# Moduli spaces of $3$-manifolds with boundary are finite

Rachael Boyd  School of Mathematics and Statistics, University of Glasgow,
Glasgow G12 8QQ, UK<EMAIL_ADDRESS>https://www.maths.gla.ac.uk/
rboyd/ ,  Corey Bregman  Department of Mathematics, Tufts University, Medford,
MA 02155, USA<EMAIL_ADDRESS>https://sites.google.com/view/cbregman
and  Jan Steinebrunner  Institut for Matematiske Fag, Københavns Universitet,
København, Denmark<EMAIL_ADDRESS>https://www.jan-steinebrunner.com

###### Abstract.

We study the classifying space $B\\!\operatorname{Diff}(M)$ of the
diffeomorphism group of a connected, compact, orientable 3-manifold $M$. In
the case that $M$ is reducible we build a contractible space parametrising the
systems of reducing spheres. We use this to prove that if $M$ has non-empty
boundary, then $B\\!\operatorname{Diff}_{\partial}(M)$ has the homotopy type
of a finite CW complex. This was conjectured by Kontsevich and appears on the
Kirby problem list as Problem 3.48. As a consequence, we are able to show that
for every compact, orientable $3$-manifold $M$, $B\\!\operatorname{Diff}(M)$
has finite type.

###### 2020 Mathematics Subject Classification:

57T20, 58D29 (primary), 57M50, 55R40, 57S05, 58D05 (secondary)

###### Contents

  1. 1 Introduction
  2. 2 Background on 3-manifolds and homotopy theory
  3. 3 Systems of spheres
  4. 4 Homotopy finiteness via the (strong) generalised Smale conjecture
  5. 5 Decomposing irreducible $3$-manifolds
  6. 6 Kontsevich’s finiteness conjecture and consequences

##  1\. Introduction

For a $3$-manifold $M$, let $\operatorname{Diff}(M)$ denote group of
diffeomorphisms, equipped with the $C^{\infty}$ topology. Its classifying
space $B\\!\operatorname{Diff}(M)$ is the moduli space of $M$: homotopy
classes of maps $X\to B\\!\operatorname{Diff}(M)$ are in bijection with
concordance classes of smooth $M$-bundles over $X$ [GRW20]. When $M$ has
boundary, we let $\operatorname{Diff}_{\partial}(M)<\operatorname{Diff}(M)$
denote the subgroup of those diffeomorphisms that fix the boundary pointwise
and consider $B\\!\operatorname{Diff}_{\partial}(M)$, which classifies smooth
bundles with trivialised boundary. We prove the following conjecture of
Kontsevich [Kir97, Problem 3.48].

Let $M$ be a compact, connected, orientable $3$-manifold with non-empty
boundary $\partial M$. Then $B\\!\operatorname{Diff}_{\partial}(M)$ is
_homotopy finite_ , i.e. it has the homotopy type of a finite CW complex.

When $M$ is irreducible, this was proven by Hatcher and McCullough [HM97] (we
restate this in 1.1) using that in this setting $B\\!\operatorname{Diff}(M)$
is aspherical. In fact we prove a slightly stronger version of the conjecture,
where we only fix a union of boundary components $F$, such that $\partial
M\setminus F$ consists of spheres and incompressible tori. This theorem
implies that the group cohomology of the topological group
$\operatorname{Diff}_{\partial}(M)$ is finitely generated and that it has
finite cohomological dimension. (Under the simplifying assumption that each
prime factor of $M$ has non-empty boundary, this (co)homological version of
Kontsevich’s conjecture was proven by Nariman [Nar21].) Our results are
effective in the sense that for any given $M$ one can read off a bound on the
cohomological dimension by going through the steps of the proof, but we have
not attempted to give a closed formula as the many case distinctions make this
quite cumbersome.

Recall that every topological 3-manifold admits an essentially unique PL and
smooth structure, by work of Cairns [Cai40] and Whitehead [Whi40], Moise
[Moi52, Moi54] and Bing [Bin54] (see also [Bin59]), and Munkres [Mun59,
Mun60a, Mun60b] and Whitehead [Whi61]. Moreover, a combination of work by Cerf
[Cer59] and Hatcher [Hat83] shows that
$\operatorname{Diff}(M)\simeq\operatorname{PL}(M)\simeq\operatorname{Homeo}(M)$
so our statements hold for all of these groups. (The equivalence of
$\operatorname{PL}(M)$ with the other two relies on a theorem attributed to
Morlet but first proven by Burghelea–Lashof [BL74], see Kirby–Siebenmann
[KS77, Essay V] for more details.)

The analogue of 6.1 for surfaces is true as a consequence of results of
Earle–Schatz [ES70] and Gramain [Gra73], but fails in dimensions $n\geq 6$:
$\pi_{0}\operatorname{Diff}_{\partial}(S^{1}\times D^{n-1})$ is not finitely
generated by work of Hatcher–Wagoner [HW73, $n\geq 7$: Part II, Corollary 5.5,
$n=6$: p230]. Budney–Gabai [BG21] prove that the mapping class group of
$S^{1}\times D^{3}$ relative to the boundary is also not finitely generated in
both the smooth and topological categories. Thus for $n=4$ or $n\geq 6$,
$B\\!\operatorname{Diff}_{\partial}(S^{1}\times D^{n-1})$ is not even of
finite type. (Recall that a space is said to be of _finite type_ if it is
homotopy equivalent to a CW complex with finite $n$-skeleton for each $n$.)

When a 3-manifold $M$ is closed or when the boundary is not fixed,
$B\\!\operatorname{Diff}(M)$ rarely has the homotopy type of a finite CW
complex, but we can ask when it is of finite type. Hatcher–McCullough showed
mapping class groups of 3-manifolds are finitely presented [HM90] so, unlike
in higher dimensions, counterexamples arising from $M$ having infinitely
generated mapping class groups do not exist. Using 6.1, we deduce the
following for all 3-manifolds.

Let $M$ be a compact, orientable $3$-manifold. Then
$B\\!\operatorname{Diff}(M)$ is of finite type.

This completes the picture for all manifolds of dimension $\leq 3$. We briefly
survey for which cases we know that $B\\!\operatorname{Diff}(M)$ is of finite
type. In dimension 1, since
$\operatorname{Diff}(S^{1})\simeq\operatorname{O}(2)$, one knows that
$B\\!\operatorname{Diff}(S^{1})\simeq B\\!\operatorname{O}(2)$ is of finite
type. The corresponding result for closed orientable surfaces combines work of
Smale [Sma59], who showed $B\\!\operatorname{Diff}(S^{2})\simeq
B\\!\operatorname{O}(3)$, and Earle–Eells [EE69] for surfaces of genus $g\geq
1$. In fact, when $g\geq 2$, they prove that $\operatorname{Diff}_{0}(M)$ is
contractible, whence $B\\!\operatorname{Diff}(M)\simeq
B\pi_{0}\operatorname{Diff}(M)=K(\operatorname{Mod}_{g}^{\pm},1)$, where
$\operatorname{Mod}^{\pm}_{g}$ is the (full) mapping class group of the
surface of genus $g$. In this case, finiteness fails because of torsion in
$\operatorname{Mod}^{\pm}_{g}$, but $B\\!\operatorname{Mod}^{\pm}_{g}$ is of
finite type because $\operatorname{Mod}^{\pm}_{g}$ has a finite index subgroup
whose classifying space is homotopy finite, see Ivanov [Iva89, §6.4]. In
higher dimensions $n\neq 4,5,7$, Kupers showed that
$B\\!\operatorname{Diff}(S^{n})$ and
$B\\!\operatorname{Diff}_{\partial}(D^{n})$ are of finite type [Kup19], and in
even dimensions $2n\geq 6$, Bustamante–Krannich–Kupers [BKK23] showed that if
$\pi_{1}(M)$ is finite, then $B\\!\operatorname{Diff}(M)$ is of finite type.

###  1.1. Overview of the proof

We will prove the following generalisation of Kontesevich’s conjecture. Here,
for a union of boundary components $F\subseteq\partial M$, we let
$\operatorname{Diff}_{F}(M)\leq\operatorname{Diff}(M)$ denote the subgroup of
those diffeomorphisms that fix $F$ pointwise. Let $M$ be a compact, connected,
orientable $3$-manifold, and let $\emptyset\neq F\subseteq\partial M$ be a
non-empty union of connected components. Assume that $\partial M\setminus F$
consists of spheres and incompressible tori. Then
$B\\!\operatorname{Diff}_{F}(M)$ is homotopy finite.

Let $\smash{\widehat{M}}$ denote the _spherical closure_ of $M$: the
3-manifold obtained by filling in spherical boundary components. Kneser
[Kne29] showed that every oriented 3-manifold $M$ without 2-spheres in its
boundary admits a connected sum decomposition into oriented irreducible
factors $P_{i}$ that are not homeomorphic to $S^{3}$ and some number of copies
of $S^{1}\times S^{2}$. Milnor [Mil62] then proved this decomposition is
uniquely determined up to reordering. Therefore every oriented 3-manifold $M$
can be written as

$M\cong\left(P_{1}\\#\cdots\\#P_{n}\\#(S^{1}\times
S^{2})^{\\#g}\right)\setminus\\{\amalg_{m}\smash{\mathring{D}}^{3}\\}$

where the $P_{i}$ and $g$ copies of $S^{1}\times S^{2}$ are the prime factors
of its spherical closure $\smash{\widehat{M}}$ and $m$ is the number of
spherical boundary components.

The $S^{1}\times S^{2}$ factors arising in the prime decomposition correspond
to a maximal collection of non-separating spheres in $M$. By allowing cuts
along both separating and non-separating spheres, one can obtain a
decomposition of $\smash{\widehat{M}}$ into only irreducible components, at
the expense of possibly introducing copies of $S^{3}$ (we make the notion of
cutting along a sphere precise in Section 2). While the diffeomorphism classes
(counted with multiplicity) of irreducible factors $P_{i}\neq S^{3}$ appearing
in such a decomposition are uniquely determined, the spheres along which $M$
is cut into irreducible pieces are not, even when considered up to isotopy.
With this in mind, we build a space $\operatorname{Sep}(M)$ that parameterises
all such decompositions of a given 3-manifold $M$. A point in this space
corresponds to a collection of disjointly embedded spheres $\Sigma$ that cut
$M$ into irreducible pieces, which we call a _separating system_ for $M$. A
schematic of a 3-manifold with two examples of separating systems are shown in
Figure 1.

$\Sigma$
---
$\Sigma^{\prime}$
---
$P_{1}\setminus\amalg_{2}\smash{\mathring{D}}^{3}$
---
$P_{2}\setminus\smash{\mathring{D}}^{3}$
---
$P_{3}\setminus\amalg_{3}\smash{\mathring{D}}^{3}$
---
$P_{1}\setminus\amalg_{2}\smash{\mathring{D}}^{3}$
---
$P_{2}\setminus\amalg_{2}\smash{\mathring{D}}^{3}$
---
$P_{3}\setminus\amalg_{5}\smash{\mathring{D}}^{3}$
---

Figure 1. A schematic for the 3-manifold
$M\cong\left(P_{1}\\#P_{2}\\#P_{3}\\#(S^{1}\times
S^{2})^{\\#2}\right)\setminus\smash{\mathring{D}}^{3}$. $M$ has one spherical
boundary component and one genus 2 surface boundary component. In this case
$P_{1}$ and $P_{3}$ are closed, and $P_{2}$ has boundary a genus 2 surface.
Two separating systems $\Sigma$ and $\Sigma^{\prime}$ are shown. Note that
$\Sigma$ is obtained from $\Sigma^{\prime}$ by adding two spheres, thus
$\Sigma^{\prime}\subset\Sigma$.

We then restrict to a subspace
$\operatorname{Sep}^{\nparallel}(M)\subset\operatorname{Sep}(M)$ of separating
systems $\Sigma\subset M$ such that no two spheres in $\Sigma$ are isotopic.
This is a topological poset under inclusion of separating systems, e.g. in
Figure 1, $\Sigma^{\prime}\subset\Sigma$. A key step in the proof of Theorem
6.2 is showing that when $M\not\cong S^{1}\times S^{2}$ the geometric
realisation of the nerve of this poset is contractible.

Suppose $M\not\cong S^{1}\times S^{2}$. Then
$\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|\simeq*$.

Because $\operatorname{Diff}_{F}(M)$ acts on
$\operatorname{Sep}^{\nparallel}(M)$, there is an induced action on
$\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|$, and thus we acquire a
model for the classifying space:

$B\\!\operatorname{Diff}_{F}(M)\simeq\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|/\\!\\!/\operatorname{Diff}_{F}(M)$

where the notation $/\\!\\!/$ denotes the homotopy orbit construction. Since
no two spheres in $\Sigma\in\operatorname{Sep}^{\nparallel}(M)$ are isotopic,
the poset has finite depth, _i.e._ there is a bound on the longest non-
degenerate descending chain. We use this fact and some additional properties
of $\operatorname{Sep}^{\nparallel}_{\bullet}(M)$ to arrive at an inductive
argument to prove Theorem 6.2. The induction is over the number of irreducible
pieces into which a separating system $\Sigma$ cuts $M$, the base case being
when $\smash{\widehat{M}}$ is irreducible. When $M=\smash{\widehat{M}}$ is
itself irreducible (_i.e._ $M$ has no spherical boundary components) this is
due to Hatcher and McCullough [HM97], see Theorem 1.1. In the case where $M$
has spherical boundary, the base case further reduces to the following
theorem, equivalent to the conjecture when $\smash{\widehat{M}}$ is
irreducible and $M$ has exactly one spherical boundary component that is
fixed.

Let $M$ be an irreducible 3-manifold with either empty or incompressible
toroidal boundary, and let $D^{3}\subset\smash{\mathring{M}}$ be an embedded
disk. Then $B\\!\operatorname{Diff}_{D^{3}}(M)$ has the homotopy type of a
finite CW complex.

The proof of this theorem comprises a large portion of our work and breaks
down into many sub-cases. Via a fiber sequence argument, it is equivalent to
showing that $\operatorname{Fr}(M)/\\!\\!/\operatorname{Diff}(M)$ has the
homotopy type of a finite CW complex, where $\operatorname{Fr}(M)$ is the
frame bundle of $M$. In a similar vein to the proof of Theorem 6.2, we cut $M$
up along embedded tori and prove the result for the simpler pieces, via a
combination of the JSJ and geometric decompositions of our irreducible
3-manifold. We use this to reduce the proof to the case of hyperbolic
manifolds, non-Haken Seifert-fibered manifolds, Haken Seifert-fibered
manifolds, and $\operatorname{Sol}$ manifolds that are torus bundles over the
circle. A schematic of the proof and its breakdown into cases is depicted in
Fig. 3.

For hyperbolic or non-Haken Seifert-fibered manifolds, as well as torus
bundles admitting $\operatorname{Sol}$ geometry, the proof of 4.1 relies on
the fact that the (strong) generalised Smale conjecture holds:
$\operatorname{Isom}(M)\hookrightarrow\operatorname{Diff}(M)$ is a homotopy
equivalence. The case of Haken Seifert-fibered manifolds splits up further
into several subcases, depending on the topology of the space of Seifert
fiberings and the presence of singular fibers. The space of Seifert fiberings
of such a manifold is completely determined by work of
Hong–Kalliongis–McCullough–Rubinstein [HKMR12], and, in particular, there are
only six orientable Haken Seifert-fibered manifolds for which there does not
exist a unique Seifert fibering up to isotopy. Apart from these exceptional
cases, we use the fibering to reduce the question of finiteness to that of
diffeomorphisms of the base surface fixing a disk.

For compact manifolds $M$ such that $\partial M$ is either empty or consists
of spheres and incompressible tori, our proof that
$B\\!\operatorname{Diff}(M)$ is of finite type (6.12) follows directly from
6.2 by delooping the fiber sequence

$\operatorname{Diff}_{D^{3}}(M)\longrightarrow\operatorname{Diff}(M)\longrightarrow\operatorname{Emb}(D^{3},\smash{\mathring{M}}).$

For the general case, when $M$ has higher genus or compressible boundary, we
again use the contractibility of
$\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|$ to reduced the claim to
irreducible $M$ with $\partial M\neq\emptyset$. For these, McCullough [McC91]
showed that the mapping class group $\pi_{0}\operatorname{Diff}(M)$ is of
finite type and Hatcher and Ivanov [Hat76, Iva76] showed that
$\operatorname{Diff}_{0}(M)\simeq(S^{1})^{\times b}$ for $b\in\\{0,1,2\\}$, so
$B\\!\operatorname{Diff}(M)$ is of finite type.

###  1.2. History and previous work on the conjecture

Kontsevich formulated his conjecture in the mid 1980s after reading notes of
Thurston on the geometrisation conjecture [Kon24]. The original formulation of
the conjecture was that $B\\!\operatorname{Diff}_{D^{3}}(M)$ was homotopy
finite for closed $M$ – similar to the formulation that appears in our 4.1. In
the 1990s, Kirby and Kontsevich adapted the conjecture to the more general
setting of $M$ with non-empty boundary – the form stated in the Kirby problem
list [Kir97, Problem 3.48]. This is the conjecture we call ‘Kontsevich’s
conjecture’ and prove in 6.1.

As we noted above, Hatcher and McCullough [HM97] proved Kontsevich’s
conjecture when $M$ is an irreducible 3-manifold. In fact, as is the case for
our result, their result holds in a more general setting when only a subset of
the boundary is required to be fixed.

######  Theorem 1.1 ([HM97, Main Theorem]).

Let $M$ be an irreducible compact connected orientable 3-manifold and let $F$
be a non-empty union of components of $\partial M$, including all the
compressible ones. Then $B\\!\operatorname{Diff}_{F}(M)$ has the homotopy type
of an aspherical finite CW complex.

Any $M$ that is irreducible and has non-empty boundary is automatically Haken,
hence in this case it was known from work of Hatcher [Hat83, Hat76] and Ivanov
[Iva76] that the classifying spaces $B\\!\operatorname{Diff}_{\partial}(M)$
are aspherical. Thus, Hatcher–McCullough prove their theorem by showing that
the classifying space of the mapping class group
$\pi_{0}\operatorname{Diff}_{\partial}(M)$ has the homotopy type of a finite
CW complex. They remark that this is not true for closed irreducible
3-manifolds in general, but had been previously shown to be virtually true for
Haken manifolds by McCullough [McC91].

