a free $\mathbb{Z}$-module, so the lower horizontal map is injective. Hence
the upper horizontal map is injective, and we can conclude that
$\mathfrak{o}_{2}=0$ if and only if $j_{*}(\alpha_{*}(\mathfrak{o}_{2}))=0\in
H^{3}(X\times I,D;\pi_{2}(K))$.

We will now show that $j_{*}(\alpha_{*}(\mathfrak{o}_{2}))$ vanishes. We may
use a CW complex homotopy equivalent to $K$, with a single basepoint and no
1-cells (for instance, use the standard CW structure for
$\prod\operatorname{\mathbb{C}P}^{\infty}\simeq K$), and we may assume that
both $\varphi$ and $\Phi^{(2)}$ are cellular. As at the start of Section 5.1,
we work with cellular chain and cochain groups. The obstruction
$\mathfrak{o}_{2}$ is represented by a cocycle

$C_{3}(X\times I,D;\mathbb{Z})\to\pi_{2}(\mathcal{F});\qquad
e^{2}_{\alpha}\times e^{1}\mapsto\Phi^{(2)}(e^{2}_{\alpha}\times e^{1}).$

We may assume $q\colon B\to K$ is cellular. Write $J:=q\circ\varphi\colon D\to
K$ and $H^{(2)}:=q\circ\Phi^{(2)}\colon(X\times I)^{(2)}\to K$, which we note
are again cellular. Considering the homotopy commutative diagram (3), we
obtain a cocycle representative for
$j_{*}(\alpha_{*}(\mathfrak{o}_{2}))=q_{*}(i_{*}(\mathfrak{o}_{2}))$ as

$\eta\colon C_{3}(X\times I,D;\mathbb{Z})\to\pi_{2}(K);\qquad
e^{2}_{\alpha}\times e^{1}\mapsto H^{(2)}(\partial(e^{2}_{\alpha}\times
e^{1})).$

This is now a cocycle representative for the unique obstruction in
$H^{3}(X\times I;\pi_{2}(K))$ for the simpler extension problem of extending
$J\colon D\to K$ to a map $H\colon X\times I\to K$. We now show this
cohomology class vanishes “by hand”. This will complete the proof.

We may assume $\overline{\nu}_{X}$ is cellular. We write
$h:=q\circ\overline{\nu}_{X}\colon X\to K$ for brevity. Under the Hurewicz
isomorphism $H^{3}(X\times I,D;\pi_{2}(K))\cong H^{3}(X\times I,D;H_{2}(K))$
the obstruction $[\eta]$ is sent to a class $[\chi]$, where $\chi$ is a
cocycle $\chi\colon C_{3}(X\times I,D;\mathbb{Z})\to H_{2}(K)$ such that

$\displaystyle\chi(e_{\alpha}^{2}\times e^{1})$
$\displaystyle=[H^{(2)}(\partial(e_{\alpha}^{2}\times e^{1}))]$
$\displaystyle=\left[J(e^{2}_{\alpha}\times
e^{0}_{1})\right]-\left[J(e^{2}_{\alpha}\times
e^{0}_{0})\right]+[H^{(2)}(\partial e^{2}_{\alpha}\times e^{1})]$
$\displaystyle=\left[h(F(e^{2}_{\alpha}))\right]-\left[h(e^{2}_{\alpha})\right]+[H^{(2)}(\partial
e^{2}_{\alpha}\times e^{1})].$

By Proposition 5.1, under the isomorphism $H^{3}(X\times I,D;H_{2}(K))\cong
H^{2}(X_{1},\partial X_{1};H_{2}(K))$, the class $[\chi]$ is sent to
$[\theta]$ where

$\theta\colon C_{2}(X_{1},\partial X_{1})\to H_{2}(K);\qquad
e^{2}_{\alpha}\times e^{0}_{1}\mapsto\chi(e^{2}_{\alpha}\times e^{1}).$

Because $H_{1}(X_{1},\partial X_{1};\mathbb{Z})\cong\mathbb{Z}^{r-1}$ is
torsion-free, the universal coefficient theorem implies the evaluation map

${H^{2}(X_{1},\partial
X_{1};H_{2}(K))}$${\operatorname{Hom}(H_{2}(X_{1},\partial
X_{1};\mathbb{Z}),H_{2}(K))}$$\scriptstyle{\operatorname{ev}}$$\scriptstyle{\cong}$

is an isomorphism. Let $\sum_{\alpha}N_{\alpha}(e^{2}_{\alpha}\times
e^{0}_{1})\in C_{2}(X_{1},\partial X_{1};\mathbb{Z})$ be a cycle. We compute

$\displaystyle\operatorname{ev}([\theta])\left(\left[\textstyle{\sum_{\alpha}}N_{\alpha}(e^{2}_{\alpha}\times
e^{0}_{1})\right]\right)$ $\displaystyle={}$
$\displaystyle\theta\left(\textstyle{\sum_{\alpha}}N_{\alpha}(e^{2}_{\alpha}\times
e^{0}_{1}))\right)=\chi\left(\textstyle{\sum_{\alpha}}N_{\alpha}(e^{2}_{\alpha}\times
e^{1})\right)$ $\displaystyle={}$
$\displaystyle\left[h(F(\textstyle{\sum_{\alpha}}N_{\alpha}e^{2}_{\alpha}))\right]-\left[h(\textstyle{\sum_{\alpha}}N_{\alpha}e^{2}_{\alpha})\right]+[H^{(2)}(\textstyle{\sum_{\alpha}}N_{\alpha}(\partial
e^{2}_{\alpha}\times e^{1}))]$ $\displaystyle={}$ $\displaystyle
h_{*}\left[F(\textstyle{\sum_{\alpha}}N_{\alpha}e^{2}_{\alpha})-\textstyle{\sum_{\alpha}}N_{\alpha}e^{2}_{\alpha})\right]+[H^{(2)}(\textstyle{\sum_{\alpha}}N_{\alpha}(\partial
e^{2}_{\alpha}\times e^{1}))]$ $\displaystyle={}$ $\displaystyle
h_{*}\Delta_{F}\left(-[\textstyle{\sum_{\alpha}}N_{\alpha}e^{2}_{\alpha}]\right)+[H^{(2)}\left(\textstyle{\sum_{\alpha}}N_{\alpha}(\partial
e^{2}_{\alpha}\times e^{1})\right)]$ $\displaystyle={}$
$\displaystyle[H^{(2)}\left(\textstyle{\sum_{\alpha}}N_{\alpha}(\partial
e^{2}_{\alpha}\times e^{1})\right)]$ $\displaystyle={}$ $\displaystyle 0.$

The third-to-last equality comes from the definition of $\Delta_{F}$. The
penultimate equality comes from the assumption that $\Delta_{F}$ is the 0 map.
To see the final equality, consider that
$\partial\left(\textstyle{\sum_{\alpha}}N_{\alpha}e^{2}_{\alpha}\right)=\textstyle{\sum_{\alpha}}N_{\alpha}\partial
e^{2}_{\alpha}=0$, by definition of a cycle. This implies that
$\textstyle{\sum_{\alpha}}N_{\alpha}\left(\partial e^{2}_{\alpha}\times
e^{1}\right)=0$. From this calculation $\operatorname{ev}([\theta])=0$. Via
the preceding isomorphisms, this implies that $[\theta]=0$, thus $[\chi]=0$,
and finally we conclude that $[\eta]=0\in H^{3}(X\times I,D;\pi_{2}(K))$. It
follows that the claimed extension $H$ exists. This completes the proof of
Proposition 5.2. ∎

###  5.2. Proof of Theorem 2.16

We now have all the ingredients to complete the proof of Theorem 2.16, which
states that our invariants detect pseudo-isotopy of homeomorphisms and
diffeomorphisms, in the appropriate categories.

###### Proof of Theorem 2.16.

Let $(X,\partial X)$ be a compact, simply connected, smooth 4-manifold with
boundary and suppose that $F\in\operatorname{Diffeo}^{+}(X,\partial X)$ such
that $F_{*}\colon H_{2}(X)\to H_{2}(X)$ is the identity map. In the case that
$X$ is spin, furthermore assume that $\Theta(F)=0$. To prove the theorem, we
will show $F$ is pseudo-isotopic to $\operatorname{Id}_{X}$ rel.
$\operatorname{Id}_{\partial X}$.

Consider the smooth, closed 5-manifold

(4)
$T:=(X\times[0,1])_{\ell}\cup\left(\partial(X\times[0,1])\times[0,1]\right)\cup(X\times[0,1])_{r}.$

The subscript $\ell$ and $r$ are to indicate “left” and “right”, and are
simply a bookkeeping device. The second union in (4) is given by the map

$\partial(X\times[0,1])_{r}\mapsto\partial(X\times[0,1])\times\\{1\\};\qquad(x,t)\mapsto\left\\{\begin{array}[]{cl}(F^{-1}(x),1,1)&\text{for
$t=1$},\\\ (x,t,1)&\text{for $t\in[0,1)$}.\end{array}\right.$

and first union in (4) is given by $(x,t)\mapsto(x,t,0)$ for all
$(x,t)\in\partial(X\times[0,1])_{\ell}$. Fix a normal $2$-smoothing
$\overline{\nu}_{X}\colon X\to B$. Endow $(X\times[0,1])_{r}$ with the product
normal 2-smoothing $\overline{\nu}_{X}\circ\operatorname{pr}_{1}$. Write $M$
for the union
$(X\times[0,1])_{\ell}\cup\left(\partial(X\times[0,1])\times[0,1]\right)$. We
see that $M$ is $(X\times[0,1])_{\ell}$ with an external collar attached, and
thus there is a diffeomorphism of manifolds with corners $M\cong
X\times[0,1]$. Using this diffeomorphism, choose a normal $2$-smoothing
$(M,\Phi)$ with the properties described in Proposition 5.2. These properties
guarantee that the normal $2$-smoothings $\Phi$ and
$\overline{\nu}_{X}\circ\operatorname{pr}_{1}$ glue to form a $B$-structure
$\xi$ on $T$. By Proposition 4.9, $(T,\xi)$ is $B$-null-bordant. By Lemma 3.1,
a choice of $B$-null-bordism determines a $B$-bordism $(Z,\Xi)$ rel. boundary
between the normal 2-smoothings $((X\times[0,1])_{\ell},\Phi|)$ and
$((X\times[0,1])_{r},\overline{\nu}_{X}\circ\operatorname{pr}_{1})$. The
bordism $Z$ is a manifold with corners, where the corner structure of
$\partial Z=T$ corresponds to the decomposition in (4). By Theorem 3.5,
$(Z,\Xi)$ determines a surgery obstruction in

$\left\\{\begin{array}[]{ll}\displaystyle{L^{s}_{6}(\mathbb{Z})}&\text{if $B$
has only trivial $w_{4}$-spheres,}\\\
\displaystyle{L^{s}_{6}(\mathbb{Z},S=\mathbb{Z})}&\text{otherwise,}\end{array}\right.$

that vanishes if and only if $(Z,\Xi)$ is $B$-bordant rel. boundary to an
$h$-cobordism rel. boundary. By Proposition 3.8 we are in the latter case, and
by [Kre, Example 4.10] the obstruction group
$L^{s}_{6}(\mathbb{Z},S=\mathbb{Z})$ is trivial. Thus there is a 6-dimensional
$h$-cobordism $B$-bordant to $Z$. Let $W$ be such an $h$-cobordism. By Smale’s
$h$-cobordism theorem [Sma61] there is a commutative diagram of maps, with
vertical diffeomorphisms, identifying $W$ with a product cobordism rel.
boundary from $X\times[0,1]$ to itself:

${X\times[0,1]}$${X\times[0,1]\times[0,1]}$${X\times[0,1]}$${(X\times[0,1])_{\ell}}$${W}$${(X\times[0,1])_{r}.}$$\scriptstyle{i_{0}}$$\scriptstyle{=}$$\scriptstyle{\cong}$$\scriptstyle{G}$$\scriptstyle{i_{1}}$$\scriptstyle{\cong}$$\scriptstyle{g}$$\scriptstyle{j_{0}}$$\scriptstyle{j_{1}}$

Here $i_{k}(x,t)=(x,t,k)$ for $k=0,1$, and the maps $j_{k}$ are the inclusion
maps. Furthermore, $G$ is the identity upon restriction to
$\partial(X\times[0,1])\times[0,1]$. As $W$ is rel. boundary bordant to $Z$,
we have that $\partial W=T$. By commutativity of the diagram, we compute that
$j_{1}\circ g(x,1)=G\circ i_{1}(x,1)=(x,1)$ for all $x\in X$. But by the
definition of $T$ we have that $j_{1}(z,1)=(F^{-1}(z),1)$ for all $z\in X$,
whence $g$ restricts to $F$ on $X\times\\{1\\}$. Similarly $g$ restricts to
$\operatorname{Id}_{X}$ on $X\times\\{0\\}$. Thus $g$ is a pseudo-isotopy from
$F$ to $\operatorname{Id}_{X}$.

The proof in the $\operatorname{CAT}=\operatorname{TOP}$ case is entirely
analogous. Note that we perform topological surgery on a 6-manifold, and apply
the topological 6-dimensional $h$-cobordism theorem. To apply surgery in the
topological category one needs [KS77, Essay III, Appendix C]. The details
necessary for adapting the proof of the smooth $h$-cobordism theorem to the
topological case are explained in [KS77, Essay III, §3.4]. ∎

This completes the proof of injectivity in Theorem A. Next we move on to
surjectivity.

##  6\. Realising obstructions and computing the topological mapping class
group

In this section we realise the obstructions in Theorem 2.16 when
$\operatorname{CAT}=\operatorname{TOP}$. This completes the computation of the
topological mapping class groups promised in Theorem A. Our objective is to
prove the maps from Theorem A are surjective. We begin by discussing some
alternative interpretations of the $\Theta$ invariant.

