau}],g({\gamma_{\tau}(t)})+{\rm
d}t^{2}\big{)}\rightarrow\big{(}B_{0}\times[0,b_{\tau}],h_{0}+{\rm
d}t^{2}\big{)}$. The following proposition is an easy consequence of a well-
known rescaling technique; see for example [12, Lemma 3] or [17, Lemma 1.3].

######  Proposition 2.8.

For any constant $\tau>0$, the smooth function $\gamma_{\tau}$ above can be
chosen so that the metric $g({\gamma_{\tau}(t)})+{\rm d}t^{2}$ has positive
scalar curvature.

We will specify the choice of $\tau>0$ shortly. In the mean time, let us
assume that $\gamma_{\tau}$ has been chosen to guarantee positive scalar
curvature. Note that near $t=0$, the metric remains unchanged.

Consider now the other end of the cylinder, where $t=\tau_{b}$. Here we have a
product of psc-submersion metrics differing only from that at the $t=0$ end in
that the fibre metric is now $\tau g_{L}$. No change has been made to the base
metric and it is assumed to extend over the base bordism $\bar{B}$ as a psc-
metric $\bar{h}$ which takes the form of a product metric ${h}_{1}+{\rm
d}t^{2}$ near the boundary component $B_{1}$. For each $y$, the admissible
horizontal distribution $\bar{\mathcal{H}}(y)$ extends
$\mathcal{H}_{0}\times\mathbb{R}$ over the rest of $\bar{E}$, inducing,
outside of the collar $\bar{E}_{c_{0}}$, Riemannian submersion metrics with
base metric $\bar{h}|_{\bar{B}_{c_{0}}^{\prime}}(x)$ and fibre metric $\tau
g_{L}$. We need to ensure positive scalar curvature by choosing appropriately
small $\tau$.

Suppose for any $(x,y)$, we choose
$\tau=\bar{\tau}\big{(}\bar{h}(x),\bar{\mathcal{H}}(y)\big{)}>0$, as defined
in (2.2). This ensures, via the canonical variation formula [2, Proposition
9.70], that the submersion metric

$\displaystyle\mathcal{S}\big{(}\bar{h}(x),\bar{\mathcal{H}}(y),\bar{\tau}\big{(}\bar{h}(x),\bar{\mathcal{H}}(y)\big{)}\big{)}$

on the bundle $\bar{E}_{c_{0}}^{\prime}\to\bar{B}_{c_{0}}^{\prime}$ with base
metric $\bar{h}(x)|_{\bar{B}_{c_{0}^{\prime}}}$, fibre metric
$\bar{\tau}\big{(}\bar{h}(x),\bar{\mathcal{H}}(y)\big{)}g_{L}$ and horizontal
distribution $\bar{\mathcal{H}}(y)|_{\bar{E}_{c_{0}}^{\prime}}$ has positive
scalar curvature. Thus, we set

$\displaystyle\tau=\bar{\tau}_{\operatorname*{{\text{\rm
min}}}}:=\operatorname*{{\text{\rm
min}}}\big{\\{}\bar{\tau}\big{(}\bar{h}(x),\bar{\mathcal{H}}(y)\big{)}\colon(x,y)\in
K_{1}\times K_{2}\big{\\}},$

to obtain a single scaling factor which works for all $(x,y)$. Finally, the
metric $\bar{g}(x,y)$ is defined to be the submersion metric determined by the
base $\big{(}\bar{B},\bar{h}(x)\big{)}$, fibre $L$ with varying metric
$\gamma_{\bar{\tau}_{\operatorname*{{\text{\rm min}}}}}(t)g_{L}$ on
$E_{c_{0}}$ and $\bar{\tau}_{\operatorname*{{\text{\rm min}}}}g_{L}$ on the
rest of $\bar{E}$, as well as horizontal distribution $\bar{\mathcal{H}}(y)$.
∎

###  2.3 The case of manifolds with fibred singularities

We now return to the manifolds with fibred singularities introduced in Section
1. Recall that $X$ is assumed to be a smooth manifold with boundary $\partial
X\neq\varnothing$, which is the total space of a smooth bundle $\partial
X\to\beta X$ with the fibre $L$ and structure map $f\colon\beta X\to BG$. Here
$G$ is a subgroup of the isometry group of $g_{L}$, a Riemannian metric on
$L$. Usually the Riemannian manifold $(L,g_{L})$ is referred to as the _link_.

As explained in Section 1, we obtain a manifold with fibred singularities,
$X_{\Sigma}$, given as $X_{\Sigma}:=X\cup_{\partial X}-N(\beta X)$, by
replacing the fibre $L$ with it’s cone, $C(L)$, to obtain bundle $N(\beta
X)\to\beta X$. Such a manifold a manifold is referred to as an
$(L,G)$-manifold. For our constructions, we will require some further
constraints on the link $(L,g_{L})$, characterised by the following
definition.

######  Definition 2.9.

A link $(L,g_{L})$ is _simple_ if it satisfies one of the following
conditions:

  1. $(a)$

$(L,g_{L})$ is homogeneous with scalar curvature, $s_{g_{L}}$, a positive
constant,

  2. $(b)$

$(L,g_{L})=\big{(}S^{1},{\rm d}\theta^{2}\big{)}$,

  3. $(c)$

$L=\mathbb{Z}_{k}$.

Henceforth, we assume that $(L,g_{L})$ is a simple link. Before going forward
with our constructions, we should 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. We define the cone
metric $g_{C(L)}$ on the cone, $C(L)$, by

$\displaystyle g_{C(L)}={\rm d}t^{2}+c_{L}^{-2}t^{2}g_{L},$ (2.3)

where $c_{L}=\sqrt{\frac{\ell(\ell-1)}{s_{g_{L}}}}$ and
$\operatorname{{\text{\rm dim}}}L=\ell$. This is a warped product metric on
$(0,1]\times L$ away from the cone point (where $t=0$) and is scalar flat, as
demonstrated in the appendix.

The case $(b)$, when $L=S^{1}$, has special features. As
$\operatorname{{\text{\rm dim}}}L=\ell=1$ here, the definition of the metric
$g_{C(L)}$ above coincides with the more general cone metric construction
detailed in the appendix for the case when $g_{L}$ is scalar flat. In
particular, formula (6.2) in the appendix gives that the scalar curvature of
the metric $g_{C(S^{1})}$ is identically zero. Moreover, any smooth
$S^{1}$-bundle $p\colon Y\to B$ admits a free $S^{1}$-action on the manifold
$Y$ such that $B=Y/S^{1}$. Then, according to a result by Bérard-Bergery [1,
Theorem C], a manifold $Y$ admits an $S^{1}$-equivariant psc-metric if and
only if the orbit space, the manifold $B=Y/S^{1}$ admits a psc-metric.

In the case $(c)$ $L=\mathbb{Z}_{k}$. Then the cone $C(\mathbb{Z}_{k})$ has
the standard Euclidian metric, and we do not make any further assumptions. In
any case, we can assume that the cone metric $g_{C(L)}$ is always scalar-flat
(outside of its vertex).

We return now to the manifold $X$, which forms part a pseudo-manifold with
$(L,G)$-singularities, $X_{\Sigma}=X\cup_{\partial X}-N(\beta X)$. Recall that
$\partial X$ forms part of $G$-bundle, $p\colon\partial X\to\beta X$ with $G$
a subgroup of the isometry group of a fixed metric $g_{L}$ on the fibre $L$.
We denote by $f\colon\beta X\to BG$, a structure map for this bundle.
Throughout, $\operatorname{{\text{\rm dim}}}X=n$, $\operatorname{{\text{\rm
dim}}}L=\ell$ and we reiterate that the link $(L,g_{L})$ is assumed to be
simple.

Bearing in mind the notions described in Section 2.2, we have a map

$\displaystyle\mathcal{S}\colon\ {\mathcal{R}}^{\mathrm{psc}}(\beta
X)\times\mathcal{HD}(p)\times(0,\infty)\longrightarrow{\mathcal{R}}(\partial
X)$ (2.4)

which sends any triple $(h,\mathcal{H},\tau)$ to the unique submersion metric
on $\partial X$ with base $(\beta X,h)$, fibre $(L,\tau g_{L})$ and horizontal
distribution $\mathcal{H}$. Recall that the image of $\mathcal{S}$ is denoted
${\mathcal{R}}_{\mathcal{S}}(\partial X)$ and we further denote by
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial X)$ its subspace of psc-
metrics. Thus, elements of
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial X)$ are psc-submersion
metrics on the total space $\partial X$ with respect to some base metric in
${\mathcal{R}}^{\mathrm{psc}}(\beta X)$ and fibre metric $\tau g_{L}$ for some
$\tau>0$. Recall that, for any $h\in{\mathcal{R}}^{\mathrm{psc}}(\beta X)$,
the space ${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial X)_{h}$
consists of all psc-submersion metrics in
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial X)$ with base metric $h$.
Recall also that by specifying a parameter $\bar{\tau}(h,\mathcal{H})$, we
obtained “lifting” maps

$\displaystyle\mathcal{S}^{\mathrm{psc}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(\beta X)\times\mathcal{HD}(p)\to$ $\displaystyle\
{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial X),$
$\displaystyle(h,\mathcal{H})\mapsto$ $\displaystyle\
\mathcal{S}(h,\mathcal{H},\bar{\tau}(h,\mathcal{H})),$

and

$\displaystyle\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(\beta
X)\rightarrow{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial X),$ (2.5)

defined
$\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}(h):=\mathcal{S}^{\mathrm{psc}}(h,\mathcal{H})$,
for some fixed horizontal distribution $\mathcal{H}$. In particular, we recall
that the homotopy type of the map $\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}$
is independent of the choice of $\mathcal{H}$ and, moreover, the map
$\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}$ forms one direction in a homotopy
equivalence (Lemma 2.5 above); its homotopy inverse is the map,
$\mathcal{B}\colon{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial
X)\to{\mathcal{R}}^{\mathrm{psc}}(\beta X)$, sending submersions to their base
metrics.

