# On the mapping class groups of simply-connected smooth $4$-manifolds

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

The mapping class group $M(X)$ of a smooth manifold $X$ is the group of smooth
isotopy classes of orientation preserving diffeomorphisms of $X$. We prove a
number of results about the mapping class groups of compact, simply-connected,
smooth $4$-manifolds. We prove that $M(X)$ is non-finitely generated for
$X=2n\mathbb{CP}^{2}\\#10n\overline{\mathbb{CP}^{2}}$, where $n\geq 3$ is odd.
Let $\Gamma(X)$ denote the group of automorphisms of the intersection lattice
of $X$ that can be realised by diffeomorphisms. Then $M(X)$ is an extension of
$\Gamma(X)$ by $T(X)$, the Torelli group of isotopy classes of diffeomorphisms
that act trivially in cohomology. We prove that this extension is split for
connected sums of $\mathbb{CP}^{2}$, but is not split for
$2\mathbb{CP}^{2}\\#n\overline{\mathbb{CP}^{2}}$, where $n\geq 11$. We prove
that the Nielsen realisation problem fails for certain finite subgroups of
$M(p\mathbb{CP}^{2}\\#q\overline{\mathbb{CP}^{2}})$ whenever $p+q\geq 4$.
Lastly we study the extension $M_{1}(X)\to M(X)$, where $M_{1}(X)$ is the
group of isotopy classes of diffeomorphisms of $X$ which fix a neighbourhood
of a point. When $X=K3$ or $K3\\#(S^{2}\times S^{2})$ we prove that
$M_{1}(X)\to M(X)$ is a non-trivial extension of $M(X)$ by $\mathbb{Z}_{2}$.
Moreover, we completely determine the extension class of $M_{1}(K3)\to M(K3)$.

##  1\. Introduction

Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. Define
the mapping class group $M(X)$ to be the group of smooth isotopy classes of
orientation preserving diffeomorphisms of $X$. There is considerable interest
in the groups $M(X)$, although little is known about their structure. In this
paper we will prove a number of new results concerning the structure of
mapping class groups of smooth $4$-manifolds.

Recall that the second cohomology group $L_{X}=H^{2}(X;\mathbb{Z})$ of $X$
equipped with its intersection form is a unimodular lattice. We let
$Aut(L_{X})$ denote the automorphism group of the lattice $L_{X}$. The group
of orientation preserving diffeomorphisms of $X$ acts on $L_{X}$ via
$f\mapsto(f^{-1})^{*}$. This action depends only on this isotopy class and so
defines a homomorphism $M(X)\to Aut(L_{X})$. Denoting the image of this map by
$\Gamma(X)$ and the kernel by $T(X)$, we obtain a short exact sequence

(1.1) $1\to T(X)\to M(X)\to\Gamma(X)\to 1.$

We call $T(X)$ the Torelli group of $X$. It is the group of isotopy classes of
diffeomorphisms of $X$ that act trivially in cohomology. By a result of Quinn,
$T(X)$ can also be defined as the group of isotopy classes of diffeomorphisms
which are continuously isotopic to the identity [23]. The group $\Gamma(X)$ is
the group of automorphisms of $L_{X}$ that can be realised by diffeomorphisms
of $X$.

Understanding the group $M(X)$ necessitates an understanding of the groups
$T(X)$, $\Gamma(X)$ and the extension (1.1). The group $\Gamma(X)$ is known
for some classes of $4$-manifolds. In particular, a theorem of Wall implies
that $\Gamma(X)=Aut(L_{X})$ for a large class of $4$-manifolds [31]. In
contrast, the Torelli group $T(X)$ is poorly understood. Ruberman showed that
$T(X)$ is not finitely generated for certain $X$ [25]. However this does not
imply that $M(X)$ is not finitely generated, since a finitely generated group
can have subgroups which are not finitely generated. Our first main result
confirms that $M(X)$ is not finitely generated for certain simply-connected
$4$-manifolds.

