# Non-trivial smooth families of $K3$ surfaces

David Baraglia  School of Mathematical Sciences, The University of Adelaide,
Adelaide SA 5005, Australia<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a complex $K3$ surface, $Diff(X)$ the group of diffeomorphisms of
$X$ and $Diff(X)_{0}$ the identity component. We prove that the fundamental
group of $Diff(X)_{0}$ contains a free abelian group of countably infinite
rank as a direct summand. The summand is detected using families
Seiberg–Witten invariants. The moduli space of Einstein metrics on $X$ is used
as a key ingredient in the proof.

##  1\. Introduction

There is considerable interest in understanding the topology of diffeomorphism
groups of $4$-manifolds. While much remains unknown there has been some recent
progress.

  * •

Ruberman gave examples of simply-connected smooth $4$-manifolds for which
$\pi_{0}(Diff(X))\to\pi_{0}(Homeo(X))$ is not injective [15, 16].

  * •

Watanabe constructed many non-trivial homotopy classes in $Diff(S^{4})$,
thereby disproving the $4$-dimensional Smale conjecture [19].

  * •

Baraglia-Konno showed that $\pi_{1}(Diff(K3))\to\pi_{1}(Homeo(K3))$ is not
surjective [4].

  * •

Smirnov showed that if $X$ is a hypersurface in $\mathbb{CP}^{3}$ of degree
$d\neq 1,4$, then $\pi_{1}(Diff(X))$ is non-trivial. Note that this result
excludes $K3$, which corresponds to hypersurfaces of degree $d=4$ [17].

The main result of this paper is the following.

######  Theorem 1.1.

Let $X$ be a $K3$ surface. Then $\pi_{1}(Diff(X)_{0})$ contains a free abelian
group of countably infinite rank as a direct summand.

The direct summand in the above theorem is detected using families
Seiberg–Witten invariants. The families Seiberg–Witten invariants were
originally defined in [12]. In this paper we consider a reformulation of the
Seiberg–Witten invariants which we now outline. Let $X$ be a compact, oriented
smooth $4$-manifold with $b^{+}(X)>1$. Let $\mathfrak{s}$ be a spinc-structure
on $X$ and let

$d(X,\mathfrak{s})=\frac{c_{1}(\mathfrak{s})^{2}-\sigma(X)}{4}-1+b_{1}(X)-b^{+}(X)$

be the expected dimension of the Seiberg–Witten moduli space. If
$d(X,\mathfrak{s})\leq-2$, then we construct a map

$sw_{\mathfrak{s}}:\pi_{-d(X,\mathfrak{s})-1}(Diff(X)_{0})\to\mathbb{Z}.$

The definition, roughly, is as follows. Let
$f\in\pi_{-d(X,\mathfrak{s})-1}(Diff(X)_{0})$. Then by the clutching
construction, $f$ defines a family $E_{f}\to S^{-d(X,\mathfrak{s})}$ over the
sphere with fibres diffeomorphic to $X$. The moduli space of solutions to the
Seiberg–Witten equations on the family $E_{f}$ with spinc-structure
$\mathfrak{s}$ is compact and has expected dimension
$d(X,\mathfrak{s})-d(X,\mathfrak{s})=0$. For a generic perturbation the
families moduli space is a compact oriented $0$-manifold and
$sw_{\mathfrak{s}}(f)$ is defined as a signed count of the points of this
moduli space. If $b^{+}(X)\leq-d(X,\mathfrak{s})+1$, then one has to deal with
wall crossing phenomena. However, we show that in the above situation there is
a canonically defined chamber and we take $sw_{\mathfrak{s}}(f)$ to be the
Seiberg–Witten invariant defined with respect to this chamber. This subtlety
is crucial to this paper, since we will be concerned with the case
$b^{+}(X)=3$ and $d(X,\mathfrak{s})=-2$.

We prove a number of results concerning the invariants $sw_{\mathfrak{s}}$.

######  Theorem 1.2.

Let $X$ be a compact, oriented, smooth $4$-manifold with $b^{+}(X)>1$. Then
for each spinc-structure with $d(X,\mathfrak{s})=-(n+1)\leq-2$, the map

$sw_{\mathfrak{s}}:\pi_{n}(Diff(X)_{0})\to\mathbb{Z}$

is a group homomorphism.

######  Theorem 1.3.

Assume that $b_{1}(X)=0$. Then for any given $f\in\pi_{n}(Diff(X)_{0})$,
$sw_{\mathfrak{s}}(f)$ is non-zero for only finitely many spinc-structures
with $d(X,\mathfrak{s})=-(n+1)$.

Theorem 1.3 is essentially a consequence of the compactness properties of the
Seiberg–Witten equations. However there is a subtlety due to the chamber
structure and wall crossing that requires some non-trivial arguments to
overcome. From these two theorems it follows that (for each $n\geq 1$) we can
put the Seiberg–Witten invariants together into a single homomorphism

$sw:\pi_{n}(Diff(X)_{0})\to\bigoplus_{\mathfrak{s}|d(X,\mathfrak{s})=-(n+1)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbb{Z},\quad
x\mapsto\bigoplus_{\mathfrak{s}}sw_{\mathfrak{s}}(x).$

Now let $X$ be a $K3$ surface. Let

$\Delta=\\{\alpha\in H^{2}(X;\mathbb{Z})\;|\;\alpha^{2}=-2\\}$

be the “roots” of $X$. For each $\alpha\in\Delta$ we get a unique spinc-
structure $\mathfrak{s}_{\alpha}$ characterised by
$c_{1}(\mathfrak{s}_{\alpha})=2\alpha$. Then $d(X,\mathfrak{s}_{\alpha})=-2$
and so we have a homomorphism $sw_{\alpha}:\pi_{1}(Diff(X))\to\mathbb{Z}$.

Choose an element $v\in H^{2}(X;\mathbb{R})$ such that $\langle
v,\delta\rangle\neq 0$ for all $\delta\in\Delta$ and define
$\Delta^{\pm}=\\{\delta\in\Delta\;|\;\pm\langle v,\delta\rangle>0\\}$. Then

$\Delta=\Delta^{+}\cup\Delta^{-}$

and $\delta\in\Delta^{+}$ if and only if $-\delta\in\Delta^{-}$. The reason
for splitting up $\Delta$ this way is that the invariants
$sw_{\mathfrak{\alpha}}$ and $sw_{\mathfrak{-\alpha}}$ are related to one
another by the charge conjugation symmetry of the Seiberg–Witten equations. In
fact, $sw_{\mathfrak{\alpha}}=-sw_{\mathfrak{-\alpha}}$ (see Proposition 2.8).

In §3 we recall the construction of the moduli space $T_{Ein}$ of Einstein
metrics on $X$, which is to be thought of as an analogue of Teichmuller space
for $K3$ surfaces. Over $T_{Ein}$ is a universal family $E_{Ein}\to T_{Ein}$.
For each $\delta\in\Delta^{+}$, we construct a homotopy class of map
$g_{\delta}:S^{2}\to T_{Ein}$. Let $E_{\delta}\to S^{2}$ be the family over
$S^{2}$ obtained by pulling back the universal family under $g_{\delta}$. We
show that $E_{\delta}$ is associated to an element
$h_{\delta}\in\pi_{1}(Diff(X)_{0})$ via the clutching construction. Then,
using the geometry of $T_{Ein}$, we compute the families Seiberg–Witten
invariant of $E_{\delta}$. This gives the following result.

######  Theorem 1.4.

Let $\alpha,\delta\in\Delta^{+}$. Then

$sw_{\alpha}(h_{\delta})=\begin{cases}1&\text{if }\alpha=\delta,\\\
0&\text{otherwise}.\end{cases}$

Our main theorem follows directly from this.

A brief outline of the paper is as follows. In §2 we recall the construction
of the families Seiberg–Witten invariants. We then show the invariants can be
reformulated as maps $sw_{\mathfrak{s}}:\pi_{n}(Diff(X)_{0})\to\mathbb{Z}$ and
prove several properties of these invariants, in particular Theorems 2.6 and
2.9. In §3 we specialise to the case that $X$ is a $K3$ surface. We construct
the family $E_{Ein}\to T_{Ein}$ over the Teichmuller space $T_{Ein}$ and use
this to contruct classes $h_{\delta}\in\pi_{1}(Diff(X)_{0})$. We then compute
the Seiberg–Witten invariants of these classes and our main theorem follows.

##  2\. The families Seiberg–Witten invariant revisited

In this section we will recall the definition of the families Seiberg–Witten
invariant. We will also show that the definition of the invariant can be
extended to situations where wall-crossing phenomena is present. We show that
under certain conditions a distinguished chamber exists, hence we can still
obtain a well-defined invariant.

Our approach to the families Seiberg–Witten invariant follows [12] but with
some additional modifications as in [2]. Let $X$ be a compact smooth oriented
$4$-manifold and let $B$ be a compact smooth manifold. Suppose we have a
smooth fibrewise oriented fibre bundle $\pi:E\to B$ whose fibres are
diffeomorphic to $X$. Such a fibre bundle will be called a smooth family over
$B$ with fibres diffeomorphic to $X$. We assume throughout that $B$ is
connected. Choose a basepoint $p\in B$ and a diffeomorphism $X_{p}\cong X$,
where $X_{p}=\pi^{-1}(p)$ denotes the fibre of $E$ over $p$. Then
$\pi_{1}(B,p)$ acts by monodromy on the set of spinc-structures on $X$.
Suppose that $\mathfrak{s}$ is a monodromy invariant spinc-structure on $X$.
Then by monodromy invariance, $\mathfrak{s}$ can be uniquely extended to a
continuously varying family of spinc-structures
$\tilde{\mathfrak{s}}=\\{\mathfrak{s}_{b}\\}_{b\in B}$ on the fibres of $E$
such that $\tilde{\mathfrak{s}}|_{X_{p}}\cong\mathfrak{s}$. Note that the
existence of the continuous family $\tilde{\mathfrak{s}}$ is in general a
weaker condition than requiring the existence of a spinc-structure on the
vertical tangent bundle $T(E/B)=Ker(\pi_{*})$. However, as explained in [1],
[2], the existence of $\tilde{\mathfrak{s}}$ is sufficient to contruct a
families Seiberg–Witten moduli space.

Let

$d(X,\mathfrak{s})=\frac{c_{1}(\mathfrak{s})^{2}-\sigma(X)}{4}-1-b^{+}(X)+b_{1}(X)$

be the virtual dimension of the ordinary Seiberg–Witten moduli space of $X$.
Let $g=\\{g_{b}\\}_{b\in B}$ be a smoothly varying family of metrics on the
fibres of $E$. Equivalently, $g$ is a metric on the vertical tangent bundle
$T(E/B)$. Then we define $\mathcal{H}^{+}_{g}(X)$ to be the vector bundle on
$B$ whose fibre over $b\in B$ is the space $H^{+}_{g_{b}}(X_{b})$ of
$g_{b}$-self-dual harmonic $2$-forms. By a families perturbation $\eta$ we
mean a smoothly varying family $\eta=\\{\eta_{b}\\}_{b\in B}$ of real
$2$-forms on the fibres of $E$, such that $\eta_{b}$ is $g_{b}$-self-dual. Let
$[\eta_{b}]\in H^{+}_{g_{b}}(X_{b})$ denote the $L^{2}$-othogonal projection
of $\eta_{b}$ to the space of self-dual harmonic forms (using the
$L^{2}$-metric defined by $g_{b}$). The map $b\mapsto[\eta_{b}]$ defines a
section of $\mathcal{H}^{+}_{g}(X)$, which we denote by $[\eta]$.