Let $X$ be a compact, oriented, spin $4$-manifold with spin structure
$\mathfrak{s}$. Let $F\in\operatorname{Homeo}^{+}(X,\partial X)$. Suppose $X$
has at least two boundary components, and choose a properly embedded arc
$\gamma\subseteq X$ with endpoints on different boundary components. Let
$\mathfrak{t}$ be the spin structure on $D:=\partial(X\times[0,1])$ given by
$\mathfrak{t}(x,t)=\mathfrak{s}\circ F$ when $t=1$ and
$\mathfrak{t}=\mathfrak{s}$ otherwise. Consider the loop

$\delta:=\gamma\times\\{1\\}\cup\partial\gamma\times[0,1]\cup\gamma\times\\{0\\}\subseteq
D$

Recall $\nu_{\delta}\colon\delta\to\operatorname{BSO}$ denotes the stable
normal bundle.

######  Lemma 6.1.

The following are equivalent.

  1. (1)

$\Theta(F,\mathfrak{s})([\gamma])=0$.

  2. (2)

There exists an embedded oriented surface $\Sigma\subseteq X\times[0,1]$ with
$\partial\Sigma=\delta$, and such that $\mathfrak{t}|_{\delta}$ extends to a
spin structure $\mathfrak{S}\colon\Sigma\to\operatorname{BTOPSpin}$ on the
stable normal bundle $\nu_{\Sigma}$.

  3. (3)

The spin structure on $\nu_{\delta}$ determined by $\mathfrak{t}$ is null-
bordant.

  4. (4)

The stable framing on $\nu_{\delta}$ determined by $\mathfrak{t}$ is null-
bordant.

  5. (5)

The unique framing of the normal $3$-plane bundle $\nu(\delta\subseteq D)$,
compatible with $\mathfrak{t}$, is null-bordant.

###### Proof.

It follows from the definition of $\Theta(F,\mathfrak{s})([\gamma])$ that (1)
is equivalent to (2). It is clear that (2)$\implies$(3). Conversely, (3)
implies that we may extend the spin structure over a disc. So, using the disc
$\Sigma:=\gamma\times[0,1]$, we see (3)$\implies$(2). The equivalence of (3)
and (4) is implied by the standard one to one correspondence between spin
structures and stable framings for a closed 1-manifold, and this
correspondence is justified as follows. A choice of null homotopy of a loop
$S^{1}\to\operatorname{BSTOP}$ fixes a choice of lift
$S^{1}\to\operatorname{BTOPSpin}$. Conversely, since any map
$S^{1}\to\operatorname{BTOPSpin}$ is null homotopic, and two null homotopies
differ by an element of $\pi_{2}(\operatorname{BTOPSpin})=0$, there is (up to
homotopy) a unique null homotopy, which projects under
$\operatorname{BTOPSpin}\to\operatorname{BSTOP}$ to a framing. To see the
equivalence of (4) and (5), consider that two choices of stable framing on a
stable bundle $S^{1}\to\operatorname{BSTOP}$ differ by an element of
$\pi_{2}(\operatorname{BSTOP})=\mathbb{Z}/2$. As the stable range for
$\pi_{2}(\operatorname{BSTOP}(n))=\mathbb{Z}/2$ is $n\geq 3$, isomorphism
classes of framing for a representative $3$-plane bundle for $\nu_{\delta}$
are in one to one correspondence with the isomorphism classes of stable
framing they determine for $\nu_{\delta}$. Now take $\nu(\delta\subseteq D)$
as a representative $3$-plane bundle for $\nu_{\delta}$. ∎

We slightly expand the class of automorphisms $F\colon X\to X$ for which a
variation $\Delta_{F}$ is defined.

######  Definition 6.2.

Suppose $F\colon X\to X$ is an orientation preserving homeomorphism of a
compact, oriented, simply connected, topological $4$-manifold with connected
boundary. Suppose there exists a codimension zero submanifold
$A\subseteq\partial X$ such that the inclusion induced map $H_{r}(A)\to
H_{r}(\partial X)$ is an isomorphism for $r=1,2$ and such that
$F|_{A}=\operatorname{Id}_{A}$. Let $x\in H_{2}(X,\partial X)$ and choose a
representative cycle $\Sigma\in C_{2}(X,A)$ for $x$. Define the _variation_ of
$F$ as the homomorphism

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

The proof that this is a well-defined homomorphism, and is moreover a Poincaré
variation in the sense of Definition 2.9 is identical to the proof for when
$F|_{\partial X}=\operatorname{Id}_{\partial X}$, noting that we are using
representative cycles in $(X,A)$, where $F|_{A}=\operatorname{Id}_{A}$.

With this definition in hand we can produce a slight extension of [Sae06,
Theorem 3.7], that we will require momentarily.

######  Theorem 6.3.

Let $X$ be a compact, oriented, simply connected, topological $4$-manifold
with connected boundary and let $\Delta\in\mathcal{V}(H_{2}(X),\lambda_{X})$.
Let $f\colon\partial X\to\partial X$ be an orientation preserving
homeomorphism with $f|_{A}=\operatorname{Id}_{A}$ for some codimension zero
submanifold $A\subseteq\partial X$ such that the inclusion induced map
$H_{r}(A)\to H_{r}(\partial X)$ is an isomorphism for $r=1,2$. Then there
exists an orientation preserving homeomorphism $F\colon X\to X$ restricting to
$f$ on the boundary and such that $\Delta_{F}=\Delta$.

###### Proof.

It is proved in [Sae06, Theorem 3.7] that $\Delta$ can be realised smoothly
stably, by a diffeomorphism restricting to the identity on the boundary. In
fact, the same proof can be used to construct
$F\in\operatorname{Homeo}^{+}(X,\partial X)$ such that $\Delta_{F}=\Delta$, as
mentioned in [Sae06, Remark 3.13]. To instead produce an orientation
preserving homeomorphism that restricts to $f$ of the type we desire, the only
modification that must be made is that (in the language of that proof), one
must set $V$ to be the twisted double of the $4$-manifold, using the map $f$.
All other parts of Saeki’s proof can be used verbatim. ∎

We now describe a construction that will help us transform problems about
manifolds with more than one boundary component to problems about manifolds
with connected boundary. Let $X$ be a compact, oriented, simply connected,
topological $4$-manifold with $r>1$ connected boundary components. Enumerate
the boundary components $Y_{1},\dots,Y_{r}$. Choose disjoint properly embedded
arcs $\gamma_{1},\dots,\gamma_{r-1}\subseteq X$ so that $\gamma_{i}$ has one
end on $Y_{i}$ and the other end on $Y_{r}$. Excise tubular neighbourhoods of
the arcs to obtain a simply connected $4$-manifold
$X_{0}:=X\setminus\left(\bigcup_{i}\nu(\gamma_{i})\right)$, with connected
boundary $\partial X_{0}\cong Y_{1}\\#Y_{2}\\#\dots\\#Y_{r}$. We call any
$(X_{0},\partial X_{0})$ obtained in this way an _internal connected sum_
manifold for $(X,\partial X)$.

By a straightforward Mayer-Vietoris calculation, the inclusion of an internal
connected sum manifold $\iota\colon(X_{0},\partial X_{0})\subseteq(X,\partial
X)$ results in an isomorphism of exact sequences

(4) ${0}$${H_{2}(\partial X_{0})}$${H_{2}(X_{0})}$${H_{2}(X_{0},\partial
X_{0})}$${H_{1}(\partial X_{0})}$${0}$${0}$${H_{2}(\partial
X)}$${H_{2}(X)}$${H_{2}(X,\partial X)}$${H_{1}(\partial
X)}$${0.}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{(j_{0})_{*}}$$\scriptstyle{\cong}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\cong}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{j_{*}}$

######  Lemma 6.5.

An internal connected sum induces an isomorphism under inclusion

$\mathcal{V}(H_{2}(X_{0}),\lambda_{X_{0}})\xrightarrow{\cong}\mathcal{V}(H_{2}(X),\lambda_{X});\qquad\Delta_{0}\mapsto\iota_{*}\circ\Delta_{0}\circ(\iota_{*})^{-1}.$

###### Proof.

The inclusion induced map $\iota_{*}\colon
H_{2}(X_{0})\xrightarrow{\cong}H_{2}(X)$ is an isometry of intersection forms.
Now apply Corollary 7.5, below. ∎

######  Theorem 6.6.

Let $X$ be a compact, oriented, simply connected, topological $4$-manifold.
Let $\Delta\in\mathcal{V}(H_{2}(X),\lambda_{X})$ and fix $x\in
H^{1}(X,\partial X;\mathbb{Z}/2)$. Then there exists
$F\in\operatorname{Homeo}^{+}(X,\partial X)$ such that $\Delta_{F}=\Delta$. If
$X$ is spin, it is possible to choose $F$ so that moreover $\Theta(F)=x$.

Assuming this result for a moment, we can complete the proof of the main
theorem.

###### Proof of Theorem A.

Combine Theorem 2.16 with the Quinn-Perron result (Theorem 2.17 (1)) that
pseudo-isotopy implies isotopy for simply connected, compact 4-manifolds, to
obtain injectivity of the maps in the statement. Surjectivity is provided by
Theorem 6.6. ∎

###### Proof of Theorem 6.6.

In the case that $\partial X$ is empty, a Poincaré variation is exactly an
isometry of the intersection form, and in this case the theorem is due to
Freedman [Fre82, Theorem 1.5, Addendum]. In the case that the boundary is
nonempty and connected, this theorem is due to Saeki; see [Sae06, Theorem 3.7]
and [Sae06, Remark 3.13].

We now assume that $X$ has $r>1$ connected boundary components. Enumerating
these $Y_{1},\dots,Y_{r}$, we form an internal connected sum manifold $X_{0}$
with boundary $\partial X_{0}\cong Y_{1}\\#\dots\\#Y_{r}$ as described above.
Write $\iota\colon(X_{0},\partial X_{0})\to(X,\partial X)$ for the inclusion.
Denote by $\Delta_{0}\in\mathcal{V}(H_{2}(X),\lambda_{X})$ the variation sent
to $\Delta$ under the inclusion induced isomorphism of Lemma 6.5.

We prove the first claim of the theorem. As the case of connected boundary is
already proved, there exists $F_{0}\in\operatorname{Homeo}^{+}(X_{0},\partial
X_{0})$ such that $\Delta_{F_{0}}=\Delta_{0}$. Extend $F_{0}$ by the identity
map
$\operatorname{Id}\colon\bigcup_{i}\nu(\gamma_{i})\to\bigcup_{i}\nu(\gamma_{i})$
to obtain $F\in\operatorname{Homeo}^{+}(X,\partial X)$. We claim that
$\Delta_{F}=\Delta$, or equivalently that
$\Delta_{F}\circ\iota_{*}=\iota_{*}\circ\Delta_{F_{0}}$. To see this, let
$z\in H_{2}(X_{0},\partial X_{0})$, and represent it by a cycle $\Sigma\in
C_{2}(X_{0},\partial X_{0})$. We compute that
$\iota_{*}\circ\Delta_{F_{0}}(x)=\iota_{*}([\Sigma-
F_{0}(\Sigma)])=[\iota(\Sigma)-\iota\circ
F_{0}(\Sigma)]=[\iota(\Sigma)-F(\iota(\Sigma))]$. But $\iota(\Sigma)\in
C_{2}(X,\partial X)$ is a representative $2$-cycle for $\iota_{*}(z)$ and
hence $[\iota(\Sigma)-F(\iota(\Sigma))]=\Delta_{F}(\iota_{*}(z))$, as was
required to be shown. This completes the proof of the first claim in the
theorem.

For the second claim in the theorem, assume that $X$ is spin. For each $i$,
choose a framing $\overline{\nu}(\gamma_{i})\cong D^{3}\times[0,1]$ compatible
with the stable framing induced by the spin structure. Write
$\varphi_{t}\in\pi_{1}(\operatorname{SO}(3))$ for a smooth generating loop
based at the identity. Denote by
$\Phi\colon\bigcup_{i}\overline{\nu}(\gamma_{i})\to\bigcup_{i}\overline{\nu}(\gamma_{i})$
the self-homeomorphism that, under the identification
$\overline{\nu}(\gamma_{i})\cong D^{3}\times[0,1]$ is the map
$(p,t)\mapsto(\varphi_{t}(p),t)$ when $x([\gamma_{i}])=1$, and is the identity
otherwise. The framing identifies the sphere bundle of $\nu(\gamma_{i})$ as
$S^{2}\times[0,1]$, and in $\partial X_{0}\cong Y_{1}\\#Y_{2}\\#\dots\\#Y_{r}$
these correspond to bicollars on the connected sum spheres. Thus
$\Phi|_{S(\nu(\gamma_{i}))}$ extends by the identity map to a self-
diffeomorphism of $\partial X_{0}$ which is the identity except near the
connected sum $2$-spheres corresponding to $x([\gamma_{i}])=1$, where it is a
Dehn twist on $S^{2}\times[0,1]$. Write $f\colon\partial X_{0}\to\partial
X_{0}$ for this extended map. Writing $A\subseteq\partial X$ for the
complement of the neighbourhoods of the connected sum spheres, and noting
$H_{r}(A)\to H_{r}(X_{0})$ is an isomorphism for $r=1,2$, we may apply Theorem
6.3 to obtain an orientation preserving homeomorphism $F^{\prime}_{0}\colon
X_{0}\to X_{0}$ restricting to $f$ on the boundary and such that
$\Delta_{F^{\prime}_{0}}=\Delta$. Extend $F^{\prime}_{0}$ to a homeomorphism
$F^{\prime}\colon X\to X$ by using the map
$\Phi\colon\bigcup_{i}\overline{\nu}(\gamma_{i})\to\bigcup_{i}\overline{\nu}(\gamma_{i})$.
Because $\iota\circ F^{\prime}_{0}=F\circ\iota$, the same argument we used
above can be applied to prove that $\Delta_{F^{\prime}}=\Delta$.