######  Theorem 1.1.

Let $X=2n\mathbb{CP}^{2}\\#10n\overline{\mathbb{CP}^{2}}$, where $n\geq 3$ is
odd. Then $M(X)$ is not finitely generated. More precisely, the following
holds:

  * (1)

There is an index $2$ subgroup $M_{+}(X)$ of $M(X)$ and a surjective
homomorphism $\Phi:M_{+}(X)\to\mathbb{Z}^{\infty}$ from $M_{+}(X)$ to
$\mathbb{Z}^{\infty}$, where $\mathbb{Z}^{\infty}$ denotes a free abelian
group of countably infinite rank.

  * (2)

The mod $2$ reduction of $\Phi$ extends to a surjective homomorphism
$\Phi:M(X)\to\mathbb{Z}_{2}^{\infty}$.

As this paper was nearing completion we received a preprint by Hokuto Konno
[15] which also proves that the mapping class groups of simply-connected
$4$-manifolds can be non-finitely generated. Konno’s proof uses essentially
the same method as ours, however we obtained our proofs completely
independently.

######  Remark 1.2.

It is interesting to contrast Theorem 1.1 with finiteness results for mapping
class groups in other dimensions. Let $X$ be a compact, simply-connected
smooth manifold of dimension $d$ and $M(X)=\pi_{0}(Diff(X))$ the mapping class
group. If $d\neq 4$, then $M(X)$ is finitely generated. For $d\leq 3$, finite
generation holds for any compact oriented manifold (see [7] for $d=2$ and [13]
for $d=3$). If $d\geq 5$ then $M(X)$ is finitely generated [6, Theorem 2.6].
Note that Theorem 2.6 of [6] is only stated for $d\geq 6$, but when $X$ is
simply-connected, the proof carries over to $d=5$. In the proof of [6, Theorem
2.6], dimension $6$ only enters in the point (i) in the proof, but Cerf’s
theorem says that in the simply-connected case $\pi_{0}(C^{Diff}(X))=0$, and
in [6, Proposition 2.7] where it is not necessary. This follows from
specialising Triantafillou [30] to simply-connected manifolds, where none of
the oversights mentioned in [6, §2.2] cause a problem111I thank Alexander
Kupers for explaining why [6, Theorem 2.6] works for simply-connected
$5$-manifolds..

One may also consider the larger group $M^{\prime}(X)$ consisting is isotopy
classes of diffeomorphisms which are not necessarily orientation preserving.
Since $M(X)$ has finite index in $M^{\prime}(X)$, it follows from Schreier’s
lemma [28] that if $M(X)$ is not finitely generated then neither is
$M^{\prime}(X)$.

######  Remark 1.3.

Let $X$ be a compact, simply-connected smooth $4$-manifold and let
$M^{top}(X)=\pi_{0}(Homeo(X))$ be the topological mapping class group. By work
of Freedman [10] and Quinn [23], the natural map $M^{top}(X)\to
Aut(H^{2}(X;\mathbb{Z}))$ to the group of automorphisms of the intersection
lattice $H^{2}(X;\mathbb{Z})$ is an isomorphism. By a result of Siegel [29],
the automorphism group of any lattice is finitely generated. Hence
$M^{top}(X)$ is finitely generated.

In contrast, we do not know whether the group $\Gamma(X)\subseteq
Aut(H^{2}(X;\mathbb{Z}))$ is always finitely generated, although we conjecture
that it is.

Our next result concerns the question of whether or not the sequence (1.1)
admits a splitting.

######  Theorem 1.4.

The following holds:

  * (1)

Let $X=n\mathbb{CP}^{2}$, where $n\geq 1$. Then there exists a splitting
$\Gamma(X)\to M(X)$.

  * (2)

Let $X=(S^{2}\times S^{2})\\#X^{\prime}$, where $b_{+}(X^{\prime})=1$,
$b_{-}(X^{\prime})\geq 10$. Then there does not exist a splitting
$\Gamma(X)\to M(X)$.