Recall that the Seiberg–Witten equations for $(X_{b},\mathfrak{s}_{b},g_{b})$
with perturbation $\eta_{b}$ are:

$\displaystyle D_{A}\psi$ $\displaystyle=0,$ $\displaystyle
F^{+}_{A}+i\eta_{b}$ $\displaystyle=\sigma(\psi),$

where $A$ is a spinc-connection, $\psi$ is a positive spinor for the spinc-
structure $\mathfrak{s}_{b}$ and $\sigma(\psi)$ denotes the imaginary the
self-dual $2$-form corresponding to the trace-free part of
$\psi^{*}\otimes\psi$ under Clifford multiplication. Let
$w:B\to\mathcal{H}^{+}_{g}(X)$ be the section of $\mathcal{H}^{+}_{g}(X)$
sending $b$ to $2\pi(c_{1}(\mathfrak{s}_{b}))^{+_{g_{b}}}$, the orthogonal
projection of $2\pi c_{1}(\mathfrak{s}_{b})$ to $H^{+}_{g_{b}}(X_{b})$ using
the $L^{2}$-metric defined by $g_{b}$. Then the $\eta_{b}$-perturbed
Seiberg–Witten equations for $(X_{b},\mathfrak{s}_{b},g_{b})$ admits reducible
solutions if and only if $[\eta_{b}]=w$. We refer to $w$ as the “wall” and we
say that the families perturbation $\eta$ does not lie on the wall if for all
$b\in B$, we have $[\eta_{b}]\neq w_{b}$.

We define a chamber (of the families Seiberg–Witten equations) for
$(E,\mathfrak{s})$ to be a connected component of the space of pairs
$(g,\eta)$, where $g$ is a family of metrics and $\eta$ is a family of
perturbations not lying on the wall. In general, there are obstructions to the
existence of chambers. For instance, if $b^{+}(X)=0$, then there does not
exist a chamber. On the other hand, if $b^{+}(X)>{\rm dim}(B)+1$, then there
exists a unique chamber.

Let $\mathcal{C}$ be a chamber of $(E,\mathfrak{s})$. Then for a sufficiently
generic element $(g,\eta)\in\mathcal{C}$, the moduli space
$\mathcal{M}(E,\mathfrak{s},g,\eta)$ of gauge equivalence classes of solutions
to the Seiberg–Witten equations on the fibres of $E$ (with respect to the
spinc-structure $\tilde{\mathfrak{s}}$, metric $g$ and perturbation $\eta$) is
a smooth, compact manifold of dimension $d(X,\mathfrak{s}_{X})+{\rm dim}(B)$
(or is empty if this number is negative). Recall that a homology orientation
for $X$ is a choice of orientation of $H^{+}(X)\oplus H^{1}(X;\mathbb{R})$
(here $H^{+}(X)$ denotes the space of harmonic self-dual $2$-forms with
respect to some metric on $X$. It is straightforward to see that the homology
orientations for different choices of metrics can be canonically identified
with one another). We say that a homology orientation is monodromy invariant
if it extends to a continuously varying orientation on the family
$\\{H^{+}_{g_{b}}(X_{b})\oplus H^{1}(X_{b};\mathbb{R})\\}_{b\in B}$.

Let $\pi:\mathcal{M}(E,\mathfrak{s},g,\eta)\to B$ be the projection to $B$. A
monodromy invariant homology orientation defines an orientation on
$T_{\mathcal{M}(E,\mathfrak{s},g,\eta)}\oplus\pi^{*}(TB)$, hence a Gysin
homomorphism

$\pi_{*}:H^{j}(\mathcal{M}(E,\mathfrak{s},g,\eta);\mathbb{Z})\to
H^{j-d(X,\mathfrak{s})}(B;\mathbb{Z}).$

We define the families Seberg–Witten invariant
$SW(E,\mathfrak{s},\mathcal{C},\mathfrak{o})\in
H^{-d(X,\mathfrak{s})}(B;\mathbb{Z})$ of $E$ with respect to the monodromy
invariant spinc-structure $\mathfrak{s}$, the chamber $\mathcal{C}$ and
monodromy invariant homology orientation $\mathfrak{o}$ to be

$SW(E,\mathfrak{s},\mathcal{C},\mathfrak{o})=\pi_{*}(1)\in
H^{-d(X,\mathfrak{s})}(B;\mathbb{Z}).$

The fact that $SW(E,\mathfrak{s},\mathcal{C},\mathfrak{o})$ depends only on
the chamber $\mathcal{C}$ and not on the particular choice of pair
$(g,\eta)\in\mathcal{C}$ follows by much the same argument as in the
unparametrised case. A generic path between pairs
$(g,\eta),(g^{\prime},\eta^{\prime})$ determines a cobordism (relative $B$) of
the moduli spaces $\mathcal{M}(E,\mathfrak{s},g,\eta)$ and
$\mathcal{M}(E,\mathfrak{s},g^{\prime},\eta^{\prime})$.

Let $\pi:E\to B$ be a smooth family over $B$ with fibres diffeomorphic to $X$.
Let $\mathcal{H}^{2}(X)$ denote the local system over $B$ whose fibre over
$b\in B$ is $\mathcal{H}^{2}(X)_{b}=H^{2}(X_{b};\mathbb{R})$. Assume that $B$
is simply-connected and that $b^{+}(X)>1$. Choose a basepoint $p\in B$. Then
parallel translation defines a trivialisation

$\tau:\mathcal{H}^{2}(X)\to B\times H^{2}(X_{p};\mathbb{R}).$

Let $H\subseteq H^{2}(X_{p};\mathbb{R})$ be a maximal positive definite
subspace with respect to the intersection form and let $H^{\perp}$ denote the
orthogonal complement of $H$ (with respect to the intersection form). This
defines a decomposition $H^{2}(X_{p};\mathbb{R})\cong H\oplus H^{\perp}$. Let
$\rho_{H}:H^{2}(X_{p};\mathbb{R})\to H$ denote the projection to the first
factor.

Choose a smoothly varying family of metrics $g=\\{g_{b}\\}$ and let
$\mathcal{H}^{+}_{g}(X)$ be defined as before. Let
$\iota_{g}:\mathcal{H}^{+}_{g}(X)\to\mathcal{H}^{2}(X)$ be the inclusion. Let
$pr_{B}:B\times H^{2}(X_{p};\mathbb{R})\to B$ be the projection to $B$. Then
the composition

$\varphi_{g}=(pr_{B}\times\rho_{H})\circ\tau\circ\iota_{g}:\mathcal{H}^{+}_{g}(X)\to
B\times H$

is an isomorphism of vector bundles. This follows since
$\tau(\iota_{g}(\mathcal{H}^{+}_{g}(X)))$ is a positive definite subbundle of
$B\times H^{2}(X_{p};\mathbb{R})$, so meets the negative definite subbundle
$B\times H^{\perp}$ in the zero section.

Let $\mathfrak{s}$ be a spinc-structure on $X$. Since $B$ is simply-connected,
$\mathfrak{s}$ is automatically monodromy invariant and so continuously
extends to the fibres of $E$.

Let $w:B\to\mathcal{H}^{+}_{g}(X)$ be the wall with respect to the spinc-
structure $\mathfrak{s}$. Then $\varphi_{g}(w)$ is a section of the trivial
bundle $B\times H$. Let

$R_{g}=\sup_{b\in B}||\varphi_{g}(w_{b})||_{H}$

where $||\;||_{H}$ is the norm on $H$ induced by the restriction to $H$ of the
intersection form on $H^{2}(X_{p};\mathbb{R})$. Since $B$ is compact, $R_{g}$
is finite. Since $b^{+}(X)>1$, $H$ is a non-zero vector space and hence there
exist elements of arbitrarily large norm. Let $v$ be any element of $H$ with
$||v||_{H}>R_{g}$. Then the constant section $b\mapsto b\times v$ is disjoint
from $\varphi_{g}(w)$. Therefore, the section $v_{g}=\varphi_{g}^{-1}(v)$ of
$\mathcal{H}^{+}_{g}(X)$ is disjoint from the wall $w$ and hence defines a
chamber, depending only on $g$ and $v$ which we will denote by
$\mathcal{C}(g,v)$.

######  Lemma 2.1.

The chamber $\mathcal{C}(g,v)$ does not depend on the choice of the pair
$(g,v)$.

###### Proof.

First we show that for fixed $g$, the chamber $\mathcal{C}(g,v)$ does not
depend on the choice of $v$. Let $R_{g}=\sup_{b\in
B}||\varphi_{g}(w_{b})||_{H}$ be defined as before and let $v,v^{\prime}$ be
any two elements of $H$ with $||v||_{H},||v^{\prime}||_{H}>R_{g}$. The space
$\\{x\in H\;|\;||x||_{H}>R_{g}\\}$ is homotopic to a sphere of dimension
$b^{+}(X)-1$ and is therefore connected, since we are assuming that
$b^{+}(X)>1$. Therefore we can find a continuous path
$\\{v_{t}\\}_{t\in[0,1]}$ in $\\{x\in H\;|\;||x||_{H}>R_{g}\\}$ joining $v$ to
$v^{\prime}$. It follows that $(g,(v_{t})_{g})\in\mathcal{C}(g,v)$ for all
$t\in[0,1]$. Hence $(g,(v^{\prime})_{g})\in\mathcal{C}(g,v)$ and
$\mathcal{C}(g,v)=\mathcal{C}(g,v^{\prime})$.

Now let $g,g^{\prime}$ be two different families of metrics. We will show that
there exists a $v\in H$ for which
$\mathcal{C}(g,v)=\mathcal{C}(g^{\prime},v)$. Together with the above shown
independence of $\mathcal{C}(g,v)$ on $v$, this will show that
$\mathcal{C}(g,v)$ does not depend on the choice of pair $(g,v)$.

Choose a continuous path $\\{g_{t}\\}_{t\in[0,1]}$ of families of metrics from
$g$ to $g^{\prime}$. Let

$R=\sup_{b\in B,t\in[0,1]}||\varphi_{g_{t}}(w_{b})||_{H}.$

Compactness of $B\times[0,1]$ implies that $R$ is finite. Now choose $v\in H$
such that $||v||_{H}>R$. Then $||v||_{H}>\sup_{b\in
B}||\varphi_{g_{t}}(w_{b})||_{H}$ for each $t\in[0,1]$. It follows that the
pair $(g_{t},v)$ defines the same chamber for all $t\in[0,1]$. Hence
$\mathcal{C}(g,v)=\mathcal{C}(g^{\prime},v)$. ∎

######  Definition 2.2.

Let $\pi:E\to B$ be a smooth family over $B$ with fibres diffeomorphic to $X$
and let $\mathfrak{s}$ be a spinc-structure on $X$. Assume that $B$ is simply-
connected and that $b^{+}(X)>1$. For any pair $(g,v)$ with $v\in\\{x\in
H\;|\;||x||_{H}>R_{g}\\}$, we let $\mathcal{C}_{0}(E,\mathfrak{s})$ denote the
chamber containing $(g,v_{g})$. By Lemma 2.1, we see that
$\mathcal{C}_{0}(E,\mathfrak{s})$ does not depend on the choice of the pair
$(g,v)$. Furthermore, it is clear that $\mathcal{C}_{0}(E,\mathfrak{s})$ does
not depend on the choice of maximal positive definite subspace $H\subseteq
H^{2}(X;\mathbb{R})$, because the space of all such subspaces is connected. We
call $\mathcal{C}_{0}(E,\mathfrak{s})$ the canonical chamber for
$(E,\mathfrak{s})$.