We now show that $\Theta(F^{\prime})=x$. In the notation from the beginning of
this subsection, the arc $\gamma_{i}$ determines a loop $\delta_{i}\subseteq
D:=\partial(X\times[0,1])$. The $D^{3}$-bundle
$\overline{\nu}(\delta_{i}\subseteq D)$ is framed by $\mathfrak{t}$ (also
defined at the start of this subsection), giving rise to an identification
$\overline{\nu}(\delta_{i}\subseteq D)\cong D^{3}\times S^{1}$. By
construction of the map $\Phi$, this framing extends to a framing of the
$D^{3}$-bundle $D^{3}\times D^{2}$ when $x([\gamma_{i}])=0$. When
$x([\gamma_{i}])=1$, when defining $\Phi$ we added a twist over the arc in
$S^{1}$ corresponding to $\gamma_{i}\times\\{1\\}$, so that for these
$\delta_{i}$, this framing is nontrivial. Applying Lemma 6.1, and using the
fact that $\gamma_{1},\dots,\gamma_{r-1}$ form a basis for $H^{1}(X,\partial
X;\mathbb{Z}/2)$, we see that $\Theta(F^{\prime})=x$. ∎

##  7\. The role played by isometries of the intersection form

In this section we first describe a purely algebraic formulation for groups of
Poincaré variations. We thereby show that the isomorphism class of such a
group depends only on the underlying symmetric bilinear form. We then recall
some results of Boyer and Saeki relating Poincaré variations to isometries.
Finally, we describe their consequences in light of Theorem A.

###  7.1. An algebraic formulation of Poincaré variations

Let $P$ be a finitely generated free $\mathbb{Z}$-module, and let
$\alpha\colon P\times P\to\mathbb{Z}$ be a symmetric bilinear form. Write
$(-)^{\ast}:=\operatorname{Hom}(-,\mathbb{Z})$ and write
$\alpha^{\operatorname{ad}}\colon P\to P^{\ast}$ for the adjoint homomorphism,
defined by $\alpha^{\operatorname{ad}}(x)(y):=\alpha(x,y)$.

######  Definition 7.1.

Given $(P,\alpha)$, define the set of _form variations_ as

$\mathcal{FV}(P,\alpha):=\\{V\colon P^{\ast}\to P\mid
V+V^{\ast}=V\alpha^{\operatorname{ad}}V^{\ast}\\}.$

######  Lemma 7.2.

The set $\mathcal{FV}(P,\alpha)$ is a group under the operation $V_{1}\ast
V_{2}:=V_{1}+(\operatorname{Id}-V_{1}\alpha^{\operatorname{ad}})V_{2}$, with
identity element 0 and inverses
$V^{-1}:=-(\operatorname{Id}-V^{*}\alpha^{\operatorname{ad}})V$.

The proof of Lemma 7.2 is essentially the same as that of Lemma 2.10, and we
omit it.

######  Lemma 7.3.

Let $\psi\colon(P,\alpha)\xrightarrow{\cong}(Q,\beta)$ be an isometry of
intersection forms. Then

$\widehat{\psi}\colon\mathcal{FV}(P,\alpha)\xrightarrow{\cong}\mathcal{FV}(Q,\beta);\qquad\widehat{\psi}(V):=\psi
V\psi^{\ast}$

is an isomorphism of groups.

###### Proof.

Write $U:=\widehat{\psi}(V)=\psi V\psi^{\ast}$. Note that $U^{\ast}=\psi
V^{\ast}\psi^{\ast}$, we have that

$U+U^{\ast}=\psi(V+V^{\ast})\psi^{\ast}=\psi(V\alpha^{\operatorname{ad}}V^{\ast})\psi^{\ast}=(\psi
V\psi^{\ast})((\psi^{\ast})^{-1}\alpha^{\operatorname{ad}}\psi^{-1})(\psi
V^{\ast}\psi^{\ast})=U\beta^{\operatorname{ad}}U^{\ast},$

so $U$ lies in the codomain as claimed. Given this, $\widehat{\psi}$ is
clearly bijective with inverse $\widehat{\psi^{-1}}$. To check that
$\widehat{\psi}$ is a homomorphism, we compute

$\begin{array}[]{rcl}\widehat{\psi}(V_{1}\ast
V_{2})&=&\psi(V_{1}+(\operatorname{Id}-V_{1}\alpha^{\operatorname{ad}})V_{2})\psi^{\ast}\\\
&=&(\psi
V_{1}\psi^{\ast})+\psi(\operatorname{Id}-V_{1}(\psi^{\ast}(\psi^{\ast})^{-1}\alpha^{\operatorname{ad}})\psi^{-1}\psi
V_{2}\psi^{\ast}\\\ &=&(\psi V_{1}\psi^{\ast})+(\operatorname{Id}-\psi
V_{1}\psi^{\ast}\beta^{\operatorname{ad}})\psi V_{2}\psi^{\ast}\\\
&=&\widehat{\psi}(V_{1})\ast\widehat{\psi}(V_{2}).\end{array}$

Thus $\widehat{\psi}$ is an isomorphism as claimed. ∎

The next lemma is immediate from the definitions and we omit the proof.

######  Lemma 7.4.

Suppose that $(X,\partial X)$ is a compact, simply connected
$\operatorname{CAT}$ $4$-manifold. Then the isomorphism
$\operatorname{PD}\circ\operatorname{ev}^{-1}\colon H_{2}(X)^{\ast}\to
H_{2}(X,\partial X)$ determines an isomorphism

$\mathcal{FV}(H_{2}(X),\lambda_{X})\to\mathcal{V}(H_{2}(X),\lambda_{X});\qquad
V\mapsto V\circ\operatorname{ev}\circ\operatorname{PD}^{-1}.$

######  Corollary 7.5.

Let $(X,\partial X)$ be a compact, simply connected $\operatorname{CAT}$
$4$-manifold. The group of variations $\mathcal{V}(H_{2}(X),\lambda_{X})$
depends only on the isometry class of the intersection form. Moreover, if
$\iota\colon(X,\partial X)\to(X^{\prime},\partial X^{\prime})$ is a map of
compact, simply connected $\operatorname{CAT}$ $4$-manifolds, inducing an
isometry of the intersection forms, then there is an induced isomorphism

$\mathcal{V}(H_{2}(X),\lambda_{X})\xrightarrow{\cong}\mathcal{V}(H_{2}(X^{\prime}),\lambda_{X^{\prime}});\qquad\Delta\mapsto\iota_{*}\circ\Delta\circ(\iota_{*})^{-1}.$

###### Proof.

The first statement is an immediate corollary of Lemmas 7.3 and 7.4. For the
second statement, it is sufficient to consider the following composition from
left to right

${\mathcal{V}(H_{2}(X),\lambda_{X})}$${\mathcal{FV}(H_{2}(X),\lambda_{X})}$${\mathcal{FV}(H_{2}(X^{\prime}),\lambda_{X^{\prime}})}$${\mathcal{V}(H_{2}(X^{\prime}),\lambda_{X^{\prime}})}$${\Delta}$${\Delta\circ\operatorname{PD}\circ\operatorname{ev}^{-1}}$${\iota_{*}\circ\Delta\circ\operatorname{PD}\circ\operatorname{ev}^{-1}\circ(\iota_{*})^{{\ast}}}$${\iota_{*}\circ\Delta\circ(\iota_{*})^{-1},}$$\scriptstyle{\cong}$$\scriptstyle{\widehat{\iota}_{*}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

where in the last map we have used the fact that
$\operatorname{PD}\circ\operatorname{ev}^{-1}\circ(\iota_{*})^{{\ast}}\circ\operatorname{ev}\circ\operatorname{PD}^{-1}=(\iota_{*})^{-1}$,
which follows from naturality of the cap product. ∎

###  7.2. Isometries rel. boundary and Saeki’s exact sequence

Suppose that $(X,\partial X)$ is a compact, simply connected
$\operatorname{CAT}$ $4$-manifold. Let $\varphi\colon H_{2}(X)\to H_{2}(X)$ be
a group isomorphism. Write $(-)^{\ast}:=\operatorname{Hom}(-,\mathbb{Z})$, and
define a composite “umkehr” isomorphism

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

The long exact sequence of $(X,\partial X)$ determines the following short
exact sequences, because $X$ is simply connected.

${0}$${H_{2}(\partial X)}$${H_{2}(X)}$${H_{2}(X,\partial X)}$${H_{1}(\partial
X)}$${0}$${0}$${H_{2}(\partial X)}$${H_{2}(X)}$${H_{2}(X,\partial
X)}$${H_{1}(\partial
X)}$${0.}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{\varphi}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{\varphi^{!}}$

The reader may check that the map $\varphi$ is an isometry of
$(H_{2}(X),\lambda_{X})$ if and only if the central square of the diagram
commutes. We say $\varphi$ is an _isometry rel. boundary_ if the entire
diagram above commutes.

######  Definition 7.6.

Write $\operatorname{Aut}(H_{2}(X),\lambda_{X})$ for the group of isometries
of the intersection form, where the group operation is by composition. Write
$\operatorname{Aut}_{\partial}(H_{2}(X),\lambda_{X})\subseteq\operatorname{Aut}(H_{2}(X),\lambda_{X})$
for the subgroup consisting of isometries rel. boundary.

Given a closed oriented $3$-manifold $Y$, denote by $\operatorname{Spin}(Y)$
the set of spin structures on $Y$ up to isomorphism. Given
$\mathfrak{s}\in\operatorname{Spin}(\partial X)$, the _relative second
Steifel-Whitney class_ $w_{2}(X,\partial X;\mathfrak{s})\in H^{2}(X;\partial
X;\mathbb{Z}/2)$ is defined as the unique obstruction to solving the following
relative lifting problem in obstruction theory

${\partial
X}$${\operatorname{BTOPSpin}}$${X}$${\operatorname{BSTOP}.}$$\scriptstyle{\mathfrak{s}}$$\scriptstyle{\nu_{\partial
X}}$$\scriptstyle{\gamma}$$\scriptstyle{\nu_{X}}$

Let $\varphi\in\operatorname{Aut}(H_{2}(X),\lambda_{X})$. It is shown in
[Boy86, Lemma 2.9] that there is a unique permutation $\pi_{\varphi}$ of
$\operatorname{Spin}(\partial X)$ such that the following diagram commutes.

(7) ${\operatorname{Spin}(\partial X)}$${H^{2}(X,\partial
X;\mathbb{Z}/2)}$${H_{2}(X;\mathbb{Z}/2)}$${\operatorname{Spin}(\partial
X)}$${H^{2}(X,\partial
X;\mathbb{Z}/2)}$${H_{2}(X;\mathbb{Z}/2).}$$\scriptstyle{w_{2}(X,\partial
X;-)}$$\scriptstyle{\pi_{\varphi}}$$\scriptstyle{\operatorname{PD}}$$\scriptstyle{\varphi}$$\scriptstyle{w_{2}(X,\partial
X;-)}$$\scriptstyle{\operatorname{PD}}$

We remark that in [Boy86, Lemma 2.9], Boyer is assuming that $\partial X$ is
connected, but that the proof is entirely formal and goes through identically
in our case.

######  Definition 7.8.

Write
$\operatorname{Aut}^{\operatorname{fix}}_{\partial}(H_{2}(X),\lambda_{X})\subseteq\operatorname{Aut}_{\partial}(H_{2}(X),\lambda_{X})$
for the subgroup of isometries rel. boundary $\varphi$ such that
$\pi_{\varphi}$ is the identity permutation.

######  Remark 7.9.

We explain why $\pi_{\varphi}=\operatorname{Id}$ is a necessary condition for
$\varphi\in\operatorname{Aut}_{\partial}(H_{2}(X),\lambda_{X})$ to be induced
by a homeomorphism. Suppose $f\colon\partial X\to\partial X$ is an orientation
preserving homeomorphism. Following Boyer’s convention, we denote by
$f_{\\#}\colon\operatorname{Spin}(\partial X)\to\operatorname{Spin}(\partial
X)$ the _inverse_ of the permutation described by $\mathfrak{s}\mapsto
f^{*}\mathfrak{s}$. Boyer [Boy86, §4] proves that there exists a cohomology
class $\theta(f,\varphi)\in\operatorname{im}(H^{1}(\partial X)\to
H^{1}(\partial X;\mathbb{Z}/2))$ such that the permutation
$f_{\\#}\circ\pi_{\varphi}^{-1}$ agrees with the permutation determined by
$\theta(f,\varphi)$ under the free, transitive action of $H^{1}(\partial
X;\mathbb{Z}/2)$ on $\operatorname{Spin}(\partial X)$. We note that although
Boyer is assuming that $\partial X$ is connected, the proof of this result
works equally well in the case of non-connected boundary. By Boyer [Boy86,
Theorem 0.7], the class $\theta(f,\varphi)$ is an obstruction to the existence
of an orientation preserving homeomorphism $F\colon X\to X$, such that
$F_{*}=\varphi$ and $F|_{\partial X}=f$. We recall the details to confirm this
holds even for non-connected boundary. Indeed, if such an $F$ exists, then the
following diagram is commutative

${\operatorname{Spin}(\partial X)}$${H^{2}(X,\partial
X;\mathbb{Z}/2)}$${H_{2}(X;\mathbb{Z}/2)}$${\operatorname{Spin}(\partial
X)}$${H^{2}(X,\partial
X;\mathbb{Z}/2)}$${H_{2}(X;\mathbb{Z}/2).}$$\scriptstyle{w_{2}(X,\partial
X;-)}$$\scriptstyle{\pi_{\varphi}}$$\scriptstyle{\operatorname{PD}}$$\scriptstyle{\varphi}$$\scriptstyle{w_{2}(X,\partial
X;-)}$$\scriptstyle{\operatorname{PD}}$$\scriptstyle{F^{*}}$

Moreover it is easily checked from the definitions that for any
$\mathfrak{s}\in\operatorname{Spin}(\partial X)$, we have

$F^{*}w_{2}(X,\partial X;\mathfrak{s})=w_{2}(X,\partial X;f^{*}\mathfrak{s}).$

Hence the permutations $\pi_{\varphi}$ and $f_{\\#}$ agree, and from this we
can conclude $\theta(f,\varphi)=0$.

