# Mapping class groups of simply connected 4-manifolds with boundary

Patrick Orson  Department of Mathematics, California Polytechnic State
University, USA<EMAIL_ADDRESS>and  Mark Powell  School of Mathematics
and Statistics, University of Glasgow, UK<EMAIL_ADDRESS>
###### Abstract.

We compute the topological mapping class group of every compact, simply
connected, topological 4-manifold. This was previously only known in the
closed case. If the 4-manifold is smooth, we deduce an analogous description
of the stable smooth mapping class group, extending work of Saeki.

###### Key words and phrases:

Mapping class group, 4-manifolds

###### 1991 Mathematics Subject Classification:

57K40, 57N37, 57R52, 57S05.

##  1\. Introduction

Given an oriented, topological manifold $X$, with (possibly empty) boundary,
$\operatorname{Homeo}^{+}(X,\partial X)$ denotes the topological group of
orientation preserving self-homeomorphisms that restrict to the identity on
the boundary $\partial X$, with the compact-open topology. The set of
connected components $\pi_{0}\operatorname{Homeo}^{+}(X,\partial X)$ is the
_topological mapping class group_ of $X$, the group of isotopy classes of
orientation preserving self-homeomorphisms that fix the boundary pointwise. We
study topological mapping class groups for compact, oriented, simply connected
$4$-manifolds.

Let $\lambda_{X}\colon H_{2}(X)\times H_{2}(X)\to\mathbb{Z}$ be the
intersection pairing of $X$. When $\partial X=\emptyset$, it was shown by
Perron [Per86] and Quinn [Qui86, Theorem 1.1] (cf. Kreck [Kre79, Theorem 1]),
that if two orientation preserving self-homeomorphisms of $X$ induce the same
isometry of the intersection form then they are isotopic. Freedman [Fre82,
Theorem 1.5, Addendum] showed that every automorphism of the intersection form
is induced by a homeomorphism. Therefore the results of Perron-Quinn and
Freedman combine to compute the mapping class group of every closed, simply
connected 4-manifold, in the sense of reducing the problem to algebra:
$\pi_{0}\operatorname{Homeo}^{+}(X)\xrightarrow{\cong}\operatorname{Aut}(H_{2}(X),\lambda_{X});\;F\mapsto
F_{*}$.

When $X$ has nonempty boundary, we need to consider a refinement of
$\operatorname{Aut}(H_{2}(X),\lambda_{X})$ to capture the algebraic data of a
homeomorphism. A map $F\in\operatorname{Homeo}^{+}(X,\partial X)$ determines a
homomorphism $\Delta_{F}\colon H_{2}(X,\partial X)\to H_{2}(X)$ called a
_variation_ [Lam75, DK75, Kau74], defined by $[x]\mapsto[x-F(x)]$. Using that
$X$ has Poincaré-Lefschetz duality, Saeki [Sae06] showed that $\Delta_{F}$
satisfies an additional condition, making it what we call a _Poincaré
variation_ (Definition 2.9). There is a binary operation on the set of
Poincaré variations, together with which they form a group
$\mathcal{V}(H_{2}(X),\lambda_{X})$. The map $F\mapsto F_{*}$ factors through
this group via homomorphisms:

$\pi_{0}\operatorname{Homeo}^{+}(X,\partial
X)\xrightarrow{F\mapsto\Delta_{F}}\mathcal{V}(H_{2}(X),\lambda_{X})\xrightarrow{\Delta\mapsto\operatorname{Id}-\Delta\circ
j}\operatorname{Aut}(H_{2}(X),\lambda_{X}),$

where $j\colon H_{2}(X)\to H_{2}(X,\partial X)$ is the quotient map. In
general $\Delta_{F}$ contains more information than $F_{*}$. Saeki [Sae06]
used $\mathcal{V}(H_{2}(X),\lambda_{X})$ to describe the smooth stable mapping
class group for simply connected 4-manifolds with nonempty, connected
boundary.

When $\partial X$ has more than one connected component and $X$ admits a spin
structure, there is a further invariant that does not appear in the closed
case nor when the boundary is connected. For
$F\in\operatorname{Homeo}^{+}(X,\partial X)$ we may compare a (topological)
spin structure $\mathfrak{s}$ on $X$ with the induced spin structure
$F^{*}\mathfrak{s}$. The two agree on $\partial X$ because $F$ fixes the
boundary pointwise. There is a free, transitive action of $H^{1}(X,\partial
X;\mathbb{Z}/2)$ on the set of isomorphism classes of spin structures on $X$
that agree on $\partial X$, and we denote by $\Theta(F)\in H^{1}(X,\partial
X;\mathbb{Z}/2)$ the class representing the difference between $\mathfrak{s}$
and $F^{*}\mathfrak{s}$ (see Definition 2.5 for a more detailed discussion).

Our main result shows that these invariants describe the entire topological
mapping class group.

######  Theorem A.

Let $(X,\partial X)$ be a compact, simply connected, oriented, topological
$4$-manifold.

  1. (1)

When $X$ is spin, the map $F\mapsto(\Theta(F),\Delta_{F})$ induces a group
isomorphism

$\pi_{0}\operatorname{Homeo}^{+}(X,\partial
X)\xrightarrow{\cong}H^{1}(X,\partial
X;\mathbb{Z}/2)\times\mathcal{V}(H_{2}(X),\lambda_{X}).$

  2. (2)

When $X$ is not spin, the map $F\mapsto\Delta_{F}$ induces a group isomorphism

$\pi_{0}\operatorname{Homeo}^{+}(X,\partial
X)\xrightarrow{\cong}\mathcal{V}(H_{2}(X),\lambda_{X}).$

Our key contribution is injectivity of the maps in Theorem A. Let us outline
the proof strategy. First recall that a _topological pseudo-isotopy_ is a
homeomorphism $F\colon X\times I\to X\times I$ such that $F|_{\partial X\times
I}=\operatorname{Id}_{\partial X\times I}$. The restrictions
$F_{0}=F|_{X\times\\{0\\}}$ and $F_{1}:=F|_{X\times\\{1\\}}$ are said to be
_topologically pseudo-isotopic_. In this article we will classify
homeomorphisms of simply connected 4-manifolds with boundary, up to
topological pseudo-isotopy. The strategy builds on that of [Kre79, Proposition
2], and is explained in Section 5.2. In broad strokes, if we can find a
6-manifold with boundary the (capped off) mapping torus of $F$, such that the
6-manifold is a rel. boundary $h$-cobordism from $X\times[0,1]$ to itself,
then it follows that $F$ is pseudo-isotopic to the identity. Our proof
consists of an analysis of the obstructions to finding such an $h$-cobordism,
and uses Kreck’s modified surgery theory [Kre99] as the main technical tool in
its construction. With the pseudo-isotopy classification in hand, the proof
that the maps in Theorem A are injective concludes by appealing to Quinn’s
result [Qui86, Theorem 1.4] that topological pseudo-isotopy implies
topological isotopy for homeomorphisms of simply connected, compact
4-manifolds. We remark that we credit this to Quinn because Perron only stated
that topological pseudo-isotopy implies topological isotopy for closed
4-manifolds, and for smooth 4-manifolds with boundary with no 1-handles. But,
later, his proof for the closed case [Per86, Section 7] could be adapted using
[Boy86, Boy93] to recover Quinn’s result.

Of course, the injectivity in Theorem A can be applied to diffeomorphisms of
smooth 4-manifolds, yielding a topological isotopy. This is a important step
in the hunt for exotic diffeomorphisms, which is currently a topic of
considerable interest. For example Theorem A was applied in this way by Iida-
Konno-Mukherjee-Taniguchi [IKMT22].

When $X$ has nonempty, connected boundary, surjectivity of the map
$\pi_{0}\operatorname{Homeo}^{+}(X,\partial
X)\to\mathcal{V}(H_{2}(X),\lambda_{X})$ was already known, and is a
consequence of Boyer’s classification of simply connected compact 4-manifolds
with connected boundary, and a subsequent result of Saeki [Boy86, Boy93,
Sae06]. The proofs of Boyer and Saeki admit routine extensions to the case of
disconnected boundary, though, carefully written, these extensions are quite
lengthy. Rather than inflicting them on the reader, in this article we use a
quicker method, where we reduce the disconnected boundary case to the case of
connected boundary; see the proof of Theorem 6.6. To show moreover that the
map in Theorem A (1) is surjective, in particular to realise the $\Theta$
invariants topologically, requires a novel geometric construction, again in
combination with Boyer and Saeki’s results [Boy86, Boy93, Sae06].

If $X$ is smooth then a _smooth pseudo-isotopy_ is defined similarly to the
topological case, replacing homeomorphism with diffeomorphism. In this article
we will also classify diffeomorphisms of such manifolds up to smooth pseudo-
isotopy. Quinn proved [Qui86, Theorem 1.4] that smooth pseudo-isotopy implies
_smooth stable isotopy_. Here, we say that two self-diffeomorphisms $F$ and
$F^{\prime}$ of a smooth 4-manifold $X$ are _stably isotopic_ if, after
isotoping each of them to fix the same $D^{4}\subseteq X$, and using this
$D^{4}$ to perform connected sums, for some $g\geq 0$ they induce smoothly
isotopic diffeomorphisms

$F\\#\operatorname{Id},\,\,F^{\prime}\\#\operatorname{Id}\colon
X\\#^{g}S^{2}\times S^{2}\xrightarrow{\cong}X\\#^{g}S^{2}\times S^{2}.$

This operation (for $g=1$) determines a well-defined homomorphism

$\pi_{0}\operatorname{Diffeo}^{+}(X,\partial
X)\to\pi_{0}\operatorname{Diffeo}^{+}(X\\#S^{2}\times S^{2},\partial X)$

for every $X$. We may consider the corresponding colimit

$\underset{g\to\infty}{\operatorname{colim}}\,\pi_{0}\operatorname{Diffeo}^{+}(X\\#^{g}S^{2}\times
S^{2},\partial X),$

the _stable mapping class group of $X$_. For $4$-manifolds, complete unstable
smooth mapping class group computations are currently only known in some cases
and are in general beyond the state-of-the-art – we do not even know
$\pi_{0}\operatorname{Diffeo}^{+}(S^{4})$. But, by passing to the stable
group, we have been able to give a complete answer in the simply-connected
case. We expect this to contrast fruitfully with future investigations into
unstable smooth mapping class groups.

On the algebraic side, extending by the identity on the hyperbolic form
$\mathcal{H}:=(\mathbb{Z}\oplus\mathbb{Z},\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right))$ gives rise to a homomorphism
$\mathcal{V}(H_{2}(X),\lambda_{X})\to\mathcal{V}\big{(}(H_{2}(X),\lambda_{X})\oplus\mathcal{H}\big{)}$,
and we may again consider the corresponding colimit. Also note that we may
canonically identify $H^{1}(X,\partial X;\mathbb{Z}/2)$ and
$H^{1}(X\\#^{g}S^{2}\times S^{2},\partial(X\\#^{g}S^{2}\times
S^{2});\mathbb{Z}/2)$. Our smooth pseudo-isotopy classification combines with
[Qui86, Theorem 1.4] to prove injectivity in the following theorem, which
computes stable mapping class groups. Surjectivity follows from our Theorem A
and the Freedman-Quinn sum-stable smoothing theorem [FQ90, Theorem 8.6]. The
proof is given in Section 2.3.

######  Theorem B.

Let $X$ be a compact, simply connected, oriented, smooth 4-manifold.

  1. (1)

When $X$ is spin, the map $F\mapsto(\Theta(F),\Delta_{F})$ induces a group
isomorphism

$\underset{g\to\infty}{\operatorname{colim}}\,\pi_{0}\operatorname{Diffeo}^{+}(X\\#^{g}S^{2}\times
S^{2},\partial X)\xrightarrow{\cong}H^{1}(X,\partial
X;\mathbb{Z}/2)\times\underset{g\to\infty}{\operatorname{colim}}\,\mathcal{V}\big{(}(H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g}\big{)}.$

  2. (2)

When $X$ is not spin, the map $F\mapsto\Delta_{F}$ induces a group isomorphism

$\underset{g\to\infty}{\operatorname{colim}}\,\pi_{0}\operatorname{Diffeo}^{+}(X\\#^{g}S^{2}\times
S^{2},\partial
X)\xrightarrow{\cong}\underset{g\to\infty}{\operatorname{colim}}\,\mathcal{V}\big{(}(H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g}\big{)}.$

In particular, if $\Delta_{F}=\Delta_{F^{\prime}}$, and additionally in the
spin case if $\Theta(F)=\Theta(F^{\prime})$, then $F$ and $F^{\prime}$ are
smoothly stably isotopic rel. boundary. In the special case that the boundary
of $X$ is nonempty and connected, Theorem B exactly recovers [Sae06, Theorem
3.7].

In Section 7, we study the relationship of Poincaré variations to isometries
of the intersection form, which allows us in some cases to phrase Theorems A
and B in the more familiar language of isometries. To state the result, let
$\operatorname{Aut}^{\operatorname{fix}}_{\partial}(H_{2}(X),\lambda_{X})\subseteq\operatorname{Aut}(H_{2}(X),\lambda_{X})$
be the subgroup of the isometries of the intersection form $\lambda_{X}$ that
induce the identity map on the homology of the boundary and induce the trivial
permutation of the set of spin structures on the boundary (see Definitions 7.6
and 7.8 for the precise statement). When $X$ is spin all isometries of
$\lambda_{X}$ induce the trivial permutation on the boundary spin structures,
and so in the spin case
$\operatorname{Aut}^{\operatorname{fix}}_{\partial}(H_{2}(X),\lambda_{X})=\operatorname{Aut}_{\partial}(H_{2}(X),\lambda_{X})$;
see Lemma 7.10.

######  Corollary C.

Let $(X,\partial X)$ be a compact, simply connected, oriented, topological
$4$-manifold such that $\partial X$ has the rational homology of $S^{3}$ or of
$S^{1}\times S^{2}$.

  1. (1)

The map $F\mapsto F_{*}$ induces a group isomorphism

$\pi_{0}\operatorname{Homeo}^{+}(X,\partial
X)\xrightarrow{\cong}\operatorname{Aut}^{\operatorname{fix}}_{\partial}(H_{2}(X),\lambda_{X}).$

  2. (2)

When $(X,\partial X)$ is moreover smooth, the map $F\mapsto F_{*}$ induces a
group isomorphism

$\underset{g\to\infty}{\operatorname{colim}}\,\pi_{0}\operatorname{Diffeo}^{+}(X\\#^{g}S^{2}\times
S^{2},\partial
X)\xrightarrow{\cong}\underset{g\to\infty}{\operatorname{colim}}\,\operatorname{Aut}_{\partial}^{\operatorname{fix}}\big{(}(H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g}\big{)}.$

Item (1) is a new result but, as mentioned above, (2) also follows from
[Sae06, Theorem 3.7]; we include its statement here for completeness.

Finally, we phrase a consequence of our results in terms of the _Torelli
group_ of $(X,\partial X)$. The _topological Torelli group_
$\operatorname{Tor}^{0}(X,\partial
X)\subseteq\pi_{0}\operatorname{Homeo}^{+}(X,\partial X)$ (resp.  _smooth
Torelli group_ $\operatorname{Tor}^{\infty}(X,\partial
X)\subseteq\pi_{0}\operatorname{Diffeo}^{+}(X,\partial X)$) is defined as the
subgroup of mapping classes that induce the identity map on the homology of
$X$. For an element $F$ in either type of Torelli group, it is observed in
[Sae06, §4] that the Poincaré variation $\Delta_{F}\colon H_{2}(X,\partial
X)\to H_{2}(X)$ determines a map $H_{1}(\partial X)\to H_{2}(\partial X)\cong
H_{1}(\partial X)^{\ast}$, and that the Poincaré property of $\Delta_{F}$
implies that the induced pairing $H_{1}(\partial X)\times H_{1}(\partial
X)\to\mathbb{Z}$ is skew-symmetric. We write
$\kappa_{F}\in\wedge^{2}{H_{1}(\partial X)^{\ast}}$ for this skew-symmetric
pairing. Combining the work in Section 7 with Theorems A and B, we deduce the
following result.

######  Corollary D.

Let $(X,\partial X)$ be a compact, simply connected, oriented, topological
$4$-manifold.

  1. (1)

When $X$ is $\left\\{\begin{array}[]{ll}\text{spin}\\\ \text{not
spin}\end{array}\right.$, there is an isomorphism

$\left\\{\begin{array}[]{rcll}\operatorname{Tor}^{0}(X,\partial
X)&\xrightarrow{\cong}&\wedge^{2}{H_{1}(\partial X)^{\ast}}\times
H^{1}(X,\partial X;\mathbb{Z}/2);&F\mapsto(\kappa_{F},\Theta(F)).\\\
\operatorname{Tor}^{0}(X,\partial
X)&\xrightarrow{\cong}&\wedge^{2}{H_{1}(\partial
X)^{\ast}};&F\mapsto\kappa_{F}.\end{array}\right.$

  2. (2)

Suppose moreover that $X$ is smooth. When $X$ is
$\left\\{\begin{array}[]{ll}\text{spin}\\\ \text{not spin}\end{array}\right.$,
there is an isomorphism

$\left\\{\begin{array}[]{rcll}\underset{g\to\infty}{\operatorname{colim}}\,\operatorname{Tor}^{\infty}(X\\#^{g}S^{2}\times
S^{2},\partial X)&\xrightarrow{\cong}&\wedge^{2}{H_{1}(\partial
X)^{\ast}}\times H^{1}(X,\partial
X;\mathbb{Z}/2);&F\mapsto(\kappa_{F},\Theta(F)).\\\
\underset{g\to\infty}{\operatorname{colim}}\,\operatorname{Tor}^{\infty}(X\\#^{g}S^{2}\times
S^{2},\partial X)&\xrightarrow{\cong}&\wedge^{2}{H_{1}(\partial
X)^{\ast}};&F\mapsto\kappa_{F}.\end{array}\right.$

For the remainder of the introduction we describe consequences and
applications of our results.

###  1.1. Dehn twists

An important type of self-homeomorphism of 4-manifolds is the _Dehn twist_ ,
which arises as follows. Let $\varphi_{t}\in\pi_{1}(\operatorname{SO}(4))$ be
a generator based at the identity matrix, represented by a smooth map
$S^{1}\to\operatorname{SO}(4)$ that is constant near the basepoint. This
induces a smooth loop of self-diffeomorphisms of $S^{3}$, which generates
$\pi_{1}(\operatorname{Diffeo}^{+}(S^{3}))\cong\mathbb{Z}/2$, and thence a
self-diffeomorphism

$\Phi\colon S^{3}\times I\xrightarrow{\cong}S^{3}\times
I;\qquad(x,t)\mapsto(\varphi_{t}(x),t).$

Given an embedding of $S^{3}\times I$ into a 4-manifold, one can extend the
map $\Phi$ by the identity to obtain a self-homeomorphism of the entire
4-manifold, and we call any self-homeomorphism obtained this way a _Dehn
twist_. If $X$ is smooth to begin with, and $S^{3}\times I$ is smoothly
embedded, then the Dehn twist is a self-diffeomorphism.

Now let $X$ be a closed, simply connected 4-manifold and decompose
$X\setminus\mathring{D}^{4}$ as the union
$N\cup_{S^{3}\times\\{1\\}}S^{3}\times I$ of a collar neighbourhood of
$\partial(X\setminus\mathring{D}^{4})$ and the closure of its complement. The
diffeomorphism $\Phi$ induces a Dehn twist homeomorphism

$t_{X}\colon X\setminus\mathring{D}^{4}\to X\setminus\mathring{D}^{4};\qquad
y\mapsto\begin{cases}\Phi(x,t)&y=(x,t)\in S^{3}\times I,\\\ y&y\in
N.\end{cases}$

######  Theorem E.

For every closed, simply connected, topological manifold $X$, the Dehn twist
$t_{X}$ is topologically isotopic rel. boundary to
$\operatorname{Id}_{X\setminus\mathring{D}^{4}}$. If
$X\setminus\mathring{D}^{4}$ admits a smooth structure, then $t_{X}$ is
smoothly pseudo-isotopic to $\operatorname{Id}_{X\setminus\mathring{D}^{4}}$.