More precise information about the failure of a splitting in case (2) is
provided by Theorem 5.1.

######  Remark 1.5.

In [5] it is shown when $X$ is a $K3$ surface, there is a splitting
$\Gamma(X)\to M(X)$. It is also easy to see that splittings exist for
$S^{2}\times S^{2}$ and $\mathbb{CP}^{2}\\#\overline{\mathbb{CP}^{2}}$.

Our next result concerns the Nielsen realisation problem. Recall that the
Nielsen realisation problem for a smooth manifold $X$ asks whether a subgroup
$G$ of the mapping class group of $X$ can be lifted to a subgroup of
$Diff(X)$. Recent results of Baraglia–Konno [5], Farb–Looijenga [9], Konno
[14] show that Nielsen realisation fails for many simply-connected spin
$4$-manifolds. Arabadji–Baykur showed that there are many non-spin
$4$-manifolds with finite non-trivial fundamental group for which Nielsen
realisation fails [1] and Konno–Miyazawa–Taniguchi gave examples with simply-
connected indefinite non-spin $4$-manifolds [16].

######  Theorem 1.6.

Let $X=X^{\prime}\\#p\mathbb{CP}^{2}\\#q\overline{\mathbb{CP}^{2}}$ where
$X^{\prime}$ is a compact, smooth, simply-connected $4$-manifold and $p+q\geq
4$. Then $M(X)$ contains a subgroup isomorphic to $\mathbb{Z}_{2}^{4}$ which
can not be lifted to $Diff(X)$.

In particular, Nielsen realisation fails for $n\mathbb{CP}^{2}$ for $n\geq 4$.
As far as we are aware, these are the first examples of definite, simply-
connected $4$-manifolds where Nielsen realisation fails.

Our last main result concerns a certain extension of $M(X)$. Let $X^{(1)}$ be
obtained from $X$ by removing an open ball and let $Diff(X^{(1)},\partial
X^{(1)})$ denote the group of diffeomorphisms of $X^{(1)}$ which are the
identity in a neighbourhood of the boundary. Let
$M_{1}(X)=\pi_{0}(Diff(X^{(1)},\partial X^{(1)}))$ denote the group of
components of $Diff(X^{(1)},\partial X^{(1)})$. It is known that the map
$M_{1}(X)\to M(X)$ is surjective and that the kernel (which is either trivial
or has order $2$) is generated by a Dehn twist on the boundary (see Section 7
for more details).

In general it is difficult to determine whether the kernel of $M_{1}(X)\to
M(X)$ is trivial or non-trivial, or equivalently, whether the boundary Dehn
twist is trivial or non-trivial. The extension is known to be trivial when $X$
is a connected sum of copies of $S^{2}\times S^{2}$. In contrast we have:

######  Theorem 1.7.

Let $X^{\prime}$ be a compact, smooth, simply-connected $4$-manifold which is
homeomorphic to $K3$. Let $X$ be $X^{\prime}$ or $X^{\prime}\\#(S^{2}\times
S^{2})$. Then the boundary Dehn twist is non-trivial. Moreover, the extension
$1\to\mathbb{Z}_{2}\to M_{1}(X)\to M(X)\to 1$ does not split.

If $M_{1}(X)\to M(X)$ is a non-trivial extension, then it is given by an
extension class $\xi_{X}\in H^{2}(M(X);\mathbb{Z}_{2})$ and the above theorem
says that $\xi_{X}\neq 0$ when $X$ is of the stated form. Our final result
completely determines $\xi_{X}$ in the case that $X$ is homeomorphic to $K3$.
Let $L_{X}$ be the intersection lattice of $X$ and $Aut(L_{X})$ the group of
automorphisms. Over the classifying space $BAut(L_{X})$ we have the
tautological flat bundle $H=EAut(L_{X})\times_{Aut(L_{X})}L_{X}$. Let
$H^{+}\to BAut(L_{X})$ be a maximal positive subbundle. This defines a
characteristic class $w_{2}(H^{+})\in H^{2}(Aut(L_{X});\mathbb{Z}_{2})$.