More recently, a homological version of Kontsevich’s conjecture was shown to
be true by Nariman [Nar21], who proved that
$B\\!\operatorname{Diff}_{\partial}(M)$ has finitely many non-zero homology
groups which are all finitely generated, in the case of $M$ being a connected
sum of irreducible 3-manifolds such that each irreducible piece has nontrivial
boundary. In particular, this additional assumption means that the case of an
irreducible manifold with a fixed disk, which we cover in 4.1, is not
necessary in his paper. Nariman builds semi-simplicial spaces of separating
spheres similar to some of the intermediate spaces in our arguments, and we
utilise one of his contractibility results in Section 3.

###  1.3. Background on 3-manifold diffeomorphisms

For prime 3-manifolds, the homotopy type of $\operatorname{Diff}(M)$ has been
studied extensively. We provide a brief, by no means exhaustive, overview of
this history prior to geometrisation here, and refer the reader to Section 4
for more references and recent results. Early work focused on the Smale
conjecture (which asserts that
$\operatorname{Diff}(S^{3})\simeq\operatorname{O}(4)$) and mapping class
groups of 3-manifolds. Initially, Cerf proved that
$\pi_{0}\operatorname{Diff}(S^{3})\cong\pi_{0}(\operatorname{O}(4))\cong\mathbb{Z}/2$
[Cer61], before Hatcher confirmed the Smale conjecture in [Hat83]. Hatcher
[Hat81] also showed $\operatorname{Diff}(S^{1}\times
S^{2})\simeq\Omega\operatorname{SO}(3)\times\operatorname{O}(3)\times\operatorname{O}(2)$,
and Ivanov [Iva79, Iva82] calculated the homotopy type of
$\operatorname{Diff}(M)$ for certain classes of lens spaces. For closed Haken
3-manifolds, Waldhausen [Wal68] proved that
$\pi_{0}\operatorname{Diff}(M)\cong\operatorname{Out}(\pi_{1}(M))$, the outer
automorphism group of $\pi_{1}(M)$ (when $\partial M\neq\emptyset$, one must
also preserve the peripheral structure). The results of Hatcher [Hat76] and
Ivanov [Iva76] mentioned above imply that for Haken 3-manifolds,
$\operatorname{Diff}_{0}(M)$ is homotopy equivalent to a torus of dimension at
most 3. Combined with Waldhausen’s theorem, this largely reduces questions of
homotopy type of $\operatorname{Diff}(M)$ to those of
$\pi_{0}\operatorname{Diff}(M)$.

Mapping class groups of Haken Seifert-fibered 3-manifolds can be computed in
terms of their base orbifold surfaces, while Thurston’s proof of
hyperbolisation for Haken atoroidal 3-manifolds [Thu86] implies that in this
case $\pi_{0}\operatorname{Diff}(M)\cong\operatorname{Isom}(M)$, a finite
group. More generally, results of McCullough [McC91] imply that
$\pi_{0}\operatorname{Diff}(M)$ has finite type for any Haken 3-manifold. For
certain classes of non-Haken Seifert-fibered 3-manifolds such as lens spaces,
analogous results to Waldhausen’s theorem were obtained along with
computations of mapping class groups (e.g. [Asa78, Rub79, Bon83b, RB84, Sco85,
BO86]).

Since the Smale conjecture asserts that
$\operatorname{Diff}(S^{3})\simeq\operatorname{O}(4)=\operatorname{Isom}(\mathbb{S}^{3})$,
following Perelman’s proof of the geometrisation theorem [Per02, Per03b,
Per03a] it is natural to ask whether a similar statement holds for other
geometric 3-manifolds. We refer to the statement that
$\operatorname{Isom}(M)\hookrightarrow\operatorname{Diff}(M)$ is a homotopy
equivalence as the _strong generalised Smale conjecture_ , and to the
statement that
$\operatorname{Isom}_{0}(M)\hookrightarrow\operatorname{Diff}_{0}(M)$ is a
homotopy equivalence as the _weak generalised Smale conjecture_. The strong
form is generally false, even among irreducible geometric 3-manifolds. For
example, the isometry group of any flat metric on the torus $T^{3}$ has
finitely many connected components, while
$\pi_{0}\operatorname{Diff}(T^{3})\cong\operatorname{GL}_{3}(\mathbb{Z})$ by
Waldhausen’s theorem. However, for irreducible geometric manifolds, the weak
form always holds, and furthermore, the homotopy type of
$\operatorname{Diff}_{0}(M)$ is known. We discuss this further, with a
complete list of references, as part of our proof in Section 4.

For reducible 3-manifolds, less is known about the homotopy type of
$\operatorname{Diff}(M)$. Hatcher showed that for $M\cong P_{1}\\#P_{2}$ the
connected sum of two irreducible 3-manifolds, one can reduce the calculation
of the homotopy type of $\operatorname{Diff}(M)$ to the calculation for
$\operatorname{Diff}(P_{1})$ and $\operatorname{Diff}(P_{2})$ [Hat81]. We give
an alternative proof of this result at the end of Section 3. In general,
however, $\operatorname{Diff}_{0}(M)$ need not be homotopy equivalent to a
finite CW complex as in the irreducible case; indeed if $M$ has at least 3
prime factors, McCullough and Kalliongis show that
$\pi_{1}\operatorname{Diff}_{0}(M)$ is infinitely generated [KM96].

In foundational work, Laudenbach [Lau73] established that homotopy implies
isotopy for 2-spheres in any 3-manifold $M$ (assuming the Poincaré
conjecture). He used this to show that the mapping class group of a connected
sum $(S^{1}\times S^{2})^{\\#g}$ is an extension of
$\operatorname{Out}(F_{g})$ by $(\mathbb{Z}/2)^{g}$, where $F_{g}$ is the free
group of rank $g$. More generally, mapping class groups of Haken 3-manifolds
were shown to be finitely presented by Waldhausen [Wal68] and Grasse [Gra89],
and mapping class groups of reducible 3-manifolds were shown to be finitely
presented by Hatcher and McCullough [HM90]. As part of this proof they build a
complex whose vertices are given by isotopy classes of embedded spheres, and
prove this complex is simply connected [HM90, Proposition 2.2]. Note that in
our proof we do not define our complexes of spheres in terms of isotopy
classes – this is because the isotopy class of an embedded sphere in a
3-manifold is not contractible in general.

A program announced by César de Sá and Rourke [CdSR79], then completed by
Hendriks and Laudenbach [HL84], and Hendriks and McCullough [HM87], aimed to
study $\operatorname{Diff}_{\partial}(M)$ for reducible $M$ in terms of the
irreducible prime pieces $P_{i}$ appearing in the decomposition of $M$. They
describe $\operatorname{Diff}_{\partial}(M)$ as fitting into a fiber sequence
with base space an embedding space of punctured 3-disks in $M$ that give a
decomposition into (once-punctured) irreducible prime pieces and 1-handles,
and with fiber the product of boundary-fixing diffeomorphism groups of these
complementary pieces. We do not utilise these fiber sequences during our
proofs, but these papers and Hatcher’s unfinished draft [Hat] provided
inspiration at the outset of this project, and in particular for our upcoming
work which we now outline.

###  1.4. Motivation and upcoming work

One motivation for this work came from previous work of the first two authors
[BB22], in which the homotopy type of the embedding space of a split link is
studied. Dropping the assumption that $M$ be compact, when the 3-manifold $M$
is the complement of a link $\rho$ in $\mathbb{R}^{3}$, our space
$\operatorname{Sep}^{\nparallel}(M)$ agrees with the space
$\operatorname{Sep}(\rho)$ defined in [BB22], and the associated
semisimplicial space $\|\operatorname{Sep}(\rho)_{\bullet}\|$ is shown to be
contractible in [BB22, Theorem A].

Another motivation came from the perspective of modular $\infty$-operads,
based on the third author’s joint work with Barkan on $\infty$-properads
[BS22]. Operadic structures have been used effectively to organise collections
of moduli spaces, for example in Budney’s work on long knots [Bud07],
Costello’s work on Riemann surfaces with boundary [Cos04], and Giansiracusa’s
work on diffeomorphisms of handlebodies [Gia11]. In upcoming work, we will
explain how the moduli spaces of connected $3$-manifolds assemble into a
modular $\infty$-operad whose composition operation is given by connected sum,
and how the contractibility of
$\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|$ implies that this modular
$\infty$-operad is freely generated by irreducible manifolds.

Further to the above, inspired by the program of [CdSR79], [HL84], [HM87], and
[Hat] outlined above, in our upcoming work we will also construct a map on the
level of classifying spaces from $B\\!\operatorname{Diff}(M)$ to
$B\\!\operatorname{Diff}(\amalg_{i}P_{i})$ where the $P_{i}$ are the
irreducible prime factors of $M$. We describe the fiber of this map as a
homotopy colimit of certain framed configuration spaces on the $P_{i}$,
enabling effective computations. We illustrate this by computing the rational
cohomology ring of $B\\!\operatorname{Diff}\left((S^{1}\times
S^{2})^{\\#2}\right)$. We will discuss past work on this topic in greater
detail in this upcoming paper.

###  1.5. Outline

In Section 2 we give background on several topics, set up notation and prove
general results that we quote in some specific settings later. This includes
subsections on 3-manifolds, fiber sequences, the homotopical orbit-stabiliser
theorem, homotopy finite spaces, and topological posets. We then introduce
sphere systems in Section 3 and prove in Theorem 3.20 that the space of non-
parallel separating systems is contractible. We also reprove Hatcher’s result
on connected sums of irreducible manifolds (Theorem 3.21). In Sections 4 and
5, we prove the base case Theorem 4.1 of the conjecture for an irreducible
manifold with a disk removed. Following this, in Section 6 we prove
Kontsevich’s conjecture to be true in Theorem 6.2, and prove
$B\\!\operatorname{Diff}(M)$ is finite type in 6.12.

###  1.6. Acknowledgements

We would like to thanks Oscar Randal-Williams for many insightful
conversations, in particular at the outset of this project. We also extend
thanks to Henry Wilton, specifically for discussions and suggestions relating
to Section 4. We would also like to thank the participants of the Glasgow
reading group on mapping class groups of 3-manifolds, and the Copenhagen
question seminar for conversations related to this paper. Finally we would
like to thank Mark Powell and Nathalie Wahl for helpful comments on a draft.

During this work the first author was supported by ERC grant No. 756444, and
EPSRC Fellowship EP/V043323/1 and EP/V043323/2. The second author was
supported by NSF grant DMS-2401403. The third author was supported by the
Independent Research Fund Denmark (grant no. 10.46540/3103-00099B) and the
Danish National Research Foundation through the ‘Copenhagen Centre for
Geometry and Topology’ (grant no. CPH-GEOTOP-DNRF151).

##  2\. Background on 3-manifolds and homotopy theory

In this section we set up notation concerning $3$-manifolds and notions of
finiteness, recall some results about fiber sequences, topological posets, and
the homotopy orbit construction, and prove a homotopical orbit stabiliser
lemma.

###  2.1. 3-manifolds

All manifolds we consider are smooth. Throughout, $M$ will be a compact,
orientable 3-manifold, possibly with boundary, which we denote $\partial M$.
In general, we will not assume that $M$ is connected and instead specify when
we do. We will use the notation $\smash{\widehat{M}}$ to denote the manifold
obtained from $M$ by filling all boundary 2-spheres with 3-balls, and we call
this the _spherical closure_ of $M$.

We say $M$ is _irreducible_ if every embedded 2-sphere in ${M}$ bounds a
3-ball. $M$ is _reducible_ if it is not irreducible. A connected 3-manifold
$M$ without 2-spheres in its boundary is _prime_ if whenever we decompose
${M}$ as a nontrivial connected sum ${M}=M_{1}\\#M_{2}$, then at least one of
$M_{1}$ or $M_{2}$ is diffeomorphic to $S^{3}$. A well-known theorem due to
Kneser [Kne29] states that every connected oriented 3-manifold without
2-spheres in its boundary admits a connected sum decomposition
$M=P_{1}\\#\cdots\\#P_{n}$ where the _prime factors_ $P_{i}$ are prime and not
diffeomorphic to $S^{3}$. Moreover Milnor showed that the oriented prime
factors appearing in this decomposition are uniquely determined up to
reordering [Mil62]. By convention, $S^{3}$ has 0 prime factors. Note that
$S^{1}\times S^{2}$ is the unique connected 3-manifold without boundary which
is prime but not irreducible. We can therefore refine the decomposition to be
$M\cong P_{1}\\#\cdots\\#P_{n}\\#(S^{1}\times S^{2})^{\\#g}$ where the $P_{i}$
are the prime factors which are also irreducible. The reader is referred to
Hempel [Hem04] for proofs of these and other foundational results on
3-manifolds.

In this paper we often work in the setting where $M$ has spherical boundary
components. In this case we will never talk about prime factors of $M$ but
only discuss prime factors of the spherical closure $\smash{\widehat{M}}$. We
also extend this definition to disconnected manifolds: in this case a prime
factor of $M$ is a prime factor of one of the connected components.

If $\Sigma\subset\smash{\mathring{M}}$ is a codimension $1$ submanifold, such
as a disjoint union of embedded 2-spheres, then $M\setminus\Sigma$ is
diffeomorphic to the interior of a compact 3-manifold with boundary, which we
denote as $M\smash{\mid}\Sigma$. Intuitively, $M\smash{\mid}\Sigma$ is the
manifold obtained from _cutting_ $M$ _along_ $\Sigma$. In particular,
$(M\smash{\mid}\Sigma)^{\circ}=\smash{\mathring{M}}\setminus\Sigma$. Note that
the boundary of $M\smash{\mid}\Sigma$ contains two spheres for each sphere in
$\Sigma$. We let $2\Sigma\subset\partial(M\smash{\mid}\Sigma)$ denote the
newly created boundary components. There is a canonical map $2\Sigma\to\Sigma$
that is a double cover, and for us this will always be a trivial double-cover
because $\Sigma$ will be $2$-sided/coorientable. A schematic is shown in
Figure 2.

$\Sigma$
---
$2\Sigma$
---
$M$
---
$M\smash{\mid}\Sigma$
---
$P_{1}\setminus\amalg_{2}\smash{\mathring{D}}^{3}$
---
$P_{2}\setminus\amalg_{2}\smash{\mathring{D}}^{3}$
---
$P_{3}\setminus\amalg_{5}\smash{\mathring{D}}^{3}$
---
$P_{1}\setminus\amalg_{2}\smash{\mathring{D}}^{3}$
---
$P_{2}\setminus\amalg_{2}\smash{\mathring{D}}^{3}$
---
$P_{3}\setminus\amalg_{5}\smash{\mathring{D}}^{3}$
---

Figure 2. A disjoint union of embedded $2$-spheres $\Sigma\subset M$ and the
compact 3 manifold $M\smash{\mid}\Sigma$. We remark that the irreducible prime
factors of both manifolds are the same.

###  2.2. Fiber sequences of embedding spaces

In this section we review results on fiber sequences of embedding spaces due
to Palais [Pal60] in the closed case and independently by Cerf [Cer61] for
manifolds with with corners and in greater generality (see also Lima [Lim64],
who gave a simplified proof of Palais’ result). The treatment we give here
closely follows the exposition in §2 of Cantero Morán–Randal-Williams
[CMRW17].

For manifolds $M$ and $V$, we let $\operatorname{Emb}(V,M)$ denote the space
of embeddings of $V$ into $M$, equipped with the $C^{\infty}$-topology. The
group of diffeomorphisms of $M$ is denoted $\operatorname{Diff}(M)$. If
$V\subseteq M$ is a submanifold, we denote by $\operatorname{Diff}_{V}(M)$ the
subgroup of $\operatorname{Diff}(M)$ consisting of diffeomorphisms which fix
$V$ pointwise, and by $\operatorname{Diff}(M,V)$ the subgroup of
$\operatorname{Diff}(M)$ consisting of diffeomorphisms which preserve $V$
setwise, _i.e._ $\varphi\in\operatorname{Diff}(M,V)$ if and only if
$\varphi(V)=V$. We denote the subgroup $\operatorname{Diff}_{\partial M}(M)$
by $\operatorname{Diff}_{\partial}(M)$.

Consider a compact manifold $M$ and a compact submanifold
$V\subset\smash{\mathring{M}}$. Both $M$ and $V$ may have boundary. Palais and
Cerf showed that there is a fiber sequence

$\operatorname{Diff}_{V}(M)\hookrightarrow\operatorname{Diff}(M)\rightarrow\operatorname{Emb}(V,\smash{\mathring{M}})$

where the fibration is given by acting with $\operatorname{Diff}(M)$ on the
standard embedding $V\subset M$ [Pal60, Cer61]. To prove this, it in fact
suffices to construct a local section of the action map, which then can be
used to obtain local trivializations. This can be formalised using the concept
of $G$-locally retractile spaces, as explained in [CMRW17, §2.3].

######  Definition 2.1.

A space $X$ with an action of a topological group $G$ is _$G$ -locally
retractile_ if for each $x\in X$ there is a neighbourhood $U\subset X$ and a
map $\xi\colon U\to G$ such that $\xi(u).x=u$ holds for all $u\in U$.

These $G$-locally retractile spaces will prove useful for showing that certain
maps are locally trivial fiber bundles. We will need the following elementary
properties [CMRW17, Lemmas 2.4, 2.5, 2.6].

######  Lemma 2.2.

Let $X$ be a $G$-locally retractile space.

  1. (1)

Any $G$-equivariant map $f\colon Y\to X$ is a locally trivial fiber bundle.

  2. (2)

If a $G$-equivariant map $g\colon X\to Z$ has local sections, then $Z$ is also
$G$-locally retractile.

  3. (3)

If $X$ is locally path connected, then $X$ is also $G_{0}$-locally retractile
for $G_{0}\subset G$ the path component of the identity.

  4. (4)

If $G<H$ is a larger topological group such that the $G$-action on $X$ extends
to an $H$-action, then $X$ is also $H$-locally retractile.

Now Palais’ and Cerf’s theorem can be stated as follows. As before, $M$ and
$V$ are compact manifolds, possibly with boundary.

######  Theorem 2.3.

The embedding space $\operatorname{Emb}(V,\smash{\mathring{M}})$ is
$\operatorname{Diff}_{c}(\smash{\mathring{M}})$-locally retractile.

Here $\operatorname{Diff}_{c}(\smash{\mathring{M}})$ is the group of
diffeomorphisms of $\smash{\mathring{M}}$ with compact support, which we can
equivalently think of as the subgroup of $\operatorname{Diff}_{\partial}(M)$
containing those diffeomorphisms that fix some (unspecified) neighbourhood of
the boundary. Using 2.2(4), we see that
$\operatorname{Emb}(V,\smash{\mathring{M}})$ is also
$\operatorname{Diff}_{\partial}(M)$\- and $\operatorname{Diff}(M)$-locally
retractile. Moreover, $\operatorname{Emb}(V,\smash{\mathring{M}})$ is locally
path connected, so we may restrict to the path component of the identity in
these groups. Cerf not only considers the case where $V$ is embedded in the
interior of $M$, but also allows for more general “face constraints”, see
[CMRW17, §2.2 and Proposition 2.9]. The specific case we will need is the
following:

######  Theorem 2.4.

The embedding space $\operatorname{Emb}(V,\partial M)$ is
$\operatorname{Diff}(M)$-locally retractile. In particular
$\operatorname{Diff}(\partial M)$ is $\operatorname{Diff}(M)$-locally
retractile.