For the proof, note that $t_{X}$ lies in the Torelli group, and that for
$X\setminus\mathring{D}^{4}$ the right hand side of Corollary D(1) is trivial.
For the specific case of the $K3$ surface, it was already known the $t_{K3}$
was topologically isotopic to the identity (see [KK21, Remark 6], which uses
results of Baraglia-Konno [BK22]). This result was used recently by Lin-
Mukherjee [LM21]. For general smooth $X$, it was not previously known that
$t_{X}$ is smoothly pseudo-isotopic to the identity. A special case of our
Theorem B is Krannich-Kupers’ [KK21, Theorem B], which was proven by a totally
different method, and states that $t_{X}$ is stably smoothly isotopic to
$\operatorname{Id}_{X\setminus\mathring{D}^{4}}$. (Krannich-Kupers’ [KK21,
Theorem B] also follows directly from Saeki’s [Sae06, Theorem 3.7].)

An explicit geometric argument of Giansiracusa shows that
$t_{\operatorname{\mathbb{C}P}^{2}}$ is smoothly isotopic to the identity
[Gia08]. This result can be extended to show that $t_{X}$ is smoothly isotopic
to the identity for any non-spin, smooth, simply connected, closed
$4$-manifold $X$; this argument was communicated to us by Dave Auckly, and we
include the proof in Appendix A. On the other hand, it was shown independently
by Baraglia-Konno [BK22] and Kronheimer-Mrowka [KM20] that $t_{K3}$ is not
smoothly isotopic to the identity. In light of Theorem E, this prompts the
obvious question.

######  Question 1.1.

For which closed, spin, simply connected, smooth manifolds $X$ is $t_{X}$
_smoothly_ isotopic rel. boundary to the identity?

Arguably, the particular interest in Dehn twists comes from the fact that they
lie in the Torelli group. We now consider extensions of the idea of a Dehn
twist, with Corollary D in mind.

###  1.2. Generalisations of Dehn twists

Let $Y\neq S^{3}$ be a closed, oriented 3-manifold and suppose there is a
nontrivial loop $\varphi_{t}\in\pi_{1}(\operatorname{Diffeo}^{+}(Y))$, based
at the identity. Represent it by a smooth map
$S^{1}\to\operatorname{Diffeo}^{+}(Y)$, and write $\Phi\colon Y\times[0,1]\to
Y\times[0,1]$ for the induced self-diffeomorphism. Not all $3$-manifolds have
such a loop, for instance diffeomorphism groups of hyperbolic 3-manifolds are
homotopy equivalent to their isometry group [Gab01], and are thus discrete. On
the other hand every Seifert fibred $3$-manifold admits such a nontrivial loop
of self-diffeomorphisms, as we show in Proposition 8.8. We formulate a
generalised version of Question 1.1, using these types of diffeomorphism, the
answer to which in the topological category is “no” by Corollary D.

######  Question 1.2.

Is there a simply connected $4$-manifold $X$ with connected boundary $Y\neq
S^{3}$, and a nontrivial loop in
$\varphi_{t}\in\pi_{1}(\operatorname{Diffeo}^{+}(Y))$ such that applying the
corresponding $\Phi$ in a collar neighbourhood $Y\times[0,1]$ of the boundary
gives a self-diffeomorphism of $F$ of $X$, such that
$\kappa_{F}=0\in\wedge^{2}{H_{1}(\partial X)^{\ast}}$, but such that $F$ is
not smoothly isotopic to the identity?

While Theorem D shows that boundary Dehn twists, and their generalisations
considered above, are always topologically isotopic to the identity when the
boundary is connected, this is not the case when $\partial X$ has more than
one connected components. Indeed, Dehn twists are the prototypical example of
self-homeomorphisms with nontrivial invariant $\Theta$.

######  Example 1.3.

The Dehn twist on $S^{3}\times[0,1]$ gives a homeomorphism that is not
isotopic rel. boundary to the identity, despite necessarily acting trivially
on $H_{2}(S^{3}\times[0,1])=0$. This is detected by the nontrivial action of
the Dehn twist on the set of relative spin structures, which has cardinality
$2=|H^{1}(S^{3}\times[0,1],S^{3}\times\\{0,1\\};\mathbb{Z}/2)|$. See Example
8.4 for details.

To generalise this, suppose there is a smooth loop in
$\pi_{1}(\operatorname{Diffeo}^{+}(Y))$, based at the identity, and a spin
structure on $Y\times[0,1]$, such that the induced self-diffeomorphism of
$Y\times[0,1]$ has nontrivial $\Theta$ invariant. We call such a self-
diffeomorphism a _generalised Dehn twist_ (see Definition 8.2). We make a
preliminary study of which $3$-manifolds admit generalised Dehn twists in
Section 8, where we prove the following proposition.

######  Proposition 1.4.

Every closed, orientable 3-manifold $Y$ of Heegaard genus at most one admits a
generalised Dehn twist for every spin structure on $Y\times[0,1]$.

Recall that the 3-manifolds of Heegaard genus at most one are $S^{3}$,
$S^{1}\times S^{2}$, and the lens spaces $L(p,q)$. If all (except possibly
one) boundary components of a smooth, spin $4$-manifold admit generalised Dehn
twists, we show in Proposition 8.3 that every possible value of $\Theta$ can
be realised smoothly. However, we expect that in general, for a smooth, spin
$4$-manifold, not all values of $\Theta$ can be realised by a (boundary
fixing) self-diffeomorphism. This would contrast with the corresponding
statement from the topological category, coming from Theorem A. Our
expectation is based on the fact that not all 3-manifolds admit generalised
Dehn twists; for example as mentioned above hyperbolic 3-manifolds do not.

On the other hand, we have not proven any results limiting which values can be
taken by $\Theta(F)$, by $\Delta_{F}$, or indeed by the isometry invariant
$F_{*}\in\operatorname{Aut}_{\partial}^{\operatorname{fix}}(H_{2}(X),\lambda_{X})$,
when $F\in\operatorname{Diffeo}^{+}(X,\partial X)$. The question of which
isometries of the intersection form can be realised by diffeomorphisms has
been well-studied, although so far only in the case that the form is
nonsingular. For closed, simply connected $4$-manifolds, Wall [Wal64] gave
conditions guaranteeing all isometries can be realised. In the other
direction, there are examples where some isometries cannot be realised; see
[FM88], [Don90], [MS97], [Bar21, Remark 1.10] for closed manifold examples,
and [KT20, Remark 1.8] for examples with homology 3-sphere boundary. See
[KT20, Section 4.3] for a detailed discussion. On the other hand, the $\Theta$
and $\kappa$ invariants have not been studied in this context, and the
following questions represent new avenues for finding differences between the
smooth and topological categories.

######  Question 1.5.

Is there a smooth, spin, simply connected $4$-manifold $X$ for which the
homomorphism $\Theta\colon\pi_{0}\operatorname{Diffeo}^{+}(X,\partial X)\to
H^{1}(X,\partial X;\mathbb{Z}/2)$ is not surjective?

Similarly, the following question represents a new approach for finding
differences between the smooth and topological categories.

######  Question 1.6.

Is there a smooth, simply connected $4$-manifold $X$ for which
$\operatorname{Tor}^{\infty}(X,\partial X)\to\wedge^{2}{H_{1}(\partial
X)^{\ast}}$, given by $F\mapsto\kappa_{F}$, is not surjective?

Corollary D implies that for every smooth $(X,\partial X)$, and any pair
$(\kappa,x)\in\wedge^{2}{H_{1}(\partial X)^{\ast}}\times H^{1}(X,\partial
X;\mathbb{Z}/2)$ there exists $g\geq 0$ and
$F\in\operatorname{Diffeo}^{+}(X\\#^{g}S^{2}\times S^{2},\partial X)$
realising the pair $(\kappa,x)$. Thus one might augment Questions 1.5 and 1.6
in terms of how many copies of $S^{2}\times S^{2}$ are required to realise all
values of $\Theta$ or of $\kappa$, for a given $X$. Is one stabilisation
always sufficient?

###  1.3. Isotopy of $\mathbb{Z}$-surfaces

One way to apply knowledge of mapping class groups is to the classification of
knotted surfaces in 4-manifolds up to ambient isotopy. A programme to show
that two locally flat, embedded surfaces $\Sigma_{0}$ and $\Sigma_{1}$ in
$X^{4}$ are topologically isotopic is as follows. First show that the
exteriors $X\setminus\nu\Sigma_{0}$ and $X\setminus\nu\Sigma_{1}$ are
homeomorphic relative to the identity of $\partial X$, and restricting to a
homeomorphism
$\partial\overline{\nu}\Sigma_{0}\to\partial\overline{\nu}\Sigma_{1}$ that
extends to a bundle isomorphism
$\overline{\nu}\Sigma_{0}\to\overline{\nu}\Sigma_{1}$. This yields a
homeomorphism $F\colon X\to X$ that restricts to $\operatorname{Id}_{\partial
X}$, with $F(\Sigma_{0})=\Sigma_{1}$. Assume that $X$ is compact and simply
connected and that $\partial X$ is connected. Then we ask whether $F$ induces
the identity on $H_{2}(X)$. If so then Corollary C yields a topological
ambient isotopy carrying $\Sigma_{1}$ to $\Sigma_{0}$, as we will explain in
the proof of Theorem F below.

An immediate application of Corollary C, via this method, is to strengthen a
theorem of Conway and the second author [CP21, Theorem 1.3] on
$\mathbb{Z}$-surfaces. A _$\mathbb{Z}$ -surface_ is a compact, connected,
oriented surface with one boundary component $\Sigma\subseteq X$, locally
flatly embedded in a simply connected, compact 4-manifold $X$ with $\partial
X=S^{3}$, with $\partial\Sigma\subseteq\partial X$, such that
$\pi_{1}(X\setminus\nu\Sigma)\cong\mathbb{Z}$, generated by a meridian of
$\Sigma$. Write $X_{\Sigma}:=X\setminus\nu\Sigma$.

We need a little more setup. The inclusion induced map $\pi_{1}(\partial
X_{\Sigma})\to\pi_{1}(X_{\Sigma})$ induces a $\mathbb{Z}$-cover of $\partial
X_{\Sigma}$. Write $\Lambda:=\mathbb{Z}[\mathbb{Z}]$,
$Y:=S^{3}\setminus\nu\partial\Sigma$, and let $Z:=\overline{\partial
X_{\Sigma}\setminus Y}$. Then $\partial X_{\Sigma}=Y\cup Z$ and the homology
$H_{1}(\partial X_{\Sigma};\Lambda)$ splits correspondingly as
$H_{1}(Y;\Lambda)\oplus H_{1}(Z;\Lambda)$. An isometry of the Blanchfield
pairing $h\colon H_{1}(\partial X_{\Sigma};\Lambda)\to H_{1}(\partial
X_{\Sigma};\Lambda)$ respects this splitting, decomposing as $h=h_{Y}\oplus
h_{Z}$ [CP21, Proposition 5.7]. Given exteriors $X_{i}:=X_{\Sigma_{i}}$ for
$i=0,1$, of two $\mathbb{Z}$-surfaces in $X$, it is shown in [CP21, Lemma
5.10] that an isomorphism

$\varphi\colon H_{2}(X_{0};\Lambda)\xrightarrow{\cong}H_{2}(X_{1};\Lambda)$

inducing an isometry of the $\Lambda$-valued intersection pairings uniquely
determines an element
$\varphi_{\mathbb{Z}}\in\operatorname{Aut}_{\partial}(H_{2}(X),\lambda_{X})$.
Moreover, if $\varphi$ is induced by a homeomorphism
$F\in\operatorname{Homeo}^{+}(X,\partial X)$, then
$F_{*}=\varphi_{\mathbb{Z}}$. (That lemma shows this when $X$ is closed, but
the proof in our case is identical.)

In [CP21, Theorem 1.3], hypotheses were given which imply there is a
homeomorphism of pairs $(X,\Sigma_{0})\cong(X,\Sigma_{1})$ inducing $\varphi$
on $\Lambda$-coefficient homology. Using Corollary C, we upgrade this to
ambient isotopy, assuming in addition that $\varphi_{\mathbb{Z}}$ is the
identity.

######  Theorem F.

Let $\Sigma_{0},\Sigma_{1}\subseteq X$ be $\mathbb{Z}$-surfaces with
$\partial\Sigma_{0}=\partial\Sigma_{1}\subseteq\partial X\cong S^{3}$. Write
$X_{i}:=X_{\Sigma_{i}}$ for $i=0,1$. Let $\varphi\colon
H_{2}(X_{0};\Lambda)\xrightarrow{\cong}H_{2}(X_{1};\Lambda)$ be an isomorphism
inducing an isometry of the $\Lambda$-valued intersection pairings of the
exteriors, and write $\partial\varphi=h_{Y}\oplus h_{Z}\colon H_{1}(\partial
X_{\Sigma};\Lambda)\xrightarrow{\cong}H_{1}(\partial X_{\Sigma};\Lambda)$ with
respect to the splitting $H_{1}(\partial X_{\Sigma};\Lambda)\cong
H_{1}(Y;\Lambda)\oplus H_{1}(Z;\Lambda)$.

Suppose that $h_{Y}$ is induced by a homeomorphism $f\colon
Y\xrightarrow{\cong}Y$ and
$\varphi_{\mathbb{Z}}=\operatorname{Id}_{H_{2}(X)}$. Then $\Sigma_{0}$ and
$\Sigma_{1}$ are ambiently isotopic. If in addition $h_{Y}=\operatorname{Id}$
then $\Sigma_{0}$ and $\Sigma_{1}$ are ambiently isotopic rel. boundary.

###### Proof.

Under the additional assumption that $f|_{\partial
Y}=\operatorname{Id}_{\partial Y}$, it was proved in [CP21, Theorem 1.3] that
the hypotheses imply there is a homeomorphism of pairs
$F\colon(X,\Sigma_{0})\to(X,\Sigma_{1})$ such that $F$ induces $\varphi$, and
that in addition if $h_{Y}=\operatorname{Id}$ then $F$ may be chosen so that
$F|_{\partial X}=\operatorname{Id}_{\partial X}$. It is shown in the proof of
[CPP22, Theorem 5.6] that the additional assumption $f|_{\partial
Y}=\operatorname{Id}_{\partial Y}$ was superfluous, and so we obtain such a
homeomorphism of pairs $F$.

If $F|_{\partial X}\neq\operatorname{Id}_{\partial X}$ then we use the fact
that $\pi_{0}\operatorname{Homeo}^{+}(S^{3})$ is trivial to choose an isotopy
$\Phi\colon\partial X\times[0,1]\to\partial X$ from $\Phi(-,0)=F|_{\partial
X}$ to $\Phi(-,1)=\operatorname{Id}_{\partial X}$. Choose a boundary collar
$\partial X\times[0,1]\subseteq X$ so that $\partial X\times\\{1\\}=\partial
X$, and arrange by an isotopy rel. boundary that $F|_{\partial
X\times[0,1]}=F|_{\partial X}\times\operatorname{Id}_{[0,1]}$. Define an
isotopy $G\colon X\times[0,1]\to X$ by $G((x,s),t):=\Phi(x,s-t)$ for $t\leq
s\leq 1$ and $G((x,s),t):=(F(x),s)$ for $0\leq s\leq t$. This is an isotopy
from $F=G(-,1)$ to a new homeomorphism $F^{\prime}:=G(-,0)$ such that
$F^{\prime}|_{\partial X}=\operatorname{Id}_{\partial X}$.

Note that
$(F^{\prime})_{*}=F_{*}=\varphi_{\mathbb{Z}}=\operatorname{Id}_{H_{2}(X)}$. As
$\partial X=S^{3}$, Corollary C now implies that $F^{\prime}$ is isotopic rel.
boundary to $\operatorname{Id}_{X}$. Since $F(\Sigma_{0})=\Sigma_{1}$, we have
obtained an ambient isotopy from $\Sigma_{0}$ to $\Sigma_{1}$. If in addition
$h_{Y}=\operatorname{Id}$ then we saw above that we may assume $F|_{\partial
X}=\operatorname{Id}_{\partial X}$, so the modification from $F$ to
$F^{\prime}$ was unnecessary, and the ambient isotopy from $\Sigma_{0}$ to
$\Sigma_{1}$ fixes $\partial X$ pointwise throughout. ∎

###  1.4. Homeomorphisms not restricting to the identity on the boundary

We consider the implications of our results when we relax the assumption that
homeomorphisms must fix the boundary pointwise. Let $X$ be a compact,
oriented, simply connected 4-manifold. There is a fibre sequence

$\operatorname{Homeo}^{+}(X,\partial
X)\to\operatorname{Homeo}^{+}(X)\to\operatorname{Homeo}^{+}(\partial X)$

(see Appendix A for justification). Consequently there is an exact sequence in
homotopy groups, extending to the left,

(7) $\pi_{1}\operatorname{Homeo}^{+}(\partial
X)\to\pi_{0}\operatorname{Homeo}^{+}(X,\partial
X)\to\pi_{0}\operatorname{Homeo}^{+}(X)\to\pi_{0}\operatorname{Homeo}^{+}(\partial
X).$

Here, the first arrow can be defined by inserting the given loop of
diffeomorphisms of $\partial X$ (based at $\operatorname{Id}_{\partial X}$)
into a collar of the boundary. Taking the basepoint of each group of
homeomorphisms to be the respective identity map, the sequence (7) is an exact
sequence of groups. Here the $\pi_{0}$ terms are also groups because they are
connected components of topological groups. The sequence suggests that the
problem of whether two homeomorphisms $F_{1},\,F_{2}\colon(X,\partial
X)\to(X,\partial X)$ are isotopic in $\operatorname{Homeo}^{+}(X)$ can be
decomposed into two stages, as follows.

The first-stage question is purely about $3$-manifolds: are $F_{1}|_{\partial
X}$ and $F_{2}|_{\partial X}$ isotopic? This is a highly nontrivial question
in general, but thanks to the current spectacular understanding of
3-manifolds, we have a good chance of being able to decide. Self-
homeomorphisms of $\partial X$ must respect the prime decomposition [Kne29,
Mil62] and the JSJ decomposition [JS79, Joh79]; see also [Hat07]. Restricting
to geometric pieces it often suffices to understand the isometry groups (in
the sense of Riemannian geometry), by [Gab01, HKMR12, BK21, BK17] and the
references therein. For simple 3-manifolds their mapping class groups were
known earlier. For lens spaces the mapping class groups were computed by
Bonahon [Bon83], while for Seifert fibred spaces in general see e.g. [BO91].
For Haken 3-manifolds, Hatcher and Ivanov [Hat76, Iva79] showed that the
mapping class group equals the group of homotopy self-equivalences. So with
enough work, the first-stage question can in principle be answered with our
current knowledge.

If there is no isotopy between $F_{1}|_{\partial X}$ and $F_{2}|_{\partial
X}$, then certainly $F_{1}$ and $F_{2}$ are not isotopic. So let us assume
that the 3-manifold question has been solved affirmatively. Then, after an
isotopy of $F_{1}$ supported in a collar of $\partial X$ we can assume that
$F_{1}|_{\partial X}=F_{2}|_{\partial X}$. We may ask the second-stage
question: is $G:=F_{2}\circ F_{1}^{-1}\in\operatorname{Homeo}^{+}(X,\partial
X)$ in the image of $\pi_{1}\operatorname{Homeo}^{+}(\partial X)$? By Theorem
A, when $X$ is not spin (resp. is spin), this is equivalent to the question of
whether $\Delta_{G}$ (resp. the pair $(\Delta_{G},\Theta(G))$), can be killed
by inserting a loop of self-homeomorphisms on $\partial X$ in a boundary
collar. Equivalently: can the algebraic invariant be _realised_ by such a
collar loop? As a loop in the collar can never produce a nontrivial
automorphism of $H_{2}(X)$, if $G_{*}\colon H_{2}(X)\to H_{2}(X)$ is
nontrivial then certainly $G$ is not in the image of
$\pi_{1}\operatorname{Homeo}^{+}(\partial X)$, and thus $F_{1}$ and $F_{2}$
are not isotopic in $\operatorname{Homeo}^{+}(X)$. So, let us assume that $G$
determines an element of the topological Torelli group. By Corollary D, we
finally arrive at the question: can the invariant $\kappa_{G}$ (resp. the pair
$(\kappa_{G},\Theta(G))$, in the spin case) be realised by a loop of
homeomorphisms, or equivalently by a loop of diffeomorphisms [Cer61, Hat83],
in a collar of the boundary?

In some cases, $\pi_{1}\operatorname{Homeo}^{+}(\partial X)=0$ and so it
causes no additional complications. A general condition for this is as
follows.