In particular, $(\operatorname{Id}_{\partial X})_{\\#}=\operatorname{Id}$, and
so $\operatorname{Id}_{\partial X}$ extends to a homeomorphism inducing a
given isometry $\varphi$ only if $\pi_{\varphi}$ is the identity permutation.

######  Lemma 7.10.

When $X$ is spin,
$\operatorname{Aut}^{\operatorname{fix}}_{\partial}(H_{2}(X),\lambda_{X})=\operatorname{Aut}_{\partial}(H_{2}(X),\lambda_{X})$.

###### Proof.

Fix $\varphi\in\operatorname{Aut}_{\partial}(X,\partial X)$. We must show that
$\pi_{\varphi}=\operatorname{Id}$. Fix a spin structure $\mathfrak{s}$ on $X$
and let $\mathfrak{t}$ be the induced spin structure on $\partial X$. Then
$w_{2}(X,\partial X;\mathfrak{t})=0$, so
$\mathfrak{t^{\prime}}:=\pi_{\varphi}(\mathfrak{t})$ must be such that
$w_{2}(X,\partial X;\mathfrak{t}^{\prime})=0$, by commutativity of diagram
(7). As $X$ is simply connected, it has a unique spin structure up to
isomorphism, so only one spin structure on $\partial X$ can extend to $X$.
Hence $\mathfrak{t}=\mathfrak{t}^{\prime}$, up to isomorphism. By Remark 7.9,
the permutation $\pi_{\varphi}$ arises from the action of a class
$-\theta(\operatorname{Id}_{\partial X},\varphi)\in H^{1}(\partial
X;\mathbb{Z}/2)$. As the action of $H^{1}(\partial X;\mathbb{Z}/2)$ on
$\operatorname{Spin}(\partial X)$ is free, the fact that
$(-\theta)\cdot\mathfrak{t}=\mathfrak{t}$ implies $-\theta=0$ and hence the
permutation $\pi_{\varphi}$ is trivial. ∎

######  Remark 7.11.

We do not know an example of $(X,\partial X)$ and
$\varphi\in\operatorname{Aut}_{\partial}(X,\partial X)$ such that
$\pi_{\varphi}\neq\operatorname{Id}$.

Following [Sae06, §4], we will now precisely describe how Poincaré variations
refine isometries of the intersection form. Denote by
$\wedge^{2}{H_{1}(\partial X)^{\ast}}$ the group of (possibly degenerate)
skew-symmetric pairings $\kappa\colon H_{1}(\partial X)\times H_{1}(\partial
X)\to\mathbb{Z}$. Given $\kappa\in\wedge^{2}{H_{1}(\partial X)^{\ast}}$, write
$\kappa^{\operatorname{ad}}$ for its adjoint, and define a composite morphism

(12) $\widetilde{\kappa}\colon H_{2}(X,\partial
X)\xrightarrow{\partial}H_{1}(\partial
X)\xrightarrow{\kappa^{\operatorname{ad}}}H_{1}(\partial
X)^{\ast}\xrightarrow{\operatorname{ev}^{-1}}H^{1}(\partial
X)\xrightarrow{\operatorname{PD}}H_{2}(\partial
X)\xrightarrow{i_{*}}H_{2}(X).$

The morphism $\widetilde{\kappa}$ is a Poincaré variation, and this defines a
homomorphism

$S\colon\wedge^{2}{H_{1}(\partial
X)^{\ast}}\to\mathcal{V}(H_{2}(X),\lambda_{X});\qquad\kappa\mapsto\widetilde{\kappa};$

see [Sae06, §4]. Write $j_{*}\colon H_{2}(X)\to H_{2}(X,\partial X)$ for the
inclusion induced map. Given $\Delta\in\mathcal{V}(H_{2}(X),\lambda_{X})$, it
is shown in [Sae06, Lemma 3.4] that the induced map

$\Xi(\Delta):=\operatorname{Id}-\Delta\circ j_{*}\colon H_{2}(X)\to H_{2}(X)$

is an isometry of the intersection form $(H_{2}(X),\lambda_{X})$.

######  Theorem 7.13.

The following is a short exact sequence

${0}$${\wedge^{2}{H_{1}(\partial
X)^{\ast}}}$${\mathcal{V}(H_{2}(X),\lambda_{X})}$${\operatorname{Aut}_{\partial}^{\operatorname{fix}}(H_{2}(X),\lambda_{X})}$${0.}$$\scriptstyle{S}$$\scriptstyle{\Xi}$

###### Proof.

When $\partial X$ is connected, this result is due to Saeki, and follows from
combining [Sae06, Propositions 4.2 and 4.3].

We now assume that $X$ has $r>1$ connected boundary components. The proof of
[Sae06, Proposition 4.2] is entirely algebraic and does not use the assumption
of connected boundary. This shows there is an exact sequence

${0}$${\wedge^{2}{H_{1}(\partial
X)^{\ast}}}$${\mathcal{V}(H_{2}(X),\lambda_{X})}$${\operatorname{Aut}(H_{2}(X),\lambda_{X}).}$$\scriptstyle{S}$$\scriptstyle{\Xi}$

To complete the proof, we claim that an isometry
$\varphi\in\operatorname{Aut}(H_{2}(X),\lambda_{X})$ lies in the image of
$\Xi$ if and only
$\varphi\in\operatorname{Aut}_{\partial}^{\operatorname{fix}}(H_{2}(X),\lambda_{X})$.

First, assume that $\varphi=\Xi(\Delta)$, for some $\Delta$. By Theorem 6.6,
there exists $F\in\operatorname{Homeo}^{+}(X,\partial X)$ such that
$\Delta_{F}=\Delta$, and thus $\varphi=F_{*}$. This implies that
$\varphi\in\operatorname{Aut}_{\partial}(H_{2}(X),\lambda_{X})$. Moreover, as
discussed in Remark 7.9, the existence of $F$ implies
$\pi_{\varphi}=\operatorname{Id}$, so that moreover
$\varphi\in\operatorname{Aut}_{\partial}^{\operatorname{fix}}(H_{2}(X),\lambda_{X})$.

Conversely, suppose that
$\varphi\in\operatorname{Aut}_{\partial}^{\operatorname{fix}}(H_{2}(X),\lambda_{X})$.
Form an internal connected sum manifold $(X_{0},\partial X_{0})$ as described
in Section 6. Write $\iota\colon(X_{0},\partial X_{0})\to(X,\partial X)$ for
the inclusion. The commutative diagram (4) just above Lemma 6.5 shows that
$\varphi_{0}:=(\iota_{*})^{-1}\circ\varphi\circ\iota_{*}\in\operatorname{Aut}_{\partial}(H_{2}(X_{0}),\lambda_{X_{0}})$.
Moreover, the following diagram commutes

${\operatorname{Spin}(\partial X_{0})}$${H^{2}(X_{0},\partial
X_{0};\mathbb{Z}/2)}$${H_{2}(X_{0};\mathbb{Z}/2)}$${\operatorname{Spin}(\partial
X)}$${H^{2}(X,\partial
X;\mathbb{Z}/2)}$${H_{2}(X;\mathbb{Z}/2)}$${\operatorname{Spin}(\partial
X)}$${H^{2}(X,\partial
X;\mathbb{Z}/2)}$${H_{2}(X;\mathbb{Z}/2)}$${\operatorname{Spin}(\partial
X_{0})}$${H^{2}(X_{0},\partial
X_{0};\mathbb{Z}/2)}$${H_{2}(X_{0};\mathbb{Z}/2).}$$\scriptstyle{w_{2}(X_{0},\partial
X_{0};-)}$$\scriptstyle{\pi_{\varphi_{0}}}$$\scriptstyle{\operatorname{PD}}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\varphi_{0}}$$\scriptstyle{w_{2}(X,\partial
X;-)}$$\scriptstyle{\pi_{\varphi}}$$\scriptstyle{\iota^{*}}$$\scriptstyle{\operatorname{PD}}$$\scriptstyle{\iota^{*}}$$\scriptstyle{\varphi}$$\scriptstyle{w_{2}(X,\partial
X;-)}$$\scriptstyle{\iota^{*}}$$\scriptstyle{\operatorname{PD}}$$\scriptstyle{\iota^{*}}$$\scriptstyle{w_{2}(X_{0},\partial
X_{0};-)}$$\scriptstyle{\operatorname{PD}}$$\scriptstyle{\iota_{*}}$

As $\pi_{\varphi}=\operatorname{Id}$, this shows that
$\pi_{\varphi_{0}}=\operatorname{Id}$ and hence
$\varphi_{0}\in\operatorname{Aut}_{\partial}^{\operatorname{fix}}(H_{2}(X_{0}),\lambda_{X_{0}})$.
By [Boy86, Theorem 0.7], there exists
$F_{0}\in\operatorname{Homeo}^{+}(X,\partial X)$ such that
$(F_{0})_{*}=\varphi_{0}$. Extend $F_{0}$ by the identity to
$F\in\operatorname{Homeo}^{+}(X,\partial X)$. Note that $F_{*}=\varphi$, and
hence $\Xi(\Delta_{F})=\varphi$ as required. ∎

Combining Theorem 7.13 with Theorem A, and recalling Lemma 7.10 in the spin
case, we arrive at the following result.

For $\kappa\in\wedge^{2}{H_{1}(\partial X)^{\ast}}$, denote by
$F_{\kappa}\in\operatorname{Homeo}^{+}(X,\partial X)$ a homeomorphism arising
as the effect of applying the realisation result Theorem 6.6 to
$(0,S(\kappa))\in H^{1}(X,\partial
X;\mathbb{Z}/2)\times\mathcal{V}(H_{2}(X),\lambda_{X})$ in the spin case,
respectively $S(\kappa)\in\mathcal{V}(H_{2}(X),\lambda_{X})$ in the nonspin
case. By Theorem A we know that the class of
$F_{\kappa}\in\pi_{0}\operatorname{Homeo}^{+}(X,\partial X)$ is well-defined.

######  Theorem 7.14.

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

  1. (1)

When $X$ is spin there is a short exact sequence

$0\to\wedge^{2}{H_{1}(\partial
X)^{\ast}}\to\pi_{0}\operatorname{Homeo}^{+}(X,\partial X)\to H^{1}(X,\partial
X;\mathbb{Z}/2)\times\operatorname{Aut}_{\partial}(H_{2}(X),\lambda_{X})\to
0,$

where the first map is given by $\kappa\mapsto F_{\kappa}$ and the second map
is given by $F\mapsto(\Theta(F),F_{*})$.

  2. (2)

When $X$ is not spin there is a short exact sequence

$0\to\wedge^{2}{H_{1}(\partial
X)^{\ast}}\to\pi_{0}\operatorname{Homeo}^{+}(X,\partial
X)\to\operatorname{Aut}^{\operatorname{fix}}_{\partial}(H_{2}(X),\lambda_{X})\to
0,$

where the first map is given by $\kappa\mapsto F_{\kappa}$ and the second map
is given by $F\mapsto F_{*}$.

######  Corollary 7.15.

Let $(X,\partial X)$ be a compact, simply connected, oriented, topological
$4$-manifold such that $H_{1}(\partial X;\mathbb{Q})$ has rank at most 1.

  1. (1)

When $X$ is spin, the map $F\mapsto(\Theta(F),F_{*})$ determines an
isomorphism

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

  2. (2)

When $X$ is not spin, the map $F\mapsto F_{*}$ determines an isomorphism

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

We can also now prove Corollaries C and D from the introduction.

###### Proof of Corollary C.

When $\partial X$ is a rational homology $S^{3}$ or $S^{1}\times S^{2}$, the
free part of $H_{1}(\partial X)$ has rank at most 1. Thus the only skew-
symmetric from $H_{1}(\partial X)\times H_{1}(\partial X)\to\mathbb{Z}$ is the
0 form, i.e. $\wedge^{2}{H_{1}(\partial X)^{\ast}}=0$. Item (1) is then
clearly a special case of Theorem 7.14.

For item (2), we first consider the general behaviour of the short exact
sequence of Theorem 7.13 under algebraic stabilisation. Observe that for all
$g\geq 0$ there is a commuting diagram where the rows are short exact
sequences

${\wedge^{2}H_{1}(\partial
X)^{*}}$${\mathcal{V}((H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g})}$${\operatorname{Aut}_{\partial}^{\operatorname{fix}}((H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g})}$${\wedge^{2}H_{1}(\partial
X)^{*}}$${\mathcal{V}((H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g+1})}$${\operatorname{Aut}_{\partial}^{\operatorname{fix}}((H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g+1})}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{S}$$\scriptstyle{\Xi}$$\scriptstyle{S}$$\scriptstyle{\Xi}$

where the unlabelled downwards maps are the stabilisation maps. Exact
sequences are preserved under colimits in the category of groups. We obtain a
short exact sequence

$\wedge^{2}H_{1}(\partial
X)^{*}\to\underset{g\to\infty}{\operatorname{colim}}\,\mathcal{V}\big{(}(H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g}\big{)}\to\underset{g\to\infty}{\operatorname{colim}}\,\operatorname{Aut}_{\partial}^{\operatorname{fix}}\big{(}(H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g}\big{)}.$

Now assuming the hypotheses of Corollary C (2), the sequence above shows that
the stable $\Xi$ map is an isomorphism, which combines with Theorem B to prove
item (2). ∎

###### Proof of Corollary D.

In the nonspin case, Corollary D (1) is immediate from the nonspin case of
Theorem 7.14. The spin case of Theorem 7.14 yields a short exact sequence

$0\to\wedge^{2}{H_{1}(\partial X)^{\ast}}\to\operatorname{Tor}^{0}(X,\partial
X)\to H^{1}(X,\partial X;\mathbb{Z}/2)\to 0.$

The exact sequence of Theorem 7.13 gives rise to a splitting
$\operatorname{Tor}^{0}(X,\partial X)\to\wedge^{2}{H_{1}(\partial X)^{\ast}}$,
completing the proof of the spin case of Corollary D (1).