There is a natural right action of $\operatorname{Diff}(V)$ on
$\operatorname{Emb}(V,\smash{\mathring{M}})$ by precomposition. This action is
free and has the effect of reparametrising an embedding, hence the quotient

$\operatorname{Sub}(V,\smash{\mathring{M}}):=\operatorname{Emb}(V,\smash{\mathring{M}})/\operatorname{Diff}(V)$

is the space of _unparametrised embeddings_ of $V$ into $N$. As a set we can
identify this with the set of submanifolds of $\smash{\mathring{M}}$ that
happen to be diffeomorphic to $V$. By Binz–Fisher[BF81] and Michor [Mic80,
13.11], the quotient map
$\operatorname{Emb}(V,\smash{\mathring{M}})\rightarrow\operatorname{Sub}(V,\smash{\mathring{M}})$
is a locally trivial fiber bundles with fiber $\operatorname{Diff}(V)$. (This
is proved only when $V$ is closed but, as pointed out in [CMRW17, Propositon
2.11], the proofs adapt to the setting of manifolds with boundary.) Crucially
for us, this implies that the unparametrised embedding space is
$\operatorname{Diff}(V)$-, and hence $\operatorname{Diff}(M)$-, locally
retractile.

######  Corollary 2.5.

The space of unparametrised embeddings
$\operatorname{Sub}(V,\smash{\mathring{M}})$ is
$\operatorname{Diff}_{\partial}(M)$-locally retractile.

As before this implies that $\operatorname{Sub}(V,\smash{\mathring{M}})$ is
$\operatorname{Diff}(M)$ and also $\operatorname{Diff}_{0}(M)$-locally
retractile. The stabiliser groups of the action by $\operatorname{Diff}(M)$
are precisely $\operatorname{Diff}(M,V)$, so we have a fiber sequence

$\operatorname{Diff}(M,V)\longrightarrow\operatorname{Diff}(M)\longrightarrow\operatorname{Sub}(V,\smash{\mathring{M}}).$

We will also use the following standard results about the contractibility of
collars.

######  Theorem 2.6 ([Cer61, 5.2.1, Corollaire 1]).

The space of collars

$\operatorname{Emb}_{\partial M\times\\{0\\}}(\partial M\times
I,M)=\\{i\in\operatorname{Emb}(\partial M\times[0,1],M)\;|\;i_{|\partial
M\times\\{0\\}}=\mathrm{id}_{\partial M}\\}$

is weakly contractible. It follows that the subgroup inclusion

$\operatorname{Diff}_{U}(M)\to\operatorname{Diff}_{\partial
U}(M\setminus\smash{\mathring{U}})$

is a weak equivalence for any compact codimension $0$ submanifold
$U\subset\smash{\mathring{M}}$.

###  2.3. The homotopy orbit construction

Given a topological group $G$, we say that a $G$-space $X$ is principal if
$X\to X/G$ is a $G$-principal bundle. For each $G$, we fix a weakly
contractible $G$-principal space $EG$. Then the _homotopy orbit construction_
of a $G$-space $X$ is defined as

$X/\\!\\!/G:=(X\times EG)/G$

and we also refer to it as the _homotopy quotient_ of $X$ by $G$. This fits
into a fiber sequence

$X\longrightarrow X/\\!\\!/G\longrightarrow*/\\!\\!/G=BG.$

######  Definition 2.7.

A _principal short exact sequence_ is a short exact sequence of topological
groups

$1\to G_{1}\to G_{2}\to G_{3}\to 1$

such that the map $G_{2}\to G_{3}$ is a $G_{1}$-principal bundle.
Equivalently, we can say that $G_{3}$ is $G_{2}$-locally retractile with
stabiliser $G_{1}$.

The following lemma allows us to combine principal short exact sequences of
groups and equivariant fiber sequences to obtain a homotopy fiber sequence of
homotopy quotients. See Basualdo Bonatto [BB23, Corollary 2.11] for a detailed
proof.

######  Lemma 2.8.

Suppose we have a principal short exact sequence of topological groups $G_{i}$
as above and let $S_{i}$ be a $G_{i}$-space such that there exists a fiber
sequence of equivariant maps $S_{1}\to S_{2}\to S_{3}$. Then the maps on
quotients form a homotopy fiber sequence

$S_{1}/\\!\\!/G_{1}\longrightarrow S_{2}/\\!\\!/G_{2}\longrightarrow
S_{3}/\\!\\!/G_{3}.$

The following fact about principal bundles seems to be well-known, but we
provide a proof as we could not find a reference.

######  Lemma 2.9.

If $H\leq G$ is a topological subgroup such that $G\to G/H$ is an
$H$-principal bundle and $X$ is a principal $G$-space, then $X$ is also a
principal $H$-space.

###### Proof.

First we prove that $X$ is a principal $G$-space if and only if we can find a
family of subsets $\\{U_{i}\subset X\\}_{i\in I}$ such that the action maps
$U_{i}\times G\hookrightarrow X$ are open embeddings and jointly cover $X$. If
$p\colon X\to X/G$ is a $G$-principal bundle, then there is an open covering
$\\{U_{i}^{\prime}\\}_{i\in I}$ of $X/G$ such that there are equivariant
homeomorphisms $\alpha_{i}\colon U_{i}^{\prime}\times G\cong
p^{-1}(U_{i}^{\prime})$ over $U_{i}^{\prime}$. We can then set
$U_{i}=\alpha_{i}(U_{i}^{\prime}\times\\{1\\})$. Conversely, if we have
$\\{U_{i}\subset X\\}_{i\in I}$, then $U_{i}^{\prime}:=p(U_{i})$ is open
because it is the image of the open map $U_{i}\times G\hookrightarrow X\to
X/G$, and $p$ will be trivial over these $U_{i}^{\prime}$.

Applying this to $G\to G/H$ we find $\\{V_{j}\subset G\\}_{j\in J}$ such that
$V_{j}\times H\hookrightarrow G$ are open embeddings covering $G$. Now we can
set $W_{i,j}$ to be the image of the embedding $U_{i}\times V_{j}\subset
U_{i}\times G\hookrightarrow X$ and these will be such that

$W_{i,j}\times H\cong U_{i}\times V_{j}\times H\hookrightarrow U_{i}\times
G\hookrightarrow X$

are open embeddings covering $X$. Therefore $X\to X/H$ is an $H$-principal
bundle. ∎

Another useful source of homotopy fiber sequences will be the _homotopical
orbit stabiliser lemma_. First we consider the case of a transitive action.

######  Lemma 2.10.

If $X$ is a $G$-locally retractile space and the $G$ action on $X$ is
transitive, then

$X/\\!\\!/G\simeq B\mathrm{Stab}_{G}(x)$

for $\mathrm{Stab}_{G}(x)<G$ the stabiliser group of some $x\in X$. Moreover,
there is a homotopy fiber sequence

$X\longrightarrow B\mathrm{Stab}_{G}(x)\longrightarrow BG.$

###### Proof.

For each orbit the map $G\to X$ defined by acting on any point $x\in X$ is a
quotient map, so since the action is transitive we can identify $X$ with the
left quotient $H\setminus G$ by the stabiliser group $H=\mathrm{Stab}_{G}(x)$.
We now compute

$X/\\!\\!/G=((H\setminus G)\times EG)/G\cong H\setminus((G\times EG)/G)\cong
H\setminus EG$

Because $H=\mathrm{Stab}_{G}(x)<G$ is such that $G\to H\setminus G$ is a
principal $H$-bundle and $EG\to G\setminus EG=BG$ is also a principal bundle,
2.9 says that $EG\to H\setminus EG$ is a principal bundle. Hence we may define
$EH:=EG$ such that $X\simeq H\setminus EG=BH=B\mathrm{Stab}_{G}(x)$. The
homotopy fiber sequence is obtained from $X\to X/\\!\\!/G\to BG$ by rewriting
the middle term. ∎

######  Lemma 2.11 (Homotopical orbit-stabiliser lemma).

Let $X$ be a $G$-locally retractile space and assume that $X$ is locally path-
connected. Then there is an equivalence

$X/\\!\\!/G\simeq\coprod_{[x]\in X/G}B\mathrm{Stab}_{G}(x)$

where the coproduct runs over orbit representatives $[x]\in X/G$.

###### Proof.

Because $X$ is $G$-locally retractile and locally path-connected, each orbit
of the $G$-action is a union of path components. We further have
$X/G=\pi_{0}(X)/\pi_{0}(G)=\pi_{0}(X/\\!\\!/G)$. Writing $X=\coprod_{i}X_{i}$
as the disjoint union of its orbits we have
$X/\\!\\!/G=\coprod_{i}X_{i}/\\!\\!/G$ and the claim follows from 2.10. ∎

######  Lemma 2.12.

The homotopy orbits functor $(-)/\\!\\!/G\colon
G\mathrm{-Top}\longrightarrow\mathrm{Top}$

  1. (1)

commutes with all colimits,

  2. (2)

commutes with products with trivial $G$-spaces: if $X$ is any $G$-space and
$Y$ has the trivial $G$-action then $(X\times Y)/\\!\\!/G\cong
X/\\!\\!/G\times Y$,

  3. (3)

preserves weak equivalences,

  4. (4)

preserves homotopy pushout squares, and

  5. (5)

preserves (fat) geometric realisations.

###### Proof.

All of these properties are standard, but we recall them here for the
convenience of the reader. The homotopy orbits functor is the composite of the
functor $(-)\times EG$ and the functor $(-)/G$. In a convenient category of
topological spaces the cartesian product preserves all colimits in each
variable, and the second functor preserves colimits because colimits commute,
hence (1) follows. For (2) we note that $(-)\times Y$ certainly commutes with
the first functor and it also commutes with the second functor because
$(-)\times Y$ preserves all colimits and hence in particular preserves taking
orbits by $G$. Item (3) follows form the long exact sequence for the fiber
sequences $X\to X/\\!\\!/G\to BG$. Because we now know that $(-)/\\!\\!/G$
preserves weak equivalences, to show (4) it will suffice to prove that the
functor preserves the standard construction of the homotopy pushout as

$X\cup^{h}_{Y}Z=X\cup_{Y}(Y\times[0,1])\cup_{Y}Z.$

This indeed is preserved because we already noted that $(-)/\\!\\!/G$
preserves pushouts and products with trivial $G$-spaces. Finally, for (5), let
$X_{\bullet}$ be semi-simplicial $G$-space. Then there is a homeomorphism
$\|X_{\bullet}/\\!\\!/G\|\cong\|X_{\bullet}\|/\\!\\!/G$ because the fat
geometric realisation is a colimit of products with the topological
$n$-simplices, which are trivial $G$-spaces. ∎

###  2.4. Homotopy finite spaces

We make precise the notion of finiteness that are used throughout the paper
and prove the fiber sequence lemma that is our most-used tool for showing
finiteness.

######  Definition 2.13.

We say that a topological space $X$ is _homotopy finite_ if there is a CW
complex $C$ with finitely many cells and a weak equivalence $C\simeq X$.

######  Remark 2.14.

If $X$ is homotopy finite, then $\pi_{1}(X)$ is finitely presented, $H_{n}(X)$
is finitely generated for all $n\geq 0$, and there is a $d\in\mathbb{N}$ such
that $H_{n}(X)=0$ for $n\geq d$. A discrete group is $G$ called _type_ $F$ if
$BG$ is homotopy finite. In this case $G$ must be finitely presented and have
finite cohomological dimension.

Homotopy finite spaces are closed under finite disjoint union and homotopy
pushouts. They are also closed under extensions in the following sense.

######  Lemma 2.15.

Let $p\colon E\to B$ be a map such that $B$ is homotopy finite and the
homotopy fiber of $p$ at any $b\in B$ is homotopy finite. Then $E$ is homotopy
finite.

###### Proof.

Without loss of generality $B$ itself is a finite CW complex and $p$ is a
fibration. We proceed by induction on the dimension of $B$ and the number of
cells in $B$. If $B$ is a disk, then $E$ is equivalent to the homotopy fiber
and the claim follows. Otherwise, we can find an open decomposition $B=U\cup
V$ where $U$, $V$, and $U\cap V$ are all equivalent to finite CW complexes
that have either fewer cells than $B$ or a lower dimension. (Let, for
instance, $U\subset B$ be an open cell, $x\in U$ and $V=B\setminus\\{x\\}$.)
By induction hypothesis, we see that $p^{-1}(U)$, $p^{-1}(V)$ and
$p^{-1}(U\cap V)$ must all be equivalent to finite CW complexes, as they are
fibrations over a homotopy finite space and have homotopy finite fibers. Now
$E=p^{-1}(U)\cup_{p^{-1}(U\cap V)}p^{-1}(V)$ is a homotopy pushout of homotopy
finite spaces and as such it is homotopy finite as claimed. ∎

###  2.5. Topological posets

Recall that a topological poset $(P,\leq)$ is a poset together with a topology
on the underlying set $P$. A map of topological posets $(P,\leq)\to(Q,\leq)$
is a continuous map $f\colon P\to Q$ such that $x\leq y\Rightarrow f(x)\leq
f(y)$. We let $N(P)$ denote the nerve, _i.e._ the simplicial space defined by

$N_{n}(P)=\\{p_{0}\leq\dots\leq p_{n}\\}\subset P^{n+1}$

and we define the classifying space $BP:=\|N_{\bullet}(P)\|$ to be the fat
geometric realisation of the nerve. We use the fat geometric realisation as
opposed to the thin one as we will apply tools from Galatius–Randal-Williams
[GRW18] and Ebert–Randal-Williams [ERW19], who work with semi-simplicial
spaces. We will often use the following elementary technique for constructing
equivalences of posets, see [ERW19, Lemma 3.4] for a proof.

######  Lemma 2.16.

If $f,g:P\to Q$ are maps of topological posets such that $f(x)\geq g(x)$ for
all $x\in P$, then the induced maps on classifying spaces $B(f),B(g):BP\to BQ$
are homotopic.

The topological posets we encounter usually arise from a specific
construction. Fix $X$ a Hausdorff topological space and let $\perp$ be a
symmetric and anti-reflexive relation on $X$ such that the subspace

$\\{(x,y)\in X^{2}\mid x\perp y\\}\subset X^{2}$

defining $\perp$ is an open subset. We say that $x$ and $y$ are _orthogonal_
when $x\perp y$ and that an $n$-tuple of points $(x_{1},\dots,x_{n})\in X^{n}$
is _orthogonal_ if and only if $x_{i}\perp x_{j}\,\forall i\neq j$. As an
example, we will mainly be interested in the setting where
$X=\operatorname{Sub}(S^{2},\smash{\mathring{M}})$ and $S_{0}\perp S_{1}$ if
and only if $S_{0}\cap S_{1}=\emptyset$. In particular, $\perp$ will usually
not be transitive.

######  Definition 2.17.

Let $S(X,\perp)$ denote the (discrete) simplicial complex where vertices are
points in $X$ and $n+1$ vertices span an $n$-simplex if and only if they are
orthogonal. Let $C(X,\perp)$ denote the space of non-empty unordered
orthogonal configurations, _i.e._ the subspace of $\coprod_{k\geq
0}X^{k}/\operatorname{Sym}_{k}$ of those $\underline{x}=[x_{1},\dots,x_{k}]$
such that $x_{i}\perp x_{j}$ holds for all $i\neq j$. We make $C(X,\perp)$
into a topological poset with respect to the subset ordering $\subseteq$. For
any orthogonal $\underline{y}=(y_{1},\dots,y_{n})$ we define
$S(X,\perp)^{\perp\underline{y}}\subset S(X,\perp)$ as the full subcomplex on
those vertices $x$ that are orthogonal to all $y_{i}$. Similarly, we define
$C(X,\perp)^{\perp\underline{y}}\subset C(X,\perp)$ as the topological
subposet containing those configurations that are orthogonal to
$\underline{y}$.

The goal of this section is to prove the following useful tool, which is a
variation of a technique by Galatius–Randal-Williams [GRW18].

######  Proposition 2.18.

Suppose that $X$ is a Hausdorff space with an open, symmetric, and anti-
reflexive relation $\perp$ as above. Suppose further that

  1. (1)

the simplicial complex $S(X,\perp)$ is $n$-connected, and

  2. (2)

for all orthogonal tuples $\underline{y}=(y_{1},\dots,y_{q})$ the simplicial
complex $S(X,\perp)^{\perp\underline{y}}$ is $(n-1)$-connected.

Then the classifying space $B(C(X,\perp))$ of the poset $C(X,\perp)$ is
$n$-connected.

###### Proof.

Write $C:=C(X,\perp)$ and $S:=S(X,\perp)$, and let $C^{\delta}$ be $C$
equipped with the discrete topology. First, note that $C$ is the poset of
simplices of $S$ and hence $B(C^{\delta})=|N_{\bullet}C^{\delta}|$ is the
barycentric subdivision of $|S|$. In particular they are homeomorphic, and
similarly
$|N_{\bullet}(C^{\perp\underline{y}})^{\delta}|\cong|S^{\perp\underline{y}}|$.
From now on all geometric realisations are “fat”, _i.e._ we do not quotient by
the relations imposed by the degeneracy maps. For simplicial sets the fat and
thin geometric realisations are always equivalent [ERW19, Lemma 1.7].

We define the following bisimplicial subspace of
$(N_{\bullet}C)^{\delta}\times N_{\bullet}C$:

$D_{p,q}:=\\{((\underline{x}^{0}\subseteq\dots\subseteq\underline{x}^{p}),(\underline{y}^{0}\subseteq\dots\subseteq\underline{y}^{q}))\in
N_{p}C^{\delta}\times N_{q}C\;|\;\underline{x}^{p}\perp\underline{y}^{q}\\}.$

Note that by $\underline{x}^{p}\perp\underline{y}^{q}$ we mean that
$\underline{x}^{i}\perp\underline{y}^{j}$ for all $i$ and $j$. We will also
need the diagonal of $D_{\bullet,\bullet}$, which is the simplicial space
given by $(\Delta D)_{p}:=D_{p,p}$. Consider the diagram

${{\|N_{\bullet}C^{\delta}\|}}$${{\|(\Delta
D)_{\bullet}\|}}$${{\|\|D_{\bullet,\bullet}\|\|}}$${{\|N_{\bullet}C\|}}$$\scriptstyle{\iota}$$\scriptstyle{\alpha}$$\scriptstyle{\simeq}$$\scriptstyle{\varepsilon}$$\scriptstyle{\gamma}$

where the map $\iota$ is the canonical map from the discretisation, $\gamma$
projects to the $\underline{x}^{i}$ and $\varepsilon$ projects to the
$\underline{y}^{j}$. The map $\alpha$ is the inclusion of the diagonal and it
is a weak equivalence by [ERW19, Theorem 7.1].