######  Proposition 1.8.

Let $X$ be a compact, simply connected, oriented, topological $4$-manifold and
suppose that every connected component of $\partial X$ is irreducible but not
Seifert fibred. Then $\pi_{1}\operatorname{Homeo}^{+}(\partial X)=0$ and so
there is exact sequence of groups

$0\to\pi_{0}\operatorname{Homeo}^{+}(X,\partial
X)\to\pi_{0}\operatorname{Homeo}^{+}(X)\to\pi_{0}\operatorname{Homeo}^{+}(\partial
X).$

Theorem A describes the left group, although in the relevant cases the
formulation in Corollary C could be more useful for applications. So in the
case that every connected component of $\partial X$ is irreducible but not
Seifert fibred, the two-stage process discussed above can be completed. We
also note that the image of the right hand map was described precisely by
Boyer [Boy86, Boy93].

###### Proof of Proposition 1.8.

Let $Y$ be a connected component of $\partial X$. Since $Y$ is irreducible and
is not Seifert fibred, it is either hyperbolic, or it has a nontrivial JSJ
decomposition. In the latter case $Y$ is Haken because a JSJ torus is an
incompressible surface. We argue that the homeomorphism group of $Y$ has
trivial fundamental group. For hyperbolic 3-manifolds this follows from
[Gab01], which shows that
$\operatorname{Homeo}(Y)\simeq\operatorname{Isom}(Y)$, since isometry groups
of hyperbolic 3-manifolds are discrete. For a closed, oriented Haken
3-manifold $Y$, Hatcher and Ivanov [Hat76, Iva79] showed that
$\pi_{1}\operatorname{Homeo}^{+}(Y)$ is isomorphic to the centre of
$\pi_{1}(Y)$, and Waldhausen [Wal67] showed that in such cases if $Y$ is not
Seifert-fibred then the centre of $\pi_{1}(Y)$ is trivial. So
$\pi_{1}(\operatorname{Homeo}^{+}(\partial X))$ is trivial as asserted.
Applying this to the exact sequence (7) yields the displayed exact sequence. ∎

In Section 8, we consider Seifert fibred 3-manifold boundary components, and
study the problem of realising the $\Theta$ and $\kappa$ invariants using
loops of diffeomorphisms in a boundary collar. A consequence of that work is
the following.

######  Corollary G.

Let $X$ be a compact, simply connected, oriented, topological $4$-manifold.
Suppose that every connected component of $\partial X$ has Heegaard genus at
most 1, and at most one of the connected components is $S^{1}\times S^{2}$.
Then there is an exact sequence of groups

$0\to\operatorname{Aut}^{\operatorname{fix}}_{\partial}(H_{2}(X),\lambda_{X})\to\pi_{0}\operatorname{Homeo}^{+}(X)\to\pi_{0}\operatorname{Homeo}^{+}(\partial
X).$

###### Proof.

Our assumptions imply $H_{1}(\partial X;\mathbb{Q})$ has rank at most 1. By
Corollary 7.15 this implies $\pi_{0}\operatorname{Homeo}^{+}(X,\partial X)$ is
isomorphic to
$\operatorname{Aut}^{\operatorname{fix}}_{\partial}(H_{2}(X),\lambda_{X})$
when $X$ is not spin, and to
$\operatorname{Aut}^{\operatorname{fix}}_{\partial}(H_{2}(X),\lambda_{X})\times
H^{1}(X,\partial X;\mathbb{Z}/2)$ when $X$ is spin. When $X$ is not spin, the
corollary follows, by considering the exact sequence (7) and the fact that the
image of $\pi_{1}\operatorname{Homeo}^{+}(\partial X)$ in this sequence
necessarily produces the trivial automorphism of $H_{2}(X)$.

Now suppose $X$ is spin. Write $\mathfrak{s}$ for the spin structure. As each
boundary component has Heegaard genus at most 1, we may apply Proposition 8.7
to see that each boundary component has a loop of diffeomorphisms producing a
generalised Dehn twist with respect to the restriction of $\mathfrak{s}$. Now
Proposition 8.3 shows that any given $\Theta$ can be realised under the action
of $\pi_{1}$ in the sequence (7). Hence the cokernel of the map
$\pi_{0}\operatorname{Homeo}^{+}(X,\partial
X)\to\pi_{0}\operatorname{Homeo}^{+}(X)$ is isomorphic to
$\operatorname{Aut}^{\operatorname{fix}}_{\partial}(H_{2}(X),\lambda_{X})$.
The claimed result now follows from the exactness of the sequence (7). ∎

### Organisation

In Section 2 we define the pseudo-isotopy invariants and carefully state our
classification of homeomorphisms up to pseudo-isotopy. In Section 3 we recall
the tools from Kreck’s modified surgery that we will need, and we start to
apply them by determining the relevant normal 2-types. In Section 4 we recall
the James spectral sequence and use it make computations of bordism groups
relevant to the pseudo-isotopy classification. Section 5 contains the proof of
the pseudo-isotopy classification in both the smooth and topological
categories. In Section 6 we show how to realise the invariants topologically
and hence prove surjectivity of the maps from Theorem A. In Section 7 we
establish the relationship between Poincaré variations and isometries of the
intersection form, allowing alternative formulations of the main theorems. In
Section 8 we discuss the problem of realising generalised Dehn twists
smoothly, in particular proving Proposition 1.4.

### Acknowledgments

We both gratefully acknowledge the MPIM, where we were visitors for part of
the time this paper was written. MP warmly thanks ETH Zürich and Peter Feller
for hospitality during a productive visit. We thank Dave Auckly, Daniel
Kasprowski, Hokuto Konno, Sander Kupers, Jianfeng Lin, Andrew Lobb, Anubhav
Mukherjee, Lisa Piccirillo, and Danny Ruberman for interesting discussions or
helpful questions about these results. We thank Peter Teichner for suggesting
a quicker approach than we were previously using for improving Boyer’s
homeomorphism realisation result to the case of multiple boundary components.

PO was partially supported by the SNSF Grant 181199, and MP was partially
supported by EPSRC New Investigator grant EP/T028335/1 and EPSRC New Horizons
grant EP/V04821X/1.

##  2\. Statement of the main pseudo-isotopy classification

In this section we carefully define the objects and maps involved in our
pseudo-isotopy classification and then state the classification.

######  Definition 2.1.

Write $\operatorname{Diff}$ for the category of smooth manifolds and smooth
embeddings. Write $\operatorname{TOP}$ for the category of topological
manifolds and locally flat embeddings. We let $\operatorname{CAT}$ denote
either $\operatorname{Diff}$ or $\operatorname{TOP}$ unless explicitly
specified or the specification is clear from context.

For this section, fix $(X,\partial X)$ a compact, connected,
$\operatorname{CAT}$ $4$-manifold with (possibly empty) boundary.

######  Definition 2.2.

When $\operatorname{CAT}=\operatorname{TOP}$, let
$\operatorname{Homeo}^{+}(X,\partial X)$ denote the space of orientation
preserving self-homeomorphisms of $X$ that restrict to the identity map on the
boundary, topologised using the compact-open topology (see [Hat02, Appendix]
for the definition). When $\operatorname{CAT}=\operatorname{Diff}$, let
$\operatorname{Diffeo}^{+}(X,\partial X)$ denote the space of orientation
preserving self-diffeomorphisms of $X$ that restrict to the identity map on
the boundary, topologised using the Whitney topology (see [Hir76, § 2.1] for
the definition).

######  Remark 2.3.

Note that if $\partial X$ is nonempty and $X$ is connected, then for an
automorphism $f\colon X\to X$, we have that $f|_{\partial
X}=\operatorname{Id}_{\partial X}$ implies that $f$ is automatically
orientation preserving. Although this means the “$+$” is redundant notation in
these cases, we retain this notation so that our statements also hold in the
case that $\partial X=\emptyset$.

######  Definition 2.4.

For $i=0,1$ let $F_{i}\colon X\xrightarrow{\cong}X$ be an orientation
preserving $\operatorname{CAT}$ isomorphism restricting to the identity map on
the boundary. We say $F_{0}$ and $F_{1}$ are _$\operatorname{CAT}$ pseudo-
isotopic rel. boundary_ if there exists a $\operatorname{CAT}$ isomorphism
$\Psi\colon X\times[0,1]\to X\times[0,1]$ such that $\Psi(x,i)=(F_{i}(x),i)$
for $i=0,1$ and $\Psi(x,t)=(x,t)$ for all $x\in\partial X$ and $t\in[0,1]$.

We write

$\begin{array}[]{rclll}\widetilde{\pi}_{0}\operatorname{Homeo}^{+}(X,\partial
X)&:=&\operatorname{Homeo}^{+}(X,\partial X)&/&\text{$\operatorname{TOP}$
pseudo-isotopy rel.~{}boundary},\\\
\widetilde{\pi}_{0}\operatorname{Diffeo}^{+}(X,\partial
X)&:=&\operatorname{Diffeo}^{+}(X,\partial X)&/&\text{$\operatorname{Diff}$
pseudo-isotopy rel.~{}boundary}.\end{array}$

Note these are groups under composition; we will call them the _pseudo mapping
class groups_.

###  2.1. Spin structures relative to the boundary

Write $\gamma\colon\operatorname{BTOPSpin}\to\operatorname{BSTOP}$ for a
principal fibration corresponding to the universal cover
$\operatorname{TOPSpin}\to\operatorname{STOP}$. A _spin structure_
$\mathfrak{s}$ on an oriented stable topological bundle $v\colon
X\to\operatorname{BSTOP}$ is a map such that the following diagram

${\operatorname{BTOPSpin}}$${X}$${\operatorname{BSTOP}}$$\scriptstyle{\gamma}$$\scriptstyle{\mathfrak{s}}$$\scriptstyle{v}$

commutes. For a stable vector bundle, the definition is the same, with
$\operatorname{BSO}$ and $\operatorname{BSpin}$ replacing
$\operatorname{BSTOP}$ and $\operatorname{BTOPSpin}$. For $X$ a
$\operatorname{TOP}$ (resp. $\operatorname{Diff}$) manifold, a spin structure
_on $X$_ means a spin structure on the stable topological normal bundle (resp.
stable normal vector bundle).

Suppose $\mathfrak{s}_{0},\mathfrak{s}_{1}\colon X\to\operatorname{BTOPSpin}$
(resp. $\operatorname{BSpin}$) are two spin structures on $X$. Write
$\partial\mathfrak{s}_{0}$ and $\partial\mathfrak{s}_{1}$ for the respective
restrictions to the boundary $\partial X$. We say $\mathfrak{s}_{0}$ and
$\mathfrak{s}_{1}$ are _isomorphic_ , and write
$\mathfrak{s}_{0}\simeq\mathfrak{s}_{1}$, if there is a path of spin
structures $\mathfrak{s}_{t}$ on $v$, from $\mathfrak{s}_{0}$ to
$\mathfrak{s}_{1}$; in other words if $\mathfrak{s}_{0}$ and
$\mathfrak{s}_{1}$ are homotopic through spin structures on $v$. Note that an
isomorphism between $\mathfrak{s}_{0}$ and $\mathfrak{s}_{1}$ induces an
isomorphism between $\partial\mathfrak{s}_{0}$ and $\partial\mathfrak{s}_{1}$.
For example, when a manifold admits a single isomorphism class of spin
structure there is a unique isomorphism class of spin structure on the
boundary that extends to the interior. By a standard obstruction-theoretic
argument, there is a free, transitive action of $H^{1}(X;\mathbb{Z}/2)$ on the
set of isomorphism classes of spin structures on $X$. In particular, when $X$
is simply connected $H_{1}(X;\mathbb{Z}/2)=0$ and hence by the universal
coefficient theorem $H^{1}(X;\mathbb{Z}/2)=0$. So such a manifold has a unique
isomorphism class of spin structures.

We will also need a more restrictive notion of isomorphism of spin structure.
We say that $\mathfrak{s}_{0}$ and $\mathfrak{s}_{1}$ are _isomorphic rel.
boundary_ if $\partial\mathfrak{s}_{0}=\partial\mathfrak{s}_{1}$, and
$\mathfrak{s}_{0}$ is homotopic to $\mathfrak{s}_{1}$ through spin structures
on $v$ that all restrict to $\partial\mathfrak{s}_{0}$ on $\partial X$. This
is, in other words, a spin structure $\mathfrak{S}\colon
X\times[0,1]\to\operatorname{BTOPSpin}$ (resp. $\operatorname{BSpin}$) that
restricts to

$\partial(X\times[0,1])\to\operatorname{BTOPSpin}\text{
(resp.~{}$\operatorname{BSpin}$)};\quad(x,t)\mapsto\left\\{\begin{array}[]{ll}\mathfrak{s}_{0}(x)&\text{for
$(x,t)\in X\times\\{1\\}$,}\\\ \mathfrak{s}_{1}(x)&\text{for $(x,t)\in
X\times\\{0\\}$,}\\\ \partial\mathfrak{s}_{0}(x)&\text{for $(x,t)\in\partial
X\times(0,1)$.}\end{array}\right.$

By a standard obstruction-theoretic argument, there is a free, transitive
action of $H^{1}(X,\partial X;\mathbb{Z}/2)$ on the set of homotopy classes of
spin structures on $X$, relative to some given spin structure on $\partial X$.
In particular, when $X$ is simply connected and has $r>0$ boundary components,
it has $|H^{1}(X,\partial X;\mathbb{Z}/2)|=|(\mathbb{Z}/2)^{r-1}|=2^{r-1}$
rel. boundary spin structures up to isomorphism.

######  Definition 2.5.

Let $(X,\mathfrak{s})$ be a $\operatorname{CAT}$ manifold with a spin
structure and let $F\colon X\xrightarrow{\cong}X$ be an orientation preserving
$\operatorname{CAT}$ isomorphism restricting to the identity map on the
boundary. We write the class representing the difference between the spin
structures $\mathfrak{s}$ and $\mathfrak{s}\circ F$ rel. boundary, as

$\Theta(F,\mathfrak{s})\in H^{1}(X,\partial X;\mathbb{Z}/2).$

When $X$ is simply connected, this class is independent of the choice of spin
structure (see Lemma 2.7), and in which case we write

$\Theta(F)\in H^{1}(X,\partial X;\mathbb{Z}/2).$

When $X$ has a smooth structure, Definition 2.5 allows us the option of
working with the $\operatorname{CAT}=\operatorname{Diff}$ definition or
passing to the underlying topological manifold and working with the
$\operatorname{CAT}=\operatorname{TOP}$ definition. In this situation, the two
definitions return the same cohomology class, as we now discuss.

Recall that
$\pi_{1}(\operatorname{SO})=\mathbb{Z}/2=\pi_{1}(\operatorname{STOP})$ and
that the forgetful map $f\colon\operatorname{SO}\to\operatorname{STOP}$
induces the identity map on $\pi_{1}$. Hence there is a homotopy commutative
diagram

${\operatorname{Spin}}$${\operatorname{SO}}$${B(\mathbb{Z}/2)}$${\operatorname{BSpin}}$${\operatorname{BSO}}$${\operatorname{TOPSpin}}$${\operatorname{STOP}}$${B(\mathbb{Z}/2)}$${\operatorname{BTOPSpin}}$${\operatorname{BSTOP}}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{h}$$\scriptstyle{Bg}$$\scriptstyle{Bf}$

where $g$ is the map of universal covers induced by $f$, and $h$ is a map
homotopic to the identity map. Suppose $X$ is a simply connected, smooth
manifold and that there is a spin structure $\mathfrak{s}\colon
X\to\operatorname{BSpin}$ on the stable normal vector bundle $\nu_{X}\colon
X\to\operatorname{BSO}$. The induced map $Bf\circ\nu_{X}$ is the stable
topological normal bundle of $X$ and $Bg\circ\mathfrak{s}$ is the induced
topological spin structure on $Bf\circ\nu_{X}$. For each $x\in
H^{1}(X,\partial X;\mathbb{Z}/2)$, homotopy commutativity of the diagram
implies that $Bg\circ(x\cdot\mathfrak{s})\simeq
h_{*}(x)\cdot(Bg\circ\mathfrak{s})$, where “$\,\cdot\,$” indicates the action
of $H^{1}(X,\partial X;\mathbb{Z}/2)$ on rel. boundary isomorphism classes of
spin structure, in the respective categories. Since $h$ is homotopic to the
identity on $B(\mathbb{Z}/2)$, we conclude that
$Bg\circ(x\cdot\mathfrak{s})\simeq x\cdot(Bg\circ\mathfrak{s})$. Here is the
precise statement comparing the two definitions.

######  Lemma 2.6.

Suppose that $F\in\operatorname{Diffeo}^{+}(X,\partial X)$. Then
$\Theta(F,\mathfrak{s})=\Theta(F,Bg\circ\mathfrak{s})$, where the former uses
the definition in $\operatorname{Diff}$ and latter uses the
$\operatorname{TOP}$ definition.

###### Proof.

There is a unique $x\in H^{1}(X,\partial X;\mathbb{Z}/2)$ such that
$x\cdot\mathfrak{s}\simeq F^{*}\mathfrak{s}:=\mathfrak{s}\circ F$. There is
also a unique $y\in H^{1}(X,\partial X;\mathbb{Z}/2)$ such that
$y\cdot(Bg\circ\mathfrak{s})\simeq
F^{*}(Bg\circ\mathfrak{s}):=Bg\circ\mathfrak{s}\circ F$. Combining the facts
obtained so far, we have

$x\cdot(Bg\circ\mathfrak{s})\simeq Bg\circ(x\cdot\mathfrak{s})\simeq
Bg\circ(F^{*}\mathfrak{s})\simeq Bg\circ\mathfrak{s}\circ F\simeq
y\cdot(Bg\circ\mathfrak{s}).$

So $x=y$, and this is the class
$\Theta(F,\mathfrak{s})=\Theta(F,Bg\circ\mathfrak{s})$, using the definition
from either category. ∎

######  Lemma 2.7.

Suppose that $X$ is a spin $\operatorname{CAT}$ 4-manifold. Let $F,G\colon
X\xrightarrow{\cong}X$ be orientation preserving $\operatorname{CAT}$
isomorphisms that restrict to the identity on the boundary. The following
assertions hold.

  1. (1)

Suppose $\mathfrak{s}_{0}$ and $\mathfrak{s}_{1}$ are isomorphic spin
structures on $X$. Then
$\Theta(F,\mathfrak{s}_{0})=\Theta(F,\mathfrak{s}_{1})$.

  2. (2)

If $X$ is simply connected then $\Theta(F,\mathfrak{s})$ is independent of the
choice of $\mathfrak{s}$. $($In this case we write $\Theta(F)$.$)$

  3. (3)

If $X$ is simply connected then there is an equality

$\Theta(G\circ F)=\Theta(G)+\Theta(F)\in H^{1}(X,\partial X;\mathbb{Z}/2).$

###### Proof.

We may choose a homotopy $\mathfrak{S}$ from $\mathfrak{s}_{0}$ and
$\mathfrak{s}_{1}$, as they are isomorphic. Consider the exact sequence of
chain complexes

${C_{*}(X,\partial X)\otimes C_{*}(\operatorname{pt})}$${C_{*}(X,\partial
X)\otimes C_{*}(\\{0,1\\})}$${C_{*}(X,\partial X)\otimes C_{*}([0,1]).}$

For a singular simplex $\tau$ in $C_{*}(\operatorname{pt})$ write its image in
$C_{*}(\\{0\\})$ as $\tau_{0}$ and its image in $C_{*}(\\{1\\})$ as
$\tau_{1}$. The left map above send
$\sigma\otimes\tau\mapsto\sigma\otimes\tau_{1}-\sigma\otimes\tau_{0}$. The
right hand map is induced by inclusion. Writing $X_{i}:=X\times\\{i\\}$ for
$i=0,1$, consider the following map in the associated exact
$\mathbb{Z}/2$-coefficient cohomology sequence:

${H^{1}(X\times[0,1],\partial
X\times[0,1];\mathbb{Z}/2)}$${H^{1}(X_{0},\partial X_{0};\mathbb{Z}/2)\oplus
H^{1}(X_{1},\partial
X_{1};\mathbb{Z}/2)}$${\Theta(F\times[0,1],\mathfrak{S})}$${(\Theta(F,\mathfrak{s}_{0}),\Theta(F,\mathfrak{s}_{1}))}$

The subsequent relative term in the homology exact sequence is
$H^{1}(X,\partial X;\mathbb{Z}/2)$, and the onwards map $H^{1}(X_{0},\partial
X_{0};\mathbb{Z}/2)\oplus H^{1}(X_{1},\partial X_{1};\mathbb{Z}/2)\to
H^{1}(X,\partial X;\mathbb{Z}/2)$ sends $(a,b)\mapsto a+b$. We conclude that
$\Theta(F,\mathfrak{s}_{0})+\Theta(F,\mathfrak{s}_{1})=0$ by exactness.