######  Remark 2.10.

We emphasise that the fibre metric $g_{L}$ and structure map $f\colon\beta
X\to BG$ are assumed to be fixed throughout. Thus, while the map $\mathcal{S}$
(along with $\mathcal{S}^{\mathrm{psc}}$ and
$\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}$) depend of course on these objects,
and could be denoted by something like $\mathcal{S}_{f,g_{L}}$, we will not
feature them in the notation.

###  2.4 Defining well-adapted metrics

Returning to the manifold $X$, we consider the space of all psc-metrics,
${\mathcal{R}}^{\mathrm{psc}}(X,\partial X)$, which take a product structure
near the boundary. Consider now the restriction map

$\displaystyle\mathrm{res}\colon\ {\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)\to{\mathcal{R}}^{\mathrm{psc}}(\partial X),\qquad\mathrm{res}\colon\
g\mapsto g|_{\partial X}.$

This map is very important for us because of the following fact:

######  Theorem 2.11 ([8, 9]).

The restriction map $\mathrm{res}\colon{\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)\to{\mathcal{R}}^{\mathrm{psc}}(\partial X)$ is a Serre fibre bundle.

We will make significant use of this fact later on. For now however, we
consider the pre-image of the space
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial X)$.

######  Definition 2.12.

The space

$\displaystyle{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X):=\mathrm{res}^{-1}\big{(}{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial
X)\big{)}\subset{\mathcal{R}}^{\mathrm{psc}}(X,\partial X)$

is called the space of _well-adapted psc-metrics on $X$_.

There is a further subspace of
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial X)$ which we must
introduce. Suppose $h\in{\mathcal{R}}^{\mathrm{psc}}(\beta X)$ is a psc-metric
on the Bockstein manifold $\beta X$. Recall that
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial
X)_{h}\subset{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial X)$, denotes
the pre-image, $\mathcal{B}^{-1}(h)$. We now denote by
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial X)_{h}$, the space
defined by

$\displaystyle{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h}:=\mathrm{res}^{-1}\big{(}{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial
X)_{h}\big{)}=\mathrm{res}^{-1}\big{(}\mathcal{B}^{-1}(h)\big{)}.$

All of this extends naturally to a definition of well-adapted metric on the
pseudo-manifold $X_{\Sigma}=X\cup_{\partial X}-N(\beta X)$. A well adapted
psc-metric $g\in{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial X)$, has
associated to it a unique Riemannian submersion $p\colon\big{(}\partial
X,g^{\partial}\big{)}\to\big{(}\beta X,h^{\beta}\big{)}$ with a fibre $(L,\tau
g_{L})$ for some constant $\tau>0$. This determines

  1. $(i)$

a cone metric, $g_{C(L)}$, as defined in (2.3), although with $g_{L}$ now
replaced by $\tau g_{L}$ in that formula,

  2. $(ii)$

and an attaching metric $g_{\mathrm{att}(L)}$.

In turn, we obtain a submersion metric on $N(\beta X)$ obtained by equipping
the fibres with $g_{\mathrm{att}(L)}\cup g_{C(L)}$. This metric then attaches
to $g$ in the obvious way to yield a metric on $X_{\Sigma}$. This process can
be thought of as a map

$\displaystyle i_{\Sigma}\colon\
{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)\longrightarrow{\mathcal{R}}(X_{\Sigma}).$ (2.6)

This map is easily seen to be a homeomorphism onto its image. This leads to
the following definition.

######  Definition 2.13.

The image of the map $i_{\Sigma}$, denoted
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$, is called the space of well
adapted psc-metrics on $X_{\Sigma}$.

This space, ${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$, is the space we are
most interested in studying.

######  Remark 2.14.

It is worth emphasising that the space
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial X)$ (as well the space
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial X)$) could be empty
even though the boundary $\partial X$ has a psc-metric. A simple example is
the following. Let $\beta X=K3$. Then for any prime generator $c\in
H^{2}(K3;\mathbb{Z})=\mathbb{Z}^{22}$ a corresponding map $c\colon
K3\to\mathbb{CP}^{\infty}$ gives a bundle $Z\to\beta X$, where $Z$ is a
simply-connected spin manifold which, of course, admits a psc-metric.
Moreover, $Z$ is a spin boundary: $Z=\partial X$. However, no psc-metric on
$Z$ could be invariant under $S^{1}$-action, otherwise, according to Bérard-
Bergery [1, Theorem C], it would imply the existence of a psc-metric on $K3$.

As we mentioned above, there are two types of surgery that could be performed
on $X_{\Sigma}$: the first one on its resolution, the interior of $X$, and the
second one on the structure map $f\colon\beta X\to BG$. 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 fibred over $\beta X$ with
fibre $L$. We use the notation $(X,\beta X,f)$ for such manifold, where the
boundary $\partial X$ of $X$ is a total space of the fibre bundle from the
diagram: Here $p_{L}\colon E(L)\to BG$ is the universal fibre bundle with the
fibre $L$ and the structure group $G$.

We recall the earlier bordism of smooth compact manifolds, $\bar{B}\colon
B_{0}\rightsquigarrow B_{1}$. We assume now that this is an elementary bordism
with $B_{0}=\beta X$ and, as before, $\bar{f}\colon\bar{B}\to BG$ a map such
that $\bar{f}|_{B_{0}}=f_{0}=f$. We will use the notation
$\big{(}\bar{B},\bar{f}\big{)}\colon(B_{0},f_{0})\rightsquigarrow(B_{1},f_{1})$.
Let $\bar{p}\colon\bar{E}\to\bar{B}$ be a corresponding fibre bundle with
fibre $L$ and structure group $G$. By construction,
$\bar{E}|_{B_{0}}=E_{0}=\partial X$, and $\bar{p}|_{E_{0}}=p$. Then the
manifold $\bar{E}$ gives a bordism $\bar{E}\colon E_{0}\rightsquigarrow
E_{1}$, where $E_{1}=E|_{B_{1}}$. As before, we assume that the bordism
$\bar{B}$ is equipped with collars $c_{i}\colon B_{i}\times[0,2)\to\bar{B}$,
$i=0,1$, along the boundary $\partial\bar{B}$.

We now equip $B_{0}=\beta X$ with the metric
$h^{\beta}_{0}=h^{\beta}(=g_{\beta X})$. We assume that there is a psc-metric
$\bar{h}^{\beta}\in{\mathcal{R}}^{\mathrm{psc}}\big{(}\bar{B}\big{)}_{h^{\beta}_{0},h^{\beta}_{1}}$,
where $h^{\beta}_{1}$ is a psc-metric on the manifold $B_{1}$. In particular,
this means that $c_{i}^{*}\bar{h}^{\beta}=h^{\beta}_{i}+{\rm d}t^{2}$, with
respect to the collars $c_{i}\colon B_{i}\times[0,2)\hookrightarrow\bar{B}$,
$i=0,1$ near the boundary $\partial\bar{B}=B_{0}\sqcup B_{1}$. Thus, the
metric $\bar{h}^{\beta}$ provides a psc-bordism

$\displaystyle\big{(}\bar{B},\bar{f},\bar{h}^{\beta}\big{)}\colon\
\big{(}B_{0},f_{0},h^{\beta}_{0}\big{)}\rightsquigarrow\big{(}B_{1},f_{1},h^{\beta}_{1}\big{)}.$

The structure map $\bar{f}\colon\bar{B}\to BG$ determines a bordism
$\bar{E}\colon E_{0}\rightsquigarrow E_{1}$, where $\bar{E}$ is a pull-back of
the universal $(L,G)$-fibration: with $\bar{E}|_{B_{0}}=E_{0}$ and
$\bar{E}|_{B_{1}}=E_{1}$. Now we glue the manifolds $X$ and $\bar{E}$ (again,
making use of collars near their boundaries) to obtain the manifold
$X_{1}=X\cup_{\partial X}\bar{E}$ with boundary $\partial X_{1}=E_{1}$, which
is the total space of the $(L,G)$-fibration $p_{1}\colon\partial X_{1}\to
B_{1}$. We denote $\beta X_{1}=B_{1}$. We denote by $\mathcal{S}_{1}$, the
analogue of the map $\mathcal{S}$ in (2.4), for the bundle,
$p_{1}\colon\partial X_{1}\to B_{1}$ and by
${\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}$, the analogous space of psc-metrics on $X_{1}$ with
boundary metric a psc-submersion which restricts on the base as the psc-metric
$h^{\beta}_{1}$. We will shortly employ the psc-submersion construction from
Lemma 2.7 to obtain a map

$\displaystyle{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{0}}\longrightarrow{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}},$

an analogue of the map in (2.1). In preparation for this, there are some
objects we must re-introduce.