######  Definition 2.3.

Let $\pi:E\to B$ be a smooth family over $B$ with fibres diffeomorphic to $X$.
Assume that $B$ is simply-connected and that $b^{+}(X)>1$. Let $\mathfrak{s}$
be a spinc-structure on $X$ and let $\mathfrak{o}$ be a homology orientation
for $X$. Then we define the (canonical) families Seiberg–Witten invariant
$SW(E,\mathfrak{s},\mathfrak{o})$ of $(E,\mathfrak{s},\mathfrak{o})$ to be the
families Seiberg–Witten invariant of $(E,\mathfrak{s},\mathfrak{o})$ defined
using the canonical chamber:

$SW(E,\mathfrak{s},\mathfrak{o})=SW(E,\mathfrak{s},\mathcal{C}_{0}(E,\mathfrak{s}),\mathfrak{o})\in
H^{-d(X,\mathfrak{s})}(B;\mathbb{Z}).$

Let $X$ be a compact, oriented, smooth $4$-manifold with $b^{+}(X)>1$. Let
$Diff(X)$ the group of orientation preserving diffeomorphisms of $X$ with the
$\mathcal{C}^{\infty}$-topology and let $Diff(X)_{0}$ be the identity
component. Let $n>0$ be a positive integer. Consider an element
$f\in\pi_{n}(Diff(X)_{0})$. Using the clutching construction, $f$ defines a
topological fibre bundle $E_{f}\to S^{n+1}$ over $S^{n+1}$ with fibres
homeomorphic to $X$ and structure group $Diff(X)_{0}$. From the main theorem
of [13], it follows that $E_{f}$ can be made into a smooth fibre bundle with
fibres diffeomorphic to $X$ in a unique way. Let $\mathfrak{s}$ be a spinc-
structure on $X$ and $\mathfrak{o}$ a homology orientation. Since $n>0$,
$S^{n+1}$ is simply-connected and $b^{+}(X)>1$, the families Seiberg–Witten
invariant

$SW(E_{f},\mathfrak{s},\mathfrak{o})\in
H^{-d(X,\mathfrak{s})}(S^{n+1};\mathbb{Z})$

is defined. If $d(X,\mathfrak{s})=-(n+1)$, then we can evaluate
$SW(E_{f},\mathfrak{s},\mathfrak{o})$ against the fundamental class of
$S^{n+1}$ to obtain an integer invariant.

######  Definition 2.4.

Let $X$ be a compact, oriented, smooth $4$-manifold with $b^{+}(X)>1$. Let
$\mathfrak{s}$ be a spinc-structure such that $d(X,\mathfrak{s})=-(n+1)$ for
some $n\geq 0$. We define

$sw_{\mathfrak{s}}:\pi_{n}(Diff(X)_{0})\to\mathbb{Z}$

by setting

$sw_{\mathfrak{s}}(f)=\int_{S^{n+1}}SW(E_{r},\mathfrak{s},\mathfrak{o}),$

where we have chosen a homology orientation $\mathfrak{o}$ and an orientation
on $S^{n+1}$. We have omitted from our notation the dependence of
$sw_{\mathfrak{s}}$ on the choice of these orientations. A change in either
orientation has the effect of changing $sw_{\mathfrak{s}}$ by an overall sign.

######  Remark 2.5.

The invariant $sw_{\mathfrak{s}}(f)\in\mathbb{Z}$ can be interpreted as
follows. Choose a generic pair
$(g,\eta)\in\mathcal{C}_{0}(E_{f},\mathfrak{s})$. Then the moduli space
$\mathcal{M}(E,\mathfrak{s},g,\eta)$ is a compact, oriented $0$-manifold and
$sw_{\mathfrak{s}}(f)$ is simply the number of points of
$\mathcal{M}(E_{f},\mathfrak{s},g,\eta)$, counted with sign.

######  Theorem 2.6.

Let $X$ be a compact, oriented, smooth $4$-manifold with $b^{+}(X)>1$. Then
for each spinc-structure with $d(X,\mathfrak{s})=-(n+1)\leq-2$, the map

$sw_{\mathfrak{s}}:\pi_{n}(Diff(X)_{0})\to\mathbb{Z}$

is a group homomorphism.

###### Proof.

Let $f\in\pi_{n}(Diff(X)_{0})$ and let $E_{f}\to S^{n+1}$ be the corresponding
family built from the clutching construction. Choose a generic pair
$(g,\eta)\in\mathcal{C}_{0}(E_{f},\mathfrak{s})$. The moduli space
$\mathcal{M}(E,\mathfrak{s},g,\eta)$ is a finite set of points. Let
$p_{1},\dots,p_{m}\in S^{n+1}$ be the finitely many points over which
$\mathcal{M}(E_{f},\mathfrak{s},g,\eta)$ lies. Choose a point $p\in S^{n+1}$
and an open disc $D\subset S^{n+1}$ around $p$ such that $D$ is disjoint from
$p_{1},\dots,p_{m}$. Using cutoff functions it is possible to construct a
smooth map $\psi:S^{n+1}\to S^{n+1}$ such that $\psi(D)=\\{p\\}$ and
$\psi:S^{n+1}\setminus D\to S^{n+1}\setminus\\{p\\}$ is a diffeomorphism. Now
consider the pullback $\psi^{*}(E_{f})$ of $E_{f}$ under $\psi$. The
conditions on $\psi$ ensures that it has degree $1$ as a map of $S^{n+1}$ to
itself. Hence $\psi$ is homotopic to the identity. It follows that
$\psi^{*}(E_{f})$ is isomorphic to $E_{f}$ as topological fibre bundles over
$S^{n+1}$ with structure group $Diff(X)$. The main theorem of [13] then
implies that $\psi^{*}(E_{f})$ and $E_{f}$ are isomorphic as smooth fibre
bundles. The pullback $(\psi^{*}(g),\psi^{*}(\eta))$ is a generic pair in
$\mathcal{C}_{0}(\psi^{*}(E_{f}),\mathfrak{s})$ and the moduli space
$\mathcal{M}(\psi^{*}(E_{f}),\mathfrak{s},\psi^{*}(g),\psi^{*}(\eta))$ is
obviously obtained by pulling back $\mathcal{M}(E_{f},\mathfrak{s},g,\eta)$ by
$\psi$. Let $X_{p}$ denote the fibre of $E_{f}$ over $p$ and fix a
diffeomorphism $X_{p}\cong X$. Since $\psi$ takes the constant value $p$ on
$D$, the restriction of $\psi^{*}(E_{f})$ to $D$ is the constant family
$\psi^{*}(E_{f})|_{D}\cong D\times X_{p}\cong D\times X$. Under this
trivialisation of $\psi^{*}(E_{f})|_{D}$ we have that
$\psi^{*}(g),\psi^{*}(\eta)$ get sent to the constant pair $(g_{p},\eta_{p})$.

Now let $f^{\prime}$ be another element of $\pi_{n}(Diff(X)_{0})$ and let
$E_{f^{\prime}}\to S^{n+1}$ be the family corresponding to $f^{\prime}$.
Choose a generic pair
$(g^{\prime},\eta^{\prime})\in\mathcal{C}_{0}(E_{f^{\prime}},\mathfrak{s})$.
After possibly rotating $S^{n+1}$ we can assume that no point of the moduli
space $\mathcal{M}(E_{f^{\prime}},\mathfrak{s},g^{\prime},\eta^{\prime})$ lies
over $p$. We can further assume that $D$ was chosen small enough that no point
of the moduli space
$\mathcal{M}(E_{f^{\prime}},\mathfrak{s},g^{\prime},\eta^{\prime})$ lies over
$D$. We then obtain the pullback family $\psi^{*}(E_{f^{\prime}})$ with
generic pair $(\psi^{*}(g^{\prime}),\psi^{*}(\eta^{\prime}))$. Moreover, we
have a trivialisation of $\psi^{*}(E_{f^{\prime}})|_{D}$ in which
$\psi^{*}(g^{\prime})$, $\psi^{*}(\eta^{\prime})$ are sent to the constant
pair $(g^{\prime}_{p^{\prime}},\eta^{\prime}_{p^{\prime}})$.

Let $D^{\prime}\subset D$ be a smaller open disc around $p$ so that the
complement $D\setminus D^{\prime}$ is diffeomorphic to $S^{3}\times(0,1)$.
Attach two copies of $S^{n+1}\setminus D^{\prime}$ to each other using a neck
$S^{3}\times[0,1]$ (where we have identified the boundary of $D^{\prime}$ with
$S^{3}$). The resulting space

$(S^{n+1}\setminus
D^{\prime})\cup_{S^{3}}(S^{3}\times[0,1])\cup_{S^{3}}(S^{n+1}\setminus
D^{\prime})$

is just the connected sum of two copies of $S^{n+1}$, which of course is just
another copy of $S^{n+1}$. Using the trivialisation $\psi^{*}(E_{f})|_{D}\cong
D\times X$ and $\psi^{*}(E_{f^{\prime}})|_{D}\cong D\times X$, we can attach
$\psi^{*}(E_{f})|_{S^{n+1}\setminus D^{\prime}}$ to
$\psi^{*}(E_{f^{\prime}})|_{S^{n+1}\setminus D^{\prime}}$ by taking a constant
family $(S^{3}\times[0,1])\times X$ along the neck. Let $E$ denote the
resulting family. Since $\psi^{*}(E_{f})$ and $\psi^{*}(E_{f^{\prime}})$ are
isomorphic to $E_{f}$ and $E_{f^{\prime}}$, it is clear that $E$ is isomorphic
to the family obtained by applying the clutching construction to
$f+f^{\prime}$, where $+$ denotes the group operation on
$\pi_{n}(Diff(X)_{0})$.