The second item follows from the first, because any two spin structures on a
simply connected manifold are isomorphic.

For the third item, write $\theta(\mathfrak{u},\mathfrak{v})\in
H^{1}(X,\partial X;\mathbb{Z}/2)$ for the class representing the difference
between two spin structures $\mathfrak{u}$ and $\mathfrak{v}$ on $X$ that are
equal on $\partial X$. The action of $H^{1}(X,\partial X;\mathbb{Z}/2)$ on the
set of spin structures rel. boundary is natural, in the sense that
$\theta(J^{*}\mathfrak{u},J^{*}\mathfrak{v})=J^{*}\theta(\mathfrak{u},\mathfrak{v})$
for any orientation preserving automorphism $J$ of $X$ that fixes the boundary
pointwise. We compute

$\Theta(G\circ F)=\theta((G\circ
F)^{*}\mathfrak{s},\mathfrak{s})=\theta(F^{*}G^{*}\mathfrak{s},\mathfrak{s})=\theta(F^{*}(G^{*}\mathfrak{s}),G^{*}\mathfrak{s})+\theta(G^{*}\mathfrak{s},\mathfrak{s})=\Theta(F)+\Theta(G).$

The third equality uses the fact that the action is affine. The fourth
equality uses the fact, proved above, that the class $\Theta(F)$ can be
defined using any spin structure on $X$. ∎

######  Corollary 2.8.

Suppose $(X,\mathfrak{s})$ is a simply connected, $\operatorname{CAT}$
4-manifold with spin structure. Let $F,F^{\prime}\colon X\xrightarrow{\cong}X$
be orientation preserving $\operatorname{CAT}$ isomorphisms that restrict to
the identity on the boundary and that are $\operatorname{CAT}$ pseudo-isotopic
rel. boundary. Then $\Theta(F)=\Theta(F^{\prime})$.

###### Proof.

A pseudo-isotopy $\Psi\colon X\times[0,1]\xrightarrow{\cong}X\times[0,1]$ can
be used to pull back the product spin structure
$\mathfrak{s}\circ\operatorname{pr}_{1}$ from the target $X\times[0,1]$ to
obtain a spin structure $\mathfrak{S}$ restricting to a spin structure on
$\partial(X\times[0,1])$ that is precisely the definition of a rel. boundary
isomorphism between $F^{*}\mathfrak{s}$ and $(F^{\prime})^{*}\mathfrak{s}$.
This implies $\Theta(F)=\Theta(F^{\prime})$. ∎

###  2.2. Poincaré variations

A _variation_ is a homomorphism of homology groups associated, as follows, to
a continuous self-map $F\colon(A,B)\to(A,B)$ of a pair of topological spaces,
that restricts to the identity map on $B$; cf. [Lam75, DK75, Kau74]. For
$i\geq 0$, let $x\in H_{i}(A,B)$ and choose $\Sigma\in C_{i}(A,B)$ a
representative cycle for $x$. Because $F$ restricts to the identity on $B$,
the chain $\Sigma-F_{*}(\Sigma)$ is a cycle in $C_{i}(A)$ and thus defines a
homology class. We claim this homology class is independent of the choice of
$\Sigma$. Indeed, if $\Sigma^{\prime}$ is another choice of representative
cycle for $x$, then $\Sigma-\Sigma^{\prime}=\partial M$, for some $M\in
C_{i+1}(A,B)$, and so

$\Sigma-
F_{*}(\Sigma)-(\Sigma^{\prime}-F_{*}(\Sigma^{\prime}))=(\Sigma-\Sigma^{\prime})-F_{*}(\Sigma-\Sigma^{\prime})=\partial
M-F_{*}(\partial M)=\partial(M-F_{*}(M)).$

Again, as $F$ is the identity on $B$, any choice of lift of $M$ to
$C_{i+1}(A)$ gives the same chain $M-F_{*}(M)\in C_{i+1}(A)$, with the
boundary as just computed. Thus the homology class $[\Sigma-F_{*}(\Sigma)]$ is
independent of the choice of $\Sigma$. For each $i\geq 0$, we thus obtain a
well-defined homomorphism $\Delta_{F}\colon H_{i}(A,B)\to H_{i}(A)$ given by
$[\Sigma]\mapsto[\Sigma-F_{*}(\Sigma)]$. The variation map $\Delta_{F}$ is
invariant under homotopies of $F$ which fix $B$.

While variations are defined in this generality, we shall apply them to
4-manifolds and when $i=2$. In this setting they satisfy an extra condition,
as shown by Saeki [Sae06]. Following Saeki, we define an appropriate value
group for these invariants. Write
$(-)^{\ast}:=\operatorname{Hom}(-,\mathbb{Z})$. Given a homomorphism
$\Delta\colon H_{2}(X,\partial X)\to H_{2}(X)$, define a composite “umkehr”
homomorphism

$\Delta^{!}\colon H_{2}(X,\partial
X)\xrightarrow{\operatorname{PD}^{-1}}H^{2}(X)\xrightarrow{\operatorname{ev}}H_{2}(X)^{\ast}\xrightarrow{\Delta^{\ast}}H_{2}(X,\partial
X)^{\ast}\xrightarrow{\operatorname{ev}^{-1}}H^{2}(X,\partial
X)\xrightarrow{\operatorname{PD}}H_{2}(X).$

Write $j_{*}\colon H_{2}(X)\to H_{2}(X,\partial X)$ for the map induced by
inclusion.

######  Definition 2.9.

A homomorphism $\Delta\colon H_{2}(X,\partial X)\to H_{2}(X)$ is a _Poincaré
variation_ if

$\Delta+\Delta^{!}=\Delta\circ j_{*}\circ\Delta^{!}\colon H_{2}(X,\partial
X)\to H_{2}(X).$

Write $\mathcal{V}(H_{2}(X),\lambda_{X})$ for the set of Poincaré variations.

######  Lemma 2.10 ([Sae06, Lemma 3.5]).

The set $\mathcal{V}(H_{2}(X),\lambda_{X})$ forms a group under the operation

$\Delta_{1}\ast\Delta_{2}:=\Delta_{1}+(\operatorname{Id}-\Delta_{1}\circ
j_{*})\circ\Delta_{2},$

with identity element $0$ and inverses
$\Delta^{-1}=-(\operatorname{Id}-\Delta^{*}\circ j_{*})\circ\Delta$.

######  Definition 2.11.

Let $X$ be a compact, simply connected 4-manifold. Given
$F\in\operatorname{Homeo}^{+}(X,\partial X)$, define the _variation_ of $F$ to
be the homomorphism

$\Delta_{F}\colon H_{2}(X,\partial X)\to
H_{2}(X);\qquad[\Sigma]\mapsto[\Sigma-F(\Sigma)].$

It is shown in [Sae06, Lemma 3.2(4)] that
$\Delta_{F}\in\mathcal{V}(H_{2}(X),\lambda_{X})$, i.e. it is a Poincaré
variation in the sense of Definition 2.9.

######  Remark 2.12.

The notation $\mathcal{V}(H_{2}(X),\lambda_{X})$ is intended to convey that
this group depends only on the isometry class of the form
$(H_{2}(X),\lambda_{X})$. This assertion is proved in Section 7, where we
recast the group of Poincaré variations more algebraically.

######  Remark 2.13.

It is straightforward to see that

$\operatorname{Id}-\Delta_{F}\circ j_{*}\colon H_{2}(X)\to H_{2}(X)$

agrees with the map $F_{*}\colon H_{2}(X)\to H_{2}(X)$. We can thus view the
Poincaré variation of a self-homeomorphism as a refinement of the isometry
induced by $F$ on the intersection form. Indeed, when $\partial X=\emptyset$,
it is easily checked that the definitions of a Poincaré variation and an
isometry coincide. We study the relationship of variations to isometries in
greater depth in Section 7.

######  Lemma 2.14.

Let $(A,B)$ be a pair of spaces and let $F_{0},F_{1}\colon(A,B)\to(A,B)$ be
continuous maps each restricting to the identity on $B$. If $F_{0}\sim F_{1}$
via a homotopy restricting to the identity on $B$, then
$\Delta_{F_{0}}=\Delta_{F_{1}}$.

###### Proof.

By assumption, there exists a continuous map
$\Psi\colon(A\times[0,1],B\times[0,1])\to(A\times[0,1],B\times[0,1])$
restricting to the identity on $B\times[0,1]$, such that
$\Psi(x,i)=(F_{i}(x),i)$ for $i=0,1$. It is easily seen from the definitions
that there is a commuting diagram

${H_{2}(A\times\\{0\\},B\times\\{0\\})\oplus
H_{2}(A\times\\{1\\},B\times\\{1\\})}$${H_{2}(A\times[0,1],B\times[0,1])}$${H_{2}(A\times\\{0\\})\oplus
H_{2}(A\times\\{1\\})}$${H_{2}(A\times[0,1]),}$$\scriptstyle{(1\,\,-1)}$$\scriptstyle{\Delta_{F_{0}}\oplus\Delta_{F_{1}}}$$\scriptstyle{\Delta_{\Psi}}$$\scriptstyle{(1\,\,-1)}$

where the horizontal arrows are the inclusion induced maps. Take $(x,x)\in
H_{2}(A\times\\{0\\})\oplus H_{2}(A\times\\{1\\})$. Mapping right and then
down sends $(x,x)$ to $0$, whereas mapping down and then right sends $(x,x)$
to $\Delta_{F_{0}}(x)-\Delta_{F_{1}}(x)$. From this and commutativity we see
that $\Delta_{F_{0}}=\Delta_{F_{1}}$. ∎

######  Corollary 2.15.

The function

$\widetilde{\pi}_{0}\left(\operatorname{Homeo}^{+}(X,\partial
X)\right)\to\mathcal{V}(H_{2}(X),\lambda_{X});\qquad[F]\mapsto\Delta_{F}$

is a well-defined group homomorphism.

###### Proof.

Suppose $F,G\in\operatorname{Homeo}^{+}(X,\partial X)$ are pseudo-isotopic
rel. boundary. Then in particular they are homotopic rel. boundary, implying
$\Delta_{F}=\Delta_{G}\in\mathcal{V}(H_{2}(X),\lambda_{X})$ by Lemma 2.14.
This shows the function $[F]\mapsto\Delta_{F}$ is well-defined on the level of
sets. It is straightforward to compute that this is a group homomorphism; for
this we refer to e.g. [Sae06, Lemma 3.2], cf. [Lam75, 3.1.4]. ∎

###  2.3. The main pseudo-isotopy classification

The following is the first main technical result of the article, in which we
show that the invariants defined in the previous subsection completely detect
the pseudo mapping class groups, in both categories. The proof will be given
in Section 5.

######  Theorem 2.16.

Suppose $\operatorname{CAT}$ is $\operatorname{Diff}$ or $\operatorname{TOP}$,
and let $(X,\partial X)$ be a compact, simply connected, oriented
$\operatorname{CAT}$ $4$-manifold.

  1. (1)

When $X$ is spin, and respectively when $\operatorname{CAT}$ is
$\operatorname{TOP}$ or $\operatorname{Diff}$, the homomorphisms

$\begin{array}[]{ccl}\widetilde{\pi}_{0}\operatorname{Homeo}^{+}(X,\partial
X)&\hookrightarrow&H^{1}(X,\partial
X;\mathbb{Z}/2)\times\mathcal{V}(H_{2}(X),\lambda_{X}),\quad{and}\\\
\widetilde{\pi}_{0}\operatorname{Diffeo}^{+}(X,\partial
X)&\hookrightarrow&H^{1}(X,\partial
X;\mathbb{Z}/2)\times\mathcal{V}(H_{2}(X),\lambda_{X}),\end{array}$

both given by $F\mapsto(\Theta(F),\Delta_{F})$, are injective.

  2. (2)

When $X$ is not spin, and respectively when $\operatorname{CAT}$ is
$\operatorname{TOP}$ or $\operatorname{Diff}$, the homomorphisms

$\begin{array}[]{ccl}\widetilde{\pi}_{0}\operatorname{Homeo}^{+}(X,\partial
X)&\hookrightarrow&\mathcal{V}(H_{2}(X),\lambda_{X}),\quad{and}\\\
\widetilde{\pi}_{0}\operatorname{Diffeo}^{+}(X,\partial
X)&\hookrightarrow&\mathcal{V}(H_{2}(X),\lambda_{X}),\end{array}$

both given by $F\mapsto\Delta_{F}$, are injective.

We recall the following theorem of Quinn [Qui86] and Perron [Per86], which can
be combined with Theorem 2.16 to yield stronger statements.

######  Theorem 2.17 (Quinn-Perron).

Let $X$ be a compact, simply connected, topological 4-manifold.

  1. (1)

Let $F\colon X\xrightarrow{\cong}X$ be a homeomorphism that is pseudo-isotopic
to the identity rel. boundary. Then $F$ is topologically isotopic to the
identity rel. boundary.

  2. (2)

Suppose that $X$ admits a smooth structure, and let $F\colon
X\xrightarrow{\cong}X$ be a diffeomorphism that is smoothly pseudo-isotopic to
the identity rel. boundary. Then $F$ is smoothly stably isotopic to the
identity rel. boundary.

Theorem 2.17 (1) shows that the forgetful map
$\pi_{0}\operatorname{Homeo}^{+}(X,\partial
X)\xrightarrow{\cong}\widetilde{\pi}_{0}\operatorname{Homeo}^{+}(X,\partial
X)$ from the pseudo mapping class group to the mapping class group is an
isomorphism. Combining Theorem 2.17 (1) with Theorem 2.16 proves the
injectivity in Theorem A, which implies injectivity in Corollary C.

Next we use Theorem 2.17 (2) to deduce Theorem B from the introduction,
computing the smooth stable mapping class group.

######  Theorem 2.18.

Let $X$ be a compact, simply connected, oriented, smooth 4-manifold.

  1. (1)

When $X$ is spin, the map $F\mapsto(\Theta(F),F_{*})$ induces a group
isomorphism

$\underset{g\to\infty}{\operatorname{colim}}\,\pi_{0}\operatorname{Diffeo}^{+}(X\\#^{g}S^{2}\times
S^{2},\partial X)\xrightarrow{\cong}H^{1}(X,\partial
X;\mathbb{Z}/2)\times\underset{g\to\infty}{\operatorname{colim}}\,\mathcal{V}((H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g}).$

  2. (2)

When $X$ is not spin, the map $F\mapsto F_{*}$ induces a group isomorphism

$\underset{g\to\infty}{\operatorname{colim}}\,\pi_{0}\operatorname{Diffeo}^{+}(X\\#^{g}S^{2}\times
S^{2},\partial
X)\xrightarrow{\cong}\underset{g\to\infty}{\operatorname{colim}}\,\mathcal{V}((H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g}).$

###### Proof.

An element of the colimit in the domain is represented by a diffeomorphism $F$
of $X\\#^{g}S^{2}\times S^{2}$ for some $g$. For injectivity, assume its image
in $\mathcal{V}((H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus g})$ is
trivial, and in the spin case assume additionally that $\Theta(F)=0$. Then by
Theorem 2.16, $F$ is smoothly pseudo-isotopic to
$\operatorname{Id}_{X\\#^{g}S^{2}\times S^{2}}$ rel. boundary. By Theorem 2.17
(2), $F$ is stably smoothly isotopic rel. boundary to the identity, meaning
that for some $h$ the image of $F$ in
$\pi_{0}\operatorname{Diffeo}^{+}(X\\#^{g+h}S^{2}\times S^{2})$ is trivial. It
follows that $[F]$ is trivial in the colimit.

For surjectivity, an element of the colimit in the codomain is represented by
a Poincaré variation
$\Delta\in\mathcal{V}((H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus g})$ for
some $g$, and in the spin case together with an element $x\in H^{1}(X,\partial
X;\mathbb{Z}/2)$. By Theorem A, there is a homeomorphism $F^{\prime}$ of
$X\\#^{g}S^{2}\times S^{2}$ restricting to $\operatorname{Id}_{\partial X}$
inducing $\Delta$, respectively $(x,\Delta)$ in the spin case.

Next, by [FQ90, Theorem 8.6], there exists an $h\in\mathbb{N}$ such that
$F^{\prime}\\#\operatorname{Id}\in\operatorname{Homeo}^{+}(X\\#^{g+h}S^{2}\times
S^{2})$ is topologically isotopic to a diffeomorphism if there is a lift of
the associated map of stable tangent microbundles to a map of the
corresponding smooth stable tangent bundles. In more detail, write
$X^{\prime}:=X\\#^{g}S^{2}\times S^{2}$, let $\tau_{X^{\prime}}\colon
X^{\prime}\to\operatorname{BTOP}$ be a classifying map for the stable tangent
microbundle of $X^{\prime}$, and let $T_{X^{\prime}}\colon
X^{\prime}\to\operatorname{BO}$ classify the tangent bundle of $X^{\prime}$.
Since $F^{\prime}\colon X^{\prime}\to X^{\prime}$ is a homeomorphism,
$\tau_{X^{\prime}}\circ F$ also classifies the tangent microbundle and so
there is a homotopy $X^{\prime}\times I\to\operatorname{BTOP}$ between
$\tau_{X^{\prime}}$ and $\tau_{X^{\prime}}\circ F^{\prime}$. On
$\partial(X^{\prime}\times I)$, since $X^{\prime}$ is smooth and since
$F^{\prime}$ restricts to the identity on $\partial X$, we can lift this map
along the fibration $\operatorname{BO}\to\operatorname{BTOP}$. The homotopy
fibre of $\operatorname{BO}\to\operatorname{BTOP}$ is
$\operatorname{TOP}/\operatorname{O}$. We want to extend this lift over all of
$X^{\prime}\times I$. Since $X\times I$ is 5-dimensional and there is a
$7$-connected map $\operatorname{TOP}/\operatorname{O}\to K(\mathbb{Z}/2,3)$,
there is a unique obstruction in $H^{4}(X^{\prime}\times
I,\partial(X^{\prime}\times I);\mathbb{Z}/2)\cong H_{1}(X^{\prime}\times
I;\mathbb{Z}/2)=0$ to extending the homotopy. Thus there is in fact no
obstruction and a homotopy exists as desired. It follows that there is an
integer $h$ as above such that $F^{\prime}\\#\operatorname{Id}$ is isotopic to
a diffeomorphism, $F$ say. We have that $[F]$ maps to
$[\Delta_{F}]=[\Delta]\in\operatorname{colim}_{g\to\infty}\mathcal{V}((H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g})$, completing the proof of surjectivity. ∎

##  3\. Elements of modified surgery theory

To prove the main pseudo-isotopy classification Theorem 2.16, we will use the
ideas of Kreck’s modified surgery theory [Kre99]. We will now recall, in some
generality, relevant concepts from this theory. After this we make some
specific calculations that we will need later.

###  3.1. Definitions and theorems from modified surgery theory

In this section, all manifolds are compact and oriented.

Suppose $\operatorname{CAT}=\operatorname{Diff}$. A _cobordism rel. boundary_
between $n$-dimensional manifolds with boundary $X_{0}$ and $X_{1}$ is a
triple $(W;\varphi_{0},\varphi_{1})$. Here $W$ is an $(n+1)$-dimensional
manifold with corners, with a decomposition

$\partial W=\partial_{0}W\cup(Y\times[0,1])\cup-\partial_{1}W,$

where the interiors of $\partial_{0}W$, $Y\times[0,1]$, and $\partial_{1}W$
are the codimension 1 corners and the codimension 2 corners are
$\partial_{0}W=Y$, $\partial_{1}W=Y$. The maps $\varphi_{i}$ are
diffeomorphisms $\varphi_{i}\colon(X_{i},\partial X_{i})\to(\partial_{i}W,Y)$.
In case that $\partial X_{i}=\emptyset$ for $i=0,1$ (and thus $Y=\emptyset$),
a cobordism rel. boundary is called a _cobordism_. In a common abuse of
notation, we will often suppress the maps $\varphi_{1}$ and $\varphi_{2}$ from
the notation, however we include this level of detail in the definition
because we will use it later on. When $\operatorname{CAT}=\operatorname{TOP}$,
the same definitions are used, except it is simpler as now the decomposition
of $\partial W$ only refers to a decomposition by codimension 0 submanifolds,
without reference to the concept of corners.