Claim: The outside diagram commutes up to homotopy, _i.e._
$\iota\circ\gamma\circ\alpha\simeq\varepsilon\circ\alpha$.

Proof of claim: Consider the topological subposet $Q\subset C^{\delta}\times
C$ containing those tuples $(\underline{x},\underline{y})$ satsifying
$\underline{x}\perp\underline{y}$. The nerve of $Q$ is isomorphic to the
diagonal $\Delta D$: the homeomorphism $N_{p}Q\cong(\Delta D)_{p}=D_{p,p}$ is
given by the tautological map

$((\underline{x}^{0},\underline{y}^{0})\subseteq\dots\subseteq(\underline{x}^{p},\underline{y}^{p}))\longmapsto((\underline{x}^{0}\subseteq\dots\subseteq\underline{x}^{p}),(\underline{y}^{0}\subseteq\dots\subseteq\underline{y}^{p})).$

The two maps $\iota\circ\gamma\circ\alpha$ and $\varepsilon\circ\alpha$ are
obtained as the realisation of the two poset-maps
$\mathrm{pr}_{1},\mathrm{pr}_{2}:Q\to C$ defined by
$\mathrm{pr}_{1}(\underline{x},\underline{y}):=\underline{x}$ and
$\mathrm{pr}_{2}(\underline{x},\underline{y}):=\underline{y}$, respectively.
We can also define a third map $q:Q\to C$ by
$q(\underline{x},\underline{y})=\underline{x}\cup\underline{y}$, which is
well-defined since $\underline{x}\perp\underline{y}$ in $Q$, and therefore it
follows that $\underline{x}$ and $\underline{y}$ are disjoint as subsets of
$X$ ($\perp$ is anti-reflexive). Now, since
$\underline{x}\subseteq(\underline{x}\cup\underline{y})\supseteq\underline{y}$,
Lemma 2.16 implies that

$\iota\circ\gamma\circ\alpha=B(\mathrm{pr}_{1})\simeq B(q)\simeq
B(\mathrm{pr}_{2})=\varepsilon\circ\alpha,$

which proves the claim.

For fixed $q$ the projection map $\varepsilon_{q}:\|D_{\bullet,q}\|\to N_{q}C$
is a micro Serre fibration by [GRW18, Proposition 2.8]. (To apply this
proposition observe that $N_{q}C^{\perp\underline{x}}\subseteq N_{q}C$ is
open.) The fiber $\varepsilon^{-1}(\beta)$ of this at some
$\beta=(\underline{y}^{0}\subseteq\dots\subseteq\underline{y}^{q})\in N_{q}C$
is isomorphic to the simplicial set
$N_{\bullet}(C^{\perp\underline{y}^{q}})^{\delta}$. The realisation of this is
homeomorphic to
$B((C^{\perp\underline{y}^{q}})^{\delta})\cong\|S^{\perp\underline{y}^{q}}\|$,
which we assumed to be $(n-1)$-connected. Hence, by [GRW18, Proposition 2.6]
the map $\varepsilon_{q}:\|D_{\bullet,q}\|\to N_{q}C$ is $n$-connected.
Moreover, combining this with [GRW18, Proposition 2.7] we see that
$\varepsilon:\|\|D_{\bullet,\bullet}\|\|\to\|N_{\bullet}C\|$ is $n$-connected.

We have hence shown that the map $\varepsilon\circ\alpha:\|(\Delta
D)_{\bullet}\|\to\|N_{\bullet}C\|$ induces a surjection on $\pi_{k}$ for
$k\leq n$. But this map is homotopic to $\iota\circ\gamma\circ\alpha$, which
factors through $\|N_{\bullet}C^{\delta}\|\cong\|S^{\delta}\|$, which we
assumed to be $n$-connected. It follows that $\pi_{k}\|N_{\bullet}C\|$ is $0$
for $k\leq n$, _i.e._ $\|N_{\bullet}C\|$ is $n$-connected. ∎

The following is a special case of Quillen’s Theorem A for topological
categories [ERW19, Theorem 4.8].

######  Corollary 2.19.

Let $(D,\leq)$ be a topological poset and $C\subset D$ a sub-poset such that:

  1. (1)

For all $d\in D$ the sub-poset $C_{d\leq}:=\\{c\in C\;|\;d\leq c\\}$ has a
contractible classifying space.

  2. (2)

$D$ is left-fibrant, i.e. the map

$\\{(d_{0},d_{1})\in D^{2}\;|\;d_{0}\leq d_{1}\\}\longrightarrow
D,\qquad(d_{0},d_{1})\longmapsto d_{0}$

is a Serre fibration.

  3. (3)

$C\subset D$ is a union of path components.

Then the inclusion $BC\to BD$ is a weak equivalence.

###### Proof.

We need to verify conditions (i) and (v) of [ERW19, Theorem 4.8]. Our
condition (1) is exactly condition (i), since
$N_{\bullet}(C_{d\leq})=(d/F)_{\bullet}$ for $F:C\to D$ the inclusion. To
check condition (v) we may use [ERW19, Lemma 4.6.(vi)]. We need to show that
$C$ and $D$ are left-fibrant, $C\to D$ is a Serre fibration and $D$ is unital.
All posets are unital as $c\leq c$ provides identity morphisms. $C\to D$ is a
Serre fibration since it is the inclusion of a union of path components.
Condition (2) exactly tells us that $D$ is left-fibrant and it follows that
$C$ is left-fibrant by restricting to path components. ∎

##  3\. Systems of spheres

In this section we introduce complexes and semi-simplicial spaces of embedded
unparametrised spheres which cut a 3-manifold into irreducible pieces. In each
subsection we introduce a variant with more constraints, and show it is
contractible, using contractibility of the previously considered complexes. As
our starting point we sketch the proof of a contractibility result due to
Nariman [Nar21], adjusted to our requirements. These contractibility results
are a key input to the proof of our main theorem. We end the section by
reproving a theorem of Hatcher, in the setting where $M$ is a connected sum of
two irreducible manifolds.

###  3.1. Single spheres

First we consider a complex where vertices are single spheres.

######  Definition 3.1.

An embedded unparametrised sphere $S\subset M$ is _essential_ if it represents
a non-trivial class in $\pi_{2}({M},p)$ for some $p\in M$. We call an
essential sphere $S\subset M$ _reducing_ if each connected component of the
spherical closure $\smash{\widehat{M\smash{\mid}S}}$ of $M\smash{\mid}S$ has
fewer prime factors than the spherical closure of $M$.

######  Remark 3.2.

It follows from the Poincaré conjecture that an embedded sphere $S\subset M$
is essential if and only if it does not bound $B^{3}\subset M$. Therefore, a
sphere $S\subset M$ is reducing if and only if $S$ is essential in the
spherical closure $\smash{\widehat{M}}$.

######  Definition 3.3.

The _sphere complex_ $\mathcal{S}(M)$ of a compact $3$-manifold $M$ is the
(discrete) simplicial complex where vertices are reducing spheres in $M$ and
$p$-simplices are spanned by finite sets of vertices $\\{S_{0},\dots,S_{p}\\}$
such that the $S_{i}$ are pairwise disjoint.

Hatcher was the first to consider such a complex when he introduced and
studied the complex of isotopy classes of essential spheres in $S^{1}\times
S^{2}$ in [Hat95]. He proved this complex was contractible [Hat95, Theorem
2.1] using a surgery argument that has since been adapted by Mann–Nariman
[MN20], Nariman [Nar21, Proposition 2.7], and the first and second authors
[BB22, Theorem 3.7] among others, in various settings where embedded spheres
satisfying certain additional properties are no longer considered up to
isotopy. Our next result is another such adaptation.

######  Proposition 3.4.

For every compact $3$-manifold $M$, the complex $\mathcal{S}(M)$ is either
empty or contractible.

###### Proof.

The argument from the proof of Proposition 2.7 in [Nar21] works with some
minor changes, since we consider reducing spheres but Nariman considers the
larger class of essential spheres that are not boundary parallel. We sketch
the proof with modifications here.

Suppose $\mathcal{S}(M)$ is nonempty and let $\kappa\colon
M\rightarrow\smash{\widehat{M}}$ be the natural inclusion which caps all
2-sphere boundary components of $M$. We will show that any continuous map
$f\colon S^{k}\rightarrow\mathcal{S}(M)$ is nulhomotopic. We may assume that
$f$ is simplicial with respect to some triangulation $K$ of $S^{k}$. As
transverse embeddings are residual [Hir76, Theorem 2.1], after a simplicial
homotopy we may assume that the images of the vertices of $K$ under $f$ are
pairwise transverse. Choose a vertex $v\in K$. Then $f(v)$ is a reducing
sphere in $M$ which is transverse to all other spheres in $f(K^{(0)})$.

We now perform a sequence of surgeries on the spheres in $f(K^{(0)})$ to
homotope the image of $f$ into the star of $f(v)$. For every other vertex
$w\in K$ the intersection $f(v)\cap f(w)$ is a union of circles. Let $N$ be
total number of intersection circles in $f(v)$, i.e. the number of circles in
$f(v)\cap f(w)$, summed over all $w\neq v$. Consider all such circles, take a
maximal disjoint family, and consider an innermost circle $C$ in this family.
(Here, innermost means that $C$ bounds a disk in $f(v)$ containing no other
circle in the maximal collection.) Then $C$ is an intersection between $f(v)$
and a reducing sphere $f(w)$ for $w\in K$. For the surgery step, we need to
show that if we surger $f(w)$ at $C$ with respect to the transverse sphere
$f(v)$ to obtain two spheres $S_{0}$ and $S_{1}$ (see [Nar21, Figure 2]) then
at least one of $S_{0}$ or $S_{1}$ is still reducing. Since $f(w)$ and $f(v)$
lie in the interior of $M$, we may perform the surgery entirely in
$\smash{\mathring{M}}$. Passing to $\smash{\widehat{M}}$ via $\kappa$, we may
perform the same surgery away from the added 3-balls. By assumption, $f(w)$
and $f(v)$ are reducing, hence essential in $\smash{\widehat{M}}$ by Remark
3.2. In $\pi_{2}(\smash{\widehat{M}},p)$ we have $[S_{0}]+[S_{1}]=[f(w)]$. As
$[f(w)]$ is nontrivial, at least one of $[S_{0}]$ or $[S_{1}]$ is nontrivial.
Thus we can pick $i\in\\{0,1\\}$ such that the sphere $S_{i}$ is still
essential in $\smash{\widehat{M}}$ and therefore reducing in $M$.

The map $f$ is now replaced with a homotopic map $f^{\prime}$ that agrees with
$f$ on all vertices except $w$, but sends $w$ to $S_{i}$ instead of $f(w)$.
(The original map $f$ sends all vertices in $\mathrm{link}_{K}(w)$ to spheres
disjoint from $S_{i}$ since $C$ was chosen to be innermost, and after a small
translation $S_{i}$ is also disjoint from $f(w)$, so $f^{\prime}$ sends all of
$\mathrm{star}_{K}(w)$ to the star of $f(w)$ in $\mathcal{S}(M)$. This is what
allows us to find this homotopy – see [BB22, Figure 4] for a schematic.) Since
$f^{\prime}(w)=S_{i}$ no longer intersects $f(v)$ at $C$, for this new map
$f^{\prime}$ the number of intersections $N$ has been reduced by one, so by
induction we can reduce $N$ to zero i.e. until $f$ sends all of $K$ into the
star of $f(v)$. Hence, $f$ is nullhomotopic and $\mathcal{S}(M)$ is
contractible as required. ∎

###  3.2. Systems of spheres

We now consider posets of _systems_ of spheres which satisfy certain
properties.

######  Definition 3.5.

A _cut system_ for a $3$-manifold $M$ is a (possibly empty) collection of
spheres $\Sigma\in\operatorname{Sub}(\amalg_{k}S^{2},M)$, for
$k\in\mathbb{N}$, such that every sphere in $\Sigma$ is reducing.

We let $\operatorname{Cut}(M)$ denote the space of cut systems, topologised as
a subspace

$\operatorname{Cut}(M)\subset\coprod_{k\geq
0}\operatorname{Sub}(\amalg_{k}S^{2},M).$

This is a topological poset with respect to the partial order $\subseteq$
defined by inclusion.

Note that $\Sigma$ is allowed to be empty, which occurs when $k=0$. The empty
cut system is minimal with respect to the partial order. If
$\Sigma\neq\emptyset\in\operatorname{Cut}(M)$ then each component of
$\smash{\widehat{M\smash{\mid}\Sigma}}$ has strictly fewer prime factors than
$\smash{\widehat{M}}$. Let $\operatorname{Cut}^{\neq\emptyset}(M)$ be the
subspace of $\operatorname{Cut}(M)$ consisting of non-empty cut systems.

######  Definition 3.6.

Let $\operatorname{Cut}_{\bullet}(M)$ and
$\operatorname{Cut}^{\neq\emptyset}_{\bullet}(M)$ denote the simplicial spaces
obtained as the nerve of the poset $(\operatorname{Cut}(M),\subseteq)$ and
$(\operatorname{Cut}^{\neq\emptyset}(M),\subseteq)$ respectively. Concretely,
the space of $p$-simplices in each case is

$\operatorname{Cut}_{p}(M)=\\{\Sigma_{0}\subseteq\dots\subseteq\Sigma_{p}\in\operatorname{Cut}(M)\\}\subset\operatorname{Cut}(M)^{\times
p+1},\text{ and }$
$\operatorname{Cut}^{\neq\emptyset}_{p}(M)=\\{\emptyset\neq\Sigma_{0}\subseteq\dots\subseteq\Sigma_{p}\in\operatorname{Cut}(M)\\}\subset\operatorname{Cut}^{\neq\emptyset}(M)^{\times
p+1}.$

Since $\Sigma=\emptyset$ is minimal with respect to the partial order, it acts
as a cone point for the realisation of $\operatorname{Cut}(M)$, and therefore
$\|\operatorname{Cut}_{\bullet}(M)\|\simeq*$. The set of reducing spheres in
$M$ is empty if and only if $\smash{\widehat{M}}$ is irreducible, so in this
case $\operatorname{Cut}(M)=\\{\emptyset\\}$, and
$\operatorname{Cut}^{\neq\emptyset}(M)$ is empty.

######  Proposition 3.7.

Suppose $\smash{\widehat{M}}$ is not irreducible. Then the classifying space
of the poset of non-empty cut systems is contractible, i.e.
$\|\operatorname{Cut}^{\neq\emptyset}_{\bullet}(M)\|\simeq*$.

###### Proof.

$\operatorname{Cut}^{\neq\emptyset}_{\bullet}(M)$ is non-empty, since we
assumed that $\smash{\widehat{M}}$ was not irreducible. Following the
framework of Section 2.5, let $X$ be the subspace of
$\operatorname{Sub}(S^{2},M)$ consisting of reducing spheres, and $\perp$ be
the symmetric and anti-reflexive relation on $X$ given by $S_{0}\perp S_{1}$
iff $S_{0}\cap S_{1}=\emptyset$. Then
$\operatorname{Cut}^{\neq\emptyset}_{\bullet}(M)$ is the nerve of the
associated topological poset $C(X,\perp)$ from Definition 2.17. We now apply
the adaptation of the discretisation technique of Galatius and Randal-Williams
[GRW18] given in Proposition 2.18. First note that $S(X,\perp)$ (Definition
2.17) is contractible since it is precisely the complex $\mathcal{S}(M)$,
which is contractible by Theorem 3.4. Hence condition $(1)$ of Proposition
2.18 is satisfied for all $n$. For condition $(2)$ we fix a cut system
$\Sigma\in\operatorname{Cut}^{\neq\emptyset}_{\bullet}(M)$ (which will
function as $\underline{y}$ in the proposition) and consider the simplicial
complex $S(X,\perp)^{\perp\Sigma}$. This complex is contractible – for any map
$f:S^{k}\to S(X,\perp)^{\perp\Sigma}$, there exists a sphere $S$ disjoint from
all vertices in $\operatorname{im}(f)$ (for example, take $S$ to be a parallel
copy of a sphere in $\Sigma$). $S$ is therefore a vertex in
$S(X,\perp)^{\perp\Sigma}$ which acts as a cone point for
$\operatorname{im}(f)$ and thus $S(X,\perp)^{\perp\Sigma}\simeq*$. Therefore
$(2)$ is also satisfied for all $n$. Applying Proposition 2.18 gives that
$\|\operatorname{Cut}^{\neq\emptyset}_{\bullet}(M)\|$ is $n$-connected for all
$n$, and thus contractible. ∎

###  3.3. Separating systems

Now we are ready to restrict to cut systems that are _separating_.

######  Definition 3.8.

A cut system $\Sigma$ on a $3$-manifold $M$ is called _separating_ if each
component of $\smash{\widehat{M\smash{\mid}\Sigma}}$ is irreducible. Let
$\operatorname{Sep}(M)\subseteq\operatorname{Cut}(M)$ denote the subspace of
separating cut systems.

For $q\geq 1$ we say that a cut system $\Sigma$ is _$q$ -separating_ if each
component of $\smash{\widehat{M\smash{\mid}\Sigma}}$ has at most $q$ prime
factors. Let
$\operatorname{Cut}^{\mathrm{pf}\leq{q}}(M)\subseteq\operatorname{Cut}(M)$
denote the subspace of $q$-separating cut systems.

Then $\operatorname{Sep}(M)$ and $\operatorname{Cut}^{\mathrm{pf}\leq{q}}(M)$
are posets with respect to $\subseteq$ and we denote their nerves by
$\operatorname{Sep}_{\bullet}(M)\subset\operatorname{Cut}_{\bullet}(M)$ and
$\operatorname{Cut}^{\mathrm{pf}\leq{q}}_{\bullet}(M)\subset\operatorname{Cut}_{\bullet}(M)$.