For the proof of item (2), first consider the case that $X$ is spin. Recall
that for all $g\geq 0$, there is a canonical identification
$H^{1}(X\\#^{g}S^{2}\times S^{2},\partial X;\mathbb{Z}/2)\cong
H^{1}(X,\partial X;\mathbb{Z}/2)$. For all $g\geq 0$ there is a commutative
diagram

${\operatorname{Tor}^{\infty}(X\\#^{g}S^{2}\times S^{2},\partial
X)}$${\wedge^{2}H_{1}(\partial X)^{*}\times H^{1}(X,\partial
X;\mathbb{Z}/2)}$${\pi_{0}(\operatorname{Diffeo}^{+}(X\\#^{g}S^{2}\times
S^{2},\partial
X))}$${\mathcal{V}\big{(}(H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g}\big{)}\times H^{1}(X,\partial
X;\mathbb{Z}/2)}$${\operatorname{Aut}_{\partial}\big{(}(H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g}\big{)}}$${\operatorname{Aut}_{\partial}\big{(}(H_{2}(X),\lambda_{X})\oplus\mathcal{H}^{\oplus
g}\big{)}}$incl.$\scriptstyle{F\mapsto(\kappa_{F},\Theta(F))}$$\scriptstyle{S\times\operatorname{Id}}$$\scriptstyle{F\mapsto(\Delta_{F},\Theta(F))}$$\scriptstyle{F\mapsto
F_{*}}$$\scriptstyle{\Xi\circ\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{Id}}$

The left-hand column is an exact sequence by definition of the Torelli group,
and the right-hand column is a short exact sequence by Theorem 7.13. The
diagram is natural under stabilisation, so we may take the colimit of the
diagram. Exactness is preserved under colimits, and in the colimit, the
central horizontal map becomes an isomorphism by Theorem B. We deduce from
this that the bottom-left vertical map becomes surjective in the colimit. We
may conclude that in the colimit the diagram is an isomorphism of short exact
sequences, in particular the upper horizontal map is an isomorphism in the
colimit, as claimed.

This argument is easily modified to the nonspin case, by removing
$H^{1}(X,\partial X;\mathbb{Z}/2)$ from the diagram and replacing
$\operatorname{Aut}_{\partial}$ with
$\operatorname{Aut}_{\partial}^{\operatorname{fix}}$. ∎

##  8\. Realising the invariants $\Theta$ and $S(\kappa)$ smoothly

We consider the problem of realising the obstruction $\Theta$ of Definition
2.5 in the smooth category, and without stabilising. The main result we prove
is that this can be done in the presence of boundary components of Heegaard
genus at most one. We then consider the problem of constructing
$F\in\operatorname{Diffeo}^{+}(X,\partial X)$ such that
$F_{*}=\operatorname{Id}_{H_{2}(X)}$ but with $\Delta_{F}\neq 0$, in other
words, building diffeomorphisms $F$ such that $\Delta_{F}=S(\kappa)$ for some
$\kappa\in\wedge^{2}{H_{1}(\partial X)^{\ast}}$. Neither of these smooth
realisation questions are fully resolved.

On the other hand, the results of this section can also be viewed as computing
the action of $\pi_{1}\operatorname{Homeo}^{+}(\partial X)$ on
$\pi_{0}\operatorname{Homeo}^{+}(X,\partial X)$ in the exact sequence (7) from
Section 1.4.

###  8.1. Modifying $\Theta$ with generalised Dehn twists

Our strategy is to modify a given $\Theta$ by using a diffeomorphism supported
in a collar on a boundary component of the $4$-manifold, or more generally on
an embedded $Y\times[0,1]$, where $Y$ is a closed $3$-manifold. For this
purpose we briefly consider the behaviour of $\Theta$ when two $4$-manifolds
are glued along one or more boundary components.

Let $(X,\mathfrak{s})$ be a compact, oriented, spin, topological $4$-manifold.
Suppose that $X$ decomposes as $X=X_{1}\cup_{Y}X_{2}$, where $Y$ is a
(possibly disconnected) $3$-manifold. Write $\mathfrak{s}_{i}$ for the
restriction of $\mathfrak{s}$ to $X_{i}$. We now consider the long exact
cohomology sequence with $\mathbb{Z}/2$-coefficients associated to the map of
pairs $(X,\partial X)\to(X,X_{2}\sqcup(\partial X_{1}\setminus Y))$.

${H^{0}(X_{2},\partial X_{2}\setminus Y)}$${H^{1}(X,X_{2}\sqcup(\partial
X_{1}\setminus Y))}$${H^{1}(X,\partial X)}$${H^{1}(X_{2},\partial
X_{2}\setminus Y)}$${H^{1}(X_{1},\partial
X_{1})}$$\scriptstyle{\operatorname{exc}}$$\scriptstyle{\cong}$$\scriptstyle{\sigma}$$\scriptstyle{\psi}$

The downwards vertical map is the excision isomorphism.

######  Lemma 8.1.

With notation as above, suppose that $F\colon X\xrightarrow{\cong}X$ is a
homeomorphism that fixes $\partial X$ pointwise and restricts to the identity
map on $X_{2}$. Then
$\psi(\Theta(F|_{X_{1}},\mathfrak{s}_{1}))=\Theta(F,\mathfrak{s})$.

###### Proof.

Choose a vector space splitting $H^{1}(X,\partial
X;\mathbb{Z}/2)\cong\psi(H^{1}(X_{1},\partial X_{1};\mathbb{Z}/2))\oplus V$.
Write $\Theta_{V}$ for the projection of $\Theta(F,\mathfrak{s})$ to $V$. Our
objective is to show that $\Theta_{V}$ is trivial. By exactness of the
sequence above, $\sigma$ is injective when restricted to $V$, so it is
sufficient to show that $\sigma(\Theta(F,\mathfrak{s}))=\sigma(\Theta_{V})$ is
trivial. Because the map $\sigma$ is induced by inclusion, the class
$\sigma(\Theta_{V})$ represents the difference in the rel. boundary spin
structure $(\mathfrak{s}_{2},\mathfrak{s}_{2}|_{\partial X_{2}\setminus Y})$
and $F^{*}(\mathfrak{s}_{2},\mathfrak{s}_{2}|_{\partial X_{2}\setminus Y})$.
But by assumption $F|_{X_{2}}$ is the identity map and this class
$\sigma(\Theta_{V})$ vanishes. ∎

######  Definition 8.2.

Let $Y$ be a closed, compact, oriented $3$-manifold. Fix a spin structure on
$Y$ and write $\mathfrak{s}$ for the resulting product spin structure on
$Y\times I$. Let $\varphi_{t}\in\pi_{1}(\operatorname{Diffeo}^{+}(Y))$ be a
smooth loop based at the identity and constant near the basepoint. Define a
self-diffeomorphism

$\displaystyle\Phi\colon Y\times I$ $\displaystyle\xrightarrow{\cong}Y\times
I$ $\displaystyle(x,t)$ $\displaystyle\mapsto(\varphi_{t}(x),t).$

If $\Theta(\Phi,\mathfrak{s})=1\in\mathbb{Z}/2=H^{1}(Y\times
I,\partial;\mathbb{Z}/2)$ then we say $\Phi$ is a _generalised Dehn twist_
with respect to the spin structure $\mathfrak{s}$ on $Y$.

######  Proposition 8.3.

Let $(X,\mathfrak{s})$ be a compact, simply connected, oriented, spin, smooth
$4$-manifold with nonempty connected boundary components $\partial
X=Y_{1}\cup\dots\cup Y_{r}$. Suppose for $i=1,\dots,r-1$ that $Y_{i}$ admits a
generalised Dehn twist with respect to the restriction of $\mathfrak{s}$ to
$Y_{i}$. Let $F\in\operatorname{Diffeo}^{+}(X,\partial X)$. Then, given $x\in
H^{1}(X,\partial X;\mathbb{Z}/2)$, there exists an orientation preserving
diffeomorphism $F^{\prime}\colon X\to X$, that fixes the boundary pointwise,
agrees with $F$ outside a collar of the boundary, and is such that
$\Theta(F^{\prime})=x$.

###### Proof.

In this proof, $\mathbb{Z}/2$ coefficients are understood throughout. When
$r=1$, the lemma is vacuously true because then $H^{1}(X,\partial X)=0$.

Now assume $r>1$. Choose properly embedded arcs
$\gamma_{1},\dots,\gamma_{r-1}$ such that for each $i$, $\gamma_{i}$ connects
$Y_{r}$ to $Y_{i}$. Note that these arcs determine a preferred basis for
$H_{1}(X,\partial X)$. Consider a fixed $i\neq r$. We claim there exists a
diffeomorphism $G\colon X\to X$ that agrees with $G$ outside of the boundary
collar of $Y_{i}$, and such that $\Theta(G)$ agrees with $\Theta(F)$ except on
the $i^{th}$ basis curve, where they differ. To prove the claim, first attach
an external collar to form $X^{\prime}:=X\cup(Y_{i}\times[0,1])$. Let
$\Phi\colon Y_{i}\times[0,1]\to Y_{i}\times[0,1]$ be a generalised Dehn twist
with respect to $\mathfrak{s}|_{Y_{i}}$. We extend $\mathfrak{s}$ to a spin
structure $\mathfrak{s}^{\prime}$ on $X^{\prime}$ by using the product spin
structure on the collar. Define a self-diffeomorphism $\Phi^{\prime}\colon
X^{\prime}\to X^{\prime}$ by extending $\Phi$ using the identity on $X$. By
Lemma 8.1, we have that $\Theta(\Phi^{\prime})$ is the image of
$\Theta(\Phi,\mathfrak{s}^{\prime}|_{Y_{i}\times[0,1]})$ under the map
$\psi\colon H^{1}(Y_{i}\times[0,1],Y_{i}\times\\{0,1\\})\to
H^{1}(X^{\prime},\partial X^{\prime})$. As $r>1$, we have $\partial X\setminus
Y\neq\emptyset$, so that $H^{0}(X,\partial X\setminus Y)=0$ and $\psi$ is
injective. Hence
$\psi(\Theta(\Phi,\mathfrak{s}^{\prime}|_{Y_{i}\times[0,1]}))=\Theta(\Phi^{\prime})$
is nontrivial. Now choose the basis for
$H_{1}(Y_{i}\times[0,1],Y_{i}\times\\{0,1\\})=\mathbb{Z}/2$ given by
$y\times[0,1]$ where $y\in\gamma_{i}$ is the endpoint on $Y_{i}$. A basis for
$H_{1}(X^{\prime},\partial X^{\prime})$ is given by
$\gamma_{1},\dots,\gamma_{i-1},\gamma_{i}\cup(y\times[0,1]),\gamma_{i+1},\dots,\gamma_{r-1}$.
As the class $\Theta(\Phi,\mathfrak{s}^{\prime}|_{Y_{i}\times[0,1]})$ is
nontrivial, it is necessarily the $\mathbb{Z}/2$-dual $[y\times[0,1]]^{*}$.
Thus

$\Theta(\Phi^{\prime})=\psi(\Theta(\Phi,\mathfrak{s}^{\prime}|_{Y_{i}\times[0,1]}))=\psi([y\times[0,1]]^{*})=[\gamma_{i}\cup(y\times[0,1])]^{*}.$

Finally, as $\Theta$ is additive under composition of maps (see Lemma 2.7),
the map $G:=\Phi^{\prime}\circ F$ has the desired properties. (Implicitly, we
are using that adding an external collar produces a diffeomorphic manifold.)

With this claim proved, we now have a procedure for altering any individual
entry of $\Theta(F)$, by changing the map $F$ in a collar of the boundary. For
each $i=1,\dots,r-1$, perform this type of modification on $F$ as necessary,
so that the resulting map $F^{\prime}$ has $\Theta(F^{\prime})=x$. ∎

###  8.2. Which 3-manifolds admit generalised Dehn twists?

In this subsection we consider the question of which closed, oriented
$3$-manifolds admit generalised Dehn twists, and with respect to which of
their spin structures. We begin our analysis of Dehn-twistability with two
special cases that are easier to understand: $Y=S^{3}$ and
$Y=S^{1}\times\Sigma_{g}$. Then we consider lens spaces, before finally saying
a brief word about Seifert fibred spaces in general.

We introduce some notation for this subsection. Given $Y$ be a closed,
compact, oriented $3$-manifold, and given
$\varphi_{t}\in\pi_{1}(\operatorname{Diffeo}^{+}(Y))$ a smooth loop based at
the identity, we denote by

$\Phi\colon
Y\times[0,1]\xrightarrow{\cong}Y\times[0,1];\qquad(x,t)\mapsto(\varphi_{t}(x),t).$

the resultant self-diffeomorphism of $Y\times[0,1]$ that fixes the boundary
pointwise. Given a properly embedded arc $\gamma\subseteq Y\times[0,1]$, with
endpoints on the different boundary components, write

$\delta:=\gamma\times\\{1\\}\cup\partial\gamma\times[0,1]\cup\gamma\times\\{0\\}\subseteq\partial((Y\times[0,1])\times[0,1]).$

Given a spin structure on $Y$, let $\mathfrak{s}$ be the product spin
structure on $Y\times[0,1]$ and let $\mathfrak{t}$ be the spin structure on
$D:=\partial((Y\times[0,1])\times[0,1])$ given by
$\mathfrak{t}(x,t)=\mathfrak{s}\circ\Phi$ when $t=1$ and
$\mathfrak{t}=\mathfrak{s}$ otherwise. As detailed in the proof of Lemma 6.1,
this determines a framing for the normal $3$-plane bundle $\nu(\delta\subseteq
D)$.