In the preamble to Lemma 2.7, we considered spaces of horizontal
distributions, $\mathcal{HD}(p_{0})$ and
$\mathcal{HD}_{\mathrm{adm}}(\bar{p})$. In the latter case, we recall that
distributions satisfy a product structure near the boundary components
$E_{i}$. It will be useful for us to consider the space of maps from
$\mathcal{HD}(p_{0})$ to $\mathcal{HD}_{\mathrm{adm}}(\bar{p})$, which we
denote
$\mathcal{F}(\mathcal{HD}(p_{0}),\mathcal{HD}_{\mathrm{adm}}(\bar{p}))$, under
the usual compact-open topology. In particular, we will be interested in maps
$\xi\colon\mathcal{HD}(p_{0})\to\mathcal{HD}_{\mathrm{adm}}(\bar{p})$ which
satisfy the condition that $\xi(\mathcal{H})|_{E_{0}}=\mathcal{H}$. Such maps
are easy to construct. For example, fix a disribution $\bar{\mathcal{H}}$ on
$\bar{E}|_{\bar{B}\setminus c_{0}(B\times[0,1])}$ which takes a product
structure, $\mathcal{H}_{1}\times\mathbb{R}$, on
$\bar{E}|_{c_{0}(B\times[1,2))}$. Then define $\xi(\mathcal{H})$ as
$\mathcal{H}\times\mathbb{R}$ on $\bar{E}|_{c_{0}(B\times[0,1]}$,
$\bar{\mathcal{H}}$ on $\bar{E}|_{\bar{B}\setminus c_{0}(B\times[0,2))}$ while
continuously transitioning between $\mathcal{H}$ and $\mathcal{H}_{1}$ along
$E|_{{c_{0}(B\times[1,2))}}$. We denote by
$\mathcal{F}_{0}(\mathcal{HD}(p_{0}),\mathcal{HD}_{\mathrm{adm}}(\bar{p}))$,
the subspace of
$\mathcal{F}(\mathcal{HD}(p_{0}),\mathcal{HD}_{\mathrm{adm}}(\bar{p}))$,
defined by

$\displaystyle\mathcal{F}_{0}\big{(}\mathcal{HD}(p_{0}),\mathcal{HD}_{\mathrm{adm}}(\bar{p})\big{)}:=\big{\\{}\xi\in\mathcal{F}(\mathcal{HD}(p_{0}),\mathcal{HD}_{\mathrm{adm}}(\bar{p}))\colon\xi(\mathcal{H})|_{E_{0}}=\mathcal{H}\big{\\}}.$

The proof of the following proposition is an elementary exercise.

######  Proposition 2.15.

The spaces
$\mathcal{F}(\mathcal{HD}(p_{0}),\mathcal{HD}_{\mathrm{adm}}(\bar{p}))$ and
$\mathcal{F}_{0}(\mathcal{HD}(p_{0}),\mathcal{HD}_{\mathrm{adm}}(\bar{p}))$
are convex.

We next recall that a metric
$g\in{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial X)_{h^{\beta}_{0}}$
restricts on the boundary $\partial X$ as a submersion metric,
$g^{\partial}=g|_{\partial X}$ over the base metric $h^{\beta}_{0}$, with
fibre metric $\tau_{0}(g)g_{L}$ (for some fibre metric scaling constant
$\tau_{0}(g)>0$) and horizontal distribution $\mathcal{H}_{0}(g)$. Both the
fibre metric scaling constant and the horizontal distribution vary
continuously with respect to $g$. We now choose a map
$\xi\colon\mathcal{HD}(p)\longrightarrow\mathcal{HD}_{\mathrm{adm}}(\bar{p})$
from the space
$\mathcal{F}_{0}(\mathcal{HD}(p_{0}),\mathcal{HD}_{\mathrm{adm}}(\bar{p}))$.
Thus, for any $g\in{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{0}}$, we obtain a distribution $\xi(\mathcal{H}_{0}(g))$ on the
total space $\bar{E}$, extending $\mathcal{H}_{0}(g)$ from $E_{0}$. This
allows us to specify a map

$\displaystyle\mu_{(\bar{B},\bar{f},\bar{h}^{\beta})}\colon\
{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial X)_{h^{\beta}_{0}}$
$\displaystyle\longrightarrow{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}},$ $\displaystyle g$ $\displaystyle\longmapsto
g\cup\bar{g}^{\partial},$ (2.7)

where $\bar{g}^{\partial}$ is the metric obtained in Lemma 2.7 above with
respect to the triple of base metric
$\bar{h}^{\beta}\in{\mathcal{R}}^{\mathrm{psc}}\big{(}\bar{B}\big{)}_{h^{\beta}_{0},h^{\beta}_{1}}$,
admissible horizontal distribution $\xi(\mathcal{H}_{0}(g))$ and fibre scaling
constant $\bar{\tau}(\bar{h}^{\beta},\xi(\mathcal{H}_{0}(g)))$ (as defined in
(2.2)). The following is an easy consequence of Lemma 2.7.

######  Corollary 2.16.

The homotopy type of the map $\mu_{(\bar{B},\bar{f},\bar{h}^{\beta})}$ is
invariant of the psc-isotopy type of the base metric $\bar{h}^{\beta}$, as
well as the choice of distribution map
$\xi\in\mathcal{F}_{0}(\mathcal{HD}(p_{0}),\mathcal{HD}_{\mathrm{adm}}(\bar{p}))$.

As our work concerns only homotopy type, we feel justified in suppressing the
role of $\xi$, in the notation of the map (2.4) above.

Now we are ready to state our main technical result which is similar to
Theorem 2.2.

######  Theorem 2.17.

Let $(X,\beta X,f)$ be a manifold with fibred singularities, i.e., the
boundary $\partial X$ is a total space of an $(L,G)$-fibration $\partial
X\to\beta X$ given by the structure map $f\colon\beta X\to BG$, where
$\operatorname{{\text{\rm dim}}}X=n$, $\operatorname{{\text{\rm dim}}}L=\ell$.
Furthermore, we assume
$\big{(}\bar{B},\bar{f}\big{)}\colon(B_{0},f_{0})\rightsquigarrow(B_{1},f_{1})$
is an elementary bordism with $p,q\geq 2$, where $B_{0}=\beta X$. Then for any
psc-metric $h^{\beta}_{0}$ on $B_{0}$ there exist psc-metrics $h^{\beta}_{1}$
on $B_{1}$ and
$\bar{h}^{\beta}\in{\mathcal{R}}^{\mathrm{psc}}\big{(}\bar{B}\big{)}_{h^{\beta}_{0},h^{\beta}_{1}}$
such that the map

$\displaystyle\mu_{(\bar{B},\bar{f},\bar{h}^{\beta})}\colon\ $
$\displaystyle{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{0}}\longrightarrow{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}},\qquad g\longmapsto g\cup\bar{g}^{\partial},$
$\displaystyle X_{1}=X\cup_{\partial X=E_{0}}\bar{E},\qquad\beta X_{1}=E_{1},$

defined as in (2.4) above, is a weak homotopy equivalence.

######  Remark 2.18.

The construction of the metrics $h^{\beta}_{1}$ on $B_{1}$ and
$\bar{h}^{\beta}$ on $\bar{B}$ used in this theorem, follows from surgery
techniques pioneered by Gromov and Lawson [12], and later generalised by
others. It should be pointed out however that, given such a metric,
$h^{\beta}_{1}$, the theorem works equally well if $h^{\beta}_{1}$ is replaced
by any psc-metric which is psc-isotopic to $h^{\beta}_{1}$, i.e., lies in the
same path component of the space ${\mathcal{R}}^{\mathrm{psc}}(B_{1})$.

##  3 Proof of Theorem 2.17

###  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 $\delta>0$ and $\lambda\geq 0$. Then a _$(\delta$
-$\lambda)$-torpedo metric_ on the disk $D^{n}$, denoted
$g_{\mathrm{torp}}^{n}(\delta)_{\lambda}$, is a psc-metric which roughly takes
the form a round hemisphere of radius $\delta>0$ near the centre before
transitioning into a cylinder of radius $\delta$ and length $\lambda\geq 0$
near the boundary; see first image in Figure 3 below.