Next, since $d(X,\mathfrak{s})=-(n+1)\leq-2$, it follows the moduli space of
solutions to the Seiberg–Witten equations for a $1$-parameter family with
fibres $(X,\mathfrak{s})$ has expected dimension $d(X,\mathfrak{s})+1=-n<0$.
Therefore, for a generic path $(g_{t},\eta_{t})$ from $(g_{p},\eta_{p})$ to
$(g^{\prime}_{p},\eta^{\prime}_{p})$, there are no solutions to the
Seibeg–Witten equations for $(X,\mathfrak{s},g_{t},\eta_{t})$. Now we define a
pair $(\tilde{g},\tilde{\eta})$ for the family $E$ as follows. Restricted to
$\psi^{*}(E_{f})|_{S^{n+1}\setminus D^{\prime}}$, we take the pair to be
$(\psi^{*}(g),\psi^{*}(\eta))$. Restricted to
$\psi^{*}(E_{f^{\prime}})|_{S^{n+1}\setminus D^{\prime}}$, we take the pair to
be $(\psi^{*}(g^{\prime}),\psi^{*}(\eta^{\prime}))$. Restricted to the
constant family on the neck $S^{3}\times[0,1]$, we take the pair to be
$(g_{t},\eta_{t})$, where $t\in[0,1]$ is the coordinate for the $[0,1]$ factor
of the neck. Since there are no solutions to the Seiberg–Witten equations for
$(g_{t},\eta_{t})$, it is clear that the moduli space for
$(E,\mathfrak{s},\tilde{g},\tilde{\eta})$ is just the disjoint union

$\mathcal{M}(\psi^{*}(E_{f}),\mathfrak{s},\psi^{*}(g),\psi^{*}(\eta))\cup\mathcal{M}(\psi^{*}(E_{f^{\prime}}),\mathfrak{s},\psi^{*}(g^{\prime}),\psi^{*}(\eta^{\prime}))$

of the corresponding moduli spaces for $\psi^{*}(E_{f})$ and
$\psi^{*}(E_{f^{\prime}})$. It follows that

$\int_{S^{n+1}}SW(E,\mathfrak{s},\tilde{g},\tilde{\eta})=sw_{\mathfrak{s}}(f)+sw_{\mathfrak{s}}(f^{\prime}).$

To complete the proof, it remains to show that
$\int_{S^{n+1}}SW(E,\mathfrak{s},\tilde{g},\tilde{\eta})=sw_{\mathfrak{s}}(f+f^{\prime})$.
Since $E$ is isomorphic to the family obtained from applying the clutching
construction to $f+f^{\prime}$, we just need to show that the pair
$(\tilde{g},\tilde{\eta})$ lies in the canonical chamber
$\in\mathcal{C}_{0}(E,\mathfrak{s})$.

Let $\mathcal{H}^{+}_{\psi^{*}(g)}(X)$,
$\mathcal{H}^{+}_{\psi^{*}(g^{\prime})}(X)$ and
$\mathcal{H}^{+}_{\tilde{g}}(X)$ be the bundles of harmonic self-dual
$2$-forms for the families of metrics $\psi^{*}(g),\psi^{*}(g^{\prime})$, and
$\tilde{g}$. Then from the construction of $\tilde{g}$, we have that
$\mathcal{H}^{+}_{\tilde{g}}(X)$ is obtained by attaching
$\mathcal{H}^{+}_{\psi^{*}(g)}(X)|_{S^{n+1}\setminus D^{\prime}}$ and
$\mathcal{H}^{+}_{\psi^{*}(g^{\prime})}(X)|_{S^{n+1}\setminus D^{\prime}}$ to
the bundle $\mathcal{H}^{+}_{g_{t}}(X)$ over $S^{3}\times[0,1]$ whose fibre
over $(x,t)\in S^{3}\times[0,1]$ is the space of $g_{t}$-self-dual harmonic
$2$-forms.

Let $\mathcal{H}^{2}_{E}(X)$ denote the local system whose fibres are the
degree $2$ cohomology of the fibres of $E$ (the subscript $E$ is just to
remind us which family $\mathcal{H}^{2}_{E}(X)$ comes from). Similarly define
the local systems
$\mathcal{H}^{2}_{\psi^{*}(E_{f})}(X),\mathcal{H}^{2}_{\psi^{*}(E_{f^{\prime}})}(X)$.
Then $\mathcal{H}^{2}_{E}(X)$ is obtained by attaching
$\mathcal{H}^{2}_{\psi^{*}(E_{f})}(X)|_{S^{n+1}\setminus D^{\prime}}$ and
$\mathcal{H}^{2}_{\psi^{*}(E_{f^{\prime}})}(X)|_{S^{n+1}\setminus D^{\prime}}$
to the constant local system over $S^{3}\times[0,1]$ with fibre
$H^{2}(X_{p};\mathbb{R})$. Choose a maximal positive definite subspace
$H\subseteq H^{2}(X_{p};\mathbb{R})$ and let
$\rho_{H}:H^{2}(X_{p};\mathbb{R})\to H$ be the projection. Taking the
composition of inclusion and projection to $H$, we obtain isomorphisms

$\varphi_{\tilde{g}}:\mathcal{H}^{+}_{\tilde{g}}(X)\to B\times
H,\quad\varphi_{\psi^{*}(g)}:\mathcal{H}^{+}_{\psi^{*}(g)}(X)\to B\times
H,\quad\varphi_{\psi^{*}(g^{\prime})}:\mathcal{H}^{+}_{\psi^{*}(g^{\prime})}(X)\to
B\times H.$

Similarly, we obtain an isomorphism
$\varphi_{g_{t}}:\mathcal{H}^{+}_{g_{t}}(X)\to B\times H$. It is clear that
the restriction of $\varphi_{\tilde{g}}$ to the first copy of
$S^{n+1}\setminus D^{\prime}$ agrees with $\varphi_{\psi^{*}(g)}$, the
restriction of $\varphi_{\tilde{g}}$ to the second copy of $S^{n+1}\setminus
D^{\prime}$ agrees with $\varphi_{\psi^{*}(g^{\prime})}$ and the restriction
of $\varphi_{\tilde{g}}$ to $S^{3}\times[0,1]$ agrees with $\varphi_{g_{t}}$.

Let $w:S^{n+1}\to\mathcal{H}^{+}_{\tilde{g}}(X)$ denote the wall for the
family $E$ and set

$R=\sup_{b\in S^{n+1}}||\varphi_{\tilde{g}}(w_{b})||_{H}.$

Recall that we have assumed $(g,\eta)\in\mathcal{C}_{0}(E_{f},\mathfrak{s})$.
Fix an element $v\in H$ such that $||v||_{H}>R$. Choose an $\epsilon>0$ such
that the open ball $B(v,\epsilon)=\\{u\in H\;|\;||u-v||_{H}<\epsilon\\}$ so
that each $u\in B(v,\epsilon)$ has $||u||_{H}>R$ ($\epsilon=(||v||_{H}-R)/2$
would suffice). We will assume that $\eta$ is chosen with
$\varphi_{g_{b}}[\eta]_{b}\in B(v,\epsilon)$ for all $b\in S^{n+1}$. Then we
also have $\varphi_{\psi^{*}(g)}[\psi^{*}(\eta)]_{b}\in B(v,\epsilon)$ for all
$b\in S^{n+1}$. Similarly, we can assume that $\eta^{\prime}$ was chosen so
that $\varphi_{g^{\prime}_{b}}[\eta^{\prime}]_{b}\in B(v,\epsilon)$ for all
$b\in S^{n+1}$ and hence
$\varphi_{\psi^{*}(g^{\prime})}[\psi^{*}(\eta^{\prime})]_{b}\in B(v,\epsilon)$
for all $b\in S^{n+1}$ as well. Lastly, we can assume that the generic path
$(g_{t},\eta_{t})$ joining $(g_{p},\eta_{p})$ to
$(g^{\prime}_{p},\eta^{\prime}_{p})$ satisfies $\varphi_{g_{t}}([\eta_{t}])\in
B(v,\epsilon)$ for all $t\in[0,1]$. It follows that
$\varphi_{\tilde{g}}[\tilde{\eta}]_{b}\in B(v,\epsilon)$ for all $b\in
S^{n+1}$. Therefore we can find a homotopy from
$\varphi_{\tilde{g}}[\tilde{\eta}]$ to the constant section $v$ and hence
$(\tilde{g},\tilde{\eta})$ lies in $\mathcal{C}_{0}(E,\mathfrak{s})$. ∎

Let $Diff(X)$ act on itself by conjugation. Since the identity element is
fixed, this gives an action of $Diff(X)$ on $\pi_{n}(Diff(X)_{0})$. We write
this action as $(f,h)\mapsto fhf^{-1}$.

######  Proposition 2.7.

Let $X$ be a compact, oriented, smooth $4$-manifold with $b^{+}(X)>1$ and let
$f:X\to X$ be an orientation preserving diffeomorphism. Then for each spinc-
structure with $d(X,\mathfrak{s})=-(n+1)\leq-2$ we have

$sw_{\mathfrak{s}}(fhf^{-1})=sw_{f^{*}(\mathfrak{s})}(h).$

###### Proof.

Let $D_{+},D_{-}$ be two copies of the unit disc on $\mathbb{R}^{n+1}$.
Attaching $D_{+}$ and $D_{-}$ on their boundary gives $S^{n+1}$. Let
$h\in\pi_{n}(Diff(X)_{0})$. The family $E_{h}$ is obtained by attaching
$D_{+}\times X$ to $D_{-}\times X$ using the attaching map $\partial
D_{+}\times X\to\partial D_{-}\times X$, $(b,x)\mapsto
a_{h}(b,x)=(b,(h(b))(x))$. Similarly $E_{fhf^{-1}}$ is constructed using
$(b,x)\mapsto a_{fhf^{-1}}(b,x)=(b,(fh(b)f^{-1})(x))$. Consider the maps
$\tilde{f}_{\pm}:D_{\pm}\times X\to D_{\pm}$ given by
$\tilde{f}_{\pm}(b,x)=(b,f(x))$. One finds that

$a_{fhf^{-1}}\circ\tilde{f}_{+}=\tilde{f}_{-}\circ a_{h}.$

This says that the maps $\tilde{f}_{\pm}$ glue together to define a map
$\tilde{f}:E_{h}\to E_{fhf^{-1}}$. The map $\tilde{f}$ is an isomorphism of
smooth families over $S^{n+1}$. Let $\tilde{s}$ be the continuous extension of
$\tilde{s}$ to a family of spinc-structures on the fibres of $E_{fhf^{-1}}$.
Then clearly $\tilde{f}^{*}(\tilde{\mathfrak{s}})$ is a continuous extension
of $f^{*}(\mathfrak{s})$ to a family of spinc-structures on the fibres of
$E_{h}$. Let $(g,\eta)$ be a generic pair for $(E_{fhf^{-1}},\mathfrak{s})$
lying in the canonical chamber. Then $(\tilde{f}^{*}(g),\tilde{f}^{*}(\eta))$
is a generic pair for $(E_{h},f^{*}(\mathfrak{s})$ lying in the canonical
chamber. Clearly $\tilde{f}$ induces an isomorphic between the corresponding
moduli spaces for $(E_{fhf^{-1}},\mathfrak{s},g,\eta)$ and
$(E_{h},f^{*}(\mathfrak{s}),\tilde{f}^{*}(g),\tilde{f}^{*}(\eta))$. Hence
$sw_{\mathfrak{s}}(fhf^{-1})=sw_{f^{*}(\mathfrak{s})}(h)$. ∎

Recall that there is an involution
$\mathfrak{s}\mapsto\overline{\mathfrak{s}}$ on the set of spinc-structure
which we refer to as charge conjugation. Recall that
$c_{1}(\overline{\mathfrak{s}})=-c_{1}(\mathfrak{s})$.