For a smooth $n$-manifold $X$ we will write the stable normal vector bundle
via its classifying map $\nu_{X}\colon X\to\operatorname{BSO}$. Given a pair
$(B,p)$, where $B$ has the homotopy type of a CW complex and $p\colon
B\to\operatorname{BSO}$ is a fibration, a map $\xi\colon X\to B$ is called a
_$B$ -structure_ on $X$ if the diagram

${B}$${X}$${\operatorname{BSO}}$$\scriptstyle{p}$$\scriptstyle{\xi}$$\scriptstyle{\nu_{X}}$

commutes up to homotopy. When $X$ is a $\operatorname{TOP}$ manifold, the same
definitions and notation are used, except with $\operatorname{BSTOP}$ in place
of $\operatorname{BSO}$ and $\nu_{X}$ now representing the stable topological
normal bundle (see e.g. [FNOP19, §7] for definitions).

Two closed $n$-manifolds with $B$-structure $(X_{i},\xi_{i})$, $i=0,1$ are
_cobordant over $B$_ if there exists $(W,\Xi)$, consisting of a cobordism $W$
between $X_{0}$ and $X_{1}$, together with a $B$-structure such that
$\partial(W,\Xi)=(X_{0},\xi_{0})\sqcup-(X_{1},\xi_{1})$, so that in particular
the diagram

${W}$${B}$${X_{i}}$$\scriptstyle{\Xi}$$\scriptstyle{\xi_{i}}$$\scriptstyle{\varphi_{i}}$

commutes for $i=0,1$. The corresponding bordism group is denoted
$\Omega_{n}(B,p)$.

Fix a $B$-structure $\zeta\colon Y\to B$ on a closed $(n-1)$-manifold $Y$.
Suppose $X$ is an $n$-manifold with boundary and that $f\colon\partial X\to Y$
is a homeomorphism. A _$B$ -structure rel. $\zeta$_ on $X$ (implicitly with
respect to the chosen $f$) is a $B$-structure $(X,\xi)$ that restricts to
$(\partial X,\zeta\circ f)$ on the boundary. Given $B$-structures rel. $\zeta$
on two such manifolds, $(X_{i},\xi_{i})$, $i=0,1$, we say they are _cobordant
rel. $\zeta$_ if there exists $(W,\Xi)$, consisting of a cobordism rel.
boundary $(W;\varphi_{0},\varphi_{1})$ between $X_{0}$ and $X_{1}$, with
$\partial W=\partial_{0}W\cup(Y\times[0,1])\cup-\partial_{1}W$, and a
$B$-structure $\Xi$ on $W$ that restricts to $\xi_{0}\circ\varphi_{0}^{-1}$,
$\xi_{1}\circ\varphi_{1}^{-1}$, and $\zeta\circ\operatorname{pr}_{1}$ on
$\partial_{0}W$, $\partial_{1}W$ and $Y\times[0,1]$ respectively.

We record the following lemma for use later on; the proof is straightforward
from the definitions and we omit it (we note that a little care must be taken
in the smooth case to show a bicollar on $Y$ is equivalent to the desired
cornered structure).

######  Lemma 3.1.

Let $\operatorname{CAT}=\operatorname{Diff}$. Suppose $B$ is a space with the
homotopy type of a CW complex and let $p\colon B\to\operatorname{BSO}$ be a
fibration. Let $Y$ be a closed $(n-1)$-manifold. For $i=0,1$, suppose that
$X_{i}$ is an $n$-manifold and that $f_{i}\colon\partial X_{i}\to Y$ is a
diffeomorphism. Suppose there is a $B$-structure $(Y,\zeta)$ and
$B$-structures rel. $\zeta$ $(X_{i},\xi_{i})$. Define a closed $n$-manifold
with $B$-structure

$(X,\xi):=(X_{0}\cup-X_{1},\xi_{0}\cup\xi_{1})$

using the diffeomorphism $f_{1}^{-1}\circ f_{0}\colon\partial X_{0}\to\partial
X_{1}$ to glue along the boundary. Then $(X,\xi)\sim 0\in\Omega_{n}(B,p)$ is
null-bordant if and only if $(X_{0},\xi_{0})$ is cobordant to
$(X_{1},\xi_{1})$ rel. $\zeta$.

The analogous statement holds when $\operatorname{CAT}=\operatorname{TOP}$.

Recall that a topological space $A$ is _$m$ -connected_ if $\pi_{k}(A)=0$ for
$1\leq k\leq m$ and is _$m$ -coconnected_ if $\pi_{k}(A)=0$ for $k\geq m$. A
map of spaces $f\colon A\to B$ is _$m$ -connected_ if the homotopy cofibre
(i.e. the mapping cone) is $m$-connected; equivalently
$f_{*}\colon\pi_{k}(A)\to\pi_{k}(B)$ is an isomorphism for $k<m$ and is
surjective for $k=m$. A map of spaces $f\colon A\to B$ is _$m$ -coconnected_
if the homotopy fibre is $m$-coconnected; equivalently
$f_{*}\colon\pi_{k}(A)\to\pi_{k}(B)$ is an isomorphism for $k>m$ and is
injective for $k=m$

######  Definition 3.2.

Let $\operatorname{CAT}=\operatorname{Diff}$. A _normal $2$-type_ for a
manifold $X$ is a pair $(B,p)$, where $B$ is a space with the homotopy type of
a CW complex, $p\colon B\to\operatorname{BSO}$ is a $3$-coconnected fibration
with connected fibre, and $(B,p)$ is such that there exists a $3$-connected
$B$-structure $\overline{\nu}_{X}\colon X\to B$. Such a $B$-structure
$(X,\overline{\nu}_{X})$ is called a _normal $2$-smoothing_ of $X$. If
$(X,\overline{\nu}_{X})$ is moreover a $B$-structure rel. $\zeta$, for some
$\zeta\colon Y\to B$ with $\partial X\cong Y$, we say $(X,\overline{\nu}_{X})$
is a _normal $2$-smoothing rel. $\zeta$_ of $X$.

We make the same definitions when $\operatorname{CAT}=\operatorname{TOP}$,
after replacing $\operatorname{BSO}$ with $\operatorname{BSTOP}$.

Given a manifold $X$, the existence of a normal $2$-type follows from the
theory of Moore-Postnikov decompositions (see [Bau77]). This theory
furthermore guarantees that any two normal $2$-types for a given $X$ are fibre
homotopy equivalent to one another.

######  Definition 3.3.

An _$h$ -cobordism rel. boundary_ between $(X_{0},\partial X_{0})$ and
$(X_{1},\partial X_{1})$ is a cobordism rel. boundary
$(W;\varphi_{0},\varphi_{1})$ such that the inclusions
$\partial_{i}W\hookrightarrow W$ are homotopy equivalences for $i=0,1$.

The following definition is due to a technical component of modified surgery
theory that will show up for us later.

######  Definition 3.4.

Given a map of CW complexes $p\colon B\to\operatorname{BSO}$ (resp. $p\colon
B\to\operatorname{BSTOP}$) we say a homotopy class $\alpha\in\pi_{4}(B)$ is
_$w_{4}$ -trivial_ if the bundle $\xi$ classified by $p\circ\alpha\colon
S^{4}\to\operatorname{BSO}$ (resp. $p\circ\alpha\colon
S^{4}\to\operatorname{BSTOP}$) has $w_{4}(\xi)=0$.

In classical surgery theory [Wal99], obstructions to doing surgery to improve
certain classes of maps to simple homotopy equivalences are situated in the
_surgery obstruction groups_. These are abelian groups
$\displaystyle{L^{s}_{n}(\mathbb{Z}[\pi])}$, depending on the dimension $n$,
and fundamental group $\pi$, of the manifold. In modified surgery, Kreck
considers a slightly altered version
$\displaystyle{L^{s}_{n}(\mathbb{Z}[\pi],S)}$ of Wall’s classical surgery
obstruction groups $\displaystyle{L^{s}_{n}(\mathbb{Z}[\pi])}$ (see [Kre,
§4]). Here $S\subseteq\mathbb{Z}[\pi]$ is a subgroup, with certain properties,
called a _form parameter_ by Bak [Bak81]. Under certain circumstances, Kreck’s
modified surgery obstructions live in these alternative surgery obstruction
groups, for example in the following theorem.

######  Theorem 3.5 ([Kre, Theorem 5.2(b)]).

Let $\operatorname{CAT}$ be either $\operatorname{Diff}$ or
$\operatorname{TOP}$. Let $(M_{0},\partial M_{0})$ and $(M_{1},\partial
M_{1})$ be simply connected $5$-manifolds with boundary, each with normal
$2$-type $(B,p)$, where $B$ is homotopy equivalent to a CW complex with finite
$2$-skeleton. Let $\partial M_{0}\cong N\cong\partial M_{1}$ and fix a
$B$-structure $\zeta\colon N\to B$. Then a bordism $(Z,\Xi)$ over $B$ rel.
boundary, between normal $2$-smoothings $(M_{0},\xi_{0})$ and
$(M_{1},\xi_{1})$ rel. $\zeta$, determines a surgery obstruction

$\theta(Z,\Xi)\in\left\\{\begin{array}[]{ll}\displaystyle{L^{s}_{6}(\mathbb{Z})}&\text{if
$\alpha$ is $w_{4}$-trivial for all $\alpha\in\pi_{4}(B)$}\\\
\displaystyle{L^{s}_{6}(\mathbb{Z},S=\mathbb{Z})}&\text{otherwise.}\end{array}\right.$

The obstruction $\theta(Z,\Xi)$ vanishes if and only if $(Z,\Xi)$ is cobordant
rel. boundary to some $(Z^{\prime},\Xi^{\prime})$ over $B$, where $Z^{\prime}$
is an $h$-cobordism rel. boundary.

###  3.2. The normal 2-type of a compact oriented simply connected 4-manifold

For this subsection, fix $(X,\partial X)$, a compact, oriented, simply
connected 4-manifold with (possibly empty) boundary.

Let $K$ be a fixed model for $K(\pi_{2}(X),2)$ and choose a map $k\colon X\to
K$ that is an isomorphism on $\pi_{2}$. Let $w\in H^{2}(K;\mathbb{Z}/2)$ be
the cohomology class such that $k^{*}w=w_{2}(X)$, and let $w\colon K\to
K(\mathbb{Z}/2,2)$ be a map representing this class. Consider the following
commutative diagrams

${\operatorname{BTOPSpin}}$${B(X)^{\operatorname{TOP}}}$${K}$${\operatorname{BTOPSpin}}$${\operatorname{BSTOP}}$${K(\mathbb{Z}/2,2),}$$\scriptstyle{i}$$\scriptstyle{=}$$\scriptstyle{q}$$\scriptstyle{p}$

$\lrcorner$

$\scriptstyle{w}$$\scriptstyle{p\circ i}$$\scriptstyle{w_{2}}$
${\operatorname{BSpin}}$${B(X)^{\operatorname{Diff}}}$${K}$${\operatorname{BSpin}}$${\operatorname{BSO}}$${K(\mathbb{Z}/2,2).}$$\scriptstyle{i}$$\scriptstyle{=}$$\scriptstyle{q}$$\scriptstyle{p}$

$\lrcorner$

$\scriptstyle{w}$$\scriptstyle{p\circ i}$$\scriptstyle{w_{2}}$

The CW complexes $B(X)^{\operatorname{TOP}}$, $B(X)^{\operatorname{Diff}}$,
and maps $p,q$, are chosen to make the right-hand squares homotopy pullbacks.
The maps $w_{2}$ are chosen to be fibrations, with fibre
$\operatorname{BTOPSpin}$ and $\operatorname{BSpin}$ respectively. This
implies each map $q$ is also a fibration. We choose the spaces
$B(X)^{\operatorname{TOP}}$ and $B(X)^{\operatorname{Diff}}$ so that in each
diagram the map $i$ is the inclusion of a fibre of $q$, and so that the left-
hand downward map is indeed an equality.

######  Proposition 3.6.

When $\operatorname{CAT}=\operatorname{TOP}$ the pair
$(B(X)^{\operatorname{TOP}},p)$ is a normal $2$-type for $X$. When
$\operatorname{CAT}=\operatorname{Diff}$, the pair
$(B(X)^{\operatorname{Diff}},p)$ is a normal $2$-type for $X$.

###### Proof.

Working first in the topological category, write
$B:=B(X)^{\operatorname{TOP}}$ for brevity. A homotopy pullback diagram
determines a long exact sequence in homotopy groups

$\cdots\to\pi_{r+1}(K(\mathbb{Z}/2,2))\to\pi_{r}(B)\xrightarrow{(p_{*},q_{*})}\pi_{r}(\operatorname{BSTOP})\times\pi_{r}(K)\to\pi_{r}(K(\mathbb{Z}/2,2)\to\cdots$

Using that $\pi_{1}(\operatorname{BSTOP})$ is trivial, we deduce there is an
exact sequence

$0\to\pi_{2}(B)\xrightarrow{(p_{*},q_{*})}\pi_{2}(\operatorname{BSTOP})\times\pi_{2}(K)\xrightarrow{(w_{2})_{*}-w_{*}}\pi_{2}(K(\mathbb{Z}/2,2))\to\pi_{1}(B)\to
0,$

and for $r\geq 3$ we see that
$p_{*}\colon\pi_{r}(B)\to\pi_{r}(\operatorname{BSTOP})$ is an isomorphism,
showing $p$ is 3-coconnected. The map
$(w_{2})_{*}\colon\pi_{2}(\operatorname{BSTOP})\to\pi_{2}(K(\mathbb{Z}/2,2))\cong\mathbb{Z}/2$
is an isomorphism, the inverse of which determines a splitting
$\pi_{2}(K(\mathbb{Z}/2,2))\to\pi_{2}(\operatorname{BSTOP})\to\pi_{2}(\operatorname{BSTOP})\times\pi_{2}(K)$
of $(w_{2})_{*}-w_{*}$. We deduce that $\pi_{1}(B)=0$ and that
$q_{*}\colon\pi_{2}(B)\to\pi_{2}(K)$ is an isomorphism.

The existence of a lift $\overline{\nu}_{X}\colon X\to B$ of $\nu_{X}$ is
implied by the universal property of the pullback applied to
$(\nu_{X}\,\,k)\colon X\to\operatorname{BSTOP}\times K$. That one obtains
homotopic maps $w_{2}\circ\nu_{X}$ and $w\circ k$ from $X$ to
$K(\mathbb{Z}/2,2)$ uses that $w_{2}(X)=w_{2}(\nu_{X})$, which follows from
$w_{1}(X)=w_{1}(\nu_{X})=0$ and the Whitney sum formula. To check this lift is
$3$-connected, consider that $\pi_{1}(B)=0$, and
$\pi_{3}(B)\cong\pi_{3}(\operatorname{BSTOP})=0$, so it is automatically a
$\pi_{1}$ isomorphism and a surjection on $\pi_{3}$. On $\pi_{2}$, consider
that

$((\nu_{X})_{*}\,\,k_{*})=(p_{*}\,\,q_{*})\circ(\overline{\nu}_{X})_{*}\colon\pi_{2}(X)\to\pi_{2}(\operatorname{BSTOP})\times\pi_{2}(K),$

so $k_{*}=q_{*}\circ\overline{\nu}_{X}\colon\pi_{2}(X)\to\pi_{2}(K)$. But as
both $q_{*}$ and $k_{*}$ are isomorphisms, so is $(\overline{\nu}_{X})_{*}$.
So $\overline{\nu}_{X}$ is 3-connected, as desired. The proof in the smooth
category is entirely analogous. ∎

Recall the notation

$\gamma\colon\operatorname{BTOPSpin}\to\operatorname{BSTOP}\qquad\text{ and
}\qquad\gamma\colon\operatorname{BSpin}\to\operatorname{BSO}$

for principal fibrations corresponding to the universal covering maps
$\operatorname{TOPSpin}\to\operatorname{STOP}$ and
$\operatorname{Spin}\to\operatorname{SO}$. In the case that $X$ is spin, here
is an alternative model for the normal $2$-type.

######  Lemma 3.7.

When $X$ is spin, the following are also models for the normal $2$-type of
$X$:

$\begin{array}[]{lclcl}\gamma\circ\operatorname{pr}_{2}\colon&\operatorname{BTOPSpin}\times
K&\to&\operatorname{BSTOP}&\qquad\text{when
$\operatorname{CAT}=\operatorname{TOP}$,}\\\
\gamma\circ\operatorname{pr}_{2}\colon&\operatorname{BSpin}\times
K&\to&\operatorname{BSO}&\qquad\text{when
$\operatorname{CAT}=\operatorname{Diff}$.}\end{array}$

###### Proof.

We work in the topological category, the smooth argument being once again
entirely similar. The homotopy fibre of $\gamma\circ\operatorname{pr}_{2}$ is
homotopy equivalent to $K(\mathbb{Z}/2,1)\times K$, which it is easy to see is
$3$-coconnected. Now as $X$ is spin, we may choose a spin structure
$\mathfrak{s}\colon X\to\operatorname{BTOPSpin}$ on $\nu_{X}$. We may also
always choose a $3$-connected map $h\colon X\to K$. As
$\operatorname{BTOPSpin}$ is $3$-connected, the map $\mathfrak{s}\times h$,
which is clearly a lift of $\nu_{X}$ under $\gamma\circ\operatorname{pr}_{2}$,
is also $3$-connected. This shows normal $2$-smoothings in
$(\operatorname{BTOPSpin}\times K,\gamma\circ\operatorname{pr}_{2})$ exist,
and hence it is a normal $2$-type. ∎

In the proof of Theorem 2.16 later, we will apply Theorem 3.5. This means we
will need to know whether it is the case that all classes in $\pi_{4}(B)$ are
$w_{4}$-trivial (recall Definition 3.4). In fact, they are not.

######  Proposition 3.8.

For both $\operatorname{CAT}=\operatorname{TOP}$ and
$\operatorname{CAT}=\operatorname{Diff}$, there exists a class in $\pi_{4}(B)$
that is not $w_{4}$-trivial.

###### Proof.

First, suppose $\operatorname{CAT}=\operatorname{Diff}$. By Proposition 3.6,
the map $p\colon B\to\operatorname{BSO}$ is 3-coconnected, so in particular
$p_{*}\colon\pi_{4}(B)\to\pi_{4}(\operatorname{BSO})$ is an isomorphism. The
latter is the infinite cyclic group, generated by (the stable class of) the
unique $4$-plane bundle $\sigma$ over $S^{4}$ with $p_{1}(\sigma)=2$ and
$e(\sigma)=1$ [Ste51, §22.6, §22.7, §23.6]. Hence $1=e(\sigma)\equiv
w_{4}(\sigma)\pmod{2}$. As Stiefel-Whitney classes are stable, this implies
$\pi_{4}(B)$ contains a class with nontrivial $w_{4}$.

Now suppose $\operatorname{CAT}=\operatorname{TOP}$. For the same reason as
above, $p_{*}\colon\pi_{4}(B)\to\pi_{4}(\operatorname{BSTOP})$ is an
isomorphism. The stable class of the underlying topological bundle of $\sigma$
still has $w_{4}(\sigma)$ nontrivial, so in this category $\pi_{4}(B)$ also
contains a class with nontrivial $w_{4}$. ∎

##  4\. Bordism computations and the James spectral sequence

We now discuss the _James spectral sequence_ , originally defined by Teichner
[Tei93, §II]. We will use this spectral sequence to compute bordism groups
over the normal 2-types we have developed so far. We will need to know some of
the differentials in the spectral sequence, for computational purposes. For
this we follow Teichner’s proof of [Tei93, Proposition 1], adapting it to our
situation and filling in some details.