Letting $D_{+}^{n}$ denote the upper hemi-disk, we obtain the metric
$g_{\mathrm{torp}+}^{n}(\delta)_{\lambda}:=g_{\mathrm{torp}}^{n}(\delta)_{\lambda}|_{D_{+}^{n}}$;
see second image in Figure 3. We call
$g_{\mathrm{torp}+}^{n}(\delta)_{\lambda}$ a _half-torpedo metric_. Let
$\lambda_{2}>0$ be some constant. Next, we consider the cylinder
$D^{n-1}\times[0,\lambda_{2}]$ equipped with the metric
$g_{\mathrm{torp}}^{n-1}(\delta)_{\lambda_{1}}+{\rm d}t^{2}$ and attach a
half-disk $D_{+}^{n}$ with half-torpedo metric
$g_{\mathrm{torp}+}^{n}(\delta)_{\lambda_{1}}$ along $D^{n-1}\times\\{0\\}$.
We denote the resulting Riemannian manifold by
$\big{(}D_{\mathrm{stretch}}^{n},\hat{g}_{\mathrm{torp}}^{n}(\delta)_{\lambda_{1},\lambda_{2}}\big{)}$.
This is depicted in the third image in Figure 3.

Typically, we will not care so much about the $\lambda_{2}$-parameter but only
$\lambda_{1}$ which we regard as the vertical height of this metric. Moreover,
we will usually be interested in the case when $\lambda_{1}=1$ and when
$\delta=1$. With this in mind we make use of the following notational
simplifications.

$\displaystyle g_{\mathrm{torp}}^{n}:=g_{\mathrm{torp}}^{n}(1)_{1}.$
$\displaystyle g_{\mathrm{torp}+}^{n}:=g_{\mathrm{torp}+}^{n}(1)_{1}.$
$\displaystyle\hat{g}_{\mathrm{torp}}^{n}(\delta)_{\lambda_{1}}:=\hat{g}_{\mathrm{torp}}^{n}(\delta)_{\lambda_{1},\lambda_{2}},\qquad\text{where}\quad\lambda_{2}\quad\text{is
arbitrary}.$
$\displaystyle\hat{g}_{\mathrm{torp}}^{n}:=\hat{g}_{\mathrm{torp}}^{n}(1)_{1}=\hat{g}_{\mathrm{torp}}^{n}(1)_{1,1}.$

The following proposition follows immediately from [19, Proposition 3.1.6].

######  Proposition 3.1.

Let $n\geq 3$, $\delta>0$ and $\lambda,\lambda_{1},\lambda_{2}\geq 0$.

  1. $(i)$

The metrics $g_{\mathrm{torp}}^{n}(\delta)_{\lambda}$,
$g_{\mathrm{torp}+}^{n}(\delta)_{\lambda}$ and
$\hat{g}_{\mathrm{torp}}^{n}(\delta)_{\lambda_{1},\lambda_{2}}$ have positive
scalar curvature.

  2. $(ii)$

For any constant $b\geq 0$ and any $\lambda,\lambda_{1},\lambda_{2}\geq 0$,
there exists $\delta>0$ so that the scalar curvature of the metrics
$g_{\mathrm{torp}}^{n}(\delta)_{\lambda}$,
$g_{\mathrm{torp}+}^{n}(\delta)_{\lambda}$ and
$\hat{g}_{\mathrm{torp}}^{{n}}(\delta)_{\lambda_{1},\lambda_{2}}$ is bounded
below by $b$.

$\lambda_{1}$$\lambda_{2}$$t_{1}$$t_{2}$$\frac{\pi}{2}\Lambda$$l_{1}$$l_{2}$$l_{3}$$l_{4}$

Figure 3: The metrics $g_{\mathrm{torp}}^{n}(\delta)_{\lambda}$,
$g_{\mathrm{torp}+}^{n}(\delta)_{\lambda}$ and
$\hat{g}_{\mathrm{torp}}^{n}(\delta)_{\lambda_{1},\lambda_{2}}$ (bottom) on
the manifolds $D^{n}$, $D_{+}^{n}$ and $D_{\mathrm{stretch}}^{n}$ (top)
followed by the boot metric $g_{\mathrm{boot}}^{n}(\delta)_{\Lambda,\bar{l}}$.

We now consider product metrics
$g_{\mathrm{torp}}^{n-1}(\delta)_{\lambda}+{\rm d}t^{2}$ on the cylinder
$D^{n-1}\times I$. It is shown in [19, Section 5], provided $n\geq 4$, that
any such product metric $g_{\mathrm{torp}}^{n-1}(\delta)_{\lambda}+{\rm
d}t^{2}$ can be moved by isotopy through psc-metrics to a particular psc-
metric called a $\delta$-boot metric. A detailed account of how such metrics
are constructed can be found in [19]. Here we will provide brief description.

  1. $(i)$

Beginning with some torpedo metric,
$g_{\mathrm{torp}}^{n-1}(\delta)_{\lambda}$, trace out a cylinder of torpedo
metrics before bending the cylinder around an angle of $\frac{\pi}{2}$ to
finish as a Riemannian cylinder (perpendicular to the first one) in the
direction suggested by the rightmost image of Figure 3. The resulting object
has two cylindrical ends, one of the form ${\rm
d}t_{1}^{2}+g_{\mathrm{torp}}^{n-1}(\delta)_{\lambda}$ and the other ${\rm
d}t_{2}^{2}+g_{\mathrm{torp}}^{n-1}(\delta)_{\lambda}$, where $t_{1}$ and
$t_{2}$ are orthogonal coordinates depicted in Figure 3.

  2. $(ii)$

In order to ensure the resulting metric has positive scalar curvature, the
bending is controlled by a paramater $\Lambda>0$. Essentially, the bending
takes place along a quarter-circle of radius $\Lambda>0$. A sufficiently large
choice of $\Lambda$ ensures that curvature arising from the bend is small and
positivity of the scalar curvature arising from the torpedo factor dominates.

  3. $(iii)$

Away from the “caps” of the torpedos, this metric takes the form ${\rm
d}t_{1}^{2}+{\rm d}t_{2}^{2}+\delta^{2}{\rm d}s_{n-1}^{2}$. This part can
easily be extended to incorporate the corner depicted in the rightmost image
of Figure 3 and so that the necks of the torpedo “ends” have any desired
lengths, $l_{1}$ and $l_{4}$ (determining along with $\Lambda$ the distances
$l_{2}$ and $l_{3}$).

  4. $(iv)$

Finally, we smoothly “cap-off” the cylindrical end which takes the form ${\rm
d}t_{1}^{2}+g_{\mathrm{torp}}^{n-1}(\delta)_{l_{1}}$, by attaching a half-
torpedo metric, $g_{\mathrm{torp}+}^{n}(\delta)_{l_{1}}$. This is the so-
called “toe” of the boot metric.

The resulting metric is denoted
$g_{\mathrm{boot}}^{n}(\delta)_{\Lambda,\bar{l}}$, where $\Lambda>0$ is the
bending constant discussed above and
$\bar{l}=(l_{1},l_{2},l_{3},l_{4})\in(0,\infty)^{4}$ determines the various
neck-lengths. As we mentioned above, $\Lambda$ may need to be chosen to be
large (although can always be found) and will depend on $\delta$. While the
choices of $l_{1}$ and $l_{4}$ are arbitrary, the constants $l_{2}$ and
$l_{3}$ are determined by $\Lambda$, $l_{1}$ and $l_{4}$.

###  3.2 Back to the proof of Theorem 2.17

The proof follows from that of [20, 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

$\displaystyle\mu_{(\bar{B},\bar{f},\bar{h}^{\beta})}\colon\
{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{0}}\longrightarrow{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}},$

into a composition of three maps as shown in the commutative diagram below:

$\textstyle{{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\mathrm{boot}}}$$\scriptstyle{\mu_{(\bar{B},\bar{f},\bar{h}^{\beta})}}$$\textstyle{{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}}$$\textstyle{{\mathcal{R}}_{\mathcal{S}-\mathrm{boot}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{\mathrm{std}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\mathrm{Estd}}}$$\textstyle{{\mathcal{R}}_{\mathcal{S}_{1}-\mathrm{Estd}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(3.5)

Here the right vertical map denotes inclusion. We will define the spaces
${\mathcal{R}}_{\mathcal{S}-\mathrm{boot}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{\mathrm{std}}}$ and
${\mathcal{R}}_{\mathcal{S}_{1}-\mathrm{Estd}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}$ and the maps $\mu_{\mathrm{boot}}$ and
$\mu_{\mathrm{Estd}}$ in due course. The point is to show that each of these
maps is a weak homotopy equivalence.