######  Proposition 2.8.

Let $X$ be a compact, oriented, smooth $4$-manifold with $b^{+}(X)>1$ and let
$\mathfrak{s}$ be a spinc-structure with $d(X,\mathfrak{s})=-(n+1)\leq-2$.
Then

$sw_{\overline{\mathfrak{s}}}=(-1)^{\frac{b_{+}(X)-b_{1}(X)-n}{2}}sw_{\mathfrak{s}}.$

###### Proof.

Recall that charge conjugation gives rise to a bijection from the
Seiberg–Witten equations for $(X,\mathfrak{s},g,\eta)$ and the Seiberg–Witten
equations for $(X,\overline{\mathfrak{s}},g,-\eta)$. Fix a homology
orientation for $X$, giving orientations on
$\mathcal{M}(X,\mathfrak{s},g,\eta)$ and
$\mathcal{M}(X,\overline{\mathfrak{s}},g,-\eta)$. The charge conjugation map
changes the orientation on the moduli space by a factor
$(-1)^{d_{\mathfrak{s}}+1-b_{1}(X)+b^{+}(X)}$ where

$d_{\mathfrak{s}}=\frac{c_{1}(\mathfrak{s})^{2}-\sigma(X)}{8}.$

Similarly, charge conjugation gives rise to a bijection of families moduli
spaces. The charge conjugation isomorphism changes the relative orientation of
the moduli space (relative to the base) by the same factor
$(-1)^{d_{\mathfrak{s}}+1-b_{1}(X)+b^{+}(X)}$. Moreover, it is clear that if
$(g,\eta)$ is in the canonical chamber for $(E,\mathfrak{s})$, then
$(g,-\eta)$ is in the canonical chamber for $(E,\overline{\mathfrak{s}})$.
Hence

$sw_{\overline{\mathfrak{s}}}=(-1)^{u}sw_{\mathfrak{s}}$

where

$u=d_{\mathfrak{s}}+1-b_{1}(X)+b^{+}(X).$

But since

$-n-1=d(X,\mathfrak{s})=2d_{\mathfrak{s}}+1-b_{1}(X)+b^{+}(X)$

we see that

$d_{\mathfrak{s}}=\frac{-n-2+b_{1}(X)-b^{+}(X)}{2}$

and hence

$u=\frac{-n-2+b_{1}(X)-b^{+}(X)}{2}+1-b_{1}(X)+b^{+}(X)=\frac{b^{+}(X)-b_{1}(X)-n}{2}.$

∎

######  Theorem 2.9.

Let $X$ be a compact, oriented, smooth $4$-manifold such that $b^{+}(X)>1$ and
$b_{1}(X)=0$. For a given $f\in\pi_{n}(Diff(X)_{0})$, we have that
$sw_{\mathfrak{s}}(f)$ is non-zero for only finitely many spinc-structures
with $d(X,\mathfrak{s})=-(n+1)$.

###### Proof.

Let $g$ be a metric on $X$ and consider the Seiberg–Witten equations on $X$
with respect to the metric $g$ and zero perturbation. Recall that the a priori
estimates for solutions $(A,\psi)$ of the Seiberg–Witten equations (after
gauge fixing) imply bounds on the norms of $A,\psi$ in a suitable Sobolev
space. A bound $M(g)$ can be chosen which depends continuously on $g$ and the
topology of $X$, but does not depend on the spinc-structure $\mathfrak{s}$.
Hence for a smooth family $E\to B$ over a compact base $B$, we obtain
compactness of the families Seiberg–Witten moduli space, taken over all spinc-
structures, with zero perturbation and a fixed family $g=\\{g_{b}\\}$ of
metrics. It follows that the families moduli space
$\mathcal{M}(E,\mathfrak{s},g,0)$ is non-empty for only finitely many spinc-
structures, say, $\mathfrak{s}_{1},\dots,\mathfrak{s}_{m}$. For any other
spinc-structure, $\mathfrak{s}$, the moduli space
$\mathcal{M}(E,\mathfrak{s},g,0)$ is empty. Hence $\eta=0$ is a regular
perturbation for $(E,\mathfrak{s})$ and $(g,0)$ defines a chamber for
$(E,\mathfrak{s})$.

Now let $f\in\pi_{n}(Diff(X)_{0})$ and take $E\to B$ to be the family
$E_{f}\to S^{n+1}$ associated to $f$. If $b^{+}(X)>n+1$, then there is only
one chamber and hence we deduce that $sw_{\mathfrak{s}}(f)=0$ for all but
finitely many spinc-structures.

If $b^{+}(X)\leq n+2$, then we have to consider chambers. Fix a family of
metrics $g$. Then we have shown that for all but finitely many spinc
structures $\mathfrak{s}$, $(g,0)$ is a regular perturbation and
$SW(E_{f},\mathfrak{s},g,0)=0$. However, $(g,0)$ might not lie in the
canonical chamber. Hence we need to consider contributions to the
Seiberg–Witten invariant from wall crossing.

For the rest of the proof, the family $E_{f}$ and metric $g$ will be fixed. To
simplify notation we will write $SW(\mathfrak{s},\eta)$ instead of
$SW(E_{f},\mathfrak{s},g,\eta)$, whenever $\eta$ is a regular perturbation for
$(E,\mathfrak{s},g)$. Fix a maximal positive definite subspace $H$ of
$H^{2}(X;\mathbb{R})$. Let $\varphi:\mathcal{H}^{+}_{g}(X)\to H$ be the map
which is the inclusion of $\mathcal{H}^{+}_{g}(X)$ into $H^{2}(X;\mathbb{R})$,
followed by projection to $H$. For each spinc-structure $\mathfrak{s}$, let
$w(\mathfrak{s})$ be the section of $H$ which sends $b\in B$ to $\varphi(2\pi
c_{1}(\mathfrak{s})^{+_{g_{b}}})$. Given a perturbation $\eta$, let $w(\eta)$
be the section of $H$ given by $b\mapsto\varphi([\eta]_{b})$. If $w(\eta)$ and
$w(\mathfrak{s})$ are disjoint, then $(g,\eta)$ defines a chamber for
$(E,\mathfrak{s})$ and hence the Seiberg–Witten invariant
$SW(\mathfrak{s},\eta)$ is defined.

Let $S(H)$ denote the unit sphere in $H$, which has dimension $b^{+}(X)-1$. If
$w(\eta)$ and $w(\mathfrak{s})$ are disjoint, then
$(w(\eta)-w(\mathfrak{s}))/||w(\eta)-w(\mathfrak{s})||_{H}$ defines a section
of $S(H)$. The wall crossing formula for the families Seiberg–Witten
invariants [12], [3] adapted to the present setting states that

$SW(\mathfrak{s},\eta_{1})-SW(\mathfrak{s},\eta_{2})=Obs\left(\frac{w(\eta_{1})-w(\mathfrak{s})}{||w(\eta_{1})-w(\mathfrak{s})||_{H}},\frac{w(\eta_{2})-w(\mathfrak{s})}{||w(\eta_{2})-w(\mathfrak{s})||_{H}}\right),$

where $Obs(\phi,\psi)\in H^{b^{+}(X)-1}(B;\mathbb{Z})$ is the primary
difference class of $\phi,\psi$ ([18, §36]), the obstruction to constructing a
homotopy of two maps $\phi,\psi:B\to S(H)$ over the $b^{+}(X)-1$ skeleton of
$B$. In our case $B=S^{n+1}$ and so
$H^{b^{+}(X)-1}(B;\mathbb{Z})=H^{b^{+}(X)-1}(S^{n+1};\mathbb{Z})$ is zero
unless $b^{+}(X)=n+2$.

If $b^{+}(X)\neq n+2$, then the primary obstruction vanishes implying that the
value of $SW(\mathfrak{s},\eta)$ does not depend on the choice of chamber. But
we have already seen that for all but finitely many spinc structures
$\mathfrak{s}$, $SW(\mathfrak{s},0)=0$. Hence $sw_{\mathfrak{s}}=0$ for all
but finitely many $\mathfrak{s}$.

It remains to consider the case $b^{+}(X)=n+2$. In this case, the primary
obstruction is valued in $H^{n+1}(S^{n+1};\mathbb{Z})\cong\mathbb{Z}$. Let
$\nu\in H^{n+1}(S^{n+1};\mathbb{Z})$ be the generator corresponding to our
chosen orientation on $S^{n+1}$. Then from [3, Proposition 5.7], we have

$Obs(\phi,\psi)=(-1)^{b^{+}(X)-1}(\phi^{*}(\nu)-\psi^{*}(\nu)).$

Integrating over $S^{n+1}$, the wall crossing formula reduces to

$\int_{S^{n+1}}SW(\mathfrak{s},\eta_{1})-\int_{S^{n+1}}SW(\mathfrak{s},\eta_{2})=(-1)^{b^{+}(X)-1}({\rm
deg}(-\phi_{\mathfrak{s},\eta_{1}})-{\rm
deg}(-\phi_{\mathfrak{s},\eta_{2}})),$

where we define

$\phi_{\mathfrak{s},\eta}=\frac{w(\mathfrak{s})-w(\eta)}{||w(\mathfrak{s})-w(\eta)||_{H}}$

for any perturbation $\eta$ such that $w(\eta)$ and $w(\mathfrak{s})$ are
disjoint.

Noting that ${\rm deg}(-\phi)=(-1)^{b^{+}(X)}{\rm deg}(\phi)$, the wall
crossing formula can be re-written as

$\int_{S^{n+1}}SW(\mathfrak{s},\eta_{1})-\int_{S^{n+1}}SW(\mathfrak{s},\eta_{2})=-{\rm
deg}(\phi_{\mathfrak{s},\eta_{1}})+{\rm deg}(\phi_{\mathfrak{s},\eta_{2}}).$

Now let us take $\eta_{1}=\eta$ to be arbitrary and choose $\eta_{2}$ such
that $w(\eta_{2})=v$ is a constant such that
$||v||_{H}>\sup_{B}||w(\mathfrak{s})||_{H}$. Then $(g,\eta)$ lies in the
canonical chamber. Now since $||w(\eta_{2})||_{H}>||w(\mathfrak{s})||_{H}$ for
all $b\in B$, we obtain a homotopy

$t\mapsto\frac{(1-t)w(\mathfrak{s})-w(\eta_{2})}{||(1-t)w(\mathfrak{s})-w(\eta_{2})||_{H}},\quad
t\in[0,1]$

from $\phi_{\mathfrak{s},\eta_{2}}$ to the constant $-v/||v||_{H}$. It follows
that ${\rm deg}(\phi_{\mathfrak{s},\eta_{2}})=0$ and therefore

$\int_{S^{n+1}}SW(\mathfrak{s},\eta)-\int_{S^{n+1}}SW(\mathfrak{s},\eta_{2})=-{\rm
deg}(\phi_{\mathfrak{s},\eta}).$

But $(g,\eta_{2})$ lies in the canonical chamber, so
$\int_{S^{n+1}}SW(\mathfrak{s},\eta_{2})=sw_{\mathfrak{s}}(f)$. Hence the
above formula reduces to

(2.1) $\int_{S^{n+1}}SW(\mathfrak{s},\eta)=sw_{\mathfrak{s}}(f)-{\rm
deg}(\phi_{\mathfrak{s},\eta}).$

Now we set $\eta=0$. Then for all but finitely many $\mathfrak{s}$, we have
that $w(\mathfrak{s})$ is non-vanishing and that $SW(\mathfrak{s},0)=0$. Hence
for all but finitely many $\mathfrak{s}$, we find

$sw_{\mathfrak{s}}(f)={\rm deg}(w(\mathfrak{s})/||w(\mathfrak{s})||_{H}).$

To finish the proof, it remains to show that when $b^{+}(X)=n+2$, there are
only finitely many $\mathfrak{s}$ such that $d(X,\mathfrak{s})=-(n+1)$,
$w(\mathfrak{s})$ is non-vanishing and
$w(\mathfrak{s})/||w(\mathfrak{s})||_{H}:S^{n+1}\to S(H)$ has non-zero degree.
For convenience, let us say that a spinc-structure $\mathfrak{s}$ is valid if
$d(X,\mathfrak{s})=-(n+1)$ and $w(\mathfrak{s})$ is non-vanishing and let us
write ${\rm deg}(w(\mathfrak{s}))$ for the degree of
$w(\mathfrak{s})/||w(\mathfrak{s})||_{H}$. Then we need to show that ${\rm
deg}(w(\mathfrak{s}))=0$ for all but finitely many valid $\mathfrak{s}$.