###  4.1. Basic properties of the James spectral sequence

The James spectral sequence is a first-quadrant spectral sequence which takes
as input a connective generalised homology theory $\mathbb{H}$, together with

${F}$${B}$${K}$and${v\colon
B}$${\operatorname{BSTOP},}$$\scriptstyle{i}$$\scriptstyle{f}$

where the left diagram is a fibration of CW complexes. (For the smooth
version, replace $v$ with a stable vector bundle $B\to\operatorname{BSO}$.) It
is furthermore required that the fibration be _$\mathbb{H}$ -orientable_ in
the sense that the monodromy-induced action of $\pi_{1}(K)$ on $\mathbb{H}(F)$
is trivial. For us, the connective homology theory $\mathbb{H}$ will always be
given by stable homotopy groups
$\mathbb{H}_{*}(-)=\pi^{\operatorname{st}}_{*}(-)$. As the homotopy groups of
spectra are by definition stable homotopy groups, we will omit the “$st$”
superscript whenever we take homotopy groups of spectra. Write $M(v)$ for the
Thom spectrum of $v$. We refer the reader to [Tei93, Theorem] for the
construction of the James spectral sequence, but we record that it has the
property

$E_{2}^{r,s}\cong H_{r}(K;\pi_{s}(M(v\circ i)))\implies\pi_{r+s}(M(v)),$

and that the spectral sequence is natural with respect to maps of fibrations,
that are commutative diagrams.

We introduce some notation. Write $B_{0}\subseteq B_{1}\subseteq\dots\subseteq
B$ for the skeletal filtration. Given $v\colon B\to\operatorname{BSTOP}$ a
stable topological bundle, the Thom spectrum $M(v)$ has $n$th space
$\operatorname{Th}(v_{n})$, where $v_{n}\colon
B_{n}\to\operatorname{BSTOP}(n)\subseteq\operatorname{BSTOP}$ is the pullback
of $v$ to $B_{n}$. Write $\rho_{n}\colon D(v_{n})\to B_{n}$ for the projection
of the associated disc bundle and write $U_{n}\in
H^{n}(\operatorname{Th}(v_{n});\mathbb{Z})$ for the Thom class. Write

$\begin{array}[]{rccccc}\Phi^{n}&=&U_{n}\cup(\rho_{n}^{*}(-))\colon&H^{*}(B;\mathbb{Z})&\xrightarrow{\cong}&H^{*+n}(\operatorname{Th}(v_{n});\mathbb{Z}),\\\
\Phi_{n}&=&(\rho_{n})_{*}(U_{n}\cap-)\colon&H_{*+n}(\operatorname{Th}(v_{n});\mathbb{Z})&\xrightarrow{\cong}&H_{*}(B;\mathbb{Z}),\end{array}$

for the unstable Thom isomorphisms. The Thom spectrum $M(v_{n})$ is homotopy
equivalent to the (desuspended) suspension spectrum
$\Sigma^{-n}\Sigma^{\infty}\operatorname{Th}(v_{n})$, which implies
$H^{n+*}(\operatorname{Th}(v_{n});\mathbb{Z})\cong H^{*}(M(v_{n});\mathbb{Z})$
and $H_{*+n}(\operatorname{Th}(v_{n});\mathbb{Z})\cong
H_{*}(M(v_{n});\mathbb{Z})$. In, respectively, the limit and colimit, we
obtain the stable Thom isomorphisms

$\begin{array}[]{rccc}\Phi^{\infty}\colon&H^{*}(B;\mathbb{Z}/2)&\xrightarrow{\cong}&H^{*}(M(v);\mathbb{Z}/2),\\\
\Phi_{\infty}\colon&H_{*}(M(v);\mathbb{Z})&\xrightarrow{\cong}&H_{*}(B;\mathbb{Z}).\end{array}$

######  Remark 4.1.

We are only stating the $\mathbb{Z}/2$-coefficient Thom isomorphism on
cohomology, as this is the only one we will need in the sequel. There are Thom
isomorphisms with other coefficients, but in these settings it may not be
possible to use a limit to define them. In general, for a ring $R$, there is a
short exact sequence

$0\to{\lim}^{1}H^{*+n-1}(\operatorname{Th}(v_{n});R)\to
H^{*}(M(v);R)\to{\lim}^{0}H^{*+n}(\operatorname{Th}(v_{n});R)\to 0;$

cf. [Swi75, Proposition 7.66]. When $R$ is a field, the ${\lim}^{1}$ term
vanishes, which is what has permitted our limit definition for cohomology of a
spectrum.

For a topological space $X$, and $y\in H^{2}(X;\mathbb{Z}/2)$, write

$\operatorname{Sq}^{2}_{y}\colon H^{*}(X;\mathbb{Z}/2)\to
H^{*+2}(X;\mathbb{Z}/2);\qquad x\mapsto\operatorname{Sq}^{2}(x)+y\cup x.$

######  Lemma 4.2.

Let $v\colon B\to\operatorname{BSTOP}$ be a stable bundle over a CW complex.
Then
$\operatorname{Sq}^{2}\circ\Phi^{\infty}=\Phi^{\infty}\circ\operatorname{Sq}^{2}_{w_{2}(v)}$.
The same statement holds using a stable vector bundle $v\colon
B\to\operatorname{BSO}$.

###### Proof.

For all $i\geq 0$, we have Thom’s formula
$\operatorname{Sq}^{i}(U_{n})=U_{n}\cup(\rho_{n}^{*}w_{i}(v_{n}))$ [Tho52]. We
combine this with the Cartan formula to calculate for any $x\in
H^{r-2}(B;\mathbb{Z}/2)$ that

$\displaystyle\operatorname{Sq}^{2}(U_{n}\cup(\rho_{n}^{*}(x)))$
$\displaystyle=\operatorname{Sq}^{0}(U_{n})\cup\operatorname{Sq}^{2}(\rho_{n}^{*}(x))+\operatorname{Sq}^{1}(U_{n})\cup\operatorname{Sq}^{1}(\rho_{n}^{*}(x))+\operatorname{Sq}^{2}(U_{n})\cup\operatorname{Sq}^{0}(\rho_{n}^{*}(x))$
$\displaystyle=U_{n}\cup\operatorname{Sq}^{2}(\rho_{n}^{*}(x))+U_{n}\cup\rho_{n}^{*}(w_{1}(v_{n}))\cup\operatorname{Sq}^{1}(\rho_{n}^{*}(x))+U_{n}\cup\rho_{n}^{*}(w_{2}(v_{n}))\cup\rho_{n}^{*}(x)$
$\displaystyle=U_{n}\cup\rho_{n}^{*}\left((\operatorname{Sq}^{2}(x))+0+w_{2}(v_{n})\cup
x)\right)$
$\displaystyle=U_{n}\cup\rho_{n}^{*}\left(\operatorname{Sq}^{2}_{w_{2}(v_{n})}\right),$

so that
$\operatorname{Sq}^{2}\circ\Phi^{n}=\Phi^{n}\circ\operatorname{Sq}^{2}_{w_{2}(v_{n})}$.
For the penultimate inequality we used that $w_{1}(v_{n})=0$. Taking the limit
of this equality we obtain the desired formula. Here we have used that
Steenrod squares are stable and natural, that $\Phi^{\infty}$ is defined as
the limit of $\Phi^{n}$, and that Stiefel-Whitney class are stable.

The case when $v$ is a stable vector bundle is proved entirely similarly. ∎

The identity map $\operatorname{Id}_{B}\colon B\to B$, considered as a
fibration with point fibres, is obviously
$\pi_{*}^{\operatorname{st}}$-orientable because the monodromy is trivial. It
is thus meaningful to consider the following James spectral sequence.

######  Lemma 4.3.

Let $B$ be a CW complex and consider the input for the James spectral sequence

${\operatorname{pt}}$${B}$${B}$and${v\colon
B}$${\operatorname{BSTOP}.}$$\scriptstyle{j}$$\scriptstyle{\operatorname{Id}}$

The James spectral sequence has the following differentials $d_{2}^{r,s}\colon
E_{2}^{r,s}\to E_{2}^{r-2,s+1}$ for all $r\geq 0$.

  1. (1)

Via the universal coefficient theorem, $d^{r,1}_{2}\colon
H_{r}(B;\mathbb{Z}/2)\to H_{r-2}(B;\mathbb{Z}/2)$ is identified with
$\operatorname{Hom}(\operatorname{Sq}^{2}_{w_{2}(v)},\mathbb{Z}/2)$. More
precisely, the following square commutes:

${H_{r}(B;\mathbb{Z}/2)}$${H_{r-2}(B;\mathbb{Z}/2)}$${\operatorname{Hom}(H^{r}(B;\mathbb{Z}/2),\mathbb{Z}/2)}$${\operatorname{Hom}(H^{r-2}(B;\mathbb{Z}/2),\mathbb{Z}/2).}$$\scriptstyle{d_{2}^{r,1}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{(\operatorname{Sq}^{2}_{w_{2}(v)})^{*}}$

  2. (2)

The map $d_{2}^{r,0}\colon H_{r}(B;\mathbb{Z})\to H_{r-2}(B;\mathbb{Z}/2)$ is
identified with reduction mod 2 followed by
$\operatorname{Hom}(\operatorname{Sq}^{2}_{w_{2}(v)},\mathbb{Z}/2)$. More
precisely, the following diagram commutes:

${H_{r}(B;\mathbb{Z})}$${H_{r-2}(B;\mathbb{Z}/2)}$${H_{r}(B;\mathbb{Z}/2)}$${\operatorname{Hom}(H^{r}(B;\mathbb{Z}/2),\mathbb{Z}/2)}$${\operatorname{Hom}(H^{r-2}(B;\mathbb{Z}/2),\mathbb{Z}/2).}$$\scriptstyle{d_{2}^{r,0}}$$\scriptstyle{\operatorname{red}_{2}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{(\operatorname{Sq}^{2}_{w_{2}(v)})^{*}}$

The same statements hold using a stable vector bundle $v\colon
B\to\operatorname{BSO}$.

###### Proof.

In this proof we write the functor
$\operatorname{Hom}(-,\mathbb{Z}/2)=:(-)^{{\ast}}$, for brevity.

Consider that $M(v\circ j)$ is homotopy equivalent to the sphere spectrum and
hence we get $\pi_{*}(M(v\circ j))\cong\pi_{*}^{\operatorname{st}}$, the
stable homotopy groups of spheres. In particular, this explains the
coefficients $\pi_{0}^{\operatorname{st}}\cong\mathbb{Z}$ and
$\pi_{1}^{\operatorname{st}},\pi_{2}^{\operatorname{st}}\cong\mathbb{Z}/2$ in
items (1) and (2).

By the construction of the James spectral sequence (see [Tei93, Theorem]), the
homology Thom isomorphisms $\Phi_{\infty}$ induce a natural isomorphism
between the Atiyah-Hirzebruch spectral sequence of the Thom spectrum

$E_{2}^{r,s}\cong
H_{r}(M(v);\pi_{s}^{\operatorname{st}})\implies\pi_{r+s}(M(v))$

and the James spectral sequence from the statement of the lemma. In the
former, it is well-known that $d^{r,1}_{2}=(\operatorname{Sq}^{2})^{\ast}$,
and $d_{2}^{r,0}=(\operatorname{Sq}^{2})^{\ast}\circ\operatorname{red}_{2}$,
where $\operatorname{red}_{2}$ denotes reduction mod 2; see e.g. [Tei93, proof
of Lemma]. We must compute what these correspond to under $\Phi_{\infty}$.

We claim the following diagram is commutative.

(4)
${H_{r}(M(v);\mathbb{Z})}$${H_{r}(B;\mathbb{Z})}$${H_{r}(M(v);\mathbb{Z}/2)}$${H_{r}(B;\mathbb{Z}/2)}$${H^{r}(M(v);\mathbb{Z}/2)^{{\ast}}}$${H^{r}(B;\mathbb{Z}/2)^{{\ast}}}$${H^{r-2}(M(v);\mathbb{Z}/2)^{{\ast}}}$${H^{r-2}(B;\mathbb{Z}/2)^{{\ast}}}$${H_{r-2}(M(v);\mathbb{Z}/2)}$${H_{r-2}(B;\mathbb{Z}/2)}$$\scriptstyle{\operatorname{red}_{2}}$$\scriptstyle{\Phi_{\infty}}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{red}_{2}}$$\scriptstyle{\operatorname{ev}}$$\scriptstyle{\cong}$$\scriptstyle{\Phi_{\infty}}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{ev}}$$\scriptstyle{\cong}$$\scriptstyle{{(\operatorname{Sq}^{2})}^{\ast}}$$\scriptstyle{{(\Phi^{\infty})}^{\ast}}$$\scriptstyle{\cong}$$\scriptstyle{{(\operatorname{Sq}^{2}_{w_{2}(v)})}^{\ast}}$$\scriptstyle{{(\Phi^{\infty})}^{\ast}}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{ev}}$$\scriptstyle{\cong}$$\scriptstyle{\Phi_{\infty}}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{ev}}$$\scriptstyle{\cong}$

Commutativity of diagram (4) is enough to complete the proof, as $d^{r,0}_{2}$
and $d_{2}^{r,1}$ in the Atiyah-Hirzebruch spectral sequence are respectively
the composition along the full left column and the lower four entries of the
left column, and these map in the right column to the respective differentials
claimed in items (1) and (2).

The top square of diagram (4) is evidently commutative. Commutativity in the
third square from the top is furnished by Lemma 4.2. To see commutativity of
the remaining two squares, fix $n\geq 0$ and consider the diagram

(5)
${H_{r+n}(\operatorname{Th}(v_{n});\mathbb{Z}/2)}$${H_{r}(D(v_{n});\mathbb{Z}/2)}$${H_{r}(B;\mathbb{Z}/2)}$${H^{r+n}(\operatorname{Th}(v_{n});\mathbb{Z}/2)^{{\ast}}}$${H^{r}(D(v_{n});\mathbb{Z}/2)^{{\ast}}}$${H^{r}(B;\mathbb{Z}/2)^{{\ast}}.}$$\scriptstyle{\operatorname{ev}}$$\scriptstyle{\cong}$$\scriptstyle{U_{n}\cap-}$$\scriptstyle{{(\rho_{n})}_{*}}$$\scriptstyle{\operatorname{ev}}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{ev}}$$\scriptstyle{\cong}$$\scriptstyle{{(U_{n}\cup-)}^{\ast}}$$\scriptstyle{{({(\rho_{n})}^{*})}^{\ast}}$

The right-hand square commutes by naturality of the Universal Coefficient
Theorem. To check commutativity of the left-hand square we compute

$(\operatorname{ev}(U_{n}\cap x))(\varphi)=\varphi(U_{n}\cap
x)=(U_{n}\cup\varphi)(x)=\operatorname{ev}(x)(U_{n}\cup\varphi)=(U_{n}\cup-)^{\ast}\operatorname{ev}(x)(\varphi)$

for any $\varphi\in H^{r}(D(v_{n});\mathbb{Z}/2)$ and $x\in
H_{r+n}(\operatorname{Th}(v_{n});\mathbb{Z}/2)$. The second equality is the
“cup/cap” formula [Hat02, p. 249]. Thus the diagram (5) commutes for all $n$.
Take the colimit of the outside rectangle of diagram (5) to obtain
commutativity for the remaining two squares of diagram (4).

The case when $v$ is a stable vector bundle is proved entirely similarly. ∎

######  Corollary 4.6.

Suppose we have a $\pi_{*}^{\operatorname{st}}$-orientable fibration and
stable topological bundle

${F}$${B}$${K}$and${v\colon
B}$${\operatorname{BSTOP},}$$\scriptstyle{i}$$\scriptstyle{f}$

such that $\operatorname{pt}\to F$ induces isomorphisms
$\pi_{s}^{\operatorname{st}}\xrightarrow{\cong}\pi_{s}(M(v\circ i))$ for
$s=0,1,2$. Suppose there exists $w\in H^{2}(K;\mathbb{Z}/2)$ such that
$f^{*}(w)=w_{2}(v)$.

  1. (1)

Suppose for some $r\in\mathbb{Z}$ that $f_{*}\colon H_{r}(B;\mathbb{Z}/2)\to
H_{r}(K;\mathbb{Z}/2)$ is surjective. Then, in the sense of Lemma 4.3,
$d^{r,1}_{2}\colon H_{r}(K;\mathbb{Z}/2)\to H_{r-2}(K;\mathbb{Z}/2)$ is
$\operatorname{Hom}(\operatorname{Sq}^{2}_{w},\mathbb{Z}/2)$.

  2. (2)

Suppose for some $r\in\mathbb{Z}$ that $f_{*}\colon H_{r}(B;\mathbb{Z})\to
H_{r}(K;\mathbb{Z})$ is surjective. Then, in the sense of Lemma 4.3,
$d_{2}^{r,0}\colon H_{r}(K;\mathbb{Z})\to H_{r-2}(K;\mathbb{Z}/2)$ is
reduction mod 2 followed by
$\operatorname{Hom}(\operatorname{Sq}^{2}_{w},\mathbb{Z}/2)$.

The same statements hold using a stable vector bundle $v\colon
B\to\operatorname{BSO}$.

###### Proof.

Consider the map of fibrations given by the commuting diagram

${\operatorname{pt}}$${B}$${B}$${F}$${B}$${K.}$$\scriptstyle{j}$$\scriptstyle{k}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{f}$$\scriptstyle{i}$$\scriptstyle{f}$

where $k$ is arbitrary and $j$ is defined to make the diagram commute. By the
construction of the James spectral sequence (see [Tei93, Theorem]), this
induces a map of spectral sequences which on the $E_{1}$-page is of the form

$f_{*}\otimes k_{*}\colon C_{r}(B)\otimes\pi^{\operatorname{st}}_{s}\to
C_{r}(K)\otimes\pi_{s}(M(v\circ i)).$

Taking homology, we see for $s=0,1,2$ and for all $r\geq 0$ the map on the
$E_{2}$ page factors as

${H_{r}(B;\pi_{s}^{\operatorname{st}})}$${H_{r}(K;\pi_{s}^{\operatorname{st}})}$${H_{r}(K;\pi_{s}(M(v\circ
i))).}$$\scriptstyle{f_{*}}$$\scriptstyle{k_{*}}$$\scriptstyle{\cong}$

When $s=1,2$, we may identify the coefficients in the above sequence with
$\mathbb{Z}/2$. When $s=0$, we choose an isomorphism
$\pi_{0}^{\operatorname{st}}\cong\mathbb{Z}$ and use $k_{*}$ to identify
$\pi_{0}(M(v\circ i))\cong\mathbb{Z}$. With these identifications, we may
identify the map of spectral sequences on the $E_{2}$ page as $f_{*}\colon
H_{r}(B;\pi_{s}^{\operatorname{st}})\to H_{r}(K;\pi_{s}^{\operatorname{st}})$,
when $s=0,1,2$.

We consider item (1). Let $x\in H_{r}(K;\mathbb{Z}/2)$. Then by hypothesis
there exists $y\in H_{r}(B;\mathbb{Z}/2)$ such that $f_{*}(y)=x$. We thus have

$\displaystyle d^{r,1}_{2}(x)=d^{r,1}_{2}(f_{*}(y))=f_{*}\circ
d^{r,1}_{2}(y)=f_{*}\circ\operatorname{Hom}(\operatorname{Sq}^{2}_{w_{2}(v)},\mathbb{Z}/2)(y)\in
H_{r-2}(K;\mathbb{Z}/2),$

where the final equality comes from Lemma 4.3. Applying the evaluation
isomorphism
$H_{r-2}(K;\mathbb{Z}/2)\cong\operatorname{Hom}(H^{r-2}(K;\mathbb{Z}/2),\mathbb{Z}/2)$,
and writing $f_{*}=\operatorname{Hom}(f^{*},\mathbb{Z}/2)$, we have shown that
$d^{r,1}_{2}(x)=\left(\operatorname{Hom}(f^{*},\mathbb{Z}/2)\circ\operatorname{Sq}^{2}_{w_{2}(v)}\right)(y)$.
Next we show that the right hand side equals
$\operatorname{Hom}(\operatorname{Sq}^{2}_{w}(x),\mathbb{Z}/2)$. To do this we
evaluate on an arbitrary class $u\in H^{r-2}(K;\mathbb{Z}/2)$:

$\displaystyle\left(\operatorname{Hom}(f^{*},\mathbb{Z}/2)\circ\operatorname{Sq}^{2}_{w_{2}(v)}(u)\right)(y)$
$\displaystyle=\left(\operatorname{Sq}^{2}_{w_{2}(v)}(f^{*}u)\right)(y)$
$\displaystyle=\operatorname{Sq}^{2}(f^{*}u)(y)+\left(w_{2}(v)\cup(f^{*}u)\right)(y)$
$\displaystyle=\operatorname{Sq}^{2}(u)(f_{*}y)+\left((f^{*}w)\cup(f^{*}u)\right)(y)$
$\displaystyle=\operatorname{Sq}^{2}(u)(f_{*}y)+\left(w\cup u\right)(f_{*}y)$
$\displaystyle=\operatorname{Sq}^{2}(u)(x)+\left(w\cup u\right)(x).$

This shows
$d^{r,1}_{2}(x)=\left(\operatorname{Hom}(f^{*},\mathbb{Z}/2)\circ\operatorname{Sq}^{2}_{w_{2}(v)}\right)(y)=\operatorname{Hom}(\operatorname{Sq}^{2}_{w}(x),\mathbb{Z}/2)$,
as claimed.