######  Example 8.4.

Elaborating on Example 1.3, we prove that the (ordinary) Dehn twist is a
generalised Dehn twist with respect to the unique spin structure on $S^{3}$.
Fix a point $p\in S^{2}\subseteq\mathbb{R}^{3}$ and let
$\varphi^{\prime}_{t}\in\operatorname{SO}(3)$, for $t\in[0,1]$, be a loop
based at the identity and rotating $\mathbb{R}^{3}$ one full turn around an
axis through the origin and $p$. Under the inclusion of the first factor
$\mathbb{R}^{3}\subseteq\mathbb{R}^{3}\times\mathbb{R}=\mathbb{R}^{4}$, and
restricting to $S^{3}\subseteq\mathbb{R}^{4}$, there is then induced a
generating loop
$\varphi_{t}\in\pi_{1}(\operatorname{Diffeo}^{+}(S^{3}))\cong\mathbb{Z}/2$.
This determines the self-diffeomorphism $\Phi\colon S^{3}\times[0,1]\to
S^{3}\times[0,1]$.

As $p$ is fixed by $\varphi_{t}$ for all $t$, the tubular neighbourhood of
$\gamma:=\\{p\\}\times[0,1]\subseteq S^{3}\times[0,1]$ is a product
$\nu(\gamma)=\nu(\\{p\\}\subseteq S^{3})\times[0,1]$. The loop $\varphi_{t}$
preserves the tubular neighbourhood $\nu(\\{p\\}\subseteq S^{3})$ set-wise; it
fixes one normal coordinate of $\nu(\\{p\\}\subseteq S^{3})$ for all $t$, and
rotates the normal 2-plane perpendicular to this fixed coordinate by one full
turn. The spin structure $\mathfrak{s}$ and $\Phi$ determine a framing
$\mathfrak{t}$ on $\nu(\delta\subseteq D)$. As $p$ was fixed, this is a
product framing $\nu(p\subseteq S^{1}\times S^{2})\times[0,1]$ over
$\gamma\times\\{0\\}$. However the framing over $\delta$ is not null-bordant
because over $\gamma\times\\{1\\}$, the map $\Phi$ adds a full twist compared
to the bundle over $\gamma\times\\{0\\}$. By Lemma 6.1,
$\Theta(\Phi,\mathfrak{s})=1$.

######  Example 8.5.

Consider the loop $\varphi_{t}\in\pi_{1}(\operatorname{Diffeo}^{+}(S^{1}\times
S^{2}))$ that fixes the $S^{1}$-coordinate, and rotates the $S^{2}$ one full
turn around an axis in $\mathbb{R}^{3}$. We claim this is a generalised Dehn
twist with respect to both of the spin structures on $S^{1}\times S^{2}$. To
see this, let $p$ be a fixed point of the $S^{2}$ rotation. Similarly to
Example 8.4, the normal bundle of
$\gamma:=\\{p\\}\times[0,1]\subseteq(S^{1}\times S^{2})\times[0,1]$ is
preserved set-wise by $\Phi$, and the normal 2-plane corresponding to the
$S^{2}$ factor is turned by one full rotation, whereas the normal direction
corresponding to the $S^{1}$ is fixed throughout. The spin structure
$\mathfrak{s}$ and $\Phi$ determine a framing $\mathfrak{t}$ on
$\nu(\delta\subseteq D)$. As $p$ was fixed, this is a product framing
$\nu(p\subseteq S^{1}\times S^{2})\times[0,1]$ over $\gamma\times\\{0\\}$.
However the framing over $\delta$ is not null-bordant because over
$\gamma\times\\{1\\}$, the map $\Phi$ adds a full twist compared to the bundle
over $\gamma\times\\{0\\}$. By Lemma 6.1, $\Theta(\Phi,\mathfrak{s})=1$.

######  Example 8.6.

In contrast to Example 8.5, it may generally be the case that a given loop of
diffeomorphisms determines a generalised Dehn twist with respect to one spin
structure but not with respect to another. For example, let
$\varphi_{t}\in\pi_{1}(\operatorname{Diffeo}^{+}(S^{1}\times S^{2}))$ be the
loop that fixes the $S^{2}$ factor and is a full rotation on the $S^{1}$
factor. Write $\mathfrak{s}_{0}$ and $\mathfrak{s}_{1}$ for the product spin
structures on $(S^{1}\times S^{2})\times[0,1]$ that arise from extending
respectively the bounding and Lie spin structures on the $S^{1}$ factor. Here,
in order to specify the bounding and Lie spin structures, we use the given
product structure of $S^{1}\times S^{2}$ to identify
$TS^{1}\oplus\varepsilon^{2}$, which has a preferred Lie framing, with
$T(S^{1}\times S^{2})|_{S^{1}\times\\{p\\}}$. We can assume that the loop
$S^{1}\times\\{p\\}$ equals the 1-skeleton in a cell decomposition of
$S^{1}\times S^{2}$. Fix a point $(q,p)\in S^{1}\times S^{2}$. We may assume
the framing of $T_{(q,p)}(S^{1}\times S^{2})\cong T_{q}S^{1}\oplus T_{p}S^{2}$
induced by both spin structures is of the form $\langle
e_{1},e_{2},e_{3}\rangle$, where $e_{1}\in T_{q}S^{1}$ and $e_{2},e_{3}\in
T_{p}S^{2}$. Since $\varphi_{t}$ rotates the $S^{1}$ factor and fixes the
$S^{2}$ factor, the induced framing on $\varphi_{t}(q,p)$ preserves this
decomposition and fixes the vectors $e_{2},e_{3}$. The induced framing on
$S^{1}\times\\{p\\}$ is identified with the non-bounding structure on
$TS^{1}\oplus\varepsilon^{2}$. Thus, using the arc
$\gamma(t)=(\varphi_{t}(x,y),t)$ to compute the invariant, we see that
$\Theta(\Phi,\mathfrak{s}_{0})=1$ and $\Theta(\Phi,\mathfrak{s}_{1})=0$.

We can similarly analyse $Y=S^{1}\times S^{1}\times S^{1}$. There are
$|H^{1}(Y;\mathbb{Z}/2)|=8$ spin structures on $Y$ characterised by whether or
not their restrictions to the three basis circles are bounding. Note that $Y$
is a Lie group, and exponentiating multiplication in each of the coordinate
directions gives rise to a Lie framing along each standard basis element of
$H_{1}(Y;\mathbb{Z}/2)$; the Lie framing is the non-bounding framing. If
$\mathfrak{s}$ is bounding for a basis circle $C$, then let $\Phi$ be induced
by the loop of diffeomorphisms that rotates $C$ and fixes the other two. As in
the example above, this gives $\Theta(\Phi,\mathfrak{s})=1$. If $\mathfrak{t}$
is the spin structure that is the Lie framing on all three basis circles, all
of the obvious circle actions on $Y$ result in $\Theta(\Phi,\mathfrak{t})=0$.

Now let $Y=S^{1}\times\Sigma_{g}$, where $g>1$ and let $\Phi$ be induced by
the loop that fixes the $\Sigma_{g}$ factor and rotates the $S^{1}$ factor by
a full rotation. As with $S^{1}\times S^{2}$, we may use the product structure
of $S^{1}\times\Sigma_{g}$ to identify $TS^{1}\oplus\varepsilon^{2}$ with
$T(S^{1}\times\Sigma_{g})|_{S^{1}\times\\{p\\}}$ along a loop
$S^{1}\times\\{p\\}$, and thence we may speak of the Lie framing and the
bounding framing of the stable tangent bundle restricted to this loop. Exactly
half of the $|H^{1}(Y;\mathbb{Z}/2)|=2^{2g+1}$ spin structures will be such
that their restriction to $S^{1}\times\\{p\\}$ is the bounding framing, and if
$\mathfrak{s}_{0}$ is such a spin structure then
$\Theta(\Phi,\mathfrak{s}_{0})=1$ by the same argument as above. The other
spin structures restrict to the Lie framing on $S^{1}\times\\{p\\}$ and if
$\mathfrak{s}_{1}$ is such a spin structure then
$\Theta(\Phi,\mathfrak{s}_{1})=0$.

Next, we generalise the method for computing $\Theta$ that was used in the
previous example. Let $Y$ be a closed, oriented 3-manifold and let
$\varphi_{t}\in\pi_{1}(\operatorname{Diffeo}^{+}(Y))$ be a smooth loop based
at the identity. This determines a self-diffeomorphism $\Phi\colon
Y\times[0,1]\to Y\times[0,1]$. Suppose we are given a spin structure
$\mathfrak{s}$ on $Y$, and fix a framing for the tangent bundle $TY$ which is
compatible with the given spin structure. We can compute
$\Theta(\Phi,\mathfrak{s})$ using any path $\gamma$ between the two boundary
components. We shall compute $\Theta$ by choosing a point $p\in Y$ and using
the particular choice of path $\gamma$ given by $t\mapsto(\varphi_{t}(p),t)$
in $Y\times[0,1]$. Write $\langle e_{1},e_{2},e_{3}\rangle$ for the standard
basis of $\mathbb{R}^{3}$, identified with $T_{p}Y$ using the framing of $TY$.
For each $t\in[0,1]$, define an element of
$\operatorname{GL}^{+}(3,\mathbb{R})$ by comparing $\langle
D\varphi_{t}(p)(e_{i})\mid i=1,2,3\rangle\in T_{\varphi_{t}(p)}Y$ with the
basis of $T_{\varphi_{t}(p)}Y$ given by the framing of $TY$. This determines a
based map $S^{1}\to\operatorname{GL}^{+}(3,\mathbb{R})$, and whence an element
$x$ of $\pi_{1}(\operatorname{GL}^{+}(3,\mathbb{R}))\cong\mathbb{Z}/2$.
Comparing with Lemma 6.1, the stable framing of $\nu_{\delta}$ (recall that
$\delta$ is defined in terms of $\gamma$) is null-bordant if and only if
$x=0\in\mathbb{Z}/2$ is trivial. Then by Lemma 6.1,
$\Theta(\Phi,\mathfrak{s})=x$.

The main result of the subsection is the following proposition, which
leverages the fact that lens spaces admit multiple Seifert fibrations to
guarantee that for every lens space there is a Seifert fibration giving rise
to nontrivial $\Theta$.

######  Proposition 8.7.

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

###### Proof.

Since Examples 8.4 and 8.5 already dealt with the cases of $S^{3}$ and
$S^{1}\times S^{2}$ respectively, it remains to consider lens spaces $L(p,q)$.
Suppose without loss of generality that $p>0$, $0<q<p$ and $(p,q)=1$. By
Bézout’s Lemma, there exist integers $r$ and $s$ such that $qs-pr=-1$. In
order to fix notation, we describe a genus one Heegaard splitting for $L(p,q)$
explicitly. Let $S_{1}$, $S_{2}$ be copies of the solid torus $S^{1}\times
D^{2}$ and write $T_{i}:=\partial S_{i}$ for $i=1,2$. Write $\mu_{i}\subseteq
T_{i}$ for $\\{1\\}\times S^{1}$ and $\lambda_{i}\subseteq T_{i}$ for
$S^{1}\times\\{1\\}$. Fixing an orientation of $S^{1}$, we will conflate this
notation for $\mu_{i}$ and $\lambda_{i}$ with the associated homology classes
in $H_{1}(T_{i})$. Also note that $\langle\mu_{i},\lambda_{i}\rangle$ is a
basis for $H_{1}(T_{i})\cong\mathbb{Z}^{2}$. Let $\psi\colon
T_{1}\xrightarrow{\cong}T_{2}$ be a diffeomorphism, determined up to isotopy
by prescribing that, with respect to the bases just defined, $\psi_{*}\colon
H_{1}(T_{1})\to H_{1}(T_{2})$ is represented by
$\left(\begin{smallmatrix}q&r\\\ p&s\end{smallmatrix}\right)$. Then
$L(p,q)=S_{1}\cup_{\psi}S_{2}$. In the Mayer-Vietoris sequence for this
decomposition, noting that $H_{1}(S_{i})\cong\mathbb{Z}$, we have the map
$\left(\begin{smallmatrix}0&1\\\
p&s\end{smallmatrix}\right)\colon\mathbb{Z}\oplus\mathbb{Z}\cong
H_{1}(T_{1})\to H_{1}(S_{1})\oplus
H_{1}(S_{2})\cong\mathbb{Z}\oplus\mathbb{Z}.$ It follows that
$H_{1}(L(p,q))\cong\mathbb{Z}/p$, generated by the core of the solid torus
$S_{2}$. The core of $S_{1}$ is identified with $s$ times the core of $S_{2}$.
In what follows we will use the core of $S_{2}$ as the preferred generator of
$\mathbb{Z}/p$ (although as $s$ is coprime to $p$, the core of $S_{1}$ is also
a generator).

Fix a spin structure on $L(p,q)$. Choose a Seifert fibration of $S_{2}$ with
multiplicity one and such that it restricts to a fibration of $T_{2}$ by
embedded curves that represent $(0,1)$ in $H_{1}(T_{2})$, with respect to the
basis $\langle\mu_{2},\lambda_{2}\rangle$. We obtain a loop
$\varphi_{t}\in\pi_{1}(\operatorname{Diffeo}^{+}(S_{2}))$. Pick a point $p\in
S_{2}$ on the core, and compute $\Theta$ by following a framing of
$T_{p}L(p,q)$ around the loop $\varphi_{t}(p)$ in the way described earlier in
this section. If $\Theta$ is trivial, we change the choice of Seifert
fibration on $S_{2}$ to one that has multiplicity 1 and restricts to a
fibration of $T_{2}$ by embedded curves that represent $(1,1)$ in
$H_{1}(T_{2})$. The additional meridional turn as the frame moves around the
core will force $\Theta$ to be nontrivial.

(Whether we had to use the $(0,1)$ or the $(1,1)$ Seifert fibration depended
on the initial given spin structure. In our applications, the spin structure
on $L(p,q)$ will be the one induced by the unique spin structure on the simply
connected 4-manifold $X$, so we cannot control it. We also remark that for $p$
odd there is a unique spin structure on $L(p,q)$ because
$H^{1}(L(p,q);\mathbb{Z}/2)=0$. We can determine this spin structure
explicitly, but since we do not need this information we omit the details.)