We denote by $k=n-\ell-1=\operatorname{{\text{\rm dim}}}\beta X=B_{0}$. We
consider carefully the elementary bordism
$\big{(}\bar{B},\bar{f}\big{)}\colon(B_{0},f_{0})\rightsquigarrow(B_{1},f_{1})$.
The manifold $\bar{B}$ is given by attaching a handle $D^{p+1}\times D^{q+1}$
to $B_{0}$ along the embeddings $\phi\colon S^{p}\times D^{q+1}\hookrightarrow
B_{0}$, where $p+q+1=k$, $q\geq 2$. We would like to have some flexibility for
the embedding $\phi$. We introduce the following family of rescaling maps:

$\displaystyle\begin{split}\sigma_{\rho}\colon\ S^{p}\times
D^{q+1}&\longrightarrow S^{p}\times D^{q+1},\\\ (x,y)&\longmapsto(x,\rho
y),\end{split}$

where $\rho\in(0,1]$. We set

$\displaystyle\phi_{\rho}:=\phi\circ\sigma_{\rho}\colon\ S^{p}\times
D^{q+1}\hookrightarrow B_{0}$

and $N_{\rho}:=\phi_{\rho}(S^{p}\times D^{q+1})$, abbreviating $N:=N_{1}$ and
$\phi:=\phi_{1}$. Let $T_{\phi}$ be the trace of the surgery on $B_{0}$ with
respect to $\phi$. We denote by
${\mathcal{R}}_{\mathrm{std}}^{\mathrm{psc}}(B_{0})$, the space defined as
follows:

$\displaystyle{\mathcal{R}}_{\mathrm{std}}^{\mathrm{psc}}(B_{0}):=\big{\\{}g\in{\mathcal{R}}^{\mathrm{psc}}(B_{0})\colon\phi_{\frac{1}{2}}^{*}g={\rm
d}s_{p}^{2}+g_{\mathrm{torp}}^{q+1}\text{ on }S^{p}\times D^{q+1}\big{\\}}.$

According to Chernysh’s theorem [8, 9], the inclusion

$\displaystyle{\mathcal{R}}_{\mathrm{std}}^{\mathrm{psc}}(B_{0})\subset{\mathcal{R}}^{\mathrm{psc}}(B_{0}),$

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 [12]) that for any psc-metric
$h^{\beta}\in{\mathcal{R}}^{\mathrm{psc}}(B_{0})$, there is an isotopy
$h_{t}^{\beta}$, $t\in I$ of metrics in ${\mathcal{R}}^{\mathrm{psc}}(B_{0})$
connecting $h_{0}^{\beta}=h^{\beta}$ to a psc-metric
$h_{\mathrm{std}}^{\beta}\in{\mathcal{R}}_{\mathrm{std}}^{\mathrm{psc}}({B_{0}})$.
By a well known argument (see [20, Lemma 2.3.2]), this isotopy gives rise to a
concordance: $\bar{h}_{\mathrm{con}}^{\beta}$ on $B_{0}\times[0,\lambda+2]$
for some $\lambda>0$ which takes the form of product metrics:

$\displaystyle h^{\beta}+{\rm d}t^{2}\quad\text{on}\quad
B_{0}\times[\lambda+1,\lambda+2]\qquad\text{and}\qquad g_{\mathrm{std}}+{\rm
d}t^{2}\quad\text{on}\quad B_{0}\times[0,1].$

Note that on the slice $N_{\frac{1}{2}}\times[0,1]$, the metric
$\bar{h}_{\mathrm{con}}^{\beta}$ pulls back to a metric of the form

$\displaystyle{\rm d}s_{p}^{2}+g_{\mathrm{torp}}^{q+1}+{\rm d}t^{2}.$ (3.6)

Making use of [20, Lemma 5.2.5] we can perform an isotopy of the metric
$\bar{h}_{\mathrm{con}}^{\beta}$, adjusting only on
$N_{\frac{1}{2}}\times[0,1]$, to replace the $g_{\mathrm{torp}}^{q+1}+{\rm
d}t^{2}$ factor in (3.6) with $g_{\mathrm{boot}}^{q+2}(1)_{\Lambda,\bar{l}}$
for some appropriately large $\Lambda>0$ and with $\bar{l}$ satisfying
$l_{1}=l_{4}=1$. We denote the resulting psc-metric $\bar{h}_{{\rm
Phys.~{}Rev.~{}E}}^{\beta}$ on $B_{0}\times[0,\lambda+2]$. We consider
$B_{0}\times[0,\lambda+2]$ as a long collar of $\bar{B}$ and assume that the
map $\bar{f}$ restricted to $B_{0}\times[0,\lambda+2]$ is given by
$\bar{f}(x,t)=f_{0}(x)$. Let $\bar{E}_{0}$ be a manifold given by pulling back
the fibre bundle where $\bar{f}_{0}$ is a restriction of $\bar{f}$.

We now use the metric $\bar{h}_{{\rm Phys.~{}Rev.~{}E}}^{\beta}$ on
$B_{0}\times[0,\lambda+2]$ to extend the metric $g^{\partial}$ from the
boundary $E_{0}=\partial X$ to a positive scalar curvature submersion metric
$\bar{g}^{\partial}_{{\rm Phys.~{}Rev.~{}E}}$ on the total space of this
bundle, $\bar{E}_{0}$. To do this, we follow the method described in Lemma 2.7
with respect to the product distribution $\mathcal{H}_{0}\times\mathbb{R}$ to
extend $g^{\partial}$ to a psc submersion metric, on $\bar{E}_{0}\cong
E_{0}\times[0,\lambda+2]$. We denote the resulting metric,
$\bar{g}^{\partial}_{{\rm Phys.~{}Rev.~{}E}}$. Note, near
$E_{0}\times\\{\lambda+2\\}$, this is a Riemannian cylinder of a psc-
submersion metric over $\big{(}B_{0},h^{\beta}_{\mathrm{std}}\big{)}$ with the
fibre $(L,\bar{\tau}g_{L})$ for $\bar{\tau}=\bar{\tau}\big{(}\bar{h}_{{\rm
Phys.~{}Rev.~{}E}}^{\beta},\mathcal{H}_{0}\times\mathbb{R}\big{)}$, as defined
in (2.2).

$\mu_{\bar{g}_{{\rm
Phys.~{}Rev.~{}E}}^{\partial}}(g)$$\mu_{\bar{g}}(\mu_{\bar{g}_{{\rm
Phys.~{}Rev.~{}E}}^{\partial}}(g))$$g$A typical element of
${\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}$

Figure 4: Representative elements of the spaces from the commutative diagram
(3.5) above in the case when $L$ is a point.

For any element $g\in{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{0}}$, the metric $g\cup_{\partial X}\bar{g}^{\partial}_{{\rm
Phys.~{}Rev.~{}E}}$ on $X\cup\bar{E}_{0}$ (obtained by the obvious gluing) is
denoted by $g_{\mathrm{std}}$ and is an element of the space
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{\mathrm{std}}}$. This gives a map

$\displaystyle\mu_{\bar{g}^{\partial}_{{\rm Phys.~{}Rev.~{}E}}}\colon\
{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{0}}\to{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{\mathrm{std}}}.$

We denote ${\mathcal{R}}_{\mathcal{S}-\mathrm{boot}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{\mathrm{std}}}:=\mathrm{Im}(\mu_{\bar{g}^{\partial}_{{\rm
Phys.~{}Rev.~{}E}}})$. This new metric is depicted in the bottom left of
Figure 4, 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 [20],
consolidating work from previous sections, shows that in the case when $L$ is
a point (and so $\partial X=\beta X$), the map $\mu_{\bar{g}_{{\rm
Phys.~{}Rev.~{}E}}}$ is a weak homotopy equivalence. This is demonstrated by
constructing certain isotopies of psc-metrics on the cylinder
$B_{0}\times[0,\lambda+2]$. In our case, where $L$ is not simply a point, we
must lift such a compact family of psc-metrics on $B_{0}\times[0,\lambda+2]$
to a corresponding compact family of psc-metrics on $\bar{E}_{0}$. This is
done using Lemma 2.7, with respect to the horizontal distribution
$\mathcal{H}_{0}\times\mathbb{R}$ and a sufficiently small scaling constant
$\bar{\tau}_{\operatorname*{{\text{\rm min}}}}>0$.

Turning our attention momentarily to the space $X_{1}=X\cup\bar{E}$, recall
that, though suppressed in the notation, the map
$\mu_{(\bar{B},\bar{f},\bar{h}^{\beta})}$ implicitly associates to the bundle,
$\bar{p}\colon\bar{E}\to\bar{B}$, an admissible horizontal distribution,
$\bar{\mathcal{H}}$ extending $\mathcal{H}_{0}$ (earlier we denoted this
$\xi(\mathcal{H}_{0})$).