######  Theorem 1.8.

Let $X$ be a smooth $4$-manifold which is homeomorphic to $K3$. Then the
extension class $\xi_{X}\in H^{2}(M(X);\mathbb{Z}_{2})$ is the pullback of
$w_{2}(H^{+})\in H^{2}(Aut(L_{X});\mathbb{Z}_{2})$ under the map $M(X)\to
Aut(L_{X})$.

###  1.1. Structure of the paper

The structure of the paper is as follows. In Section 2 we review the
Seiberg–Witten invariants for the Torelli group (as in [24, 26, 3]) and show
how these invariants can be assembled into cohomology classes on the mapping
class group. In Section 3 we use these cohomology classes to show that $M(X)$
is not finitely generated for certain $X$. In Section 4 we construct a
splitting $\Gamma(X)\to M(X)$ when $X$ is a connected sum of copies of
$\mathbb{CP}^{2}$. In Section 5 we prove the non-existence of splittings
$\Gamma(X)\to M(X)$ for certain $4$-manifolds. The proof uses families
Seiberg–Witten theory and more specifically the main result of [2]. In Section
6 we prove Theorem 1.6. Finally, in Section 7 we study boundary Dehn twists
and the extension $M_{1}(X)\to M(X)$ and we prove Theorems 1.7 and 1.8.

###  1.2. Acknowledgements

I thank Hokuto Konno for sending me his paper [15] and also for comments on a
draft of this paper. I also thank Alexander Kupers for clarifying the finite
generation of mapping class groups for simply-connected $5$-manifolds.

##  2\. Seiberg–Witten invariants for the mapping class group

In this section we define Seiberg–Witten invariants for the mapping class
group, extending the Seiberg–Witten invariants on the Torelli group which have
previously been considered in [24, 26, 3]. These invariants will be used to
show that certain simply-connected $4$-manifolds have non-finitely generated
mapping class group.

Let $X$ be a compact, smooth, simply connected $4$-manifold and let
$\mathfrak{s}$ be a spinc-structure with $d(\mathfrak{s})=-1$, where

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

is the expected dimension of the Seiberg–Witten moduli space for
$\mathfrak{s}$. Let $\mathcal{S}(X)$ denote the set of all isomorphism classes
of spinc-structures on $X$ for which $d(\mathfrak{s})=-1$. Since $X$ is
assumed to be simply connected, $\mathcal{S}(X)$ can be identified with the
set of characteristic elements $c\in L=H^{2}(X;\mathbb{Z})$ for which
$(c^{2}-\sigma(X))/4-b_{+}(X)=0$.

Let $\Pi$ denote the space of pairs $(g,\eta)$ where $g$ is a Riemannian
metric on $X$ and $\eta$ is a $2$-form which is self-dual with respect to $g$.
For any $h\in\Pi$ and any $\mathfrak{s}\in\mathcal{S}(X)$ we may consider the
Seiberg–Witten equations on $X$ with respect to the metric $g$, spinc-
structure $\mathfrak{s}$ and $2$-form perturbation $\eta$. Let
$\mathcal{M}(X,\mathfrak{s},h)$ denote the moduli space of gauge equivalence
classes of solutions to the Seiberg–Witten equations for $(X,\mathfrak{s},h)$.
Assume $b_{+}(X)>2$. We will say that $h\in\Pi$ is regular if
$\mathcal{M}(X,\mathfrak{s},h)$ is empty for all $\mathfrak{s}\in S(X)$. Since
$b_{+}(X)>0$ and the expected dimension of $\mathcal{M}(X,\mathfrak{s},h)$ is
negative, the regular elements form a subset of $\Pi$ of Baire second category
with respect to the $\mathcal{C}^{\infty}$ topology. Let
$\Pi^{reg}\subseteq\Pi$ denote the set of regular elements.