It remains to check that our choice of Seifert fibration on $S_{2}$ extends
over $S_{1}$, therefore inducing an $S^{1}$ action on all of $L(p,q)$ as
required. We have that $\psi^{-1}\colon T_{2}\to T_{1}$ induces
$\left(\begin{smallmatrix}q&r\\\
p&s\end{smallmatrix}\right)^{-1}=\left(\begin{smallmatrix}-s&r\\\
p&-q\end{smallmatrix}\right)\colon\mathbb{Z}\oplus\mathbb{Z}\cong
H_{1}(T_{2})\to H_{1}(T_{1})\cong\mathbb{Z}\oplus\mathbb{Z}.$ A $(0,1)$-curve
in $T_{2}$ is identified with an $(r,-q)$-curve in $T_{1}$, while a
$(1,1)$-curve in $T_{2}$ is identified with an $(r-s,p-q)$-curve in $T_{1}$.
In both cases, the second coordinate is nonzero, by our assumption that
$0<q<p$. The induced Seifert fibrations of $T_{1}$ therefore extend to
standard Seifert fibrations of $S_{1}$, with singular fibres at the core of
$S_{1}$ of multiplicities $q$ and $p-q$ respectively. Thus in both cases we
obtain a Seifert fibration of $L(p,q)$ with exactly one singular fibre, such
that, with respect to the given spin structure, rotating around the Seifert
fibres gives rise to a generalised Dehn twist, as desired. ∎

We also considered the question of which Seifert fibred spaces, in general,
admit a generalised Dehn twist. As we now show, every orientable Seifert
fibred space $Y$ admits a homotopically nontrivial loop of self-
diffeomorphisms, so there is the potential for a generalised Dehn twist to
exist. We already discussed that
$\pi_{1}(\operatorname{Diffeo}^{+}(S^{3}))\cong\mathbb{Z}/2$, so we focus on
the case $Y\neq S^{3}$.

######  Proposition 8.8.

Let $Y\neq S^{3}$ be a closed, orientable Seifert fibred $3$-manifold. Then
the rotation around the Seifert fibres determines a nontrivial loop
$\varphi_{t}\in\pi_{1}(\operatorname{Diffeo}^{+}(Y))$.

###### Proof.

For each $p\in Y$ there is a map
$\pi_{1}(\operatorname{Diffeo}^{+}(Y))\to\pi_{1}(Y,p)$ that sends
$\varphi_{t}$ to the loop $\varphi_{t}(p)\subseteq Y$. It is sufficient to
show that for some $p\in Y$ the map is nontrivial. Assume the converse, for a
contradiction. This implies that all fibres of the Seifert fibration are
homotopically trivial. A presentation for $\pi_{1}(Y)$ is computed in [JN83,
Theorem 6.1]. In the language of that proof, $\alpha_{j}$ is the multiplicity
of the $j^{\text{th}}$ singular fibre, and there are integers
$\alpha_{j}^{\prime},\beta_{j},\beta_{j}^{\prime}$ coming from a specific
description of the neighbourhoods of the singular fibres (see [JN83, §1]),
such that $\alpha_{j}\beta_{j}^{\prime}-\alpha_{j}^{\prime}\beta_{j}=1$, where
$\alpha_{j}$. In the language of that theorem, and its proof, our assumption
that the classes of the regular fibre $h$ and the singular fibres
${q_{j}}^{\alpha_{j}^{\prime}}h^{\beta^{\prime}_{j}}$ are trivial, implies
${q_{j}}^{\alpha_{j}^{\prime}}=1$. Already in the presentation we have
${q_{j}}^{\alpha_{j}}h^{\beta_{j}}=1$, so that also ${q_{j}}^{\alpha_{j}}=1$.
For all $j$, we thus have

$q_{j}=q_{j}^{\alpha_{j}\beta_{j}^{\prime}-\alpha_{j}^{\prime}\beta_{j}}=(q_{j}^{\alpha_{j}})^{\beta^{\prime}_{j}}(q_{j}^{\alpha^{\prime}_{j}})^{-\beta_{j}}=1.$

Setting $h=1$ and $q_{j}=1$, for all $j$, in the presentation of [JN83,
Theorem 6.1], we have that $\pi_{1}(Y)\cong\pi_{1}(\Sigma)$, where $\Sigma$ is
the base surface of the Seifert fibration.

The only closed oriented $3$-manifolds $Y$ with surface fundamental group are
$Y\cong S^{3}$ and $Y\cong\operatorname{\mathbb{R}P}^{3}$. To see this, note
first that $Y$ is irreducible as a surface group is not a non-trivial free
product nor isomorphic to $\mathbb{Z}$. Suppose $\pi_{1}(Y)$ is infinite. Then
$\pi_{2}(Y)=0$, by the Sphere Theorem, and hence $Y\simeq
K(\pi_{1}(\Sigma),1)$. But $H_{3}(Y;\mathbb{Z})=\mathbb{Z}$ whereas
$H_{3}(\pi_{1}(\Sigma);\mathbb{Z})=0$ for every closed surface $\Sigma$ with
infinite fundamental group, which is a contradicition. Thus we are left with
the case that $\pi_{1}(Y)$ is finite. The only finite surface groups are the
trivial group and $\mathbb{Z}/2$, implying respectively either $Y\cong S^{3}$
or $Y\cong\operatorname{\mathbb{R}P}^{3}$.

By assumption, it is not the case that $Y\cong S^{3}$, so suppose
$Y\cong\operatorname{\mathbb{R}P}^{3}$. In this case, the projection of the
Seifert fibration
$f\colon\operatorname{\mathbb{R}P}^{3}\to\operatorname{\mathbb{R}P}^{2}$
induces an isomorphism
$\pi_{1}(\operatorname{\mathbb{R}P}^{3})\to\pi_{1}(\operatorname{\mathbb{R}P}^{2})$,
and hence also an isomorphism $f^{*}\colon
H^{1}(\operatorname{\mathbb{R}P}^{2};\mathbb{Z}/2)\to
H^{1}(\operatorname{\mathbb{R}P}^{3};\mathbb{Z}/2)$. But for $x\in
H^{1}(\operatorname{\mathbb{R}P}^{2};\mathbb{Z}/2)$ a generator we have
$0=f^{*}(0)=f^{*}(x^{3})=f(x)^{3}\neq 0\in
H^{3}(\operatorname{\mathbb{R}P}^{3};\mathbb{Z}/2)$, a contradiction. ∎

Regarding the $\Theta$ invariant for a general Seifert fibred space, with
$\Phi$ the self-diffeomorphism induced by rotating the fibres, we were able to
show the following.

######  Claim.

Let $Y$ be an oriented Seifert fibred space. Write the Euler number of the
Seifert fibred structure as $a/b$, where $b$ is the least common multiple of
the multiplicities of the exceptional fibres. If $a$ and $b$ are odd, then
$\Theta(\Phi,\mathfrak{s})=0$ with respect to any spin structure
$\mathfrak{s}$ on $Y$.

We judge our proof of this result to be too lengthy to include. The result
does not actually preclude any given value of $\Theta$ from being realised by
a diffeomorphism (it merely shows this particular diffeomorphism will not be
useful).

However, it does show that if $X$ is spin and every boundary component of $X$
is homeomorphic to such a $Y$, then the action of
$\pi_{1}\operatorname{Homeo}^{+}(\partial X)$ on
$\pi_{0}\operatorname{Homeo}^{+}(X,\partial X)\cong H^{1}(X,\partial
X;\mathbb{Z}/2)\times\mathcal{V}(H_{2}(X),\lambda_{X})$ does not affect the
$H^{1}(X,\partial X;\mathbb{Z}/2)$ component.

It remains an interesting outstanding problem to fully determine which Seifert
fibred 3-manifolds admit generalised Dehn twists.

###  8.3. Realising the kernel of $\Xi$ smoothly

Recall the maps $S$ and $\Xi$ from Theorem 7.13. We investigate the
possibility of realising elements
$S(\kappa)\in\mathcal{V}(H_{2}(X),\lambda_{X})$ by using smooth self-maps of
boundary collars, extended by the identity to the rest of the $4$-manifold. We
prove the following partial result on this question. Suppose that
$Y\subseteq\partial X$ is a Seifert fibred connected component of the boundary
of a simply connected, compact, smooth 4-manifold $X$, and suppose that the
Seifert fibred structure on $Y$ has oriented base surface of genus $g\geq 1$.
Let $\\{\alpha_{i},\beta_{i}\\}$ be a symplectic basis of curves on the base
surface. By [JN83, Corollary 6.2] these curves are all homologically essential
and infinite order in $H_{1}(Y)$. Define

$\omega:=\sum_{i=1}^{g}\alpha_{i}^{*}\wedge\beta_{i}^{*}\in\wedge^{2}{H_{1}(Y)^{\ast}}\subseteq\wedge^{2}{H_{1}(\partial
X)^{\ast}}.$

As above, rotating the fibres determines a smooth loop of diffeomorphisms of
$Y$, which gives rise to a diffeomorphism $F\colon X\to X$ supported in a
collar of $Y$.

######  Proposition 8.9.

The Poincaré variation of the diffeomorphism $F$ satisfies

$\Delta_{F}=S(\omega)\in\mathcal{V}(H_{2}(X),\lambda_{X}).$

###### Proof.

Represent the curves $\alpha_{i},\beta_{i}$, for $i=1,\dots,g$, by simple
closed curves on the base surface disjoint from exceptional fibres. Take the
product with the fibres to obtain tori $T_{i}^{\alpha},T^{\beta}_{i}$, again
for $i=1,\dots,g$. The torus $T_{i}^{\alpha}$ intersects the curve $\beta_{i}$
transversely in one point. Since $\beta_{i}$ is infinite order in $H_{1}(Y)$,
by Poincaré duality $T_{i}^{\alpha}$ is nontrivial in $H_{2}(Y)$, and
represents $\beta_{i}^{*}\in H_{1}(Y)^{*}$ under the isomorphisms
$H_{2}(Y)\cong H^{1}(Y)\cong H_{1}(Y)^{*}$. The same holds with $\alpha$ and
$\beta$ reversed, so $T_{i}^{\beta}$ is nontrivial in $H_{2}(Y)$ and
represents $\alpha_{i}^{*}$.

Let $\Phi\colon Y\times[0,1]\to Y\times[0,1]$ be the diffeomorphism induced by
the loop of diffeomorphisms of $Y$ that rotates the fibre. Identifying
$Y\times[0,1]$ with a collar of this boundary component in $X$, extend $\Phi$
to a diffeomorphism $F\colon X\to X$ by the identity map. We have that
$F_{*}=\operatorname{Id}_{H_{2}(X)}$, so $\Delta_{F}=S(\kappa)$ for some
$\kappa\in\wedge^{2}{H_{1}(\partial X)^{\ast}}$. Since $F$ is supported in a
collar of $Y$ it follows that $\kappa$ lies in the summand
$\wedge^{2}{H_{1}(Y)^{\ast}}$ of $\wedge^{2}{H_{1}(\partial X)^{\ast}}$.

We now compute $\Delta_{F}$. Let $\gamma$ denote a regular fibre. By [JN83,
Corollary 6.2], $\gamma$ is infinite order in $H_{1}(Y)$ if and only if the
Euler number of the Seifert fibration is zero. Since $H_{2}(X,\partial X)\to
H_{1}(\partial X)$ is surjective, for $i=1,\dots,g$ we may choose
homologically essential, properly embedded surfaces $\Sigma_{i}\subseteq X$,
for $i=1,\dots,2g$ with boundaries $\partial\Sigma_{2i-1}=\alpha_{i}$ and
$\partial\Sigma_{2i}=\beta_{i}$ for $i=1,\dots,g$. If the Euler number of the
Seifert fibred structure is zero then additionally let $\Sigma_{2g+1}$ be a
homologically essential, properly embedded surface in $X$ with
$\partial\Sigma_{2g+1}=\gamma$.

For each $i$, the surface $F(\Sigma_{i})$ is the effect of replacing a
boundary collar of $\Sigma_{i}$ with the track of the isotopy of
$\partial\Sigma_{i}$ under the loop of diffeomorphisms defining $\Phi$. This
collar isotopy rotates $h$, but fixes it setwise, and so
$[F(\Sigma_{2g+1})-\Sigma_{2g+1}]=0\in H_{2}(X)$. The tracks of $\alpha_{i}$
and $\beta_{i}$ under this isotopy are the tori $T^{\alpha}_{i}$ and
$T^{\beta}_{i}$ respectively. It follows that
$[F(\Sigma_{2i-1})-\Sigma_{2i-1}]=[T_{i}^{\alpha}]$ and
$[F(\Sigma_{2i})-\Sigma_{2i}]=[T_{i}^{\beta}]$. This gives $\Delta_{F}$.

It remains to deduce that $\kappa=\omega$. Use that $T_{i}^{\alpha}$ is dual
to $\beta_{i}^{*}$ and $T_{i}^{\beta}$ is dual to $\alpha_{i}^{*}$ to see that
for $\kappa=\omega$ the composition $\widetilde{\kappa}$ in (12) sends
$\Sigma_{2i-1}\mapsto T^{\alpha}_{i}$ and $\Sigma_{2i}\mapsto T^{\beta}_{i}$
for $i=1,\dots,g$, and sends $\Sigma_{2g+1}\mapsto 0$. We thus compute that
$\Delta_{F}=S(\kappa)=S(\omega)$ as desired. Using injectivity of $S$, in
fact, $\kappa=\omega$. ∎

######  Remark 8.10.

Combining Proposition 8.9 with Example 8.6 we observe there exist examples of
collar twisting maps that simultaneously realise nontrivial elements
$\kappa\in\wedge^{2}H_{1}(\partial X)^{*}$ and $\Theta\in H^{1}(X,\partial
X)$. As discussed in the introduction, it is an interesting problem to
determine in general which combinations of
$(\kappa,x)\in\wedge^{2}H_{1}(\partial X)^{*}\times H^{1}(X,\partial
X;\mathbb{Z}/2)$ can be realised smoothly, and also in the context of Section
1.4, to determine how $\pi_{1}\operatorname{Homeo}^{+}(\partial X)$ acts on
$\wedge^{2}H_{1}(\partial X)^{*}\times H^{1}(X,\partial X;\mathbb{Z}/2)$.