Consider now an element of the space
${\mathcal{R}}_{\mathcal{S}-\mathrm{boot}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{\mathrm{std}}}:=\mathrm{Im}(\mu_{\bar{g}^{\partial}_{{\rm
Phys.~{}Rev.~{}E}}})$. Such an element has, near the boundary, a “standard
piece” where the submersion metric $\bar{g}_{{\rm
Phys.~{}Rev.~{}E}}^{\partial}$ restricts over a region where the base metric
takes the form ${\rm
d}s_{p}^{2}+g_{\mathrm{boot}}^{q+2}(1)_{\Lambda,\bar{l}}$. Replacing this
piece, on the base, with $g_{\mathrm{torp}}^{p+1}+g_{\mathrm{torp}}^{q+1}$
near the boundary, determines a base metric on $\bar{B}$. This is exactly what
is done in [20, Theorem A], determining a homeomorphism onto its image in the
relevant spaces of psc-metrics on the bases. Using Lemma 2.7, with respect to
the distribution $\bar{\mathcal{H}}$ allows us to lift the resulting base
metric on $\bar{B}$ to a psc-submersion metric on $\bar{E}$. This determines a
map

$\displaystyle\mu_{\mathrm{Estd}}\colon\
{\mathcal{R}}_{\mathcal{S}-\mathrm{boot}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{\mathrm{std}}}\to{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}.$

Replacing the codomain of this map by its image, denoted

$\displaystyle{\mathcal{R}}_{\mathcal{S}_{1}-\mathrm{Estd}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}\subset{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}},$

we obtain the lower horizontal map in diagram (3.5). A typical element in the
image of this map is depicted in the lower right of Figure 4. As with the base
case in [20, Theorem A], this lower horizontal map is demonstrably a
homeomorphism.

It remains to show that the inclusion
${\mathcal{R}}_{\mathcal{S}_{1}-\mathrm{Estd}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}\subset{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}$ is a weak homotopy equivalence. Note that the notaiton
“$\mathrm{Estd}$” used in describing the former space (originating in [20]) is
intended to convey the fact that these metrics take a standard form on a much
larger region than typical metrics in
${\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}$ and are thus “Extra-standard”.

A typical element of
${\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}$ (in the case when $L$ is a point and so $\partial
X_{1}=\beta X_{1}$) is depicted in the upper right of Figure 4. Showing that,
in the case when $L$ is a point, a compact family of metrics in
${\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}$ could be continuously moved to a compact family of
extra standard metrics in
${\mathcal{R}}_{\mathcal{S}_{1}-\mathrm{Estd}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}$, without moving already extra-standard metrics out of
that space, is the most technically difficult part of the proof of [20,
Theorem A]. This is done in [20, Section 6.6].

In our case, $L$ is not simply a point. All of the adjustment takes place off
of $X$ and so we restrict our attention to $X_{1}\setminus X$. Once again
however, the argument in [20, Theorem A] involves continuous deformations of
compact families of psc-metrics on the base. Thus, applying Lemma 2.7, for the
one admissible horizontal distribution $\bar{\mathcal{H}}$ and a sufficiently
small scaling constant $\bar{\tau}_{\operatorname*{{\text{\rm min}}}}>0$, all
such (compact) deformations of psc-metrics lift to appropriate deformations on
the total space $X_{1}\setminus X$. This completes the proof of Theorem 2.17.

##  4 Proof of Theorem 1.8

We now come to the proof of Theorem 1.8. We noted earlier (see Remark 1.9)
that part $(i)$ of Theorem 1.8 follows from the main result of Chernysh in
[8]. Thus, our focus is on part $(ii)$.

Let $X_{\Sigma}=X\cup_{\partial X}-N(\beta X)$ be a pseudo-manifold as above,
where $X$ is a manifold with boundary $\partial X$, the total space of a
bundle, $p\colon\partial X\to\beta X$ with fibre $L$. This bundle has a
structure map $f\colon\beta X\to BG$. Recall that $G$ is a subgroup of the
isometry group of the simple link $(L,g_{L})$. We begin by recalling the
restriction map

$\displaystyle\mathrm{res}\colon\ {\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)\to{\mathcal{R}}^{\mathrm{psc}}(\partial X),\qquad\mathrm{res}\colon\
g\mapsto g|_{\partial X}.$

As we mentioned earlier, from work of Chernysh [8] and Ebert–Frenck [9], we
know that this map is a Serre fibre bundle.

Now we consider two pseudo-manifolds $X_{\Sigma}=X\cup_{\partial X}-N(\beta
X)$ and $X_{\Sigma,1}=X_{1}\cup_{\partial X}-N(\beta X_{1})$, where
$X_{1}=X\cup_{\partial X}Z$, and the manifold $Z$ is given by an elementary
psc-bordism $\bar{B}\colon\beta X\rightsquigarrow\beta X_{1}$ and a structure
map $\bar{f}\colon\bar{B}\to BG$, so that $f=\bar{f}|_{\beta X}$ and
$f_{1}=\bar{f}|_{\beta X_{1}}$. Namely, the manifold $Z$ is a total space of
the following smooth bundle: Consider a pair of psc-submersion metrics,
$g^{\partial}_{0}\in{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial X)$
and
$g^{\partial}_{1}\in{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(\partial
X_{1})$, where $\mathcal{S}_{1}$ is the analogue of the map $\mathcal{S}$ for
the bundle $\partial X_{1}\to\beta X_{1}$. Thus, $g^{\partial}_{0}$ and
$g^{\partial}_{1}$ are psc-submersion metrics over $\beta X$ (respectively,
over $\beta X_{1}$), with fibre metrics $\tau_{0}g_{L}$ and $\tau_{1}g_{L}$
for some constants $\tau_{0},\tau_{1}>0$. We denote by
$h_{0}^{\beta}\in{\mathcal{R}}^{\mathrm{psc}}(\beta X)$ and
$h_{1}^{\beta}\in{\mathcal{R}}^{\mathrm{psc}}(\beta X_{1})$, the respective
base metrics $\mathcal{B}\big{(}g^{\partial}\big{)}=h^{\beta}=:h_{0}^{\beta}$
and $\mathcal{B}_{1}\big{(}g^{\partial}_{1}\big{)}=:h_{1}^{\beta}$.

Now we notice that the spaces
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial X)_{h^{\beta}_{0}}$ and
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}$ coincide with the fibres

$\displaystyle{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}_{0}}=\mathrm{res}^{-1}_{0}\big{(}g_{0}^{\partial}\big{)},\qquad{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}=\mathrm{res}^{-1}_{1}\big{(}g_{1}^{\partial}\big{)},$

of the corresponding restriction maps:

$\displaystyle\mathrm{res}_{0}\colon\ {\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)\to{\mathcal{R}}^{\mathrm{psc}}(\partial X),\qquad\mathrm{res}_{1}\colon\
{\mathcal{R}}^{\mathrm{psc}}(X_{1},\partial
X_{1})\to{\mathcal{R}}^{\mathrm{psc}}(\partial X_{1}).$

We consider the inclusion map

$\displaystyle{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial
X)\hookrightarrow{\mathcal{R}}^{\mathrm{psc}}(\partial X).$

Now, by definition, we obtain the space
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$ as a pull-back in the following
diagram where the right vertical map is inclusion. The left vertical map sends
a psc-metric on $X_{\Sigma}$ to its restriction to $X$.

There are two more maps we must define. Recall we have a projection map,

$\displaystyle\mathcal{B}\colon\
{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}\to{\mathcal{R}}^{\mathrm{psc}}(\beta
X),$

sending each submersion metric to its base. In Lemma 2.5 we showed that
$\mathcal{B}$ forms part of a homotopy equivalence. In particular the lifting
map (2.5)

$\displaystyle\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(\beta
X)\to{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}},$

for any choice of horizontal distribution, $\mathcal{H}$, on the bundle
$p\colon\partial X\to\beta X$, is a homotopy inverse. Composing, for some
choice of $\mathcal{H}$, the map $\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}$
with the inclusion map above yields the map

$\displaystyle\pi_{\mathcal{H}}\colon\ {\mathcal{R}}^{\mathrm{psc}}(\beta
X)\xrightarrow[]{\mathcal{S}_{\mathcal{H}}^{\mathrm{psc}}}{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial
X)\hookrightarrow{\mathcal{R}}^{\mathrm{psc}}(\partial X).$

The restriction map
$r\colon{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\xrightarrow[]{}{\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)$ above is in fact a homeomorphism onto its image,
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial X)$, the space of well
adapted psc-metrics on $X$. Replacing the codomain
${\mathcal{R}}^{\mathrm{psc}}(X,\partial X)$ with
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial X)$, we then compose
$r$ with the restrictions $\mathrm{res}$ and $\mathcal{B}$ to obtain the map

$\displaystyle{\mathrm{res}}_{\Sigma}\colon\
{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\xrightarrow[]{r}{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)\xrightarrow[]{\mathrm{res}}{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(\partial
X)\xrightarrow[]{\mathcal{B}}{\mathcal{R}}^{\mathrm{psc}}(\beta X).$

As $r$ (as deployed above) is now a homeomorphism, $\mathrm{res}$ is a Serre
fibre bundle and ${\mathcal{B}}$ is a homotopy equivalence, we conclude that
the composition map, $\mathrm{res}_{\Sigma}$, is a Serre fibre bundle.
Recalling that the spaces ${\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)_{g_{0}^{\partial}}=\mathrm{res}_{0}^{-1}\big{(}g_{0}^{\partial}\big{)}$ and
${\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial X)_{h_{0}^{\beta}}$ are
in fact the same, we obtain the following diagram of fibre bundles: Here, the
top left horizontal map $i_{\Sigma}$ is the metric attaching map defined in
2.6.