A similar calculation to prove item (2) is left to the reader. The case when
$v$ is a stable vector bundle is proved entirely similarly. ∎

###  4.2. Bordism over the normal 2-type of a simply connected 4-manifold

From now on, let $(B,p)$ denote either $(B(X)^{\operatorname{TOP}},p)$ or
$(B(X)^{\operatorname{Diff}},p)$, unless one is specified. Refer to the start
of Section 3.2 to recall the notation. We will be interested in the following
inputs for the James spectral sequence. When
$\operatorname{CAT}=\operatorname{TOP}$, we will use

${\operatorname{BTOPSpin}}$${B}$${K}$and${p\colon
B}$${\operatorname{BSTOP.}}$$\scriptstyle{i}$$\scriptstyle{q}$

When $\operatorname{CAT}=\operatorname{Diff}$, we use as input

${\operatorname{BSpin}}$${B}$${K}$and${p\colon
B}$${\operatorname{BSO.}}$$\scriptstyle{i}$$\scriptstyle{q}$

Note that as $K$ is simply connected, these fibrations are both
$\pi_{*}^{\operatorname{st}}$-orientable, so the James spectral sequence
applies. We also not note that the maps $p\circ i$ classify the universal
bundle over $\operatorname{BTOPSpin}$ and $\operatorname{BSpin}$ respectively
(cf. beginning of Section 3.2). With these inputs, we have $E_{2}$ page:

$E_{2}^{r,s}\cong H_{r}(K;\pi_{s}(M(p\circ
i)))\cong\left\\{\begin{array}[]{ll}H_{r}(K;\Omega_{s}^{\operatorname{TOPSpin}})&\text{when
$\operatorname{CAT}=\operatorname{TOP}$,}\\\\[3.0pt]
H_{r}(K;\Omega_{s}^{\operatorname{Spin}})&\text{when
$\operatorname{CAT}=\operatorname{Diff}$.}\end{array}\right.$

The second isomorphism follows from the Pontryagin-Thom theorem. Again by the
Pontryagin-Thom theorem, the spectral sequence converges to
$\pi_{r+s}(M(p))\cong\Omega_{r+s}(B,p)$.

We will need to reference the following well-known computation.

######  Lemma 4.7.

The groups $\pi_{n}(\operatorname{STOP})$ are
$\mathbb{Z}/2,0,\mathbb{Z}\oplus\mathbb{Z}/2,0$ for $n=1,2,3,4$ respectively.

###### Proof.

We explain briefly how these groups are computed. First, there is a pullback
square

${\operatorname{BSO}}$${\operatorname{BSTOP}}$${\operatorname{BO}}$${\operatorname{BTOP}.}$

Because it is a pullback square, the respective horizontal homotopy fibres
$\operatorname{STOP}/\operatorname{SO}$ and
$\operatorname{TOP}/\operatorname{O}$ are (weakly) homotopy equivalent to one
another. Thus the homotopy groups of $\operatorname{STOP}/\operatorname{SO}$
satisfy
$\pi_{i}(\operatorname{STOP}/\operatorname{SO})\cong\pi_{i}(\operatorname{TOP}/\operatorname{O})\cong\pi_{i}(K(\mathbb{Z}/2,3))$
for $i=1,2,3,4$, with the latter isomorphism coming from [KS77, Annex C, p.
317]. It follows from the long exact sequence of the fibration
$\operatorname{STOP}/\operatorname{SO}\to\operatorname{BSO}\to\operatorname{BSTOP}$
that $\pi_{i}(\operatorname{BSTOP})\cong\pi_{i}(\operatorname{BSO})$ for
$i=2,3,5$. We also have a short exact sequence

$\underbrace{\pi_{4}(\operatorname{STOP}/\operatorname{SO})}_{=\,0}\to\underbrace{\pi_{4}(\operatorname{BSO})}_{\cong\,\mathbb{Z}}\to\pi_{4}(\operatorname{BSTOP})\to\underbrace{\pi_{3}(\operatorname{STOP}/\operatorname{SO})}_{\cong\,\mathbb{Z}/2}\to\underbrace{\pi_{3}(\operatorname{BSO})}_{=\,0}.$

Siebenmann [KS77, Annex C, p. 318] showed that this sequence splits, so
$\pi_{4}(\operatorname{BSTOP})\cong\mathbb{Z}\oplus\mathbb{Z}/2$. The desired
computation then follows from
$\pi_{n}(\operatorname{STOP})\cong\pi_{n+1}(\operatorname{BSTOP})$. ∎

######  Proposition 4.8.

The following assertions hold.

  1. (1)

For $r$ odd or $r\leq 4$, the differential $d_{2}\colon
H_{r}(K;\Omega_{1}^{\operatorname{BTOPSpin}})\to
H_{r-2}(K;\Omega_{2}^{\operatorname{BTOPSpin}})$ is
$\operatorname{Hom}(\operatorname{Sq}^{2}_{w},\mathbb{Z}/2)$.

  2. (2)

For $r$ odd or $r\leq 6$, the differential $d_{2}\colon
H_{r}(K;\Omega_{0}^{\operatorname{BTOPSpin}})\to
H_{r-2}(K;\Omega_{1}^{\operatorname{BTOPSpin}})$ is reduction modulo 2,
followed by $\operatorname{Hom}(\operatorname{Sq}^{2}_{w},\mathbb{Z}/2)$.

The analogous statements in the smooth category are also true.

###### Proof.

By the definition of $B$, there is guaranteed to be a class $w\in
H^{2}(K;\mathbb{Z}/2)$ such that $q^{*}w=w_{2}(p)$. Further,
$\operatorname{pt}\to\operatorname{BTOPSpin}$ induces isomorphisms
$\pi_{s}^{\operatorname{st}}\cong\pi_{s}(M(\gamma))\cong\Omega^{\operatorname{TOPSpin}}_{s}$
for $s=0,1,2$. Thus Corollary 4.6 applies, so we are interested in analysing
for which $r$ the maps $q_{*}\colon H_{r}(B;\mathbb{Z}/2)\to
H_{r}(K;\mathbb{Z}/2)$ and $q_{*}\colon H_{r}(B;\mathbb{Z})\to
H_{r}(K;\mathbb{Z})$ are surjective.

A model for $K$ is given by $\prod^{n}\operatorname{\mathbb{C}P}^{\infty}$,
where $n=b_{2}(X)$ is the second Betti number. This model has a cell structure
with only even-dimensional cells. In particular
$H_{r}(K;\mathbb{Z})=0=H_{r}(K;\mathbb{Z}/2)$ for $r$ odd, and the maps in
question are then necessarily surjective.

Since $\operatorname{TOPSpin}$ is the universal cover of
$\operatorname{STOP}$, and using Lemma 4.7, it follows that the groups
$\pi_{n}(\operatorname{TOPSpin})$ are $0,0,\mathbb{Z}\oplus\mathbb{Z}/2,0$ for
$n=1,2,3,4$. This implies that the groups $\pi_{n}(\operatorname{BTOPSpin})$
are $0,0,0,\mathbb{Z}\oplus\mathbb{Z}/2,0$ for $n=1,2,3,4,5$. From this, and
the long exact sequence of the fibration $q\colon B\to K$, we see that $q$ is
$4$-connected. The Hurewicz theorem now implies that $q_{*}\colon
H_{r}(B;\mathbb{Z})\to H_{r}(K;\mathbb{Z})$ is surjective for $p=0,1,2,3,4$.

We show surjectivity when $r=6$ by analysing the Leray-Serre spectral sequence
for $q\colon B\to K$, which recall has differentials $d_{n}^{r,s}\colon
E_{n}^{r,s}\to E_{n}^{r-n,s+n-1}$ and $E_{2}^{r,s}\cong
H_{r}(K;H_{s}(F))\implies H_{r+s}(B)$. The Hurewicz Theorem shows that the map
$\pi_{n}(\operatorname{BTOPSpin})\to
H_{n}(\operatorname{BTOPSpin};\mathbb{Z})$, is an isomorphism for $n=1,2,3,4$
and a surjection for $n=5$. But as $\pi_{5}(\operatorname{BTOPSpin})=0$, this
implies $H_{5}(\operatorname{BTOPSpin};\mathbb{Z})=0$. Thus
$H_{n}(\operatorname{BTOPSpin};\mathbb{Z})=0$ for $n=1,2,3,5$. Looking at the
$r+s=6$ line, for the Leray-Serre spectral sequence and noting that
$E_{2}^{1,4}\cong H_{1}(K;\mathbb{Z}\oplus\mathbb{Z}/2)=0$, we see that
$E_{\infty}^{6,0}=E_{2}^{6,0}\cong H_{6}(K;\mathbb{Z})$, from which we deduce
that $q_{*}\colon H_{6}(B;\mathbb{Z})\to H_{6}(K;\mathbb{Z})$ is surjective.

The proof in the smooth category is entirely similar except we replace
$\operatorname{TOPSpin}$ with $\operatorname{Spin}$, and so work with the
homotopy groups $\pi_{n}(\operatorname{Spin})=0,0,\mathbb{Z},0$, for
$n=1,2,3,4$. ∎

We now apply the James spectral sequence to compute.

######  Proposition 4.9.

When $\operatorname{CAT}=\operatorname{TOP}$, the $2$-types described in
Proposition 3.6 have bordism groups

$\begin{array}[]{lcl}\Omega_{4}(B(X)^{\operatorname{TOP}},p)&\cong&\left\\{\begin{array}[]{ll}\Omega^{\operatorname{TOPSpin}}_{4}\oplus
E_{3}^{4,0}&\text{for $X$ spin,}\\\ \Omega^{\operatorname{TOPSpin}}_{4}\oplus
E_{3}^{4,0}\oplus E_{3}^{2,2}&\text{for $X$ non spin,}\end{array}\right.\\\
\Omega_{5}(B(X)^{\operatorname{TOP}},p)&=&0.\end{array}$

The summand $\mathbb{Z}/2\cong E_{3}^{2,2}\subseteq H_{2}(K;\mathbb{Z}/2)$ is
generated by the $\operatorname{Hom}$-dual to $w\in H^{2}(K;\mathbb{Z}/2)$.
Writing $\langle x_{1},\dots,x_{n}\rangle$ for generators of $H^{2}(K)$, the
summand $E_{3}^{4,0}$ is isomorphic to

$\ker\left(\operatorname{red}_{2}\circ\operatorname{Hom}(\operatorname{Sq}_{w}^{2},\mathbb{Z}/2)\right)\cong
2\mathbb{Z}^{n}\oplus\mathbb{Z}^{n(n-1)}\subseteq H_{4}(K;\mathbb{Z}),$

where $\mathbb{Z}^{n(n-1)}$ is generated by the $\operatorname{Hom}$-duals to
$x_{i}\cup x_{j}$ for $1\leq i<j\leq n$, and $2\mathbb{Z}^{n}$ is generated by
the $\operatorname{Hom}$-duals to $x_{i}\cup x_{i}$ for $1\leq i\leq n$.

When $\operatorname{CAT}=\operatorname{Diff}$, the corresponding statements
hold, with $\Omega^{\operatorname{TOPSpin}}_{4}$ replaced by
$\Omega^{\operatorname{Spin}}_{4}$.

###### Proof.

We work initially in the topological category, then indicate which parts of
the proof change in the smooth category.

A model for $K(\mathbb{Z}^{n},2)$ is given by
$\prod^{n}_{i=1}\operatorname{\mathbb{C}P}^{\infty}$, where here $n=b_{2}(X)$
is the second Betti number. Here are the relevant terms on the $E_{2}$ page of
the James spectral sequence.

$0$$1$$2$$3$$4$$5$$q$$0$$1$$2$$3$$4$$5$$6$$p$$0$$0$$H_{4}(K;\mathbb{Z})$$0$$H_{6}(K;\mathbb{Z})$$0$$H_{2}(K;\mathbb{Z}/2)$$0$$H_{4}(K;\mathbb{Z}/2)$$0$$0$$H_{2}(K;\mathbb{Z}/2)$$0$$0$$0$$0$$0$$0$$0$$0$$0$$\Omega_{4}^{\operatorname{TOPSpin}}$$0$$0$$0$$\Omega_{5}^{\operatorname{TOPSpin}}$$0$$0$$0$$0$$0$$0$$\boldsymbol{d_{2}^{\,6,0}}$$\boldsymbol{d_{2}^{\,4,1}}$$\boldsymbol{d_{2}^{\,4,0}}$

In the case that $X$ is spin, choose any basis $\langle
y_{1},\dots,y_{n}\rangle$ of $\pi_{2}(X)$. In the case that $X$ is not spin,
we choose a basis more carefully, as follows. Let $y_{1}\in\pi_{2}(X)$ be such
that $w_{2}(X)(y_{1})=1$. Extend to a basis $\langle y_{1},\dots,y_{n}\rangle$
of $\pi_{2}(X)$. If $w_{2}(X)(y_{i})\neq 0$ for some $i=2,\dots,n$, we replace
$y_{i}$ with $y_{1}+y_{i}$, so that without loss of generality it may be
assumed $w_{2}(X)(y_{i})=0$ for all $i=2,\dots,n$.

Recall that in the definition of $(B,p)$ there is a reference map $k\colon
X\to K$ that is an isomorphism on $\pi_{2}$. Thus in both the spin and non
spin case, $\langle y_{1},\dots,y_{n}\rangle$ determines a basis $\langle
k_{*}(y_{1}),\dots,k_{*}(y_{n})\rangle$ of $\pi_{2}(K)$. Write $e^{0}_{i}\cup
e^{2}_{i}\cup e^{4}_{i}\cup\cdots$ for the standard cell structure of the
$i^{\text{th}}$ $\operatorname{\mathbb{C}P}^{\infty}$ factor of
$\prod_{i=1}^{n}\operatorname{\mathbb{C}P}^{\infty}$. Choose a homotopy
equivalence $f\colon
K\simeq\prod_{i=1}^{n}\operatorname{\mathbb{C}P}^{\infty}$, such that
$f_{*}(k_{*}(y_{k}))=[e^{2}_{i}]$. Write $x_{i}\in
H^{2}(\prod_{i=1}^{n}\operatorname{\mathbb{C}P}^{\infty};\mathbb{Z}/2)$ for
the class that is $\mathbb{Z}/2$-dual of $[e^{2}_{i}]$. There are isomorphisms
of graded algebras

${(\mathbb{Z}/2)[x_{1},\dots,x_{n}]}$${H^{*}(\prod_{i=1}^{n}\operatorname{\mathbb{C}P}^{\infty};\mathbb{Z}/2)}$${H^{*}(K;\mathbb{Z}/2),}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{f^{*}}$

where the first isomorphism is given by the Künneth theorem. In an abuse of
notation, we will refer to the images of the $x_{i}$ in
$H^{*}(K;\mathbb{Z}/2)$ by the same name. Thus, we have bases

$\displaystyle H^{2}(K;\mathbb{Z}/2)$ $\displaystyle\cong\langle
x_{i}\,|\,1\leq i\leq n\rangle,$ $\displaystyle H^{4}(K;\mathbb{Z}/2)$
$\displaystyle\cong\langle x_{i}\cup x_{j}\,|\,1\leq i,j\leq n\rangle,$
$\displaystyle H^{6}(K;\mathbb{Z}/2)$ $\displaystyle\cong\langle x_{i}\cup
x_{j}\cup x_{k}\,|\,1\leq i,j,k\leq n\rangle.$

We now split the calculations into the case $X$ is spin and the case $X$ is
not spin.

Firstly, assume $X$ is spin, so that $w=0$. By Proposition 4.8, the
differentials $d_{2}^{4,1}$ and $d_{2}^{4,0}$ are each the dual to the map
$x_{i}\mapsto\operatorname{Sq}^{2}(x_{i})=x_{i}\cup x_{i}$. From this we see
that $d_{2}^{4,1}$ is surjective and thus $E_{3}^{2,2}=0$. Looking at the
$r+s=4$ line and observing that $\Omega_{4}^{\operatorname{TOPSpin}}$ survives
to $E_{\infty}$, we have shown that
$\Omega_{4}(B(X)^{\operatorname{TOP}},p)\cong\Omega^{\operatorname{TOPSpin}}_{4}\oplus
E_{3}^{2,2}\oplus E_{3}^{4,0}$, as claimed. We also see that
$\ker(d_{2}^{4,1})$ has basis $\langle[x_{i}\cup x_{j}]^{*}\,|\,1\leq i<j\leq
n\rangle$. By Proposition 4.8, the differential $d_{2}^{6,0}$ is reduction
module 2 followed by the dual to the map $\operatorname{Sq}^{2}$. We apply the
Cartan formula:

$\displaystyle x_{i}\cup x_{j}\mapsto\operatorname{Sq}^{2}(x_{i}\cup x_{j})$
$\displaystyle=\operatorname{Sq}^{0}(x_{i})\cup\operatorname{Sq}^{2}(x_{j})+\operatorname{Sq}^{1}(x_{i})\cup\operatorname{Sq}^{1}(x_{j})+\operatorname{Sq}^{2}(x_{i})\cup\operatorname{Sq}^{1}(x_{j})$
$\displaystyle=x_{i}\cup x_{j}\cup x_{j}+x_{i}\cup x_{i}\cup x_{j}.$

Fixing $i,j$ with $1\leq i<j\leq n$, our calculation shows that both
$d_{2}^{6,0}([x_{i}\cup x_{j}\cup x_{j}]^{*})$ and $d_{2}^{6,0}([x_{i}\cup
x_{i}\cup x_{j}]^{*})$ project nontrivially to the span of $[x_{i}\cup
x_{j}]^{*}$. By inspection, we see that the set $\\{a,b\\}=\\{i,j\\}$ is the
unique set such that $\operatorname{Sq}^{2}(x_{a}\cup x_{b})$ projects
nontrivially to the span of $x_{i}\cup x_{i}\cup x_{j}$. Hence
$d_{2}^{6,0}([x_{i}\cup x_{i}\cup x_{j}]^{*})=[x_{i}\cup x_{j}]^{*}$. (A
similar argument can be made for $x_{i}\cup x_{j}\cup x_{j}$, but it is not
needed to show that $[x_{i}\cup x_{j}]^{*}\in\operatorname{im}d_{2}^{6,0}$.)
Applying this for every $1\leq i<j\leq n$, we can conclude that
$\operatorname{im}(d_{2}^{6,0})=\ker(d_{2}^{4,1})$, so that $E_{3}^{4,1}=0$.
By [KS77, Theorem 13.1], we have that $\Omega_{5}^{\operatorname{TOPSpin}}=0$,
and hence the $r+s=5$ line dies already on the $E_{3}$ page and we have shown
that $\Omega_{5}(B(X)^{\operatorname{TOP}},p)=0$ in the spin case.