Suppose that $h_{0},h_{1}\in\Pi^{reg}$. If $h:[0,1]\to\Pi$ is a path in $\Pi$
from $h_{0}$ to $h_{1}$, we can consider the families moduli space, which is
the union over $t\in[0,1]$ of the Seiberg–Witten moduli spaces for each
$h_{t}\in\Pi$. For a sufficiently generic path $h_{t}$, the moduli space is a
compact, smooth, $0$-dimensional manifold. A choice of orientation on a
maximal positive definite subspace of $H^{2}(X;\mathbb{R})$ determines an
orientation on the moduli space and hence we can count with sign the number of
points in the moduli space. Fix a choice of such an orientation. It can be
shown [24] that the number of solutions depends on the endpoints
$h_{0},h_{1}$, but not on the choice of generic path $h_{t}$. Hence we may
denote by $SW_{\mathfrak{s}}(h_{0},h_{1})\in\mathbb{Z}$ the signed count of
points in the moduli space. From the definition it is clear that this count of
points satisfies the following properties:

  * (1)

$SW_{\mathfrak{s}}(h_{0},h_{1})+SW_{\mathfrak{s}}(h_{1},h_{2})=SW_{\mathfrak{s}}(h_{0},h_{2})$,

  * (2)

$SW_{\mathfrak{s}}(h_{0},h_{1})=sgn_{+}(f)SW_{f(\mathfrak{s})}(f(h_{0}),f(h_{1}))$
for any orientation preserving diffeomorphism $f$.

In (2), $sgn_{+}(f)$ is defined as follows. The space of oriented, maximal
positive definite subspaces of $H^{2}(X;\mathbb{R})$ has two connected
components. For an isometry $\varphi$ of $H^{2}(X;\mathbb{R})$ we let
$sgn_{+}(\varphi)$ equal $1$ or $-1$ according to whether $\varphi$ preserves
or exchanges the two components. If $f$ is an orientation preserving
diffeomorphism of $X$, then $sgn_{+}(f)$ denotes $sgn_{+}(f_{*})$, where
$f_{*}=(f^{-1})^{*}$ is the isometry of $H^{2}(X;\mathbb{R})$ induced by $f$.

Property (1) follows by concatenating a path from $h_{0}$ to $h_{1}$ with a
path from $h_{1}$ to $h_{2}$. Property (2) follows from diffeomorphism
invariance of the Seiberg–Witten equations. In addition,
$SW_{\mathfrak{s}}(h_{0},h_{1})$ obeys a symmetry with respect to charge
conjugation:

  * (3)

$SW_{\mathfrak{s}}(h_{0},h_{1})=(-1)^{b_{+}(X)/2+1}SW_{\bar{\mathfrak{s}}}(\bar{h}_{0},\bar{h}_{1})$,

where $\bar{\mathfrak{s}}$ denotes the charge conjugate of $\mathfrak{s}$ and
for $h=(g,\eta)\in\Pi$, we set $\bar{h}=(g,-\eta)$. Property (3) is an
immediate consequence of the charge conjugation symmetry of the Seiberg–Witten
equations.

Let $f\in T(X)$ be an element of the Torelli group. Fix a spinc-structure
$\mathfrak{s}$ with $d(\mathfrak{s})=-1$. The mapping cylinder of $f$ defines
a smooth family $E\to S^{1}$ over $S^{1}$ with fibres diffeomorphic to $X$.
Since $f$ acts trivially on cohomology, it preserves the isomorphism class of
$\mathfrak{s}$. It follows easily that there is a unique spinc-structure on
the vertical tangent bundle of $E$ which restricts to $\mathfrak{s}$ on each
fibre. Since $b_{+}(X)>2$, there is a single chamber for the families
Seiberg–Witten equations for $E$. Furthermore, the families moduli space is
oriented and so we obtain an integer-valued invariant
$SW_{\mathfrak{s}}(f)\in\mathbb{Z}$ which depends only on $(X,\mathfrak{s})$
and the isotopy class of $f$ (see [3] for more details). From the definition
of $SW_{\mathfrak{s}}(f)$, it is easy to see that