We first show that there is a constant $\kappa$ such that ${\rm
deg}(w(\mathfrak{s}))=\kappa$ for all but finitely many valid $\mathfrak{s}$.
We will then argue that $\kappa=0$.

Note that if $b^{+}(X)=n+2$ and $d(X,\mathfrak{s})=-(n+1)$, then

$\frac{c_{1}(\mathfrak{s})^{2}-\sigma(X)}{4}-n-3=-n-1$

(where we used $b_{1}(X)=0$) and hence if $\mathfrak{s}$ is valid, then

$c_{1}(\mathfrak{s})^{2}=\sigma(X)+8.$

We set $N=\sigma(X)+8$. If $N\geq 0$, then for every non-zero $c\in
H^{2}(X;\mathbb{R})$ such that $c^{2}=N$, the projection $c^{+_{g}}$ is non-
zero. This is because

$(c^{+_{g}})^{2}\geq(c^{+_{g}})^{2}-|(c^{-_{g}})^{2}|=c^{2}=N\geq 0$

and equality $(c^{+_{g}})^{2}=0$ can only occur if $c=0$. So if $N\geq 0$,
then every non-zero $c\in H^{2}(X;\mathbb{R})$ defines a non-zero map
$w(c):S^{n+1}\to H$ by taking $w(c)=\varphi(2\pi c^{+_{g}})$. The set $\\{c\in
H^{2}(X\;\mathbb{R})\;|\;c\neq 0,\;c^{2}=N\\}$ is clearly connected if $N>0$,
since $b^{+}(X)>1$. Also if $N=0$, then $\sigma(X)=-8$ and so
$b^{+}(X),b^{-}(X)>1$. It follows that $\\{c\in H^{2}(X\;\mathbb{R})\;|\;c\neq
0,\;c^{2}=0\\}$ is connected. Therefore the degree of $w(c)/||w(c)||_{H}$ is a
constant $\kappa$. Now there are only finitely many spinc structures
$\mathfrak{s}$ for which $c_{1}(\mathfrak{s})=0$, hence
$c_{1}(\mathfrak{s})\in\\{c\in H^{2}(X;\mathbb{R})\;|\;c\neq N\\}$ for all but
finitely many valid $\mathfrak{s}$. So ${\rm deg}(w(\mathfrak{s}))=\kappa$ for
all but finitely many valid $\mathfrak{s}$.

Now we suppose $N<0$. So $\sigma(X)<-8$ and in particular
$b^{-}(X)>b^{+}(X)>1$. Let us define

$C_{N}=\\{c\in H^{2}(X;\mathbb{R}\;|\;c^{2}=N\\}.$

Then $C_{N}$ is homotopy equivalent to a sphere of dimension $b^{-}(X)-1$. For
each $b\in B$, consider

$S_{b}=\\{c\in C_{N}\;|\;c^{+_{g_{b}}}=0\\}.$

The condition $c^{+_{g_{b}}}=0$ means that $c$ lies in the negative definite
subspace $H^{-}_{g_{b}}(X)$. Therefore $S_{b}$ is a sphere of dimension
$b^{-}(X)-1$. In particular $S_{b}$ is compact. Similarly, let

$S=\bigcup_{b\in B}S_{b}=\\{c\in C_{N}\;|\;c^{+_{g_{b}}}=0\text{ for some
}b\in B\\}.$

Then $S$ is a compact subset of $C_{N}$ (by compactness of $B$). Let
$||\;||_{E}$ denote an arbitrary Euclidean norm on $H^{2}(X;\mathbb{R})$. Then
by compactness of $S$, we have that $S$ is contained in some ball
$B_{R}=\\{x\in H^{2}(X;\mathbb{R})\;|\;||x||_{E}\leq R\\}$ of sufficiently
large radius $R>0$. Since $b^{+}(X),b^{-}(X)>1$, it is easy to see that
$C_{N}\setminus(C_{N}\cap B_{R})$ is connected. For any $c\in
C_{N}\setminus(C_{N}\cap B_{R})$, define $w(c):S^{n+1}\to H$ as
$w(c)=\varphi(2\pi c^{+_{g}})$. Then since $c\notin B_{R}$, we have that
$w(c)$ is non-vanishing and hence the degree of $w(c)/||w(c)||_{H}$ is
defined. Since $C_{N}\setminus(C_{N}\cap B_{R})$ is connected, the degree of
$w(c)/||w(c)||_{H}$ is equal to a constant, $\kappa$, for every $c\in
C_{N}\setminus(C_{N}\cap B_{R})$.

Let $\mathfrak{s}$ be a valid spinc-structure. Then $c_{1}(\mathfrak{s})\in
C_{N}$. If $c_{1}(\mathfrak{s})\notin B_{R}$, then it follows that ${\rm
deg}(w(\mathfrak{s}))=\kappa$. Next, we note that if $c_{1}(\mathfrak{s})\in
B_{R}$, then $c_{1}(\mathfrak{s})\in B_{R}\cap H^{2}(X;\mathbb{Z})$. But
$B_{R}\cap H^{2}(X;\mathbb{Z})$ is finite because $B_{R}$ is compact and
$H^{2}(X;\mathbb{Z})$ is discrete. It follows that for all but finitely many
valid $\mathfrak{s}$, we have ${\rm deg}(w(\mathfrak{s}))=\kappa$.

Now we argue that $\kappa=0$. We showed that for all $c\in C_{N}$ outside some
ball $B_{R}$, the degree of $w(c)$ is $\kappa$. Let
$\psi:H^{2}(X;\mathbb{R})\to H^{2}(X;\mathbb{R})$ be an isometry of the
intersection form on $H^{2}(X;\mathbb{R})$ that sends $H$ to itself and
reverses orientation on $H$. Then $B_{R}\cup\psi(B_{R})$ is compact so there
exists a $c\in C_{N}$ such that $c\notin B_{R}\cup\psi(B_{R})$. Hence
$c,\psi(c)\in C_{N}\setminus(C_{N}\cap B_{R})$. Therefore

${\rm deg}(w(c))={\rm deg}(w(\psi(c)))=\kappa.$

On the other hand, since $\psi$ reverses orientation on $H$, we have

${\rm deg}(w(\psi(c)))=-{\rm deg}(w(c))=-\kappa.$

This gives $\kappa=-\kappa$, hence $\kappa=0$. Thus ${\rm
deg}(w(\mathfrak{s}))=0$ for all but finitely many valid $\mathfrak{s}$ and
the proof is complete. ∎

##  3\. The Einstein family

Let $X$ be the underlying oriented smooth $4$-manifold of a complex $K3$
surface. We will use the moduli space of Einstein metrics on $X$ to construct
non-trivial families. The construction of this moduli space follows [8], [9]
and [4].

Let $Ein$ denote the space of all Einstein metrics on $X$ with unit volume
given the $\mathcal{C}^{\infty}$-topology. Every Einstein metric on $X$ is a
hyper-kähler metric [10] and in particular, Ricci flat. Then $Diff(X)$ acts on
$Ein$ by pullback. That is, for each $\varphi\in Diff(X)$ we define

$\varphi_{*}:Ein\to Ein,\quad\varphi_{*}(g)=(\varphi^{-1})^{*}(g).$

Let $TDiff(X)$ denote the subgroup of $Diff(X)$ consisting of those
diffeomorphisms which act trivially on $H^{2}(X;\mathbb{Z})$. Let
$Aut(H^{2}(X;\mathbb{Z}))$ be the group of automorphisms of the intersection
form. Then we have a short exact sequence

$1\to TDiff(X)\to Diff(X)\to\Gamma\to 1,$

where $\Gamma\subset Aut(H^{2}(X;\mathbb{Z}))$ is the subgroup of
automorphisms that are induced by diffeomorphisms of $X$. Note that since
$\Gamma$ is discrete, the identity components of $TDiff(X)$ and $Diff(X)$ are
the same

$TDiff(X)_{0}=Diff(X)_{0}.$

As a consequence of the global Torelli theorem for $K3$ surfaces, one finds
that $TDiff(X)$ acts freely and properly on $Ein$. Let

$T_{Ein}=Ein/TDiff(X)$

be the quotient. Over $Ein$ we have the constant family $X\times Ein\to Ein$.
We can equip the vertical tangent space of $X\times Ein$ with the tautological
metric, namely the metric on the fibre $X\times\\{g\\}$ is $g$ itself. The
action of $Diff(X)$ on $Ein$ lifts to $X\times Ein$ by setting

$\varphi_{*}(x,g)=(\varphi(x),\varphi_{*}(g)).$

It is easily checked that this action preserves the fibrewise metric. Let
$E_{Ein}=(X\times Ein)/TDiff(X)$ be the quotient of $X\times Ein$ by the
action of $TDiff(X)$. As a consequence of the Ebin slice theorem [7] [6], and
the deformation theory of Einstein metrics around Kähler–Einstein metrics
[11], the quotient $E_{Ein}$ is a locally trivial smooth family over $T_{Ein}$
with fibres diffeomorphic to $X$. Moreover, the tautological metric descends
to a metric $g_{Ein}$ on the vertical tangent bundle such that the restriction
of $g_{Ein}$ to the fibre of $E_{Ein}$ over $[g]\in T_{Ein}$ is a
representative of the isomorphism class of Einstein metrics $[g]$. Thus
$E_{Ein}$ is a family of Einstein metrics on $X$.