Now assume $X$ is not spin, so that $w\neq 0$. Our construction of a basis was
such that $x_{1}=w$. By Proposition 4.8, the differential $d_{2}^{4,1}$ is
dual to the map

$\operatorname{Sq}^{2}_{w}\colon
x_{i}\mapsto\operatorname{Sq}^{2}(x_{i})+x_{i}\cup
x_{1}=\left\\{\begin{array}[]{ll}0&\text{when $i=1$,}\\\ x_{i}\cup
x_{i}+x_{i}\cup x_{1}&\text{when $i\neq 1$}.\end{array}\right.$

From this we see that $\operatorname{im}(d_{2}^{4,1})=\langle
x_{2}^{*},\dots,x_{n}^{*}\rangle$ and so $E_{3}^{2,2}\cong\langle
x_{1}^{*}\rangle$. Looking at the $r+s=4$ line and observing that
$\Omega_{4}^{\operatorname{TOPSpin}}$ survives to $E_{\infty}$, we have shown
that
$\Omega_{4}(B(X)^{\operatorname{TOP}},p)\cong\Omega^{\operatorname{TOPSpin}}_{4}\oplus
E_{3}^{2,2}\oplus E_{3}^{4,0}$, as claimed. We also see that
$\ker(d_{2}^{4,1})$ has basis $\langle[x_{1}\cup x_{1}]^{*},[x_{i}\cup
x_{j}]^{*}\,|\,1<i<j\leq n\rangle$. By Proposition 4.8, the map $d_{2}^{6,0}$
is reduction modulo 2 followed by the dual to $\operatorname{Sq}^{2}_{w}$. As
$H_{6}(K;\mathbb{Z})$ is free, reduction modulo 2 is surjective to
$H_{6}(K;\mathbb{Z}/2)$. We now show that the dual of
$\operatorname{Sq}_{w}^{2}$ is surjective to $\ker(d_{2}^{4,1})$. Let
$1<i<j\leq n$ and use our Cartan formula computation above:

$\begin{array}[]{ccl}\operatorname{Sq}^{2}_{w}\colon x_{i}\cup
x_{j}&\mapsto&\operatorname{Sq}^{2}(x_{i}\cup x_{j})+(x_{i}\cup x_{j})\cup
x_{1}\\\ &&=x_{i}\cup x_{j}\cup x_{j}+x_{i}\cup x_{i}\cup x_{j}+x_{i}\cup
x_{j}\cup x_{1},\\\ \operatorname{Sq}^{2}_{w}\colon x_{1}\cup
x_{1}&\mapsto&\operatorname{Sq}^{2}(x_{1}\cup x_{1})+(x_{1}\cup x_{1})\cup
x_{1}\\\ &&=x_{1}\cup x_{1}\cup x_{1}+x_{1}\cup x_{1}\cup x_{1}+x_{1}\cup
x_{1}\cup x_{1}\\\ &&=x_{1}\cup x_{1}\cup x_{1}.\end{array}$

There is no $1<i<j\leq n$ such that $(d_{2}^{6,0})^{*}(x_{i}\cup x_{j})$
projects nontrivially to $x_{1}\cup x_{1}\cup x_{1}$ so we deduce that
$d_{2}^{6,0}([x_{1}\cup x_{1}\cup x_{1}]^{*})=[x_{1}\cup x_{1}]^{*}$. Now fix
$i,j$ with $1<i<j\leq n$. By inspection, the set $\\{a,b\\}=\\{i,j\\}$ is the
unique set such that $\operatorname{Sq}^{2}(x_{a}\cup x_{b})+(x_{a}\cup
x_{b})\cup x_{1}$ projects nontrivially to the span of $(x_{i}\cup x_{j})\cup
x_{1}$. This shows that $d_{2}^{6,0}([x_{i}\cup x_{j}\cup
x_{1}]^{*})=[x_{i}\cup x_{j}]^{*}$. Applying this for every $1<i<j\leq n$, we
can conclude that $\operatorname{im}(d_{2}^{6,0})=\ker(d_{2}^{4,1})$, so that
again $E_{3}^{4,1}=0$. From this and $\Omega_{5}^{\operatorname{TOPSpin}}=0$,
hence the $r+s=5$ line dies already on the $E_{3}$ page and we have shown that
$\Omega_{5}(B(X)^{\operatorname{TOP}},p)=0$ in the spin case.

In the smooth category, the proof is entirely similar except we replace
$\Omega_{5}^{\operatorname{TOPSpin}}$ with $\Omega_{5}^{\operatorname{Spin}}$,
which vanishes [Mil63]. ∎

##  5\. Proof of the main pseudo-isotopy classification

We establish an elementary technical result which we then use to extend a
$B$-structure from $\partial(X\times[0,1])$ to $X\times[0,1]$. With this
extension in hand, we then prove Theorem 2.16, the main pseudo-isotopy
classification.

###  5.1. Extending a $B$-structure from the boundary

Let $(X,\partial X)$ be a compact 4-manifold. Write $I:=[0,1]$,
$X_{i}:=X\times\\{i\\}$ for $i=0,1$, and

$D:=X_{0}\cup(\partial X\times I)\cup X_{1},$

i.e. $D=\partial(X\times I)$. Choose a finite cell complex homotopy equivalent
to $X$, relative to a finite cell complex homotopy equivalent to $\partial X$.
For each $r\in\mathbb{Z}$, write the $r$-cells of $X$ relative to $\partial X$
as $e^{r}_{\alpha}$, for $1\leq\alpha\leq N_{r}$ some choice of ordering.
Write $I=e^{0}_{0}\cup e^{1}\cup e^{0}_{1}$ for the standard cell structure of
$I$. Take the corresponding product cell complex for $X\times I$. Let
$C_{r}(-)$ denote the $r$th _reduced_ cellular chain group. Fix an abelian
group $G$ and write $C^{r}(-):=\operatorname{Hom}(C_{r}(-),G)$. Our choice of
cell complex for $X\times I$ means

$\begin{array}[]{lcl}C_{r}(X\times I,D)&=&\mathbb{Z}\langle
e^{r-1}_{\alpha}\times e^{1}\mid 1\leq\alpha\leq N_{r-1}\rangle,\\\
C_{r}(D,X_{0}\cup(\partial X\times I))&=&\mathbb{Z}\langle e^{r}_{\beta}\times
e^{0}_{1}\mid 1\leq\beta\leq N_{r}\rangle,\\\ C_{r}(X\times
I,X_{0}\cup\partial X_{0}\times I)&=&\mathbb{Z}\langle e^{r-1}_{\alpha}\times
e^{1},e^{r}_{\beta}\times e^{0}_{1}\mid 1\leq\alpha\leq N_{r-1},1\leq\beta\leq
N_{r}\rangle.\end{array}$

We note that in the long exact reduced cohomology sequence of the triple
$(X\times I,D,X_{0}\cup(\partial X\times I))$, the connecting map $\delta$ is
an isomorphism, because the inclusion map $X_{0}\cup(\partial X\times I)\to
X\times I$ is a homotopy equivalence.

######  Proposition 5.1.

Consider the composition of maps

${H^{r}(X\times I,D;G)}$${H^{r-1}(D,X_{0}\cup(\partial X\times
I);G)}$${H^{r-1}(X_{1},\partial
X_{1};G),}$$\scriptstyle{\delta^{-1}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{exc}}$

where $\delta$ denotes the connecting map in the long exact cohomology
sequence of the triple $(X\times I,D,X_{0}\cup(\partial X\times I))$ and
$\operatorname{exc.}$ denotes the (inverse of the) excision isomorphism. Given
a cocycle $\chi\in C^{r}(X\times I,D;G)$, the image
$\operatorname{exc}(\delta^{-1}([\chi]))$ is represented by the cocycle

$\varphi\colon C_{r-1}(X_{1},\partial X_{1})\to G;\qquad
e^{r-1}_{\alpha}\times e^{0}_{1}\mapsto\chi(e^{r-1}_{\alpha}\times e^{1}).$

###### Proof.

We first analyse the map $\delta^{-1}$. Fix $r\in\mathbb{Z}$ and consider the
short exact sequence

${0}$${C^{r}(X\times I,D)}$${C^{r}(X\times I,X_{0}\cup(\partial X\times
I))}$${C^{r}(D,X_{0}\cup(\partial X\times
I))}$${0.}$$\scriptstyle{j^{*}}$$\scriptstyle{i^{*}}$

This is the short exact sequence of cochain complexes which induces the long
exact sequence of the triple in cohomology. Define a cochain $\psi\colon
C_{r-1}(X\times I,X_{0}\cup(\partial X\times I))\to G$ by extending linearly
from

$\psi(e^{r-1}_{\beta}\times e^{0}_{1}):=\chi(e^{r-1}_{\beta}\times
e^{1})\text{ and }\psi(e^{r-2}_{\alpha}\times e^{1}):=0.$

From this we compute the codifferential

$d^{*}(\psi)(e^{r-1}_{\alpha}\times e^{1})=\psi(e^{r-1}_{\alpha}\times
e^{0}_{1})+\psi(\partial e^{r-1}_{\alpha}\times
e^{1})=\chi(e^{r-1}_{\alpha}\times e^{1})+0=\chi(e^{r-1}_{\alpha}\times
e^{1}).$

Now for $1\leq\beta\leq N_{r}$, write
$d_{*}(e^{r}_{\beta})=\sum_{\gamma}n_{\gamma}e^{r-1}_{\gamma}\in
C_{*}(X,\partial X)$ for $n_{\gamma}\in\mathbb{Z}$. We compute the
codifferential

$\displaystyle d^{*}(\psi)(e^{r}_{\beta}\times e^{0}_{1})$
$\displaystyle=\psi(d_{*}(e^{r}_{\beta}\times
e^{0}_{1}))=\psi\left(\displaystyle{\sum}_{\gamma}n_{\gamma}e^{r-1}_{\gamma}\times
e^{0}_{1}\right)=\displaystyle{\sum}_{\gamma}n_{\gamma}\chi(e^{r-1}_{\gamma}\times
e^{1})$
$\displaystyle=\chi\left(\displaystyle{\sum}_{\gamma}n_{\gamma}e^{r-1}_{\gamma}\times
e^{1}\right)=\chi(d_{*}(e^{r}_{\beta}\times
e^{1}))=d^{*}(\chi)(e^{r}_{\beta}\times e^{1})=0,$

where in the final equality we have used our assumption that $\chi$ is a
cocycle. These computations confirm that $d^{*}(\psi)=j^{*}(\chi)$ as these
two cochains agree on generators. From this it follows from the definition of
$\delta$ that $i^{*}(\psi)\in C^{r-1}(D,X_{0}\cup(\partial X\times I))$ is a
representative cocycle for $\delta^{-1}([\chi])$.

Finally, due to our choice of cell structure, it is easy to see that
$i^{*}(\psi)$ is sent to the cochain

$\varphi\colon C_{r-1}(X_{1},\partial X_{1})\to G;\qquad
e^{r-1}_{\alpha}\times e^{0}_{1}\mapsto\chi(e^{r-1}_{\alpha}\times e^{1}).$

under the excision map. We check that $\varphi$ is a cocycle. To do this we
use that $\chi$ is 0 on $D$, and therefore $\chi(e^{r}_{\alpha}\times
e^{0}_{0})=0=\chi(e^{r}_{\alpha}\times e^{0}_{1})$ for every $\alpha$. Then we
compute:

$\displaystyle d^{*}(\varphi)(e^{r}_{\alpha}\times e^{0}_{1})$
$\displaystyle=\varphi(\partial e^{r}_{\alpha}\times e^{0}_{1})=\chi(\partial
e^{r}_{\alpha}\times e^{1})=\chi(\partial e^{r}_{\alpha}\times
e^{1})-\chi(e^{r}_{\alpha}\times e^{0}_{0})+\chi(e^{r}_{\alpha}\times
e^{0}_{1})$ $\displaystyle=\chi(\partial(e^{r}_{\alpha}\times
e^{1}))=d^{*}(\chi)(e^{r}_{\alpha}\times e^{1})=0.$

Therefore $\varphi$ is a cocycle as desired. ∎

######  Proposition 5.2.

Let $X$ be a simply connected, $\operatorname{TOP}$ $4$-manifold and suppose
that $F\in\operatorname{Homeo}^{+}(X,\partial X)$, with $\Delta_{F}\colon
H_{2}(X,\partial X)\to H_{2}(X)$ the zero map. Fix a normal $2$-smoothing
$\overline{\nu}_{X}\colon X\to B$. In the case that $X$ is spin, assume
furthermore that $\Theta(F)=0$. Then there is a normal 2-smoothing

$\Phi\colon X\times[0,1]\to B$

that restricts on the boundary to the $B$-structure

$\varphi\colon\partial(X\times[0,1])\to
B;\qquad(x,t)\mapsto\left\\{\begin{array}[]{ll}\overline{\nu}_{X}\circ
F(x)&\text{when $t=1$,}\\\
\overline{\nu}_{X}(x)&\text{otherwise.}\end{array}\right.$

The analogous statement holds in the smooth category, where $X$ is a smooth
manifold and $F\in\operatorname{Diffeo}^{+}(X,\partial X)$ with
$\Delta_{F}\colon H_{2}(X,\partial X)\to H_{2}(X)$ the zero map.

###### Proof.

We proceed in the topological category; the proof in the smooth category is
entirely similar, after replacing $\operatorname{BSTOP}$ by
$\operatorname{BSO}$ and stable topological normal bundles with stable normal
vector bundles.

We first observe that if we can produce a map $\Phi\colon X\times[0,1]\to B$
restricting to the specified map $\varphi$ on the boundary then it will
automatically be a normal $2$-smoothing of $X\times[0,1]$. To see this, let
$\Phi$ be such a map and write $i\colon X\times\\{0\\}\hookrightarrow
X\times[0,1]$ for the inclusion map. Noting that
$i\circ\operatorname{pr}_{1}\simeq\operatorname{Id}_{X\times[0,1]}$, we have
that $\Phi\circ i=\overline{\nu}_{X}$ implies
$\Phi\simeq\overline{\nu}_{X}\circ\operatorname{pr}_{1}$. From this, we see
$p\circ\Phi\simeq(p\circ\overline{\nu}_{X})\circ\operatorname{pr}_{1}\simeq\nu_{X}\circ\operatorname{pr}_{1}\simeq\nu_{X\times[0,1]}$,
where recall $p\colon B\to\operatorname{BSTOP}$ is the $2$-type. This shows
that $\Phi$ is a lift of $\nu_{X\times[0,1]}$ as required. Moreover, as
$\operatorname{pr}_{1}$ is a homotopy equivalence and $\overline{\nu}_{X}$ is
$3$-connected, it follows from the homotopy
$\Phi\simeq\overline{\nu}_{X}\circ\operatorname{pr}_{1}$ that $\Phi$ is
$3$-connected as required.

We turn to the problem of extending the given $B$-structure
$\varphi\colon\partial(X\times[0,1])\to B$ to a $B$-structure on the whole of
$X\times[0,1]$. Write $D=\partial(X\times[0,1])$. This is standard relative
lifting problem in obstruction theory:

${D}$${B}$${X\times[0,1]}$${\operatorname{BSTOP}}$$\scriptstyle{\varphi}$$\scriptstyle{\nu_{D}}$$\scriptstyle{p}$$\scriptstyle{\nu_{X\times[0,1]}}$

determining a sequence of obstructions $\mathfrak{o}_{r+1}\in
H^{r+1}(X\times[0,1],D;\pi_{r}(\mathcal{F}))$, where $\mathcal{F}$ denotes the
homotopy fibre of $p$. For these obstructions to be well-defined, the action
of $\pi_{1}(\mathcal{F})$ on $\pi_{r}(\mathcal{F})$ must be trivial. However,
in the non-spin case, we have already noted that $\pi_{1}(\mathcal{F})=0$ so
the action is trivial. In the spin case we derived an alternative $2$-type in
Lemma 3.7, where it is clear that the action is trivial. As all $2$-types are
fibre homotopy equivalent, this shows that $\pi_{r}(\mathcal{F})=0$ has the
trivial $\pi_{1}(\mathcal{F})=0$ action in this case as well.

To compute the homotopy groups of $\mathcal{F}$, recall the fibration $w\colon
K\to K(\mathbb{Z}/2,2)$ defined in Subsection 3.2, and recall that $B$ was
defined by a homotopy pullback, and a general property of this construction is
that there is a weak homotopy equivalence
$\alpha\colon\mathcal{F}\to\mathcal{F}(w)$ to the homotopy fibre of $w$, and a
homotopy commutative diagram

(3)
${\mathcal{F}}$${\mathcal{F}(w)}$${B}$${K}$${\operatorname{BSTOP}}$${K(\mathbb{Z}/2,2)}$$\scriptstyle{\alpha}$$\scriptstyle{\simeq_{\text{weak}}}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{w}$

where $i$ and $j$ denote the respective inclusions of homotopy fibres. The
long exact sequence associated with the right hand column contains
$0\to\pi_{r}(\mathcal{F}(w))\to 0$ for $r\geq 3$, and ends with

$\displaystyle
0\to\pi_{2}(\mathcal{F}(w))\xrightarrow{j_{*}}\pi_{2}(K)\xrightarrow{w_{*}}\pi_{2}(K(\mathbb{Z}/2,2))\to\pi_{1}(\mathcal{F}(w))\to
0.$

We deduce that $\pi_{r}(\mathcal{F}(w))=0$ for $r\geq 3$. The map $w_{*}$ is
surjective if and only if $X$ is not spin. We deduce that

$\pi_{1}(\mathcal{F})=\left\\{\begin{array}[]{ll}\mathbb{Z}/2&\text{when $X$
is spin,}\\\ 0&\text{when $X$ is not
spin,}\end{array}\right.\quad\text{and}\quad
j_{*}\circ\alpha_{*}\colon\pi_{2}(\mathcal{F})\hookrightarrow\pi_{2}(K)$

is injective. (When $X$ is spin, in fact
$\pi_{2}(\mathcal{F})\cong\pi_{2}(K)$, but we will not use this.)

In the case that $X$ is spin, there is a potentially nontrivial obstruction
$\mathfrak{o}_{1}\in H^{2}(X\times I,D;\mathbb{Z}/2)$. We claim that under the
sequence of maps in Proposition 5.1, this obstruction is sent to $\Theta(F)\in
H^{1}(X,\partial X;\mathbb{Z}/2)$. To see this, recall by Lemma 3.7 there is
an alternative model for the normal $2$-type of $X$ given by
$(\operatorname{BTOPSpin}\times K,\gamma\circ\operatorname{pr}_{2})$. Moore-
Postnikov theory [Bau77, Theorem 1.5.8] shows there is a fibre homotopy
equivalence $(B,p)\simeq(\operatorname{BTOPSpin}\times
K,\gamma\circ\operatorname{pr}_{2})$. Under this, the $B$-structure
$\varphi\colon D\to B$ corresponds to a pair $(\mathfrak{s},h)$, where
$\mathfrak{s}$ is a spin structure on $\nu_{D}$ and $h\colon D\to K$ is a map.
The obstruction $\mathfrak{o}_{1}\in H^{2}(X\times I,D;\mathbb{Z}/2)$ is
identified with the primary obstruction to extending $(\mathfrak{s},h)$ to a
$(\operatorname{BTOPSpin}\times K)$-structure on all of $X\times I$. But this
is just the obstruction to extending the spin structure $\mathfrak{s}$ to a
spin structure $\mathfrak{S}$ on $\nu_{X\times I}$. In turn, this is exactly
the obstruction to the existence of a rel. boundary homotopy from
$(\mathfrak{s},\partial\mathfrak{s})$ to $(\mathfrak{s}\circ
F,\partial\mathfrak{s})$. In other words $\mathfrak{o}_{1}$ is identified with
$\Theta(F)\in H^{1}(X_{1},\partial X_{1};\mathbb{Z}/2)$ under the isomorphisms
$H^{1}(X\times I,D;\mathbb{Z}/2)\cong H^{1}(D,X_{0}\cup(\partial X\times
I);\mathbb{Z}/2)\cong H^{1}(X_{1},\partial X_{1};\mathbb{Z}/2)$ from
Proposition 5.1. By assumption, this latter obstruction vanishes.

From here on, we use the cell complex and notation from Proposition 5.1. In
both the case that $X$ is spin and the case it is not, we may now assume we
have extended $\varphi$ to the relative $2$-skeleton $\Phi^{(2)}\colon(X\times
I)^{(2)}\to B$, lifting the restriction $\nu_{X\times I}|_{(X\times
I)^{(2)}}$. In each case, the class $\mathfrak{o}_{2}\in
H^{3}(X\times[0,1],D;\pi_{2}(\mathcal{F}))$ is now the unique remaining
obstruction to extending the lift $\Phi^{(2)}$ to the whole of $X\times[0,1]$,
relative to $\varphi$ on $D$.

Recall that $j_{*}\circ\alpha_{*}\colon\pi_{2}(\mathcal{F})\to\pi_{2}(K)$ is
an injective map of free abelian groups. As free abelian groups are flat over
$\mathbb{Z}$, there is a commutative diagram

${H^{3}(X\times I,D;\pi_{2}(\mathcal{F}))}$${H^{3}(X\times
I,D;\pi_{2}(K))}$${H^{3}(X\times
I,D;\mathbb{Z})\otimes\pi_{2}(\mathcal{F})}$${H^{3}(X\times
I,D;\mathbb{Z})\otimes\pi_{2}(K)}$$\scriptstyle{(j_{*}\circ\alpha_{*})_{*}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{Id}\otimes(j_{*}\circ\alpha_{*})}$

Now $j_{*}\circ\alpha_{*}$ is injective and $H^{3}(X\times
I,D;\mathbb{Z})\cong H_{2}(X\times I;\mathbb{Z})\cong H_{2}(X;\mathbb{Z})$ is