Let $\big{(}\bar{B},\bar{h}_{\beta}\big{)}\colon\big{(}\beta
X,h_{0}^{\beta}\big{)}\rightsquigarrow\big{(}\beta X_{1},h_{1}^{\beta}\big{)}$
be an elementary psc-bordism (with $p,q\geq 2$), which is given together with
a map $\bar{f}\colon\bar{B}\to BG$ such that $f=\bar{f}|_{\beta X}$ and
$f_{1}=\bar{f}|_{\beta X_{1}}$. In turn, the psc-bordism
$\big{(}\bar{B},\bar{h}_{\beta}\big{)}$ determines a corresponding psc-bordism
$\big{(}Z,\bar{g}^{\partial}\big{)}\colon\big{(}\partial
X,g^{\partial}_{0}\big{)}\rightsquigarrow\big{(}\partial
X_{1},g^{\partial}_{1}\big{)}$ by means of Lemma 2.7. In particular, the psc-
submersion metrics $g^{\partial}_{0}$ and $g^{\partial}_{1}$ have base metrics
$h_{0}^{\beta}$ and $h_{1}^{\beta}$. Theorem 2.2 and Theorem 2.17 give us the
following homotopy equivalences:

$\displaystyle\mu_{\bar{B},\bar{g}^{\beta}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(\beta
X)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\mathcal{R}}^{\mathrm{psc}}(\beta
X_{1}),$ $\displaystyle\mu_{Z,\bar{g}^{\beta}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(\partial
X)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\mathcal{R}}^{\mathrm{psc}}(\partial
X_{1}),$ $\displaystyle\mu_{Z,\bar{g}^{\partial}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)_{g^{\partial}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\mathcal{R}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{g^{\partial}_{1}},$
$\displaystyle\mu_{\bar{B},\bar{f},\bar{h}^{\beta}}\colon\
{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}.$

We obtain the following commutative diagram:

---
$\textstyle{{\mathcal{R}}_{\mathcal{S}}^{\mathrm{psc}}(X,\partial
X)_{h^{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\Sigma}}$$\scriptstyle{\mu_{\bar{B},\bar{f},\bar{h}^{\beta}}\\!\\!\\!\\!}$$\textstyle{{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{res}_{\Sigma}}$$\textstyle{\simeq}$$\textstyle{{\mathcal{R}}^{\mathrm{psc}}(\beta
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathcal{H}_{0}}}$$\scriptstyle{\mu_{\bar{B},\bar{g}^{\beta}}\\!\\!\\!\\!}$$\textstyle{{\mathcal{R}}_{\mathcal{S}_{1}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{h^{\beta}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\ \ \ \
i_{\Sigma}}$$\textstyle{{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma,1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\ \ \ \ \ \
\mathrm{res}_{\Sigma}}$$\textstyle{{\mathcal{R}}^{\mathrm{psc}}(\beta
X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathcal{H}_{1}}}$$\textstyle{{\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)_{g^{\partial}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\ \
i}$$\scriptstyle{\mu_{Z,\bar{g}^{\partial}}\\!\\!\\!\\!}$$\textstyle{{\mathcal{R}}^{\mathrm{psc}}(X,\partial
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{res}}$$\textstyle{\simeq}$$\textstyle{{\mathcal{R}}^{\mathrm{psc}}(\partial
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{Z,\bar{g}^{\beta}}\\!\\!\\!\\!\\!}$$\textstyle{{\mathcal{R}}^{\mathrm{psc}}(X_{1},\partial
X_{1})_{g^{\partial}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{{\mathcal{R}}^{\mathrm{psc}}(X_{1},\partial
X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{res}}$$\textstyle{{\mathcal{R}}^{\mathrm{psc}}(\partial
X_{1})}$

where all horizontal rows are Serre fibre bundles, and the maps
$\pi_{\mathcal{H}_{0}}$ and $\pi_{\mathcal{H}_{1}}$ are defined as above.
Recall in particular that the left most vertical arrows (front and back)
represent equality. Commutativity of this diagram makes it evident that the
circled maps above are weak homotopy equivalences. This proves Theorem 1.8.
$\Box$

##  5 Some further developments

In this section we would like to emphasize that recent results concerning
homotopy groups of the spaces, ${\mathcal{R}}^{\mathrm{psc}}(M)$, of psc-
metrics on a smooth closed manifold $M$ (see [4, 10, 11, 13, 14]) 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

$\displaystyle\mathsf{inddiff}_{g_{0}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(M)\to\Omega^{\infty+n+1}\mathbf{KO},$ (5.1)

which is defined as follows. Let $g_{0}\in{\mathcal{R}}^{\mathrm{psc}}(M)$ 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 $D_{g_{t}}$ starts and ends at the subspace
$\big{(}\mathbf{Fred}^{n}\big{)}^{\times}\subset\mathbf{Fred}^{n}$ of
invertible Dirac operators. Since the subspace
$\big{(}\mathbf{Fred}^{n}\big{)}^{\times}$ is contractible, the curve
$D_{g_{t}}$ is a loop in the space $\mathbf{Fred}^{n}$ of all Dirac operators.
This space, in turn, is homotopy equivalent to the loop space
$\Omega^{\infty+n}\mathbf{KO}$ representing a shifted $KO$-theory, i.e.,
$\pi_{q}\big{(}\Omega^{\infty+n}\mathbf{KO}\big{)}=KO_{n+q}$. Thus the curve
$D_{g_{t}}$ gives an element in $\Omega^{\infty+n+1}\mathbf{KO}$, well-defined
up to homotopy, to determine the map (5.1).

######  Theorem 5.1 (Botvinnik–Ebert–Randal-Williams [4], and Perlmutter [13,
14]).

Assume $M$ is a spin manifold with $\operatorname{{\text{\rm dim}}}M\geq 5$
and ${\mathcal{R}}^{\mathrm{psc}}(M)\neq\varnothing$ with a base point
$g_{0}\in{\mathcal{R}}^{\mathrm{psc}}(M)$. Then the index-diffence map (5.1)
induces a non-trivial homomorphism in the homotopy groups

$\displaystyle(\mathsf{inddiff}_{g_{0}})_{*}\colon\
\pi_{q}({\mathcal{R}}^{\mathrm{psc}}(M))\to KO_{q+n+1},$

when the target group $KO_{q+n+1}$ is non-trivial.

###  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 [10, 11]. 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 $X_{\Sigma}$ be a spin $(\left<k\right>\\!\mbox{-}\mathrm{fb})$-manifold.
Assume $\operatorname{{\text{\rm dim}}}X\geq 7$ and $X$ and $\beta
X\neq\varnothing$ are simply-connected and
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\neq\varnothing$ with a base point
$g_{0}\in{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$. Then there is an index-
difference map

$\displaystyle\mathsf{inddiff}_{g_{0}}^{\left<k\right>}\colon\
{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\to\Omega^{\infty+n+1}\mathbf{KO}^{\left<k\right>},$

which induces a non-trivial homomorphism in the homotopy groups

$\displaystyle\big{(}\mathsf{inddiff}_{g_{0}}^{\left<k\right>}\big{)}_{*}\colon\
\pi_{q}({\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma}))\to
KO^{\left<k\right>}_{n+q+1}$

when the target group $KO^{\left<k\right>}_{n+q+1}$ $(KO$ with
$\mathbb{Z}_{k}$-coefficients$)$ is non-trivial.

######  Conjecture 5.3.

Let $X_{\Sigma}$ be a spin manifold with
$(\eta\mbox{-}\mathrm{fb})$-singularity of dimension $n\geq 9$. Assume $\beta
X\neq\varnothing$, and
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\neq\varnothing$ with a base point
$g_{0}\in{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$. Then there is an index-
difference map

$\displaystyle\mathsf{inddiff}_{g_{0}}^{\eta\mbox{-}\mathrm{fb}}\colon\
{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\to\Omega^{\infty+n+1}\mathbf{KO}^{\eta\mbox{-}\mathrm{fb}},$

which induces a non-trivial homomorphism in the homotopy groups

$\displaystyle\big{(}\mathsf{inddiff}_{g_{0}}^{\eta}\big{)}_{*}\colon\
\pi_{q}({\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma}))\to
KO^{\eta\mbox{-}\mathrm{fb}}_{q+n+1}$

when the target group
$KO^{\eta\mbox{-}\mathrm{fb}}_{q+n+1}=KO_{q+n+1}(\mathbb{CP}^{\infty})$ 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
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$. Let $X_{\Sigma}=X\cup_{\partial
X}-N(\beta X)$ be a spin manifold with $(L,G)$-singularities. Let
$g\in{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$ be a well-adapted metric. Then
$g$ determines the metrics $g_{\partial
X}\in{\mathcal{R}}^{\mathrm{psc}}(\partial X)$ and $g_{\beta
X}\in{\mathcal{R}}^{\mathrm{psc}}(\beta X)$ such that the bundle $\partial
X\to\beta X$ is a Riemannian submersion. We fix the metric $g_{\beta X,0}$.
This gives rise to a Serre fibre bundle