##  Appendix A Dehn twists for non-spin smooth manifolds

Let $M$ be a closed, connected, oriented, $\operatorname{CAT}$ $n$-manifold
that decomposes into $n$-manifolds as $M=N\cup_{\partial N}P$. Then there are
fibre sequences [Las76], [HKMR12, Corollary 3.1]:

(1) ${\operatorname{Diffeo}^{+}(N,\partial
N)}$${\operatorname{Diffeo}^{+}(M)}$${\operatorname{Emb}^{+}(P,M)}$when
$\operatorname{CAT}=\operatorname{Diff}$;${\operatorname{Homeo}^{+}(N,\partial
N)}$${\operatorname{Homeo}^{+}(M)}$${\operatorname{Emb}^{+}_{\operatorname{TOP}}(P,M)}$when
$\operatorname{CAT}=\operatorname{TOP}$.$\scriptstyle{r}$$\scriptstyle{r}$

Here, the space $\operatorname{Emb}^{+}_{\operatorname{TOP}}(P,M)$ (resp.
$\operatorname{Emb}^{+}(P,M)$) denotes the topological group of locally flat
(resp. smooth) orientation-preserving embeddings $P\hookrightarrow M$. In each
case, the map $r$ denotes the map that restricts a morphism to the copy of $P$
in the decomposition $M=N\cup_{\partial N}P$.

We will use the second fibre sequence now, to justify a statement we used in
the introduction.

######  Proposition A.2.

Let $X$ be a connected, oriented manifold. Then there is a fibre sequence

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

where $r^{\prime}$ denotes the restriction map.

###### Proof.

Certainly the fibre of the identity map is
$\operatorname{Homeo}^{+}(X,\partial X)$. We must prove that $r^{\prime}$ has
the homotopy lifting property. Let $F\colon Y\to\operatorname{Homeo}^{+}(X)$
be a map from a space $Y$ and let $H\colon Y\times
I\to\operatorname{Homeo}^{+}(\partial X)$ be a homotopy starting at
$r^{\prime}\circ F$. Write $i\colon\partial X\to-X\cup_{\partial X}X$ for the
inclusion map of the boundary, which is a separating codimension one
submanifold.

Apply (1) when $M=-X\cup_{\partial X}X$ and $P=\partial X$ to see that
$r\colon\operatorname{Homeo}^{+}(-X\cup_{\partial
X}X)\to\operatorname{Emb}_{\operatorname{TOP}}^{+}(\partial X,-X\cup_{\partial
X}X)$ is a Serre fibration, where $r(f)=f\circ i$. Doubling a homeomorphism
yields a map
$D\colon\operatorname{Homeo}^{+}(X)\to\operatorname{Homeo}^{+}(-X\cup_{\partial
X}X)$. Consider the following commuting diagram:

${Y}$${\operatorname{Homeo}^{+}(X)}$${\operatorname{Homeo}^{+}(-X\cup_{\partial
X}X)}$${Y\times I}$${\operatorname{Homeo}^{+}(\partial
X)}$${\operatorname{Emb}^{+}_{\operatorname{TOP}}(\partial X,-X\cup_{\partial
X}X).}$$\scriptstyle{F}$$\scriptstyle{y\mapsto(y,0)}$$\scriptstyle{r^{\prime}}$$\scriptstyle{D}$$\scriptstyle{r}$$\scriptstyle{H}$$\scriptstyle{i_{*}}$

Let $H^{\prime}:=i_{*}\circ H\colon Y\times
I\to\operatorname{Emb}^{+}_{\operatorname{TOP}}(\partial X,-X\cup_{\partial
X}X)$ be the composition along the bottom row. Since $r$ satisfies the
homotopy lifting property, we obtain a lift $\widetilde{H}^{\prime}\colon
Y\times I\to\operatorname{Homeo}^{+}(-X\cup_{\partial X}X)$ of $H^{\prime}$
such that $r\circ\widetilde{H}^{\prime}=H^{\prime}$ and
$\widetilde{H}^{\prime}|_{Y\times\\{0\\}}=D\circ F$.

Now note that for all $(y,t)\in Y\times I$, we have that
$r\circ\widetilde{H}^{\prime}(y,t)=i\circ
H(y,t)\in\operatorname{Emb}^{+}_{\operatorname{TOP}}(\partial
X,-X\cup_{\partial X}X)$ sends $\partial X$ to $\partial
X\subseteq-X\cup_{\partial X}X$. Since $\partial X$ is separating in
$-X\cup_{\partial X}X$ and since $r\circ\widetilde{H}^{\prime}(y,0)=D\circ
F(y)\in\operatorname{Homeo}^{+}(-X\cup_{\partial X}X)$ preserves each copy of
$X$, we have that $\widetilde{H}^{\prime}(y,t)$ restricts to the desired lift
$\widetilde{H}^{\prime\prime}\colon Y\times I\to\operatorname{Homeo}^{+}(X)$.
Thus $r^{\prime}$ is a Serre fibration as desired. ∎

We are grateful to Dave Auckly for informing us of the following theorem and
proof, which resulted from conversations between him, Peter Kronheimer, and
Danny Ruberman.

######  Theorem A.3 (Auckly-Kronheimer-Ruberman).

Let $n\geq 4$. Let $X$ be a connected, oriented, smooth $n$-manifold with
boundary $\partial X\cong S^{n-1}\sqcup Y$ and let $\tau$ be a Dehn twist on a
sphere parallel to the $S^{n-1}$ boundary component. If there exists a class
$\sigma\in\pi_{2}(X)$ with $w_{2}(TX)[\sigma]\neq 0$, then $\tau$ is smoothly
isotopic to the identity relative to $S^{n-1}$.

###### Proof.

Let $\widehat{X}=X\cup_{S^{n-1}}-D^{n}$ and let
$\operatorname{Diffeo}^{+}(\widehat{X},D^{n})$ be the space of diffeomorphisms
that restrict to the identity on $D^{n}$. Consider the fibration
$\operatorname{Diffeo}^{+}(\widehat{X},D^{n})\to\operatorname{Diffeo}^{+}(\widehat{X})\to\operatorname{Emb}^{+}(D^{n},\widehat{X})$
of (1) and the connecting homomorphism in the associated long exact sequence
in homotopy groups

$\pi_{1}\operatorname{Emb}^{+}(D^{n},\widehat{X})\xrightarrow{\partial}\pi_{0}\operatorname{Diffeo}^{+}(\widehat{X},D^{n}).$

Write $\widehat{\tau}\in\operatorname{Diffeo}^{+}(\widehat{X})$ for the
extension of $\tau$ to $\widehat{X}$ by the identity on $D^{n}$. The
restriction map induces a homeomorphism
$\operatorname{Diffeo}^{+}(\widehat{X},D^{n})\cong\operatorname{Diffeo}^{+}(X,S^{n-1})$,
so the twist $\tau\in\operatorname{Diffeo}^{+}(X,S^{n-1})$ is isotopic to the
identity $\operatorname{Id}_{X}$ rel. $S^{n-1}$ if and only if
$[\widehat{\tau}]$ is trivial. We now aim to show this is the case.

Let $\varepsilon$ be the loop of self-diffeomorphisms of $D^{n}$ that is a
full rotation in the first two coordinates and fixes the remaining
coordinates. This determines a generator for
$\pi_{1}\operatorname{Diffeo}^{+}(D^{n})=\mathbb{Z}/2$ since $n\geq 3$. Write
$i\colon D^{n}\to\widehat{X}$ for the inclusion map. Define a loop
$\gamma^{\prime}:=i\circ\varepsilon$ in
$\operatorname{Emb}^{+}(D^{n},\widehat{X})$. Associated to the fibration (1),
is an action of loops in $\operatorname{Emb}^{+}(D^{n},\widehat{X})$ on
$\operatorname{Diffeo}^{+}(\widehat{X},D^{n})$; the connecting homomorphism
$\partial$ is defined by looking at the effect of this action on the basepoint
$\operatorname{Id}_{\widehat{X}}\in\operatorname{Diffeo}^{+}(\widehat{X},D^{n})$.
The effect of the action of $\gamma^{\prime}$ on
$\operatorname{Id}_{\widehat{X}}$ is a diffeomorphism in
$\operatorname{Diffeo}^{+}(\widehat{X},D^{n})$ isotopic to the Dehn twist
$\widehat{\tau}$. In other words, $[\gamma^{\prime}]$ maps to to
$[\widehat{\tau}]$ under the connecting map.

Write $\operatorname{Fr}^{+}(T\widehat{X})$ for the
$\operatorname{SO}(n)$-principal bundle of oriented orthonormal frames on
$T\widehat{X}$. Consider the map
$\operatorname{Emb}^{+}(D^{n},\widehat{X})\to\operatorname{Fr}^{+}(T\widehat{X})$,
given by sending $f\in\operatorname{Emb}^{+}(D^{n},\widehat{X})$ to the image
of the standard frame on $T_{0}D^{n}$ under the derivative $Df\colon
T_{0}D^{n}\to T_{f(0)}\widehat{X}$. This map determines a homotopy equivalence
$\operatorname{Emb}^{+}(D^{n},\widehat{X})\simeq\operatorname{Fr}^{+}(T\widehat{X})$.
Under this, $\gamma^{\prime}$ is sent to a loop $\gamma$ in
$\operatorname{Fr}^{+}(T\widehat{X})$ that is a full rotation in the first two
coordinates of the frame above $0\in D^{n}\subseteq\widehat{X}$ and fixes the
other coordinates. Consider the long exact sequence

$\cdots\to\pi_{2}(\widehat{X})\xrightarrow{\partial}\pi_{1}(\operatorname{SO}(n))\to\pi_{1}\operatorname{Fr}^{+}(T\widehat{X})\to\pi_{1}(\widehat{X})\to\cdots$

coming from the principal fibration. The loop $[\gamma]$ is clearly in the
image of $\pi_{1}(\operatorname{SO}(n))=\mathbb{Z}/2$ and is therefore trivial
if there exists a class $\widehat{\sigma}\in\pi_{2}(\widehat{X})$ such that
$\partial\widehat{\sigma}\neq 0$. Write $h\colon\pi_{2}(\widehat{X})\to
H_{2}(\widehat{X})$ for the Hurewicz map and $\operatorname{ev}\colon
H^{2}(\widehat{X};\pi_{1}(\operatorname{SO}(n)))\to\operatorname{Hom}(H_{2}(\widehat{X}),\pi_{1}(\operatorname{SO}(n)))$
for the evaluation map. Comparing the definition of the connecting
homomorphism $\partial$ with the interpretation of $w_{2}$ as the obstruction
to reducing the structure group of $\operatorname{Fr}^{+}(T\widehat{X})$ to
$\operatorname{Spin}(n)$, we have $\operatorname{ev}(w_{2}(T\widehat{X}))\circ
h=\partial$. The inclusion $i\colon X\to\widehat{X}$ induces an isomorphism on
$\pi_{2}$, and we define $\widehat{\sigma}:=(i_{*})^{-1}(\sigma)$. Then

$\partial\widehat{\sigma}=(\operatorname{ev}(w_{2}(T\widehat{X}))\circ
h)((i_{*})^{-1}(\sigma))=i^{*}(\operatorname{ev}(w_{2}(T\widehat{X}))\circ
h)(\sigma)=w_{2}(TX)[\sigma]\neq 0.$

Therefore $[\gamma]=0\in\pi_{1}\operatorname{Fr}^{+}(T\widehat{X})$ and so
$[\gamma^{\prime}]=0\in\pi_{1}\operatorname{Emb}^{+}(D^{n},\widehat{X})$.
Since
$\partial[\gamma^{\prime}]=[\widehat{\tau}]\in\pi_{0}\operatorname{Diffeo}^{+}(\widehat{X},D^{n})$,
which in turn maps to $\tau$ under the isomorphism
$\pi_{0}\operatorname{Diffeo}^{+}(\widehat{X},D^{n})\cong\pi_{0}\operatorname{Diffeo}^{+}(X,S^{n-1})$.
This completes the proof. ∎

######  Remark A.4.

We have stated Theorem A.3 with the assumption $n\geq 4$ because for
$n=1,2,3$, the statement is vacuous. When $n=1,2$, this is because
$\pi_{2}(X)=0$. When $n=3$, consider a spherical class $\sigma\colon S^{2}\to
X$ and observe that $w_{2}(TX)[\sigma]=w_{2}(\sigma^{*}TX)\in\mathbb{Z}/2$. We
may assume $\sigma$ is represented by an immersion, so that $\sigma^{*}TX\cong
TS^{2}\oplus\nu({\sigma(S^{2})\subseteq X})$. We compute that
$w_{2}(\sigma^{*}TX)=w_{2}(TS^{2}\oplus\nu\sigma)=w_{2}(TS^{2})+w_{1}(TS^{2})w_{1}(\nu\sigma)+w_{2}(\nu\sigma)$.
Since $\nu\sigma$ is a line bundle, the last term vanishes, while the first
two terms vanish because $TS^{2}$ is stably trivial. Hence
$w_{2}(TX)[\sigma]=0$ for all spherical classes when $n=3$.

######  Corollary A.5.

Let $n\geq 4$. Let $X$ be a simply connected, oriented, smooth manifold with
boundary $\partial X\cong S^{n-1}\sqcup M$ and let $\tau$ be a Dehn twist on a
sphere parallel to the $S^{n-1}$ boundary component. If $X$ is non-spin, then
$\tau$ is smoothly isotopic to the identity map, relative to $S^{n-1}$.

###### Proof.

For a simply connected manifold, the Hurewicz map determines an isomorphism
$\pi_{2}(X)\otimes\mathbb{Z}/2\cong H_{2}(X;\mathbb{Z}/2)$. For a non-spin
manifold, $w_{2}(TX)$ is nontrivial. Hence there must exist a homology class
$x\in H_{2}(X;\mathbb{Z}/2)$ such that $w_{2}(TX)[\sigma]\neq 0$. The result
now follows immediately from Theorem A.3. ∎

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