Let $Gr_{3}(\mathbb{R}^{3,19})$ denote the Grassmannian of positive definite
$3$-planes in $\mathbb{R}^{3,19}$. There is a period map

$P:T_{Ein}\to Gr_{3}(\mathbb{R}^{3,19}).$

Defined as follows. Fix an isometry
$H^{2}(X;\mathbb{R})\cong\mathbb{R}^{3,19}$. Then $P$ sends an Einstein metric
$g$ to the $3$-plane $H^{+}_{g}(X)$. Let

$\Delta=\\{\delta\in H^{2}(X;\mathbb{Z})\;|\;\delta^{2}=-2\\}$

and set

$W=\\{H\in Gr_{3}(\mathbb{R}^{3,19})\;|\;H^{\perp}\cap\Delta=\emptyset\\}.$

The Grassmannian $Gr_{3}(\mathbb{R}^{3,19})$ is a contractible manifold and
for each $\delta\in\Delta$, the subset

$A_{\delta}=\\{H\in Gr_{3}(\mathbb{R}^{3,19})\;|\;\delta\in H^{\perp}\\}$

is a codimension $3$ embedded submanifold and
$W=Gr_{3}(\mathbb{R}^{3,19})\setminus\bigcup_{\delta\in\Delta}A_{\delta}$. It
follows from the global Torelli theorem for $K3$ that the period map $P$ is a
homeomorphism of $T_{Ein}$ to the set $W$ [5, Chapter 12, K].

Choose an element $v\in H^{2}(X;\mathbb{R})$ such that $\langle
v,\delta\rangle\neq 0$ for all $\delta\in\Delta$ and define
$\Delta^{\pm}=\\{\delta\in\Delta\;|\;\pm\langle v,\delta\rangle>0\\}$. Then

$\Delta=\Delta^{+}\cup\Delta^{-}$

and $\delta\in\Delta^{+}$ if and only if $-\delta\in\Delta^{-}$.

Let $\delta\in\Delta^{+}$ and choose a point $p\in A_{\delta}$ such that $p$
does not lie on any $A_{\delta^{\prime}}$ for $\delta^{\prime}\in\Delta^{+}$
other than $\delta$. Let $H\subset H^{2}(X;\mathbb{R})$ be the positive
definite $3$-plane corresponding to $p$. Choose a basis
$\theta_{1},\theta_{2},\theta_{3}$ for $H$ satisfying
$\langle\theta_{i},\theta_{j}\rangle=\delta_{ij}$. Since $p\in A_{\delta}$, we
have $\langle\theta_{j},\delta\rangle=0$ for $j=1,2,3$. Moreover, since $p$
does not lie on $A_{\delta^{\prime}}$ for any $\delta^{\prime}\in\Delta^{+}$
not equal to $\delta$, we have $\langle\theta_{j},\delta^{\prime}\rangle\neq
0$ for some $j$.

Let
$B=S^{2}=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\;|\;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\\}\subset\mathbb{R}^{3}$
be the unit $2$-sphere and choose an $\epsilon\in(0,1)$. Consider the map

$f_{\delta}:S^{2}\to Gr_{3}(\mathbb{R}^{3,19})$

defined by

$f_{\delta}(x_{1},x_{2},x_{3})={\rm span}(\omega_{1},\omega_{2},\omega_{3})$

where for $i=1,2,3$, we set

$\omega_{i}=\theta_{i}-\epsilon x_{1}\delta/2.$

We choose $\epsilon$ sufficiently small so that
$f_{\delta}(x_{1},x_{2},x_{3})$ is a positive definite subspace of
$H^{2}(X;\mathbb{R})$. Moreover, if $\epsilon$ is sufficiently small then
$f_{\delta}(x_{1},x_{2},x_{3})$ is close to $p$ for all
$(x_{1},x_{2},x_{3})\in S^{2}$ and hence $f_{\delta}(x_{1},x_{2},x_{3})$ does
not lie on $A_{\delta^{\prime}}$ for any $\delta^{\prime}\in\Delta^{+}$ not
equal to $\delta$. Moreover, we have

$(\langle\omega_{1},\delta\rangle,\langle\omega_{2},\delta\rangle,\langle\omega_{3},\delta\rangle)=\epsilon(x_{1},x_{2},x_{3}).$

Then since $(x_{1},x_{2},x_{3})\in S^{2}$, we see that at least one of
$\langle\omega_{1}$, $\delta\rangle,\langle\omega_{2},\delta\rangle$, and
$\langle\omega_{3},\delta\rangle$ must be non-zero. So $f(x_{1},x_{2},x_{3})$
does not lie on $A_{\delta}$. It follows that $f$ maps to $W$. So there is a
map $g_{\delta}:B\to T_{Ein}$ such that $f_{\delta}=P\circ g_{\delta}$.

######  Definition 3.1.

Let $E_{\delta}=g_{\delta}^{*}(E_{Ein})$ be the pullback of the family
$E_{Ein}\to T_{Ein}$ by the map $g_{\delta}:S^{2}\to T_{Ein}$. This is a
smooth family of $K3$ surfaces over $S^{2}$.

######  Remark 3.2.

In addition to $\delta$, the map $g_{\delta}:S^{2}\to T_{Ein}$ depends on a
choice of $p\in
A_{\delta}\setminus(A_{\delta}\cap(\cup_{\delta^{\prime}\in\Delta^{+}\setminus\\{\delta\\}}A_{\delta^{\prime}})$
and a sufficiently small $\epsilon>0$. However the space of such pairs
$(p,\epsilon)$ is connected, and moving $(p,\epsilon)$ along a path in this
space only changes $g_{\delta}$ by isotopy. Hence the family $E_{\delta}$ is
well defined up to isomorphism.

######  Remark 3.3.

By construction, the family $E_{Ein}\to T_{Ein}$ has structure group
$TDiff(X)$, hence the same is true of the pullback family $E_{\delta}$. To say
that $E_{\delta}$ has structure group $TDiff(X)$ amounts to saying that
$E_{\delta}$ is equipped with a trivialisation $\mathcal{H}^{2}(X)\cong
H^{2}(X;\mathbb{R})\times B$ of the local system $\mathcal{H}^{2}(X)$.

For each $\delta\in\Delta^{+}$ we have constructed a homotopy class of map
$g_{\delta}:S^{2}\to T_{Ein}$. As $T_{Ein}$ is simply connected [4], there is
a bijection between unbased homotopy classes of maps $S^{2}\to T_{Ein}$ and
the homotopy group $\pi_{2}(T_{Ein})$. Hence $g_{\delta}$ defines a class
$[g_{\delta}]\in\pi_{2}(T_{Ein})$. Now since $T_{Ein}=Ein/TDiff(X)$, the long
exact sequence of homotopy groups gives a map

$\partial:\pi_{2}(T_{Ein})\to\pi_{1}(TDiff(X)_{0})=\pi_{1}(Diff(X)_{0}).$

In particular, we may define
$h_{\delta}=\partial[g_{\delta}]\in\pi_{1}(Diff(X)_{0})$. Applying the
clutching construction to $h_{\delta}$, we recover the family $E_{\delta}$.

Since $T_{Ein}$ is simply connected, the Hurwitz theorem gives an isomorphism
$\pi_{2}(T_{Ein})=H_{2}(T_{Ein};\mathbb{Z})$. From [8, Lemma 5.3], we have

$H_{2}(T_{Ein};\mathbb{Z})=\bigoplus_{\delta\in\Delta^{+}}\mathbb{Z}[g_{\delta}].$

Hence $\pi_{2}(T_{Ein})$ is a free abelian group with generators the maps
$\\{[g_{\delta}]\\}_{\delta\in\Delta^{+}}$.

Recall that a $K3$ surface has

$b^{+}(X)=3,\;\sigma(X)=-16,\;b_{1}(X)=0.$

Since $X$ is spin and simply-connected, for each $u\in H^{2}(X;\mathbb{Z})$,
there is a uniquely determined spinc-structure $\mathfrak{s}_{u}$ for which
$c_{1}(\mathfrak{s}_{u})=2u$. Then

$d(X,\mathfrak{s}_{u})=\frac{(2u)^{2}+16}{4}-1-3=u^{2}.$

Let $\alpha\in\Delta$. Then $\alpha^{2}=-2$ and hence
$d(X,\mathfrak{s}_{\alpha})=-2$. Therefore we have the Seiberg–Witten
invariant

$sw_{\mathfrak{s}_{\alpha}}:\pi_{1}(Diff(X)_{0})\to\mathbb{Z}.$

To simplify notation we will write $sw_{\alpha}$ for
$sw_{\mathfrak{s}_{\alpha}}$. If $\alpha\in\Delta$, then from Proposition 2.8,
we have

$sw_{-\alpha}=-sw_{\alpha}.$

For this reason, it suffices to only consider the homomorphisms $sw_{\alpha}$
for $\alpha\in\Delta^{+}$.

From Theorem 2.9, we have that for each $f\in\pi_{1}(Diff(X))$,
$sw_{\alpha}(f)$ is non-zero for only finitely many $\alpha\in\Delta^{+}$.
Therefore, we obtain a homomorphism

$sw:\pi_{1}(Diff(X)_{0})\to\bigoplus_{\alpha\in\Delta^{+}}\mathbb{Z},\quad
sw(f)=\oplus_{\alpha}sw_{\alpha}(f).$

Recall that we constructed classes
$h_{\delta}=\partial[g_{\delta}]\in\pi_{1}(Diff(X)_{0})$ such that the family
$E_{\delta}$ is obtained from the clutching construction applied to
$h_{\delta}$.

######  Theorem 3.4.

Let $\alpha,\delta\in\Delta^{+}$. Then

$sw_{\alpha}(h_{\delta})=\begin{cases}1&\text{if }\alpha=\delta,\\\
0&\text{otherwise}.\end{cases}$

###### Proof.

By definition, $sw_{\alpha}(h_{\alpha})$ is the Seiberg–Witten invariant of
$(E_{\delta},\mathfrak{s}_{\alpha})$ with respect to the canonical chamber,
where $E_{\delta}$ is the family obtained by the clutching construction
applied to $h_{\delta}$.

We recall the construction of $E_{\delta}$. Choose a point $p\in A_{\delta}$
such that $p$ does not lie on any $A_{\delta^{\prime}}$ for
$\delta^{\prime}\in\Delta^{+}$ other than $\delta$. Let $H\subset
H^{2}(X;\mathbb{R})$ be the positive definite $3$-plane corresponding to $p$.
Choose a basis $\theta_{1},\theta_{2},\theta_{3}$ for $H$ satisfying
$\langle\theta_{i},\theta_{j}\rangle=\delta_{ij}$. Since $p\in A_{\delta}$, we
have $\langle\theta_{j},\delta\rangle=0$ for $j=1,2,3$. Let $B=S^{2}$ be the
unit sphere in $\mathbb{R}^{3}$. We take $E_{\delta}\to S^{2}$ to be the
pullback of the family $E_{Ein}\to T_{Ein}$ by a map $g_{\delta}:S^{2}\to
T_{Ein}$. Let $P:T_{Ein}\to Gr_{3}(\mathbb{R}^{3,19})$ be the period map. Then
$f_{\delta}=P\circ g_{\delta}:S^{2}\to Gr_{3}(\mathbb{R}^{3,19})$ is defined
as

$f_{\delta}(x_{1},x_{2},x_{3})={\rm span}(\omega_{1},\omega_{2},\omega_{3}),$

where $\epsilon>0$ is sufficiently small and

$\omega_{i}=\theta_{i}-\epsilon x_{i}\delta/2,\text{ for }i=1,2,3.$

Let $\rho_{H}:H^{2}(X;\mathbb{R})\to H$ be the projection to $H$ with kernel
$H^{\perp}$. Then since $\theta_{1},\theta_{2},\theta_{3}\in H$ and $\delta\in
H^{\perp}$, we have