There are inclusions

$\operatorname{Sep}(M)\subseteq\operatorname{Cut}^{\mathrm{pf}\leq{1}}(M)\subset\dots\subset\operatorname{Cut}^{\mathrm{pf}\leq{q-1}}(M)\subset\operatorname{Cut}^{\mathrm{pf}\leq{q}}(M)=\operatorname{Cut}(M)$

where $q$ is the number of prime factors of $M$. Note that the first inclusion
is strict exactly when $\smash{\widehat{M}}$ has $S^{1}\times S^{2}$ as a
prime factor, since $S^{1}\times S^{2}$ is the only closed 3-manifold which is
prime but not irreducible. Since every reducing sphere cuts $M$ into
components with strictly fewer prime factors than $M$, it follows that
$\operatorname{Cut}^{\mathrm{pf}\leq{q-1}}(M)=\operatorname{Cut}^{\neq\emptyset}(M)$.

######  Example 3.9.

When $\smash{\widehat{M}}=S^{1}\times S^{2}$,
$\operatorname{Sep}(M)=\operatorname{Cut}^{\neq\emptyset}(M)$ and
$\operatorname{Cut}^{\mathrm{pf}\leq{1}}(M)=\operatorname{Cut}(M)$.

We record the following for later use.

######  Lemma 3.10.

The map

$\varphi_{n}\colon\operatorname{Sep}_{n}(M)\longrightarrow\operatorname{Sep}_{0}(M)=\operatorname{Sep}(M),\qquad(\Sigma_{0}\subseteq\dots\subseteq\Sigma_{n})\mapsto\Sigma_{n}$

is a finite covering.

###### Proof.

The map is a locally trivial fiber bundle because it is
$\operatorname{Diff}(M)$-equivariant and $\operatorname{Sep}(M)$ is locally
$\operatorname{Diff}(M)$-retractile, so apply 2.2(1). Therefore it will
suffice to check that the fiber $\varphi_{n}^{-1}(\Sigma)$ is a finite set.
(It then automatically has the discrete topology because
$\operatorname{Sep}_{n}(M)$ is Hausdorff.) Indeed, this fiber is the set of
sequences of cut systems
$\Upsilon_{0}\subseteq\dots\subseteq\Upsilon_{n}=\Sigma$ such that each
$\Upsilon_{i}$ is a separating system. As there are only finitely cut systems
contained in $\Sigma$, this set is finite. ∎

######  Proposition 3.11.

The spaces $\|\operatorname{Cut}^{\mathrm{pf}\leq{p}}_{\bullet}(M)\|$ are
contractible for all $p\geq 1$ and all $M$.

###### Proof.

Let $q$ be the number of prime factors in $M$. For $p\geq q$ the poset
$\operatorname{Cut}^{\mathrm{pf}\leq{p}}(M)$ contains the empty cut system,
which is initial and hence acts as a cone point for
$\|\operatorname{Cut}^{\mathrm{pf}\leq{p}}_{\bullet}(M)\|$. We show that
$\|\operatorname{Cut}^{\mathrm{pf}\leq{p}}_{\bullet}(M)\|\to\|\operatorname{Cut}^{\mathrm{pf}\leq{p+1}}_{\bullet}(M)\|$
is a homotopy equivalence for all $p\geq 1$.

We invoke Quillen’s Theorem A for topological categories [ERW19, Theorem 4.8]
as recalled in Corollary 2.19, noting that 2.19(3) is true since connected
components correspond to isotopy classes of cut systems, and 2.19(2) is true
because the map

$\operatorname{Cut}_{1}(M)=\\{\Sigma_{0}\subseteq\Sigma_{1}\in\operatorname{Cut}(M)\\}\to\operatorname{Cut}(M),\qquad(\Sigma_{0}\subseteq\Sigma_{1})\mapsto\Sigma_{0}$

is a $\operatorname{Diff}(M)$-equivariant map into a
$\operatorname{Diff}(M)$-locally retractile space and hence a fiber bundle by
2.2(1).

It therefore suffices to show 2.19(1): the slice category of the inclusion
functor at every object
$\Upsilon\in\operatorname{Cut}^{\mathrm{pf}\leq{p+1}}(M)$ has a contractible
classifying space. In the case at hand this slice category is the poset

$\operatorname{Cut}^{\mathrm{pf}\leq{p}}(M)_{\Upsilon\subseteq}=\\{\Sigma\in\operatorname{Cut}^{\mathrm{pf}\leq{p}}(M)\;|\;\Upsilon\subseteq\Sigma\\}.$

Any separating system $\Sigma$ containing $\Upsilon$ decomposes as the
disjoint union $\Sigma=\Upsilon\sqcup\Upsilon^{c}$ and it will suffice to keep
track of $\Upsilon^{c}$. Note that while $\Upsilon^{c}$ is $p$-separating in
$M\smash{\mid}\Upsilon$, it may contain spheres which are not reducing in
$M\smash{\mid}\Upsilon$ (for example, they may be boundary parallel, if they
were parallel to a sphere in $\Upsilon$).

We introduce the following modification of
$\operatorname{Cut}^{\mathrm{pf}\leq{p}}(M)$ to account for this. Let $N$ be a
3-manifold and let $B\subset\partial N$ be a union of 2-sphere boundary
components. Let $N_{\smash{\widehat{B}}}$ denote be the manifold obtained by
filling each 2-sphere of $B$ with a disk. Define $C^{\mathrm{pf}\leq p}(N,B)$
to be collections of disjointly embedded spheres in $N$ that are essential in
$N_{\smash{\widehat{B}}}$ and $p$-separating, and let
$C_{\bullet}^{\mathrm{pf}\leq p}(N,B)$ be the nerve of the associated poset.
In particular, $\operatorname{Cut}^{\mathrm{pf}\leq{p}}(N)=C^{\mathrm{pf}\leq
p}(N,\partial N)\subseteq C^{\mathrm{pf}\leq p}(N,B)$ but the essential
spheres appearing in collections in $C^{\mathrm{pf}\leq p}(N,B)$ are not
necessarily reducing.

Let $U$ be a connected component of $M\smash{\mid}\Upsilon$. Each boundary
sphere of $M$ lies in a unique such component $U$. Let $B_{U}$ be the union of
all 2-spheres in $\partial U$ that come from $\partial M$. Then there is a
homeomorphism

$\operatorname{Cut}^{\mathrm{pf}\leq{p}}(M)_{\Upsilon\subseteq}\cong\prod_{U\in\pi_{0}(M\smash{\mid}\Upsilon)}C^{\mathrm{pf}\leq
p}(U,B_{U}).$

Since $\Upsilon\in\operatorname{Cut}^{\mathrm{pf}\leq{p+1}}(M)$, the
components of $\smash{\widehat{M\smash{\mid}\Upsilon}}$ have at most $p+1$
prime factors. Therefore, for each $U$ there are two possibilities:

  * •

if $\smash{\widehat{U}}$ has at most $p$ prime factors, then
$C^{\mathrm{pf}\leq p}({U},B_{U})$ contains the empty sphere system, which is
an initial object and hence $\|C_{\bullet}^{\mathrm{pf}\leq p}({U},B_{U})\|$
is contractible.

  * •

If $\smash{\widehat{U}}$ has exactly $p+1$ prime factors, then every
$\Sigma\in C^{\mathrm{pf}\leq p}({U},B_{U})$ must contain at least one
reducing sphere. Therefore the inclusion
$\operatorname{Cut}^{\neq\emptyset}({U})\subseteq C^{\mathrm{pf}\leq
p}({U},B_{U})$ admits a retraction $F:C^{\mathrm{pf}\leq
p}(U,B_{U})\to\operatorname{Cut}^{\neq\emptyset}({U})$ that forgets spheres
which are not reducing. By Lemma 2.16 this is homotopic to the identity map,
so $\operatorname{Cut}^{\neq\emptyset}_{\bullet}({U})$ is a deformation
retract of $C^{\mathrm{pf}\leq p}({U},B_{U})$. Since
$\|\operatorname{Cut}^{\neq\emptyset}_{\bullet}({U})\|$ is contractible by
Proposition 3.7, it follows that $\|C_{\bullet}^{\mathrm{pf}\leq
p}({U},B_{U})\|$ is contractible.

Therefore, $\|C_{\bullet}^{\mathrm{pf}\leq p}({U},B_{U})\|$ is always
contractible. The fat geometric realisation of a product of simplicial spaces
is equivalent to the product of the fat geometric realisations [ERW19, Theorem
7.1], so
$\|\operatorname{Cut}^{\mathrm{pf}\leq{p}}_{\bullet}(M)_{\Upsilon\subseteq}\|$
is contractible, and applying 2.19 completes the proof. ∎

We now extend this one step further.

######  Proposition 3.12.

The space $\|\operatorname{Sep}_{\bullet}(M)\|$ is contractible for all $M$.

###### Proof.

Paralleling 3.11, we will show that
$\|\operatorname{Sep}_{\bullet}(M)\|\to\|\operatorname{Cut}^{\mathrm{pf}\leq{1}}_{\bullet}(M)\|$
is weak equivalence by proving that for every
$\Upsilon\in\operatorname{Cut}^{\mathrm{pf}\leq{1}}(M)$ the poset

$\operatorname{Sep}(M)_{\Upsilon\subseteq}=\\{\Sigma\in\operatorname{Sep}(M)\;|\;\Upsilon\subseteq\Sigma\\}.$

has a contractible classifying space.

We define $C_{\bullet}^{\operatorname{Sep}}(N,B)\subset
C_{\bullet}^{\mathrm{pf}\leq 1}(N,B)$ as the subspace of those $\Sigma\subset
N$ such that every component of $\smash{\widehat{N\smash{\mid}\Sigma}}$ is
irreducible. In particular $\operatorname{Sep}(N)\subseteq
C^{\operatorname{Sep}}(N,B)$ but the latter contains systems with essential
spheres that are not necessarily reducing. As before we have a product

$\operatorname{Sep}(M)_{\Upsilon\subseteq}\cong\prod_{U\in\pi_{0}(M\smash{\mid}\Upsilon)}C^{\operatorname{Sep}}({U},B_{U}).$

Now, since $\Upsilon\in\operatorname{Cut}^{\mathrm{pf}\leq{1}}(M)$, each ${U}$
has at most one prime factor. There are two cases:

  * •

if $U$ admits no reducing spheres then $\smash{\widehat{U}}$ must be
irreducible and $C^{\operatorname{Sep}}({U},B_{U})$ contains the empty sphere
system, which is an initial object and hence
$\|C^{\operatorname{Sep}}_{\bullet}({U},B_{U})\|$ is contractible.

  * •

If $U$ admits a reducing sphere, then because $\smash{\widehat{U}}$ is prime,
$U$ must be $S^{1}\times S^{2}\setminus\amalg_{k}\smash{\mathring{D}}^{3}$. As
in the proof of Proposition 3.11, $C^{\operatorname{Sep}}({U},B_{U})$ has a
deformation retraction to $\operatorname{Sep}(U)$. But in this case
$\operatorname{Sep}(U)=\operatorname{Cut}^{\neq\emptyset}(U)$, and thus
$\|C^{\operatorname{Sep}}_{\bullet}({U},B_{U})\|$ is contractible by
Proposition 3.7.

As in Proposition 3.11, assembling these results together completes the proof.
∎

###  3.4. Removing parallel spheres

We now turn our attention to the presence of unnecessary ‘parallel’ spheres in
our separating systems, and show these can be removed.

######  Definition 3.13.

We say that two spheres $S_{0},S_{1}\subset M$ are _parallel_ if they bound an
embedded $S^{2}\times I$, _i.e._ if there is an embedding
$e:S^{2}\times[0,1]\hookrightarrow M$ with $e(S^{2}\times\\{i\\})=S_{i}$.

Let $\operatorname{Sep}^{\nparallel}(M)\subset\operatorname{Sep}(M)$ denote
the sub-poset consisting of those separating systems
$\Sigma\in\operatorname{Sep}(M)$ such that no two spheres in $\Sigma$ are
parallel.

Note that there is a maximum possible number $N$ of spheres in any given
separating system $\Sigma\in\operatorname{Sep}^{\nparallel}(M)$. (In 6.5 we
give an upper bound for $N$ in terms of the prime decomposition of $M$.)
Therefore all chains of length $>N$ in the poset must include an equality,
_i.e._ when $i>N$, all simplices in $\operatorname{Sep}^{\nparallel}_{i}(M)$
are degenerate. This is our motivation for passing to the space
$\operatorname{Sep}^{\nparallel}(M)$.

It is useful to have the following alternative characterisation of parallel
spheres.

######  Lemma 3.14.

Two essential spheres $S_{0},S_{1}\subset M$ are parallel if and only if they
are disjoint and (unparametrised) isotopic.

###### Proof.

By [Lau73, Lemma 2.1], if two disjoint essential spheres are homotopic, they
bound an $h$-cobordism in $M$. As a consequence of the Poincaré conjecture any
such $h$-cobordism is trivial, _i.e._ an embedded $S^{2}\times[0,1]$, so the
spheres are parallel. ∎

The goal of this section is to prove the following result.

######  Proposition 3.15.

For $M\not\cong S^{1}\times S^{2}$ the inclusion
$\operatorname{Sep}^{\nparallel}(M)\subset\operatorname{Sep}(M)$ induces an
equivalence

$\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|\xrightarrow{\ \simeq\
}\|\operatorname{Sep}_{\bullet}(M)\|.$

In order to show that
$\|\operatorname{Sep}_{\bullet}^{\nparallel}(M)\|\to\|\operatorname{Sep}_{\bullet}(M)\|$
is an equivalence we will need to choose a way of ordering parallel spheres.
To do so we make a non-canonical choice and use this to define some auxiliary
posets encoding the ordering. To show this choice is well-defined we include
the following lemma.

We would like to thank the Copenhagen “homotopy theory problem seminar” who
worked out the following proof and showed more generally that if $M$ is a
closed oriented $n$-manifold ($n\neq 4$) such that there is an $S^{n-1}\subset
M$ that admits an orientation-reversing isotopy, then $M$ is an exotic
$n$-sphere.

######  Lemma 3.16.

Essential spheres do not admit orientation-reversing self-isotopies: if
$i,j\colon S^{2}\hookrightarrow M$ are isotopic embeddings of an essential
sphere such that $j(S^{2})=i(S^{2})$, then $j^{-1}\circ
i\in\operatorname{Diff}(S^{2})$ is orientation preserving.

###### Proof.

Suppose such an orientation-reversing isotopy exists. By isotopy extension, we
can find an isotopy from $\textrm{Id}_{M}$ to a diffeomorphism $h$ reversing
the coorientation on $S$. Since $M$ is orientable, $h$ must reverse the
orientation on $S$ as well. We first claim $S$ separates $M$. If not, there is
a simple closed curve $\gamma\subset M$ that intersects $S$ in exactly one
point. Hence $\gamma$ and $S$ are duals for the intersection pairing on $M$,
so $[S]\in H_{2}(M)$ generates a nontrivial free factor. But then
$h_{*}[S]=-[S]$, contradicting $h\simeq\textrm{Id}_{M}$.

So we may assume $S$ separates $M$. Since $h$ reverses the coorientation on
$S$, it exchanges the two (necessarily homeomorphic) components $M_{0}$ and
$M_{1}$ of $M\smash{\mid}S$. If neither $M_{0}$ nor $M_{1}$ are simply
connected, then $\pi_{1}(M_{0})\cong\pi_{1}(M_{1})\cong G$ for some nontrivial
$G$. Moreover, $\pi_{1}(M)\cong G*G$ and $h_{*}$ exchanges the two factors.
Since $h_{*}$ preserves the kernel of the natural homomorphism $G*G\rightarrow
G\times G$, it induces an automorphism of $G\times G$, which cannot be not
inner, since no nontrivial element of $G\times\\{1\\}$ is conjugate to an
element of $\\{1\\}\times G$. Thus, $h_{*}\colon\pi_{1}(M)\to\pi_{1}(M)$ is
not inner, again contradicting $h\simeq\textrm{Id}_{M}$. Finally, if $M_{0}$
and $M_{1}$ are simply connected, they are each homeomorphic to
$S^{3}\setminus(\coprod_{k}\smash{\mathring{D}}^{3})$, and since $S$ is
essential, we must have $k\geq 2$. But then $h$ would have to permute the
boundary spheres, and it cannot because it is isotopic to the identity. ∎

For each essential sphere $S\subset M$ we now choose a coorientation and use
it to define (locally around $S$) a left side and a right side of $S$. To be
precise, we make this choice once for one representative in each isotopy
class, and then define the coorientation on any essential sphere by
transporting it along an isotopy from the representative in its isotopy class.
By 3.16 self-isotopies of essential spheres cannot reverse coorientation, so
this procedure is well-defined.

For two parallel spheres $S_{0},S_{1}\subset M$ we say that $S_{0}$ is _to the
left of $S_{1}$_ if the part of $M\smash{\mid}(S_{0}\cup S_{1})$ that is right
of $S_{0}$ and left of $S_{1}$ is diffeomorphic to $S^{2}\times[0,1]$.
Conversely we define _to the right of_. Note that if $S_{0}$ is both left of
and right of $S_{1}$, then $M\cong S^{2}\times S^{1}$. From now on we will
assume that $M\not\cong S^{2}\times S^{1}$.

Within an isotopy class of reducing spheres, we will need to understand the
subspace of all spheres to the left of a fixed sphere $S_{0}$. Equivalently,
we can regard this as a space of spheres parallel to a fixed boundary
component of $M\smash{\mid}S_{0}$. For this, we have the following lemma.

######  Lemma 3.17.

Let $M$ be a compact $3$-manifold and $S_{0}\subset\partial M$ a preferred
boundary sphere. Then the space of unparametrised embeddings

$P(S_{0};M):=\\{S\subset\smash{\mathring{M}}\;|\;S\text{ is parallel to
}S_{0}\\}\subset\operatorname{Sub}(S^{2},\smash{\mathring{M}})$

is contractible.