$SW_{\mathfrak{s}}(f)=SW_{\mathfrak{s}}(h,f(h))$

for any $h\in\Pi^{reg}$. It is instructive to see why $SW_{\mathfrak{s}}(f)$
is independent of the choice of $h\in\Pi^{reg}$. Let $h^{\prime}\in\Pi^{reg}$.
Then

$\displaystyle SW_{\mathfrak{s}}(h^{\prime},f(h^{\prime}))$
$\displaystyle=-SW_{\mathfrak{s}}(h,h^{\prime})+SW_{\mathfrak{s}}(h,f(h))+SW_{\mathfrak{s}}(f(h),f(h^{\prime}))$
$\displaystyle=-SW_{\mathfrak{s}}(h,h^{\prime})+SW_{\mathfrak{s}}(h,f(h))+SW_{f^{-1}(\mathfrak{s})}(h,h^{\prime})$
$\displaystyle=SW_{\mathfrak{s}}(h,f(h)),$

where the last line follows from $f^{-1}(\mathfrak{s})=\mathfrak{s}$, which
holds since $f\in T(X)$.

For $f,g\in T(X)$, we have that $SW_{\mathfrak{s}}(f\circ
g)=SW_{\mathfrak{s}}(f)+SW_{\mathfrak{s}}(g)$ (this is a special case of
Proposition 2.1, proven below). Therefore $SW_{\mathfrak{s}}$ defines a
homomorphism

$SW_{\mathfrak{s}}:T(X)\to\mathbb{Z}$

or equivalently, a cohomology class $SW_{\mathfrak{s}}\in
H^{1}(T(X);\mathbb{Z})$. These cohomology classes generally do not extend to
the full mapping class group $M(X)$, because $\Gamma(X)$ acts non-trivially on
the set of spinc-structures.

Recall that the compactness of the Seiberg–Witten moduli space follows from a
priori bounds. These bounds depend on the pair $h\in\Pi$, but not on the
spinc-structure. This argument also works for families over a compact base
space, hence for fixed $f\in T(X)$, $SW_{\mathfrak{s}}(f)$ is non-zero for
only finitely many $\mathfrak{s}\in S(X)$. Therefore, we can collect the
homomorphisms $SW_{\mathfrak{s}}$ into a single invariant

$\displaystyle SW:T(X)$ $\displaystyle\to\bigoplus_{\mathfrak{s}\in
S(X)}\mathbb{Z}$ $\displaystyle f$
$\displaystyle\mapsto\bigoplus_{\mathfrak{s}}SW_{\mathfrak{s}}(f).$

In what follows, we will see that $SW$ can be extended from $T(X)$ to the full
mapping class group $M(X)$ as a cohomology class valued in a certain
$\Gamma(X)$-module.

Recall that each $\mathfrak{s}\in S(X)$ is determined by the corresponding
characteristic element $c(\mathfrak{s})\in L$. Therefore the group $\Gamma(X)$
acts on $\mathcal{S}(X)$ and hence on $\mathbb{Z}[\mathcal{S}(X)]$, the free
abelian group with basis $\mathcal{S}(X)$. Let $\widehat{\mathbb{Z}}$ denote
$\mathbb{Z}$ equipped with the action of $\Gamma(X)$ such that $f\in\Gamma(X)$
acts as multiplication by $sgn_{+}(f)$. Let
$\widehat{\mathbb{Z}}[\mathcal{S}(X)]=\widehat{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Z}[\mathcal{S}(X)]$.
It will be convenient to regard $\widehat{\mathbb{Z}}[\mathcal{S}(X)]$ as the
group of functions $\phi:\mathcal{S}(X)\to\mathbb{Z}$ with finite support.
Then the action of $f\in\Gamma(X)$ is given by
$(f\phi)(\mathfrak{s})=sgn_{+}(f)\phi(f^{-1}(\mathfrak{s}))$. We will show
that the families Seiberg–Witten invariant for $1$-dimensional families (where
$b_{+}(X)>2$) can be viewed as an element of
$H^{1}(M(X);\widehat{\mathbb{Z}}[\mathcal{S}(X)])$.