$\rho_{H}(\theta_{i})=\theta_{i},\quad\rho_{H}(\delta)=0.$

In particular, this gives

$\rho_{H}(\omega_{i})=\theta_{i}.$

Let $\mathcal{H}^{+}_{g}(X)\to S^{2}$ be the bundle of harmonic self-dual
$2$-forms. Then $\omega_{1},\omega_{2},\omega_{3}$ is a frame for
$\mathcal{H}^{+}_{g}(X)$. Let $\varphi:\mathcal{H}^{+}_{g}(X)\to H$ be
inclusion into $H^{2}(X;\mathbb{R})$ followed by $\rho_{H}$. Define
$w(\alpha)$ to be the section of $H$ given by

$w(\alpha)(b)=\varphi(2\pi
c_{1}(\mathfrak{s}_{\alpha})^{+_{g_{b}}})=\varphi(4\pi\alpha^{+_{g_{b}}}).$

Let $\omega_{1}^{*},\omega_{2}^{*},\omega_{3}^{*}$ be the dual frame of
$\mathcal{H}^{+}_{g_{b}}$, defined by the condition

$\langle\omega_{i},\omega_{j}^{*}\rangle=\delta_{ij}.$

From $\omega_{i}=\theta_{i}-\epsilon x_{i}\delta/2$, one finds

$\langle\omega_{i},\omega_{j}\rangle=\delta_{ij}-\epsilon^{2}x_{i}x_{j}/2.$

One can then directly check that the dual frame is given by

(3.1) $\omega^{*}_{i}=\omega_{i}+\mu
x_{i}(x_{1}\omega_{1}+x_{2}\omega_{2}+x_{3}\omega_{3}),$

where

$\mu=\frac{\epsilon^{2}/2}{1-\epsilon^{2}/2}.$

We have

$\alpha^{+_{g}}=\langle\alpha,\omega^{*}_{1}\rangle\omega_{1}+\langle\alpha,\omega^{*}_{2}\rangle\omega_{2}+\langle\alpha,\omega^{*}_{3}\rangle\omega_{3}.$

Applying $\rho_{H}$, we get

$w(\alpha)=4\pi\left(\langle\alpha,\omega^{*}_{1}\rangle\theta_{1}+\langle\alpha,\omega^{*}_{2}\rangle\theta_{2}+\langle\alpha,\omega^{*}_{3}\rangle\theta_{3}\right).$

We use the basis $\theta_{1},\theta_{2},\theta_{3}$ to identify $H$ with
$\mathbb{R}^{3}$. Then

$w(\alpha)=4\pi\left(\langle\alpha,\omega^{*}_{1}\rangle,\langle\alpha,\omega^{*}_{2}\rangle,\langle\alpha,\omega^{*}_{3}\rangle\right).$

Suppose that $\alpha\neq\delta$. Then since $p$ does not lie on $A_{\alpha}$,
we have $\langle\theta_{j},\alpha\rangle\neq 0$ for some $j$. From (3.1), we
get

$w(\alpha)=4\pi\left(\langle\alpha,\omega_{1}\rangle,\langle\alpha,\omega_{2}\rangle,\langle\alpha,\omega_{3}\rangle\right)+O(\epsilon)$

where $O(\epsilon)$ denotes terms of order $\epsilon$. Then (for sufficiently
small $\epsilon$) it follows that $w(\alpha)$ is non-vanishing and that ${\rm
deg}(w(\alpha))=0$.

Suppose instead that $\alpha=\delta$. In this case we find

$w(\alpha)=w(\delta)=4\pi\epsilon(x_{1},x_{2},x_{3})+O(\epsilon^{2}).$

So (for sufficiently small $\epsilon$) $w(\alpha)$ is non-vanishing and ${\rm
deg}(w(\alpha))=1$ (since the map $(x_{1},x_{2},x_{3})\mapsto
4\pi\epsilon(x_{1},x_{2},x_{3})$ has degree $1$). In all cases, we see that
$w(\alpha)$ is non-vanishing.

The family $E_{Ein}$ has a fibrewise metric $g_{Ein}$ which is a Ricci flat
Einstein metric on each fibre. Let $g=g_{\delta}^{*}(g_{Ein})$ be the
fibrewise metric on $E_{\delta}$ obtained by pullback. Then for each $b\in
S^{2}$, the metric $g_{b}$ is Ricci flat and in particular has zero scalar
curvature. Now consider the families Seiberg–Witten moduli space
$\mathcal{M}(E_{\delta},\mathfrak{s}_{\alpha},g,0)$ for the zero perturbation
$\eta=0$.

Suppose that $(A,\psi)\in\mathcal{M}(E_{\delta},\mathfrak{s}_{\alpha},g,0)$ is
a solution to the Seiberg–Witten equations in the family. Then $(A,\psi)$ is
the solution of the Seiberg–Witten equations on some fibre $X_{b}$ with metric
$g_{b}$ and zero perturbation. The Weitzenböck formula together with the
Seiberg–Witten equations and the fact that $g_{b}$ has zero scalar curvature
implies that $\psi=0$ ([14, Corollary 2.2.6]). So every solution in
$\mathcal{M}(E_{\delta},\mathfrak{s}_{\alpha},g,0)$ is reducible.

On the other hand, since $w(\alpha)$ is non-vanishing, the perturbation
$\eta=0$ does not lie on the wall, that is, there are no reducible solutions.
So the moduli space $\mathcal{M}(E_{\delta},\mathfrak{s}_{\alpha},g,0)$ is
empty. Recall that in the proof of Theorem 2.9, we used the wall crossing
formula to deduce the identity

$\int_{S^{n+1}}SW(\mathfrak{s},\eta)=sw_{\mathfrak{s}}(f)-{\rm
deg}(\phi_{\mathfrak{s},\eta})$

(see Equation (2.1)). Taking $\mathfrak{s}=\mathfrak{s}_{\alpha}$,
$f=h_{\delta}$ and $\eta=0$, we obtain

$\int_{S^{n+1}}SW(\mathfrak{s}_{\alpha},0)=sw_{\alpha}(h_{\delta})-{\rm
deg}(w(\alpha)).$

But $\mathcal{M}(E_{\delta},\mathfrak{s}_{\alpha},g,0)$ is empty, so
$SW(\mathfrak{s}_{\alpha},0)=0$ and hence

$sw_{\alpha}(h_{\delta})={\rm deg}(w(\alpha)).$

Further, we have already shown that

${\rm deg}(w(\alpha))=\begin{cases}1&\text{if }\alpha=\delta,\\\
0&\text{otherwise}.\end{cases}$

∎

As an immediate consequence of Theorem, 3.4, we have:

######  Theorem 3.5.

The homomorphism

$sw:\pi_{1}(Diff(X)_{0})\to\bigoplus_{\alpha\in\Delta^{+}}\mathbb{Z}$

is surjective, hence $\pi_{1}(Diff(X)_{0})$ contains
$\bigoplus_{\alpha\in\Delta^{+}}\mathbb{Z}$ as a direct summand.

######  Theorem 3.6.

The boundary map

$\partial:\pi_{2}(T_{Ein})\to\pi_{1}(TDiff(X)_{0})=\pi_{1}(Diff(X)_{0})$

induced by the fibration $Ein\to Ein/TDiff(X)=T_{Ein}$ admits a left inverse,
given by

$\pi_{1}(Diff(X)_{0})\to\pi_{2}(T_{Ein}),\quad
x\mapsto\bigoplus_{\alpha}sw_{\alpha}(x)[g_{\alpha}].$

###### Proof.

Recall that

$\pi_{2}(T_{Ein})\cong
H_{2}(T_{Ein};\mathbb{Z})\cong\bigoplus_{\alpha\in\Delta^{+}}\mathbb{Z}[g_{\alpha}]$

and that $\partial[g_{\alpha}]=h_{\alpha}$. Hence, if we define
$t:\pi_{1}(Diff(X)_{0})\to\pi_{2}(T_{Ein})$ to be given by
$t(x)=\bigoplus_{\alpha\in\Delta^{+}}sw_{\alpha}(x)[g_{\alpha}]$. Then from
Theorem 3.4, it follows that $t\circ\partial=id$, so that $t$ is a left
inverse of $\partial$, as claimed. ∎

From the homeomorphism $P:T_{Ein}\to
W=Gr_{3}(\mathbb{R}^{3,19})\setminus\bigcup_{\delta\in\Delta}A_{\delta}$, we
see that $T_{Ein}$ is connected. Then since $T_{Ein}=Ein/TDiff(X)$, it follows
that $TDiff(X)$ acts transitively on the connected components of $Ein$ and
that the components of $Ein$ are all homeomorphic to each other. Choose
arbitrarily a basepoint $p\in Ein$. Since the components of $Ein$ are all
homeomorphic, $\pi_{1}(Ein,p)$ does not depend on the choice of $p$ and we
simply write $\pi_{1}(Ein)$. From the long exact sequence in homotopy groups
associated to $Ein\to T_{Ein}$ we get an exact sequence

$\cdots\to\pi_{2}(T_{Ein})\buildrel\partial\over{\longrightarrow}\pi_{1}(TDiff(X)_{0})\to\pi_{1}(Ein)\to\pi_{1}(T_{Ein}).$

We have also seen that $T_{Ein}$ is simply-connected and that $\partial$
admits a left inverse, so we obtain an isomophism

$\pi_{1}(TDiff(X)_{0})=\pi_{1}(Diff(X)_{0})\cong\left(\bigoplus_{\alpha\in\Delta^{+}}\mathbb{Z}\right)\oplus\pi_{1}(Ein),$

where the summand $\bigoplus_{\alpha\in\Delta^{+}}\mathbb{Z}$ is detected by
the Seiberg–Witten invariants $sw_{\mathfrak{\beta}}$.

######  Remark 3.7.

Smooth families over $S^{2}$ with fibres diffeomorphic to $X$ correspond, via
the clutching construction, to elements of $\pi_{1}(Diff(X)_{0})$ considered
modulo the conjugation action of $Diff(X)$. For this reason we are interested
in the action of $Diff(X)$ on $\pi_{1}(Diff(X)_{0})$. The Seiberg–Witten
invariants are compatible with this action in the following sense. Let
$\\{e_{\alpha}\\}_{\alpha\in\Delta^{+}}$ denote the standard basis for
$\bigoplus_{\alpha\in\Delta^{+}}\mathbb{Z}$. For $f\in Diff(X)$ and $x\in
H^{2}(X;\mathbb{R})$, let us write $f_{*}(x)=(f^{-1})^{*}(x)$ so that
$(f,x)\mapsto f_{*}(x)$ is a left action. Let $Diff(X)$ act on
$\bigoplus_{\alpha\in\Delta^{+}}\mathbb{Z}$ by setting

$f\cdot e_{\alpha}=\begin{cases}e_{f_{*}\alpha}&\text{if
}f_{*}\alpha\in\Delta^{+},\\\ -e_{-f_{*}\alpha}&\text{if
}f_{*}\alpha\in\Delta^{-}.\end{cases}$

Then it follows easily from Propositions 2.7 and 2.8 that the map
$sw:\pi_{1}(Diff(X)_{0})\to\bigoplus_{\alpha\in\Delta^{+}}\mathbb{Z}$ is
$Diff(X)$-equivariant.

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