###### Proof.

The $\operatorname{Diff}_{\partial}(M)$-equivariant map

$p\colon\operatorname{Emb}_{S_{0}\times\\{0\\}}(S_{0}\times[0,1],M)\to\operatorname{Sub}(S_{0},\smash{\mathring{M}}),\qquad
i\mapsto i(S_{0}\times\\{1\\})$

is a fiber bundle because $\operatorname{Sub}(S_{0},M)$ is
$\operatorname{Diff}_{\partial}(M)$-locally retractile by 2.5. Moreover, by
the definition of parallel, the image of this map is exactly the connected
component $P(S_{0};M)\subset\operatorname{Sub}(S_{0},\smash{\mathring{M}})$.
Let $S_{1}\in P(S_{0};M)$ be some unparametrised sphere parallel to $S_{0}$.
The fiber $p$ at some unparametrised sphere $S_{1}\in P(S_{0};M)$, which is
parallel to $S_{0}$, is the space of those embeddings $\iota\colon
S_{0}\times[0,1]\hookrightarrow M$ that are the identity on
$S_{0}\times\\{0\\}$ and send $S_{0}\times\\{1\\}$ to $S_{1}$. The image of
such an embedding is always exactly the connected component $U\subset
M\smash{\mid}S_{1}$ that lies between $S_{0}$ and $S_{1}$. $U$ is necessarily
diffeomorphic to $S^{2}\times[0,1]$. Specifying another embedding
$\iota^{\prime}\in p^{-1}(S_{1})$ is hence equivalent to specifying a
diffeomorphism $S_{0}\times[0,1]\cong U$ that is the identity on
$S_{0}\times\\{0\\}$. Therefore we have a fiber sequence

$\operatorname{Diff}_{S_{0}\times\\{0\\}}(S_{0}\times[0,1])\to\operatorname{Emb}_{S_{0}\times\\{0\\}}(S_{0}\times[0,1],M)\to
P(S_{0};M).$

The left term is the group of pseudo-isotopies of $S^{2}$, which is
contractible by Hatcher’s proof of the Smale conjecture [Hat83]. The middle
term is the space of collars of $S_{0}$ in $M$, which is also contractible by
[Cer61], as recalled in 2.6. Therefore $P(S_{0};M)$ is contractible. ∎

######  Definition 3.18.

For a sphere system $\Sigma$ we let $L(\Sigma)\subseteq\Sigma$ denote the
sphere system that selects the left-most sphere from each isotopy class
represented in $\Sigma$. Then the topological poset
$\operatorname{Sep}^{\mathrm{L}}(M)$ is defined as follows. Its underlying
space is the subspace
$\operatorname{Sep}^{\mathrm{L}}(M)\subseteq\operatorname{Sep}(M)$ of those
separating systems where each isotopy class contains either no sphere or at
least two spheres. The relation is defined as $\Sigma\leq\Sigma^{\prime}$ if
and only if $\Sigma\subseteq\Sigma^{\prime}$ and $L(\Sigma)\subseteq
L(\Sigma^{\prime})$. Define
$\operatorname{Sep}^{\mathrm{L},2}(M)\subset\operatorname{Sep}^{\mathrm{L}}(M)$
as the full subposet on those separating systems where each isotopy class has
either no sphere or exactly two spheres.

By construction, $L$ defines a map of posets
$L\colon\operatorname{Sep}^{\mathrm{L}}(M)\longrightarrow\operatorname{Sep}^{\nparallel}(M),$
and we can also define
$R\colon\operatorname{Sep}^{\mathrm{L}}(M)\longrightarrow\operatorname{Sep}(M),$
via $R(\Sigma):=\Sigma\setminus L(\Sigma).$ Thus the map $R$ removes the left-
most sphere from each isotopy class. This is still a separating system as we
assumed that every non-empty isotopy class of
$\Sigma\in\operatorname{Sep}^{\mathrm{L}}(M)$ contains at least two spheres.
The following shows that forgetting the left-most sphere does not change the
homotopy type of the space of separating systems.

######  Lemma 3.19.

Assume $M\not\cong S^{2}\times S^{1}$. Then the map

$R\colon\operatorname{Sep}^{\mathrm{L}}(M)\longrightarrow\operatorname{Sep}(M)$

that discards the left-most sphere from each isotopy class is a trivial Serre
fibration. As a consequence, the maps
$R\colon\|\operatorname{Sep}^{\mathrm{L}}_{\bullet}(M)\|\to\|\operatorname{Sep}_{\bullet}(M)\|$
and $L\colon\|\operatorname{Sep}^{\rm
L,2}_{\bullet}(M)\|\to\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|$ are
weak equivalences.

###### Proof.

By 2.5, $\operatorname{Sep}(M)$ is a locally
$\operatorname{Diff}(M)$-retractile space and so by 2.2(1) any
$\operatorname{Diff}(M)$-equivariant map into it is a locally trivial
fibration.

For an isotopy class $a\in\pi_{0}\operatorname{Sub}(S^{2},M)$, let
$\operatorname{Sep}^{a}(M)\subset\operatorname{Sep}(M)$ denote the subspace of
those separating systems $\Sigma$ such that the number of spheres of $\Sigma$
that are in the isotopy class $a$ is at least $2$. This only depends on the
isotopy class of $\Sigma$, hence $\operatorname{Sep}^{a}(M)$ is a union of
path components of $\operatorname{Sep}(M)$. Then we can define a map

$R_{a}\colon\operatorname{Sep}^{a}(M)\to\operatorname{Sep}(M)$

that forgets the left-most sphere from the isotopy class $a$. This map is
again a fiber bundle because $\operatorname{Sep}(M)$ is locally
$\operatorname{Diff}(M)$-retractile. Each path component of
$\operatorname{Sep}^{\mathrm{L}}(M)$ belongs to $\operatorname{Sep}^{a}(M)$
for only finitely many $a$, hence restricted to each path component the map
$R$ is equal to a finite composite of maps of the form $R_{a}$. Thus, it will
suffice to show that each $R_{a}$ is a trivial Serre fibration.

We therefore have to show that the fiber $R_{a}^{-1}(\Sigma)$ is contractible
at any $\Sigma\in\operatorname{Sep}(M)$ in the image of $R_{a}$. A point in
this fiber is of the form $\Sigma\sqcup S$ for some
$S\in\operatorname{Sub}(S^{2},M)$ such that $S$ is disjoint from $\Sigma$, $S$
is in the isotopy class $a$, and (within $\Sigma\sqcup S$) $S$ is the left-
most sphere in its isotopy class. Let $S_{0}$ be the left-most sphere in
$\Sigma$ that is in the isotopy class $a$. Then we can equivalently write the
fiber as the subspace of those unparametrised spheres that are parallel to
$S_{0}\subset\partial(M\smash{\mid}\Sigma)$. (Here there are two ways of
thinking of $S_{0}$ as a sphere in the boundary of $M\smash{\mid}\Sigma$. We
choose the one that corresponds to the left side of $S_{0}\subset M$.)

$R_{a}^{-1}(\Sigma)\cong\\{S\in\operatorname{Sub}(S^{2},(M\smash{\mid}\Sigma)^{\circ})\;|\;S\text{
parallel to }S_{0}\\}$

Contractibility now follows from 3.17.

To prove the final claim, we first consider the commutative square

${\operatorname{Sep}^{\mathrm{L}}_{n}(M)}$${\operatorname{Sep}_{n}(M)}$${\operatorname{Sep}^{\mathrm{L}}_{0}(M)}$${\operatorname{Sep}_{0}(M)}$$\scriptstyle{R_{n}}$$\scriptstyle{R_{0}}$

where the vertical maps send $(\Sigma_{0}\subseteq\dots\subseteq\Sigma_{n})$
to $\Sigma_{n}$. These vertical maps are finite coverings by 3.10 and the
horizontal map induces a bijection on fibers, so the square is a pullback and
hence $R_{n}$ is also a trivial Serre fibration. Fat geometric realisation
preserves weak equivalences [ERW19, Theorem 2.2] so $\|R\|$ is a weak
equivalence. The proof for $L$ is entirely analogous. ∎

Using these intermediate spaces we can now prove 3.15.

###### Proof of 3.15.

Consider the following (non-commutative!) diagram of maps of topological
posets.

${\operatorname{Sep}^{\mathrm{L},2}(M)}$${\operatorname{Sep}^{\mathrm{L}}(M)}$${\operatorname{Sep}^{\nparallel}(M)}$${\operatorname{Sep}(M)}$$\scriptstyle{I}$$\scriptstyle{L}$$\scriptstyle{L}$$\scriptstyle{R}$$\scriptstyle{I}$

Here the maps labelled $I$ are inclusions and $L$ and $R$ are as defined
above. The top triangle in this diagram commutes. Consider the three maps

$L,F,R\colon\operatorname{Sep}^{\mathrm{L}}(M)\longrightarrow\operatorname{Sep}(M)$

where $F$ is the forgetful map (_i.e._ the inclusion on underlying spaces).
For every $\Sigma$ we have inclusions $L(\Sigma)\subset
F(\Sigma)=\Sigma\supset R(\Sigma)$. Therefore Lemma 2.16 provides us with
homotopies $|L|\simeq|F|\simeq|R|$, so the following diagram commutes up to
homotopy.

${{\|\operatorname{Sep}^{\mathrm{L},2}_{\bullet}(M)\|}}$${{\|\operatorname{Sep}^{\mathrm{L}}_{\bullet}(M)\|}}$${{\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|}}$${{\|\operatorname{Sep}_{\bullet}(M)\|}}$$\scriptstyle{\|I\|}$$\scriptstyle{\|L\|}$$\scriptstyle{\|L\|}$$\scriptstyle{\|R\|}$$\scriptstyle{\|I\|}$

By 3.19 the two vertical maps in the above square are weak equivalences. Now
it follows from the $2$-out-of-$6$-property that all maps in the diagram are
weak equivalences. (Alternatively, one can apply the functor $\pi_{k}(-)$ to
the diagram and then show that the diagonal map is both injective and
surjective.) ∎

By combining 3.15 and 3.12 we obtain the following theorem, which we will use
in the proof of our main theorem in Section 6.

######  Theorem 3.20.

Suppose $M\not\cong S^{1}\times S^{2}$. Then
$\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|\simeq*$.

###  3.5. The connected sum of two irreducible manifolds

Theorem 3.20 allows us to reprove the following result of Hatcher’s, which is
given in the final remark of [Hat81], and also discussed after the main
theorem in a revision of that same article [Hat03].

######  Theorem 3.21 ([Hat81]).

Let $M\cong M_{1}\\#M_{2}$, where the $M_{i}$ are irreducible and not $S^{3}$.
Let $S\subset M$ be a reducing 2-sphere. Then the inclusion
$\operatorname{Diff}(M,S)\hookrightarrow\operatorname{Diff}(M)$ is an
equivalence.

###### Proof.

Note that since the $M_{i}$ are irreducible, the prime decomposition of $M$
contains no $S^{1}\times S^{2}$ factors and therefore any essential sphere is
separating and induces a connected sum decomposition of $M$ into two pieces.
We claim that any two disjoint reducing spheres in $M$ must be parallel.
Indeed if $S_{0}$ and $S_{1}$ are disjoint embedded spheres, then they induce
a connected sum decomposition of $M$ into three pieces: $M\cong
N_{1}\\#_{S_{0}}N_{2}\\#_{S_{1}}N_{3}$. Since $M$ has two prime factors, it
follows that one of the $N_{i}$ is homeomorphic to $S^{3}$, and it cannot be
$N_{1}$ or $N_{3}$, since the $S_{i}$ are reducing. Therefore $N_{2}$ is
homeomorphic to $S^{3}$, and $N_{2}\setminus(S_{0}\sqcup S_{1})\cong
S^{2}\times I$ determines an embedding with boundary components $S_{0}$ and
$S_{1}$, so $S_{0}$ and $S_{1}$ are parallel. Consequently, any
$\Sigma\in\operatorname{Sep}^{\nparallel}(M)$ consists of a single sphere, in
other words
$\operatorname{Sep}^{\nparallel}(M)\subset\operatorname{Sub}(S^{2},\smash{\mathring{M}})$
is the space of essential spheres. In particular, there are no non-degenerate
chains in $(\operatorname{Sep}^{\nparallel}(M),\subseteq)$ and it follows that
$\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|\simeq\operatorname{Sep}^{\nparallel}(M)$,
so Theorem 3.20 implies that $\operatorname{Sep}^{\nparallel}(M)\simeq*$.

By the discussion following 2.5 we have the fibration sequence

$\operatorname{Diff}(M,S)\hookrightarrow\operatorname{Diff}(M)\rightarrow\operatorname{Sub}(S^{2},\smash{\mathring{M}}).$

The fibration lands in the subspace
$\operatorname{Sep}^{\nparallel}(M)\subset\operatorname{Sub}(S^{2},\smash{\mathring{M}})$
of essential spheres (because the action of $\operatorname{Diff}(M)$ preserves
these) which we have shown is contractible. Therefore the inclusion of
$\operatorname{Diff}(M,S)\to\operatorname{Diff}(M)$ must be an equivalence as
claimed. ∎

The above theorem is a powerful computational tool, because when the $M_{i}$
are distinct we can describe $\operatorname{Diff}(M,S)$ using the homotopy
fiber sequence

$\operatorname{Diff}_{\partial}(M_{1}\setminus\smash{\mathring{D}}^{3})\longrightarrow\operatorname{Diff}(M,S)\longrightarrow\operatorname{Diff}(M_{2}\setminus\smash{\mathring{D}}^{3}),$

which can be used to describe $\operatorname{Diff}(M)$ in terms of
$\operatorname{Diff}(M_{i})$ for the irreducible pieces $M_{i}$.

##  4\. Homotopy finiteness via the (strong) generalised Smale conjecture

In Section 6 we will use contractibility of
$\|\operatorname{Sep}^{\nparallel}_{\bullet}(M)\|$ to prove Theorem 6.1 via an
inductive argument. The base case of our argument is homotopy finiteness for
$B\\!\operatorname{Diff}_{\partial}(M)$ where $\smash{\widehat{M}}$ is an
irreducible manifold. In the case that $M$ itself is irreducible (_i.e._ has
no spherical boundary components) this is a theorem of Hatcher–McCullough
[HM97], as recalled in 1.1. We will show in 6.4 that the general case can be
reduced to either Hatcher–McCullough’s result or the following theorem, which
is equivalent to the case in which $M$ has one fixed spherical boundary
component.

######  Theorem 4.1.

Let $M$ be an irreducible 3-manifold with either empty or incompressible
toroidal boundary, and let $D^{3}\subset\smash{\mathring{M}}$ be an embedded
disk. Then $B\\!\operatorname{Diff}_{D^{3}}(M)$ has the homotopy type of a
finite CW complex.

The proof of this theorem will occupy Sections 4 and 5. For its proof, we
exploit the JSJ and geometric decompositions of $M$, which we recall now. Let
$M$ be a prime orientable 3-manifold with empty or incompressible toroidal
boundary. Then there exists a minimal collection $T_{\operatorname{JSJ}}$ of
embedded, incompressible tori and annuli such that each of the components of
$M\setminus T_{\operatorname{JSJ}}$ are either Seifert-fibered or atoroidal,
and this collection is unique up to isotopy. (Recall that an irreducible
manifold is called _atoroidal_ if every essential immersed torus is homotopic
to the boundary.) This decomposition of $M$ is known as the _JSJ
decomposition_ , having been discovered independently by Jaco–Shalen [JS78]
and Johannson [Joh79a]. Moreover, since $\partial M$ consists only of tori, no
annuli appear in this decomposition (see e.g. [Jac80]).

######  Definition 4.2.

A prime 3-manifold with empty or incompressible toroidal boundary has _trivial
JSJ decomposition_ if it is either Seifert fibered or atoroidal (_i.e._ if
$T_{\operatorname{JSJ}}$ is empty).

On the other hand, Thurston’s geometrisation conjecture states that there
exists a collection $T_{G}$ of incompressible 2-sided tori and 1-sided Klein
bottles embedded in $M$ such that each component of $M\setminus T_{G}$ admits
a finite-volume Riemannian metric locally isometric to one of 8 model
geometries: $\mathbb{S}^{3}$, $\mathbb{E}^{3}$, $\mathbb{H}^{3}$,
$\mathbb{S}^{2}\times\mathbb{R}$, $\mathbb{H}^{2}\times\mathbb{R}$,
$\operatorname{Nil}$, $\operatorname{Sol}$, or
$\widetilde{\operatorname{PSL}}_{2}(\mathbb{R})$, unless $M=T^{2}\times I$ or
$K\widetilde{\times}I$, the twisted $I$-bundle over the Klein bottle. This
collection $T_{G}$ is unique up to isotopy. Except for $\mathbb{H}^{3}$ and
$\operatorname{Sol}$, the remaining geometries are all Seifert-fibered.
Thurston proved his conjecture in the case of Haken 3-manifolds, showing in
particular that every closed atoroidal Haken 3-manifold admits a finite-volume
metric based on $\mathbb{H}^{3}$ [Thu86]. (Recall an orientable manifold is
_Haken_ if it is irreducible and contains an orientable incompressible
surface.) The full conjecture was proven by Perelman [Per02, Per03b, Per03a]
(see also [CZ06, BBM+10, MT14]).

######  Definition 4.3.

A compact, connected 3-manifold $M$ is called _geometric_ if
$\smash{\mathring{M}}$ admits a complete, finite-volume metric locally modeled
on one of the 8 Thurston geometries.

Geometric 3-manifolds necessarily have incompressible boundary, which is
diffeomorphic to a (possibly empty) disjoint union of tori (see, _e.g._
[Bon02, §2]). The decomposition of $M$ along $T_{G}$ is called the _geometric
decomposition_ of $M$, and is nearly the same as the JSJ decomposition, except
that $T_{G}$ may contain 1-sided Klein bottles, and that closed manifolds
supporting $\operatorname{Sol}$-geometry have a trivial geometric
decomposition but a nontrivial JSJ decomposition.

If $M$ is an orientable $T^{2}$-bundle over $S^{1}$ that is not Seifert-
fibered, then $M$ is called an _Anosov bundle_ and admits
$\operatorname{Sol}$-geometry. Although not all $\operatorname{Sol}$-manifolds
are of this form, such bundles will be a special case of Theorem 4.1 that we
will not be able to treat in the same way we treat other manifolds with
nontrivial JSJ decomposition. Conversely, if $M$ is orientable, irreducible
and has trivial JSJ-decomposition then $M$ is geometric unless $M=T^{2}\times
I$ or $M=K\tilde{\times}I$. Thus when decomposing manifolds, we cannot
completely exclude the case of manifolds with non-trivial JSJ decomposition
nor the case of non-geometric manifolds. However, if $M$ has nontrivial JSJ
decomposition, and is not an Anosov bundle, the components of
$M|T_{\operatorname{JSJ}}$ all have nonempty toroidal boundary and thus are
either Haken Seifert-fibered (and not $T^{2}\times I$) or hyperbolic.

The flowchart in Fig. 3 explains how the proof of 4.1 breaks up into cases
depending on the geometry of $M$. Each leaf contains a reference to the result
in which we prove $B\\!\operatorname{Diff}_{D^{3}}(M)$ is homotopy finite for
such $M$. Arrows correspond to straightforward case distinctions, except in
one case where a proposition is required to reduce to the case where $M$ has
trivial JSJ decomposition. In order to prove 4.1 when $M$ has nontrivial JSJ
decomposition and is not an Anosov bundle, we will need a stronger finiteness
condition on $B\\!\operatorname{Diff}$ for components of
$M\smash{\mid}T_{\operatorname{JSJ}}$. We introduce this condition, called
_hereditary finiteness_ (5.2) in the next section and prove it holds in the
(fiber-rigid) Haken Seifert-fibered and hyperbolic cases, as only these can
arise in $M\smash{\mid}T_{\operatorname{JSJ}}$. The leaves for which we prove
hereditary finiteness are coloured in blue in Fig. 3.