Recall that for a group $G$ and a $G$-module $M$, the group $H^{1}(G;M)$ can
be viewed as the set of equivalence classes of twisted homomomorphisms $G\to
M$. A twisted homomorphism is a map $\phi:G\to M$ such that
$\phi(gh)=\phi(g)+g\phi(h)$. A trivial twisted homomorphism is a twisted
homomorphism of the form $\phi(g)=gm-m$ for some $m\in M$. Two twisted
homomorphisms are considered equivalent if they differ by a trivial twisted
homomorphism.

Let $h\in\Pi^{reg}$. Define a map
$\phi_{h}:Diff_{+}(X)\to\widehat{\mathbb{Z}}[\mathcal{S}(X)]$ from the group
of orientation preserving diffeomorphisms to
$\widehat{\mathbb{Z}}[\mathcal{S}(X)]$ by setting

$(\phi_{h}(f))(\mathfrak{s})=SW_{\mathfrak{s}}(h,f(h)).$

Suppose that $f_{0},f_{1}\in Diff_{+}(X)$ are isotopic. Choose an isotopy
$f_{t}$. Then

$\displaystyle\phi_{h}(f_{1})$ $\displaystyle=SW_{\mathfrak{s}}(h,f_{1}(h))$
$\displaystyle=SW_{\mathfrak{s}}(h,f_{0}(h))+SW_{\mathfrak{s}}(f_{0}(h),f_{1}(h))$
$\displaystyle=\phi_{h}(f_{0})+SW_{\mathfrak{s}}(f_{0}(h),f_{1}(h)).$

Consider the path $h_{t}=f_{t}(h)$ from $f_{0}(h)$ to $f_{1}(h)$. By
diffeomorphism invariance of the Seiberg–Witten equations, the Seiberg–Witten
moduli space for $h_{t}$ is empty for each $t\in[0,1]$, hence
$SW_{\mathfrak{s}}(f_{0}(h),f_{1}(h))=0$ and
$\phi_{h}(f_{1})=\phi_{h}(f_{2})$. This shows that $\phi$ only depends on the
underlying isotopy class and so we may view it as a map
$\phi_{h}:M(X)\to\widehat{\mathbb{Z}}[\mathcal{S}(X)]$.

######  Proposition 2.1.

The map $\phi_{h}:M(X)\to\widehat{\mathbb{Z}}[\mathcal{S}(X)]$ is a twisted
homomorphism. Furthermore, the underlying cohomology class $[\phi_{h}]\in
H^{1}(M(X);\widehat{\mathbb{Z}}[\mathcal{S}(X)])$ does not depend on the
choice of $h\in\Pi^{reg}$.

###### Proof.

Let $f,g\in M(X)$. Then

$\displaystyle\phi_{h}(gf)(\mathfrak{s})$
$\displaystyle=SW_{\mathfrak{s}}(h,gf(h))$
$\displaystyle=SW_{\mathfrak{s}}(h,g(h))+SW_{\mathfrak{s}}(g(h),g(f(h)))$
$\displaystyle=SW_{\mathfrak{s}}(h,g(h))+sgn_{+}(g)SW_{g^{-1}\mathfrak{s}}(h,f(h))$
$\displaystyle=\phi_{h}(g)(\mathfrak{s})+sgn_{+}(g)\phi_{h}(f)(g^{-1}\mathfrak{s})$
$\displaystyle=(\phi_{h}(g)+(g\phi_{h})(f))(\mathfrak{s}).$