$\displaystyle\mathrm{res}_{\Sigma}\colon\
{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\to{\mathcal{R}}^{\mathrm{psc}}(\beta
X)$

with fibre ${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})_{g_{\beta X},0}$, where
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})_{g_{\beta X}}$ is the space of all
metrics $g\in{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})$ which restrict to
$g_{\beta X,0}$ on ${\mathcal{R}}^{\mathrm{psc}}(\beta X)$. Since the metric
$g_{\partial X,0}$ on $\partial X$ is determined by the metric $g_{\beta
X,0}$, the fibre ${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})_{g_{\beta X},0}$
coincides with the space ${\mathcal{R}}^{\mathrm{psc}}(X)_{g_{\partial X},0}$.
Here is the result we need:

######  Theorem 5.4 (see [6, Theorem 6.1]).

Let $X_{\Sigma}$ be an $(L,G)$-fibred compact pseudo-manifold with $L$ a
simply connected homogeneous space of a compact semisimple Lie group. Assume
${\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\neq\varnothing$. Then there exists a
section $s\colon\mathcal{R}^{\mathrm{psc}}(\beta
X)\to\mathcal{R}^{\mathrm{psc}}(X_{\Sigma})$ to $\mathrm{res}_{\Sigma}$. In
particular, there is a split short exact sequence:

$\displaystyle 0\to\pi_{q}(\mathcal{R}_{w}^{+}(X_{\Sigma})_{g_{\beta
X}})\xrightarrow{i_{*}}\pi_{q}(\mathcal{R}_{w}^{+}(X_{\Sigma}))\xrightarrow{(\mathrm{res}_{\Sigma})_{*}}\pi_{q}(\mathcal{R}^{+}(\beta
X))\to 0,\qquad q=0,1,\ldots.$

Here is one of the conclusions we would like to emphasize:

######  Corollary 5.5 (see [5, Corollary 6.7]).

Let $X_{\Sigma}$ be an $(L,G)$-fibred compact pseudo-manifold with $L$ a
simply connected homogeneous space of a compact semisimple Lie group, and
$n-\ell-1\geq 5$, where $\operatorname{{\text{\rm dim}}}X=n$,
$\operatorname{{\text{\rm dim}}}L=\ell$. Let
$g_{0}\in{\mathcal{R}}^{\mathrm{psc}}(X_{\Sigma})\neq\varnothing$ be a base
point giving corresponding base points, the metrics $g_{\beta
X,0}\in{\mathcal{R}}^{\mathrm{psc}}(\beta X)$, $g_{\partial
X,0}\in{\mathcal{R}}^{\mathrm{psc}}(\partial X)$ and
$g_{X,0}\in{\mathcal{R}}^{\mathrm{psc}}(X)_{g_{\partial X,0}}$.

If $M_{\Sigma}$ is spin and simply connected, then we have the following
commutative diagram: where the homomorphisms $\mathrm{inddiff}_{g_{X,0}}$ and
$\mathrm{inddiff}_{g_{\beta X,0}}$ are both nontrivial whenever the target
groups are. In particular, the homomorphism

$\displaystyle\mathrm{inddiff}_{g_{0}}\colon\
\pi_{q}\mathcal{R}^{\mathrm{psc}}(X_{\Sigma})\to KO_{q+n+1}\oplus
KO_{q+n-\ell}$

is surjective rationally and surjective onto the torsion of $KO_{q+n+1}\oplus
KO_{q+n-\ell}$.

There are much more general results concerning the homotopy groups of the
space $\pi_{q}\mathcal{R}^{\mathrm{psc}}(X_{\Sigma})$ if $X_{\Sigma}$ is not
simply-connected; see [5, Section 6].

##  6 Appendix: cone metrics

###  6.1 Cone metric of $\boldsymbol{(L,g_{L})}$ when $\boldsymbol{g_{L}}$ has
constant non-negative
scalar curvature

Let $L$ be smooth closed manifold of dimension $l$ and $g_{L}$, a metric on
$L$ with constant non-negative scalar curvature. We wish to extend $g_{L}$ to
a scalar flat metric on the cone $C(L)$. The resulting metric will be denoted
$g_{C(L)}$ and is constructed as follows. Consider the warped product metric
${\rm d}t^{2}+\phi(t)^{2}g_{L}$ on the cylinder $(0,b)\times L$ for some
smooth function $\phi\colon[0,b]\rightarrow(0,\infty)$. The scalar curvature,
$s$, of such a metric is given by the formula

$\displaystyle
s=\frac{-4l}{(l+1)}\left(\frac{u^{\prime\prime}}{u}\right)+s_{g_{L}}u^{\frac{-4}{l+1}},$
(6.1)

where $l:=\operatorname{{\text{\rm dim}}}{L}$, $u$ is the function satisfying
$u^{\frac{2}{l+1}}=\phi$ and $s_{g_{L}}$ is the scalar curvature of the metric
$g_{L}$.

Let us first consider the case when $s_{g_{L}}=0$. An easy calculation shows
$s=0$ precisely when

$\displaystyle u(t)=At+B,$

for various constants $A,B$. Indeed, in the case when $A=0$, we obtain a
Riemannian cylinder. In our case, we set $A=1$, $B=0$ and $b=\frac{1}{2}$.
This gives rise to a metric

$\displaystyle{g_{C(L)}}={\rm d}t^{2}+t^{\frac{4}{l+1}}g_{L},$

on the cylinder $\big{[}0,\frac{1}{2}\big{]}\times L$, which collapses at
$t=0$. The result is a scalar-flat metric (with a singularity) on the cone
$C(L)$ obtained from $\big{[}0,\frac{1}{2}\big{]}\times L$ by collapsing
$\\{0\\}\times L$ to a point.

Now consider the case that the constant $s_{g_{L}}>0$. In this case, we set
$\phi(t)=\frac{1}{c_{L}}t$, where the constant $c_{L}$ satisfies

$\displaystyle c_{L}=\sqrt{\frac{l(l-1)}{s_{g_{L}}}}.$

Thus $u=t^{\frac{l+1}{2}}$ and the resulting metric

$\displaystyle{g_{C(L)}}={\rm d}t^{2}+\frac{1}{c_{L}^{2}}t^{2}g_{L},$

has scalar curvature given by

$\displaystyle s_{g_{C(L)}}=\big{(}c_{L}^{2}s_{g_{L}}-l(l-1)\big{)}t^{-2}=0,$
(6.2)

given our choice of $c_{L}$. Again, we may assume that
$t\in\big{[}0,\frac{1}{2}\big{]}$.

######  Remark 6.1.

Regarding the earlier setting when $g_{L}$ is scalar flat, we will typically
only be interested in the case when $L=S^{1}$ and so $l=1$. Thus, metric
formulae in both cases coincide.

###  6.2 Attaching the cone to the cylinder

In constructing certain metrics on $X_{\Sigma}$, it is necessary to attach the
above conical metric, ${g_{C(L)}}$, to a cylindrical metric ${\rm
d}t^{2}+g_{L}$. This requires an intermediary “attaching” metric to transition
smoothly between the two forms. To do this, we first specify a smooth
transition function, $a\colon[0,1]\rightarrow\big{[}0,\frac{1}{2}\big{]}$,
satisfying:

  1. $(i)$

$a(t)=\frac{1}{2}+t$ when $t$ is near $0$,

  2. $(ii)$

$a(t)=1$ when $t$ is near $1$,

  3. $(iii)$

for all $t\in[0,1]$, $0\leq a^{\prime}(t)\leq 1$ and $a^{\prime\prime}(t)\leq
0$.

The existence of such functions is obvious. We choose one and define the
attaching metric, ${g_{\mathrm{att}(L)}}$, on the cylinder $[0,1]\times L$ by
the formula:

$\displaystyle{g_{\mathrm{att}(L)}}=\begin{cases}{\rm
d}t^{2}+a^{\frac{4}{l+1}}(t)g_{L},&s_{g_{L}}=0,\\\ {\rm
d}t^{2}+a^{2}(t)g_{L},&s_{g_{L}}>0.\end{cases}$

######  Proposition 6.2.

Attaching $\big{(}\big{[}0,\frac{1}{2}\big{]}\times L,g_{C(L)}\big{)}$ to
$\big{(}[0,1]\times L,g_{\mathrm{att}(L)}\big{)}$ by identifying
$\big{\\{}\frac{1}{2}\big{\\}}\times L$ in the former with $\\{0\\}\times L$
in the latter determines a smooth metric of non-negative scalar curvature
$g_{\mathrm{att}(L)}\cup g_{C(L)}$.

###### Proof.

Smoothness is immediate from the properties of the transition function $a$.
The component, $g_{C(L)}$, has zero scalar curvature and that
$g_{\mathrm{att}(L)}$ has non-negative scalar curvature is easily seen from
the formula (6.1). ∎

### Acknowledgments

BB was partially supported by Simons collaboration grant 708183. We thank the
referees for their careful reading and thoughtful comments. The second author
would like to thank David Wraith for some productive conversations.

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