4.1: $B\\!\operatorname{Diff}_{D^{3}}(M)$ homotopy finite Anosov $T^{2}$
bundle over $S^{1}$ 4.20 5.11 Trivial JSJ decomposition Seifert-fibered
Seifert-fibered, non-Haken 4.8 Hyperbolic 4.8 Seifert-fibered Haken Flexible
$T^{3}$ 4.16 $M_{HW}$ 4.17 $T^{2}\times I$ 4.15 Fiber-rigid Singular 5.19 Non-
singular 5.17

Figure 3. Flowchart of the proof of 4.1

Recall that we have a fiber sequence

$\operatorname{Diff}_{D^{3}}(M)\longrightarrow\operatorname{Diff}(M)\longrightarrow\operatorname{Emb}(D^{3},\smash{\mathring{M}})$

whence $B\\!\operatorname{Diff}_{D^{3}}(M)$ is equivalent to a connected
component of
$\operatorname{Emb}(D^{3},\smash{\mathring{M}})/\\!\\!/\operatorname{Diff}(M)$.
The quotient
$\operatorname{Emb}(D^{3},\smash{\mathring{M}})/\\!\\!/\operatorname{Diff}(M)$
is connected if $M$ admits an orientation reversing diffeomorphism, and
otherwise it has two connected components.

Let $\operatorname{Fr}(M)$ denote the frame bundle of $M$. Since $M$ is
parallelisable, $\operatorname{Fr}(M)\cong
M\times\operatorname{GL}_{3}(\mathbb{R})$. We will also consider the oriented
orthonormal frame bundle $\operatorname{Fr}^{\perp}(M)$, which is homotopy
equivalent to $\operatorname{Fr}(M)$. Again by parallelisability,
$\operatorname{Fr}^{\perp}(M)\cong M\times\operatorname{O}(3)$. Since
$\operatorname{Emb}(D^{3},\smash{\mathring{M}})\simeq\operatorname{Fr}(M)$, we
obtain that $B\\!\operatorname{Diff}_{D^{3}}(M)$ is equivalent to a connected
component of $\operatorname{Fr}(M)/\\!\\!/\operatorname{Diff}(M)$. The overall
strategy for proving 4.1 when $M$ has trivial JSJ decomposition will be to
exploit the geometry on $M$ to find a suitable subgroup
$G<\operatorname{Diff}(M)$ such that $G\hookrightarrow\operatorname{Diff}(M)$
is a homotopy equivalence and $\operatorname{Fr}(M)/\\!\\!/G$ is homotopy
finite. In this case the basepoint component of
$\operatorname{Fr}(M)/\\!\\!/G$, which we can also write as
$\operatorname{Fr}^{+}(M)/\\!\\!/G^{+}$ by restricting to oriented frames and
oriented diffeomorphisms, will be a model for
$B\\!\operatorname{Diff}_{D^{3}}(M)$. The simplest case will be when $G$ is a
Lie group acting freely on $\operatorname{Fr}(M)$. For example, if
$G=\operatorname{Isom}(M)$ for some Riemannian metric on $M$, we learnt the
idea of the following proof from Sam Nariman.

######  Lemma 4.4.

Suppose $(M,g)$ is a Riemannian 3-manifold such that the inclusion
$\operatorname{Isom}(M)\rightarrow\operatorname{Diff}(M)$ is a homotopy
equivalence. Then $B\\!\operatorname{Diff}_{D^{3}}(M)$ is homotopy finite.

###### Proof.

As argued above $B\\!\operatorname{Diff}_{D^{3}}(M)$ is a connected component
of

$\operatorname{Fr}(M)/\\!\\!/\operatorname{Diff}(M)\simeq\operatorname{Fr}(M)/\\!\\!/\operatorname{Isom}(M)\simeq\operatorname{Fr}^{\perp}(M)/\\!\\!/\operatorname{Isom}(M).$

The isometry group $\operatorname{Isom}(M)$ is a compact Lie group acting
freely on the compact manifold $\operatorname{Fr}^{\perp}(M)$, so
$\operatorname{Fr}^{\perp}(M)/\\!\\!/\operatorname{Isom}(M)\simeq\operatorname{Fr}^{\perp}(M)/\operatorname{Isom}(M)$
and this quotient is a compact manifold [Lee13, Theorem 7.10], and hence is
homotopy finite. ∎

Recall from the introduction that we refer to the statement that
$\operatorname{Isom}(M)\hookrightarrow\operatorname{Diff}(M)$ is a homotopy
equivalence as the _strong_ generalised Smale conjecture, and to the statement
that $\operatorname{Isom}_{0}(M)\hookrightarrow\operatorname{Diff}_{0}(M)$ is
a homotopy equivalence as the _weak_ generalised Smale conjecture. For the
rest of Section 4, we will prove all cases of 4.1 (corresponding to leaves in
the flowchart in Fig. 3) where there exists a Lie group
$G<\operatorname{Diff}(M)$ acting freely on $\operatorname{Fr}(M)$ and such
that $G\hookrightarrow\operatorname{Diff}(M)$ is a homotopy equivalence.
Except for $T^{3}$ and $T^{2}\times I$, the strong generalised Smale
conjecture will hold, allowing us to take $G=\operatorname{Isom}(M)$ and apply
4.4.

###  4.1. The generalised Smale conjecture

The strong form of the generalised Smale conjecture is false, even among
irreducible geometric 3-manifolds. However, for irreducible geometric
manifolds, the weak form always holds.

######  Theorem 4.5 (Weak generalised Smale conjecture).

Let $(M,g)$ be an irreducible geometric 3-manifold. Then the inclusion
$\operatorname{Isom}_{0}(M)\hookrightarrow\operatorname{Diff}_{0}(M)$ is a
homotopy equivalence.

The proof of Theorem 4.5 combines the results of several authors spanning
decades. For Haken geometric 3-manifolds, it follows from results of Hatcher
[Hat76] and, independently, Ivanov [Iva76] on spaces of incompressible
surfaces. Hatcher and Ivanov’s work covers all Haken cases and thus all cases
when $M$ has non-empty boundary, all cases with $\operatorname{Sol}$-geometry,
and many of the hyperbolic and Seifert-fibered cases except possibly when the
base-orbifold is homeomorphic to $S^{2}$.

For $M=S^{3}$, Theorem 4.5 follows from Hatcher [Hat83]. It was proven for
quotients of $S^{3}$ containing one-sided incompressible Klein bottles
(prismatic and quaternionic) and lens spaces other than
$\mathbb{R}\mathrm{P}^{3}$ using topological techniques [Iva79, Iva82,
HKMR12]. Bamler–Kleiner [BK23b, BK19] used Ricci flow techniques to prove the
remaining cases (tetrahedral, octahedral, icosahedral, and
$\mathbb{R}\mathrm{P}^{3}$) and gave a unified proof for all spherical
3-manifolds. In particular, their approach offers a new proof of the Smale
conjecture.

In the $\mathbb{H}^{3}$-case, Gabai [Gab01] proved that
$\operatorname{Diff}_{0}(M)$ is contractible, and the techniques of [BK23b]
also reprove this case. Since $(M,g)$ is negatively curved,
$\operatorname{Isom}(M)$ is discrete (see e.g. [KN63, Corollary 5.4]) so
Theorem 4.5 follows. The non-Haken cases of $\mathbb{H}^{2}\times\mathbb{R}$\-
and $\widetilde{\operatorname{PSL}}_{2}(\mathbb{R})$-geometry were proven by
McCullough and Soma [MS13]. Recently, Bamler–Kleiner [BK23a] proved the last
remaining non-spherical case, namely closed manifolds admitting
$\operatorname{Nil}$-geometry and whose base-orbifold is homeomorphic to
$S^{2}$.

We note that in each case, the actual homotopy type of
$\operatorname{Diff}_{0}$ is known. For irreducible 3-manifolds that are
spherical, see Tables 1 and 2 in [HKMR12]. For non-spherical geometric
3-manifolds, the homotopy type of $\operatorname{Diff}_{0}$ is simpler to
describe, and since we will refer to it later we state it here.

######  Corollary 4.6 (Haken [Hat76, Iva76, Iva79], Non-Haken [MS13, BK23a]).

Let $M$ be an irreducible geometric 3-manifold that is not covered by $S^{3}$.

  1. (1)

If $M$ is not Seifert-fibered, then $\operatorname{Diff}_{0}(M)\simeq*$.

  2. (2)

If $M$ is Seifert-fibered then
$\displaystyle\operatorname{Diff}_{0}(M)\simeq\prod_{b}\operatorname{SO}(2)$,
where $b$ is the rank of the center $Z(\pi_{1}(M))$.

Theorem 4.5 tells us that $\operatorname{Isom}(M)$ and
$\operatorname{Diff}(M)$ can only differ with respect to $\pi_{0}$, the group
of path components. The group $\pi_{0}\operatorname{Diff}(M)$ is known as the
_mapping class group_ of $M$. We briefly survey some results on mapping class
groups of geometric manifolds. The computation of the mapping class group for
$S^{3}$ is a consequence of work of Cerf [Cer68], for $S^{1}\times S^{2}$ it
is due to Gluck [Glu62], and for lens spaces it is due to Bonahon [Bon83b],
although some cases were known [Rub79, Iva79, Iva82, RB84]. More generally,
the results for all spherical 3-manifolds are surveyed in [HKMR12], and
include work of Asano [Asa78], Rubinstein [Rub78, Rub79], Boileau–Otal [BO86],
and Birman–Rubinstein [RB84]. Johannson [Joh79b] showed that Haken 3-manifolds
with incompressible boundary that are atoroidal and anannular have finite
mapping class groups, and McCullough [McC91] showed more generally that
mapping class groups of Haken 3-manifolds are finitely presented, and
investigated other finiteness properties.

In many cases, the strong form of the generalised Smale conjecture is also
known to hold.

######  Theorem 4.7.

Suppose $M$ is an irreducible geometric 3-manifold that is either hyperbolic
or non-Haken Seifert-fibered. Then the inclusion
$\operatorname{Isom}(M)\hookrightarrow\operatorname{Diff}(M)$ is a homotopy
equivalence.

The non-Haken Seifert-fibered manifolds in the above theorem include all those
with $\mathbb{S}^{3}$-geometry completed by [HKMR12, BK19, BK23b], see
references therein for a comprehensive history. All other irreducible non-
Haken Seifert-fibered cases carry either the geometry of
$\mathbb{H}^{2}\times\mathbb{R}$,
$\widetilde{\operatorname{PSL}}_{2}(\mathbb{R})$, or $\operatorname{Nil}$, and
fiber over the 2-sphere with 3 cone points. Theorem 4.7 was proven for the
$\mathbb{H}^{2}\times\mathbb{R}$ and
$\widetilde{\operatorname{PSL}}_{2}(\mathbb{R})$ cases in [MS13], and for the
$\operatorname{Nil}$ case in [BK23a]. (In fact, Theorem 4.7 also holds when
$M$ fibers over $S^{2}$ with 3 cone points and carries
$\mathbb{E}^{3}$-geometry [MS13], although these are Haken.) The hyperbolic
case follows from Mostow–Prasad rigidity (see e.g. [Gro81]) and the work of
Gabai [Gab01] and Gabai–Meyerhoff–Thurston [GMT03]. Combining Theorem 4.7 with
Lemma 4.4 we obtain the following corollary.

######  Corollary 4.8.

Let $M$ be an irreducible geometric 3-manifold that is either hyperbolic or
non-Haken Seifert-fibered. Then $B\\!\operatorname{Diff}_{D^{3}}(M)$ is
homotopy finite.

In Proposition 4.17 and Theorem 4.20 we show that the generalised Smale
conjecture also holds for the Hantzsche-Wendt manifold $M_{\rm HW}$ (see
Section 4.3 below) and $T^{2}$-bundles over the circle that have Anosov
monodromy.

###  4.2. Haken Seifert-fibered manifolds – overview

The strong form of the generalised Smale conjecture often does not hold for
Haken Seifert-fibered 3-manifolds, essentially because the mapping class group
of the base surface may be infinite. Our proof of Theorem 4.1 for these
manifolds will rely on understanding the space of all Seifert fiberings.

We say two Seifert fiberings of $M$ are _equivalent_ if there is a
diffeomorphism from $M$ to itself taking the fibers of one to the fibers of
the other. Given a fixed Seifert fibering $\varphi$ of $M$, let
$\operatorname{Diff}^{f}(M;[\varphi])$ denote the subgroup of _fiber-
preserving diffeomorphisms_ of $\varphi$, _i.e._ those diffeomorphisms which
map fibers of $\varphi$ diffeomorphically to fibers. On the other hand, in
Section 5 we also study the subgroup of _vertical diffeomorphisms_
$\operatorname{Diff}^{v}(M;[\varphi])$, consisting of diffeomorphisms
$f\in\operatorname{Diff}(M)$ that satisfy $\varphi\circ f=\varphi$, _i.e._
they preserve each fiber setwise.

If $\varphi$ and $\varphi^{\prime}$ are equivalent, then
$\operatorname{Diff}^{f}(M;[\varphi])$ and
$\operatorname{Diff}^{f}(M;[\varphi^{\prime}])$ are conjugate subgroups of
$\operatorname{Diff}(M)$. Following [HKMR12, Definition 3.13], we define the
_space of Seifert fiberings isomorphic to $\varphi$_ to be the coset space
$\operatorname{SF}(M;[\varphi])=\operatorname{Diff}(M)/\operatorname{Diff}^{f}(M,[\varphi])$.
Theorems 3.12 and 3.14 of [HKMR12] state that the quotient map
$\operatorname{Diff}(M)\rightarrow\operatorname{SF}(M;[\varphi])$ is a locally
trivial fiber bundle and that the components of
$\operatorname{SF}(M;[\varphi])$ are contractible.

For many Haken Seifert-fibered 3-manifolds, $\operatorname{SF}(M;[\varphi])$
is also connected, which is to say that any diffeomorphism of $M$ is isotopic
to a fiber-preserving diffeomorphism. Indeed, by results of Waldhausen [Wal67,
Theorem 10.1] (see also Jaco [Jac80, Theorem VI.18] for the case where $M$ has
boundary), there are only finitely many orientable Haken Seifert-fibered
3-manifolds for which there does not exist a unique Seifert fibering up to
isotopy. These six manifolds, which we call _exceptional_ , are:

  * •

$S^{1}\times D^{2}$,

  * •

$T^{3}$, the 3-torus,

  * •

$T^{2}\times I$, the trivial $I$-bundle over the 2-torus,

  * •

$M_{HW}$, the Hantzsche-Wendt manifold (first studied in [HW35]),

  * •

$K\widetilde{\times}I$, the orientable $I$-bundle over the Klein bottle $K$,
and

  * •

$D(K\widetilde{\times}I)$ the double of $K\widetilde{\times}I$.

We remark that although $S^{1}\times D^{2}$ and its fiberings will play a
major role in Section 5, we do not need to prove 4.1 for $S^{1}\times D^{2}$,
as it has compressible boundary. Indeed, $S^{1}\times D^{2}$ is the only Haken
Seifert-fibered 3-manifold with compressible boundary.

######  Definition 4.9.

A Haken Seifert-fibered manifold $M$ is _fiber-rigid_ if it admits a Seifert
fibering $\varphi\colon M\rightarrow S$ such that the inclusion
$\operatorname{Diff}^{f}(M;[\varphi])\hookrightarrow\operatorname{Diff}(M)$ is
a homotopy equivalence. (Or equivalently such that
$\operatorname{SF}(M;[\varphi])$ is contractible.) Otherwise we say $M$ is
_flexible_.

######  Theorem 4.10.

Let $M$ be orientable and Haken Seifert-fibered. If $M$ is not $S^{1}\times
D^{2}$, $T^{3}$, $T^{2}\times I$, or $M_{HW}$, then $M$ is fiber-rigid.

###### Proof.

By Theorem 3.15 of [HKMR12] all orientable Haken Seifert-fibered manifolds
that are not exceptional are fiber-rigid. In fact, their theorem says that for
non-exceptional cases $\operatorname{SF}(M;[\varphi])$ is contractible for any
choice of $\varphi$. Thus it remains to prove that for $K\widetilde{\times}I$
and $D(K\widetilde{\times}I)$ there exists a fibering $\varphi$ for which
$\operatorname{Diff}^{f}(M;[\varphi])\to\operatorname{Diff}(M)$ is surjective
on $\pi_{0}$.

If $M=K\widetilde{\times}I$, then $M\simeq K$ so $\pi_{1}(M)\cong\pi:=\langle
a,t\mid tat^{-1}=a^{-1}\rangle$. By Waldhausen’s theorem [Wal68],
$\pi_{0}\operatorname{Diff}(M)$ is isomorphic to the subgroup of
$\operatorname{Out}(\pi)$ preserving $\pi_{1}(\partial M)$. In the case at
hand $\pi_{1}(\partial M)\cong\mathbb{Z}^{2}$ is the subgroup $\langle
a,t^{2}\rangle\leq\pi$, which is characteristic, so
$\pi_{0}\operatorname{Diff}(M)\cong\operatorname{Out}(\pi)\cong\mathbb{Z}/2\times\mathbb{Z}/2$.
The generators of $\operatorname{Out}(\pi)$ are $\sigma_{1}\colon t\mapsto
t^{-1}$, $a\mapsto a$ and $\sigma_{2}\colon t\mapsto ta$, $a\mapsto a$. There
are two distinct Seifert fiberings $\varphi_{1}$, $\varphi_{2}$ of $M$
corresponding to the two maximal normal infinite cyclic subgroups of $\pi$,
namely $\langle a\rangle$ and $\langle t^{2}\rangle$, respectively. The base
surface of $\varphi_{1}$ is Möbius strip and there are no singular fibers. The
base orbifold of $\varphi_{2}$ is a disk with two cone points of order 2.
These two cone points correspond to the conjugacy classes of $t$ and $ta$
respectively. Both $\sigma_{1}$ and $\sigma_{2}$ can be realised by
diffeomorphisms of $K\widetilde{\times}I$ that preserve both fiberings:
$\sigma_{1}$ is a reflection that reverses the orientation of each fiber of
$\varphi_{2}$, preserving each setwise, while $\sigma_{2}$ rotates each fibers
of $\varphi_{1}$ by $180^{\circ}$ and permutes the fibers of $\varphi_{2}$,
exchanging the two singular fibers (see Figure 4). Thus
$\pi_{0}\operatorname{Diff}^{f}(M;[\varphi_{i}])$ surjects onto
$\pi_{0}\operatorname{Diff}(M)$ for $i=1,2$. Using either $\varphi_{1}$ or
$\varphi_{2}$ proves the theorem in this case.

Finally, consider $M=D(K\widetilde{\times}I)$, the double of
$K\widetilde{\times}I$. Then $M$ supports a flat metric, and by [CV73, Theorem
1], the group of affine automorphisms $\operatorname{Aff}(M)$ fits into a
short exact sequence

$1\rightarrow\operatorname{Aff}_{0}(M)\rightarrow\operatorname{Aff}(M)\rightarrow\operatorname{Aff}(M)/\operatorname{Aff}_{0}(M)\rightarrow
1,$

where $\operatorname{Aff}_{0}(M)=\operatorname{Isom}_{0}(M)$, and
$\operatorname{Aff}(M)/\operatorname{Aff}_{0}(M)\cong\operatorname{Out}(\pi_{1}(M))$.
Therefore by 4.5 and Waldhausen’s theorem [Wal68], the inclusion
$\operatorname{Aff}(M)\rightarrow\operatorname{Diff}(M)$ is a homotopy
equivalence. On the other hand, the quotient map $\varphi_{0}\colon
M\rightarrow M/\operatorname{Aff}_{0}(M)$ is a Seifert fibering with base
surface a sphere with 4 cone points of order 2. Since
$\operatorname{Aff}_{0}(M)$ is normal in $\operatorname{Aff}(M)$, every
element of
$\operatorname{Out}(\pi_{1}(M))\cong\operatorname{Aff}(M)/\operatorname{Aff}_{0}(M)$
is realised by a fiber-preserving diffeomorphism of $M$. Thus,
$\pi_{0}\operatorname{Diff}(M;[\varphi_{0}])\rightarrow\operatorname{Out}(\pi_{1}(M))\cong\pi_{0}\operatorname{Diff}(M)$
is surjective. ∎

$\varphi_{1}$$\varphi_{2}$

1 0 0.176326288187507532 1$A$

1 0 0.176326288187507532 1$A$

1 0 0 1$B$

1 0 0 1$B$

Figure 4. $K\widetilde{\times}I$ is formed by identifying sides of a cube as
indicated above. The vertical axis is the $I$-direction while the central
square is the core $K$. Projecting onto the $B$-face is the nonsingular
fibering $\varphi_{1}$ over the Möbius band, and the projection $\varphi_{2}$
onto the $A$-face has two singular fibers of order 2, shaded in blue and red
respectively. Reflection in the plane perpendicular to the red line induces
$\sigma_{1}$, while the rotation of $K$ that exchanges the blue and red curves
extends to $K\widetilde{\times}I$ as $\sigma_{2}$. The top and bottom faces of
the cube glue up to form a torus $T^{2}$, and the projection onto the core $K$
is the orientation double cover $T^{2}\to K$.

######  Remark 4.11.

For a Seifert fibering $\varphi\colon M\to S$ where $M$ has non-trivial
boundary and $M\not\cong T^{2}\times I$ and $M\not\cong S^{1}\times D^{2}$, we
have $\operatorname{Diff}^{f}(M;[\varphi])\simeq\operatorname{Diff}(M)$. This
is because any irreducible $M$ with boundary is Haken in which case the only
exceptional cases we need to consider are $T^{2}\times I$ and
$K\tilde{\times}I$, but for the latter we just checked that both the non-
singular fibering and singular fibering is rigid.

For the sake of completeness, in the next subsection we will show that the
converse to Theorem 4.10 is also true. That is, the four excluded manifolds
$S^{1}\times D^{2}$, $T^{3}$, $T^{2}\times I$ and $M_{HW}$ are the only
flexible Haken Seifert-fibered 3-manifolds. As alluded to above, the proof of
Theorem 4.1 for $T^{3}$, $T^{2}\times I$ and $M_{HW}$ fits into the same
schema as those addressed in Section 4.1, while we deal with the fiber-rigid
case in Section 5.

###  4.3. Flexible Seifert-fibered manifolds

We now deal with $T^{3}$, $T^{2}\times I$, and $M_{HW}$. First, we recall
basic facts about their fiberings and show that all three of these are
flexible. For $M=T^{2}\times I$ or $M=T^{3}$, there infinitely many Seifert
fiberings, each one corresponding to an infinite cyclic free factor of
$\pi_{1}(M).$ For $T^{3}$, the base surface is always $T^{2}$ and there are no
singular fibers, while for $T^{2}\times I$ the base surface is always
$S^{1}\times I$, again without singular fibers. In each case, any two
fiberings are equivalent, since the mapping class group acts transitively on
free factors of $\pi_{1}(M)$. Thus, for any fibering $\varphi$ the components
of $\operatorname{SF}(T^{3};[\varphi])$ (respectively
$\operatorname{SF}(T^{2}\times I;[\varphi])$) are in one-to-one correspondence
with the cosets
$\operatorname{GL}_{3}(\mathbb{Z})/\operatorname{GL}_{3}^{\langle
e_{1}\rangle}(\mathbb{Z})$ (respectively
$\operatorname{GL}_{2}(\mathbb{Z})/\operatorname{GL}_{2}^{\langle
e_{1}\rangle}(\mathbb{Z})$ ), where $\operatorname{GL}_{n}^{\langle
e_{1}\rangle}(\mathbb{Z})$ denotes the
$\operatorname{GL}_{n}(\mathbb{Z})$-stabiliser of the subgroup $\langle
e_{1}:=(1,0,\ldots,0)\rangle\leq\mathbb{Z}^{n}$. In particular, both $T^{3}$
and $T^{2}\times I$ are flexible.

To see that $M_{HW}$ is flexible, we first recall its construction (see
[HKMR12, Section 3.9] for more details). The $I$-bundle $K\tilde{\times}I$ can
be thought of as the mapping cylinder of the oriented double cover
$T^{2}\rightarrow K$, and as such its boundary is $T^{2}$ (see Fig. 4). Take
two copies of $K\widetilde{\times}I$ and let $T_{1}$ and $T_{2}$ be the two
boundary tori. While the double $D(K\tilde{\times}I)$ identifies longitude to
longitude and meridian to meridian, $M_{HW}$ is formed by identifying the
meridian of $T_{1}$ with the longitude of $T_{2}$, and vice versa. There are
two Seifert fiberings of $M_{HW}$, one where the meridians of $T_{1}$ are
fibers, and the other where the longitudes of $T_{1}$ are fibers. However,
there is a diffeomorphism of $M_{HW}$ which exchanges the two copies of
$K\widetilde{\times}I$ and simultaneously swaps the meridian and longitude of
$T_{1}$. This is not fiber-preserving, so $M_{HW}$ is flexible.

To prove $B\\!\operatorname{Diff}_{D^{3}}(M)$ is homotopy finite for these
three manifolds, we exhibit a subgroup $G\leq\operatorname{Diff}(M)$ such that
the inclusion $G\to\operatorname{Diff}(M)$ is a homotopy equivalence and $G$
acts freely on $\operatorname{Fr}(M)$ with homotopy finite quotient. When $M$
is $M_{HW}$, we will exhibit a metric on $M$ such that
$G=\operatorname{Isom}(M)$, and apply Lemma 4.4. When $M$ is $T^{2}\times I$
or $T^{3}$, $\pi_{0}\operatorname{Diff}(M)$ is infinite, so no such metric
exists; nevertheless, $\pi_{0}\operatorname{Diff}(M)$ does lift to a group of
diffeomorphisms.

Let us first proceed with the proof for $T^{2}\times I$. Note that the results
in this case will also be needed in Section 5. We begin with some finiteness
results for diffeomorphisms of surfaces.

######  Lemma 4.12.

Let $S$ be a surface (not necessarily orientable). Then
$B\\!\operatorname{Diff}_{D^{2}}(S)$ is homotopy finite.

###### Proof.

Let $S^{\prime}=S\setminus\smash{\mathring{D}}^{2}$, and let $b$ be the
boundary component of $S^{\prime}$ corresponding to $\partial D^{2}$. Then
$B\\!\operatorname{Diff}_{D^{2}}(S)\simeq
B\\!\operatorname{Diff}_{b}(S^{\prime})$. By [Gra73, Proposition 6], the
connected components of $B\\!\operatorname{Diff}_{b}(S^{\prime})$ are
contractible, so it suffices to show that
$\pi_{0}\operatorname{Diff}_{b}(S^{\prime})$ has finite classifying space.
This is proved in [HM97, Lemma 1.2]. ∎

Given a closed, compact manifold $M$, the _pseudo-isotopy group_ is defined to
be

$\operatorname{PI}(M):=\operatorname{Diff}_{M\times\\{0\\}}(M\times I).$

For surfaces, Hatcher proved the following.

######  Theorem 4.13 ($S\neq S^{2}$ [Hat76], $S=S^{2}$ [Hat83]).

Let $S$ be an orientable surface. Then $\operatorname{PI}(S)$ is contractible.

######  Corollary 4.14.

Let $S$ be an orientable surface. Then $\operatorname{Diff}(S\times
I)\simeq\operatorname{Diff}(S)\times\mathbb{Z}/2$.

###### Proof.

Let $S_{0}=S\times\\{0\\}$ and $S_{1}=S\times\\{1\\}$.
$\operatorname{Diff}(S\times I)$ may permute the boundary components, so we
have principal short exact sequence

$1\rightarrow\operatorname{Diff}(S\times
I,S_{0})\rightarrow\operatorname{Diff}(S\times
I)\rightarrow\mathbb{Z}/2\rightarrow 1.$

Letting $r\colon S\times I\rightarrow S\times I$ be the reflection
$r(x,t)=(x,1-t)$ we see this sequence splits. On the other hand, by
restricting to $S_{0}$ we get another (split) principal short exact sequence

$1\rightarrow\operatorname{PI}(S\times
I)\rightarrow\operatorname{Diff}(S\times
I,S_{0})\rightarrow\operatorname{Diff}(S_{0})\rightarrow 1.$

with splitting given by $f\mapsto f\times\operatorname{id}_{I}$. By Theorem
4.13, the fiber is contractible, so $\operatorname{Diff}(S\times
I,S_{0})\simeq\operatorname{Diff}(S_{0})=\operatorname{Diff}(S)$. Finally, $r$
commutes with $f\times\operatorname{id}_{I}$, so the inclusion
$\operatorname{Diff}(S)\times\mathbb{Z}/2\rightarrow\operatorname{Diff}(S\times
I)$ is a homotopy equivalence. ∎

We apply the previous corollary to the case where $S=T^{2}$.

######  Proposition 4.15.

Let $D^{3}\subset(T^{2}\times I)^{\circ}$. Then
$B\\!\operatorname{Diff}_{D^{3}}(T^{2}\times I)$ is homotopy finite.

###### Proof.

We know that
$\operatorname{Diff}(T^{2}\times[0,1])\simeq\mathbb{Z}/2\times\operatorname{Diff}(T^{2})$,
so we can write

$B\\!\operatorname{Diff}_{D^{3}}(T^{2}\times[0,1])\simeq\operatorname{Fr}(T^{2}\times[0,1])/\\!\\!/\operatorname{Diff}(T^{2}\times[0,1])\simeq\operatorname{Fr}(T^{2}\times[0,1])/\\!\\!/(\mathbb{Z}/2\times\operatorname{Diff}(T^{2})).$

There also is a homotopy fiber sequence

$\operatorname{Fr}(T^{2})\times\mathbb{Z}/2\longrightarrow\operatorname{Fr}(T^{2}\times[0,1])\longrightarrow\mathbb{R}\mathrm{P}^{2}$

where the fibration is defined by recording the line spanned by last vector in
the frame. The first map is
$\operatorname{Diff}(T^{2})\times\mathbb{Z}/2$-equivariant and the second map
$\operatorname{Diff}(T^{2})\times\mathbb{Z}/2$-invariant, so by 2.8 we get a
homotopy fiber sequence

$(\operatorname{Fr}(T^{2})\times\mathbb{Z}/2)/\\!\\!/(\operatorname{Diff}(T^{2})\times\mathbb{Z}/2)\longrightarrow\operatorname{Fr}(T^{2}\times[0,1])/\\!\\!/(\operatorname{Diff}(T^{2})\times\mathbb{Z}/2)\longrightarrow\mathbb{R}\mathrm{P}^{2}.$

The base is a finite CW complex and the fiber is homotopy equivalent to
$\operatorname{Fr}(T^{2})/\\!\\!/\operatorname{Diff}(T^{2})\simeq
B\\!\operatorname{Diff}_{D^{2}}(T^{2})$, which is homotopy finite by Lemma
4.12. ∎

To prove Theorem 4.1 for $T^{3}$, we will use the fact that
$\pi_{0}\operatorname{Diff}(T^{3})\cong\operatorname{GL}_{3}(\mathbb{Z})$ acts
on $T^{3}$ by diffeomorphisms.

######  Proposition 4.16.

Let $D^{3}\subset\smash{\mathring{T}}^{3}$ be an embedded disk. Then
$B\\!\operatorname{Diff}_{D^{3}}(T^{3})$ is homotopy finite.

###### Proof.

Since $T^{3}$ is closed and Haken, Waldhausen’s theorem [Wal68] implies that
the natural map
$\pi_{0}\operatorname{Diff}(T^{3})\rightarrow\operatorname{Out}(\pi_{1}(T^{3}))\cong\operatorname{GL}_{3}(\mathbb{Z})$
is an isomorphism. The standard action of $\operatorname{GL}_{3}(\mathbb{Z})$
on $\mathbb{R}^{3}$ preserves the $\mathbb{Z}^{3}$ lattice, hence descends to
a group of diffeomorphisms of $T^{3}=\mathbb{R}^{3}/\mathbb{Z}^{3}$. By 4.5,
the group of translations of $T^{3}$, which is equal to
$\operatorname{Isom}_{0}(T^{3})$ and which we denote by $\mathbb{T}^{3}$, is
homotopy equivalent to $\operatorname{Diff}_{0}(T^{3})$. Hence the inclusion
$\mathbb{T}^{3}\rtimes\operatorname{GL}_{3}(\mathbb{Z})\hookrightarrow\operatorname{Diff}(T^{3})$
is a homotopy equivalence.

Let $\operatorname{Fr}(T^{3})$ be the space of frames on $T^{3}$. There is a
diffeomorphism $\operatorname{Fr}(T^{3})\cong
T^{3}\times\operatorname{GL}_{3}(\mathbb{R})$, and using the standard flat
metric on $T^{3}$, this diffeomorphism is compatible with the action of
$\mathbb{T}^{3}\rtimes\operatorname{GL}_{3}(\mathbb{Z})$: the translation
action is trivial on the $\operatorname{GL}_{3}(\mathbb{R})$ factor and freely
transitive on $T^{3}$, while $\operatorname{GL}_{3}(\mathbb{Z})$ acts on
$\operatorname{GL}_{3}(\mathbb{R})$ via the standard right action. In
particular, the action is free with quotient
$\operatorname{GL}_{3}(\mathbb{R})/\operatorname{GL}_{3}(\mathbb{Z})$. By the
preceding discussion, we have

$\displaystyle
B\\!\operatorname{Diff}_{D^{3}}(T^{3})\simeq\operatorname{Fr}(T^{3})/\\!\\!/\operatorname{Diff}(T^{3})$
$\displaystyle\simeq(T^{3}\times\operatorname{GL}_{3}(\mathbb{R}))/\\!\\!/(\mathbb{T}^{3}\rtimes\operatorname{GL}_{3}(\mathbb{Z}))$
$\displaystyle\cong\operatorname{GL}_{3}(\mathbb{R})/\operatorname{GL}_{3}(\mathbb{Z}).$

To see that the latter is homotopy finite, recall that there is a fiber bundle
$\operatorname{O}(3)\hookrightarrow\operatorname{GL}_{3}(\mathbb{R})\rightarrow\mathcal{Q}_{3}$,
where $\mathcal{Q}_{3}$ is the space of positive definite $3\times 3$ matrices
over $\mathbb{R}$. By a result of Soulé [Sou78], $\mathcal{Q}_{3}$ admits a
$\operatorname{GL}_{3}(\mathbb{Z})$-equivariant deformation retraction onto a
cell complex $X_{3}\subset\mathcal{Q}_{3}$ on which the action of
$\operatorname{GL}_{3}(\mathbb{Z})$ is cocompact. Lifting this deformation
retraction to $\operatorname{GL}_{3}(\mathbb{R})$, we find a subspace
$\widetilde{X}_{3}\subset\operatorname{GL}_{3}(\mathbb{R})$ on which the
action of $\operatorname{GL}_{3}(\mathbb{Z})$ is free, properly discontinuous
and cocompact. ∎

To complete the flexible cases of 4.1, for $M_{HW}$ we will show there exists
a metric satisfying the strong generalised Smale conjecture.

######  Proposition 4.17.

Let $D^{3}\subset M_{HW}$ be an embedded disk. Then
$B\\!\operatorname{Diff}_{D^{3}}(M)$ is homotopy finite.

###### Proof.

Let $M=M_{HW}$ and $\pi=\pi_{1}(M_{HW})$. Recall that $M$ admits a flat
metric, and let $\operatorname{Aff}(M)$ denote the group of affine
diffeomorphisms of $M$. By [CV73, Theorem 1, p474], the identity component
$\operatorname{Aff}_{0}(M)$ equals
$\operatorname{Isom}_{0}(M)\cong\prod_{b}\operatorname{SO}(2)$ for any flat
metric on $M$, where $b$ is the rank of the center of $\pi$. Moreover,
$\operatorname{Aff}(M)/\operatorname{Aff}_{0}(M)\cong\operatorname{Out}(\pi)$.
The center of $\pi$ is trivial, hence
$\operatorname{Aff}(M)\cong\operatorname{Out}(\pi)$, which is a finite group
[CV73, Example 1] (see [Zim90] for a correct calculation of
$\operatorname{Out}(\pi)$). Moreover, by Corollary 4.6 and Waldhausen’s
theorem, this implies that
$\operatorname{Aff}(M)\hookrightarrow\operatorname{Diff}(M)$ is a homotopy
equivalence.

By a result of Zieschang–Zimmermann [ZZ79, Satz 3.17], for any flat manifold
$M$ and any finite subgroup $F\leq\operatorname{Aff}(M)$ there exists a flat
metric on $M$ realising $F$ by isometries. Applying this to
$\operatorname{Out}(\pi)$ itself, we find a flat metric on $M$ such that
$\operatorname{Isom}(M)\cong\operatorname{Aff}(M)\simeq\operatorname{Diff}(M)$.
Applying Lemma 4.4 completes the proof. ∎

###  4.4. Anosov torus bundles over the circle