Hence $\phi_{h}$ is a twisted homomorphism. Next we show that the underlying
cohomology class of $\phi_{h}$ does not depend on the choice of $h$. Let
$h^{\prime}\in\Pi^{reg}$ be another generic pair. Choose a path $h_{t}$ from
$h=h_{0}$ to $h^{\prime}=h_{1}$. Then

$\displaystyle\phi_{h^{\prime}}(f)(\mathfrak{s})$
$\displaystyle=SW_{\mathfrak{s}}(h_{1},f(h_{1}))$
$\displaystyle=SW_{\mathfrak{s}}(h_{0},f(h_{0}))-SW_{\mathfrak{s}}(h_{0},h_{1})+SW_{\mathfrak{s}}(f(h_{0}),f(h_{1}))$
$\displaystyle=\phi_{h}(f)(\mathfrak{s})+sgn_{+}(f)SW_{f^{-1}\mathfrak{s}}(h_{0},h_{1})-SW_{\mathfrak{s}}(h_{0},h_{1})$
$\displaystyle=\phi_{h}(f)(\mathfrak{s})+(fm-m)(\mathfrak{s})$

where $m(\mathfrak{s})=SW_{\mathfrak{s}}(h_{0},h_{1})$. Hence $\phi_{h}$ and
$\phi_{h^{\prime}}$ define the same cohomology class. ∎

######  Definition 2.2.

We denote by

$SW\in H^{1}(M(X);\widehat{\mathbb{Z}}[\mathcal{S}(X)])$

the cohomology class $SW=[\phi_{h}]$ for any $h\in\Pi^{reg}$.

Let $M_{+}(X)$ denote the subgroup of $M(X)$ consisting of all $f\in M(X)$ for
which $sgn_{+}(f)=1$. Then $M_{+}(X)$ has index $1$ or $2$ in $M(X)$. Observe
that $\widehat{\mathbb{Z}}|_{M_{+}(X)}=\mathbb{Z}$ and thus $SW|_{M_{+}(X)}\in
H^{1}(M(X);\mathbb{Z}[\mathcal{S}(X)])$. From $SW|_{M_{+}(X)}$ we can extract
$\mathbb{Z}$-valued cohomology classes as follows: let
$\mathcal{O}\subseteq\mathcal{S}(X)$ be a $\Gamma(X)$-invariant subset of
$\mathcal{S}(X)$. Then we obtain a morphism
$p_{\mathcal{O}}:\mathbb{Z}[\mathcal{S}(X)]\to\mathbb{Z}$ given by
$\phi\mapsto\sum_{\mathfrak{s}\in\mathcal{O}}\phi(\mathfrak{s})$. We define
$SW_{\mathcal{O}}\in H^{1}(M_{+}(X);\mathbb{Z})$ by setting
$SW_{\mathcal{O}}=p_{\mathcal{O}}(SW|_{M_{+}(X)})$. From this definition it
follows that

$SW_{\mathcal{O}}|_{T(X)}=\sum_{\mathfrak{s}\in\mathcal{O}}SW_{\mathfrak{s}}.$

Furthermore, for any $f\in M_{+}(X)$, we have

$SW_{\mathcal{O}}(f)=\sum_{\mathfrak{s}\in\mathcal{O}}SW_{\mathfrak{s}}(h,f(h))$

where $h\in\Pi^{reg}$.

######  Remark 2.3.

In [26], Ruberman defined an invariant $SW_{\mathfrak{s}}^{tot}$ which is
similar to the definition of $SW_{\mathcal{O}}$ given above. Namely
$SW_{\mathfrak{s}}^{tot}:M_{+}(X)\to\mathbb{Z}$ is given by
$SW_{\mathfrak{s}}^{tot}(f)=\sum_{\mathfrak{s}^{\prime}}SW_{\mathfrak{s}^{\prime}}(h,f(h))$
where the sum is over all spinc-structures $\mathfrak{s}^{\prime}$ such that
$\mathfrak{s}^{\prime}=f^{m}(\mathfrak{s})$ for some $m\in\mathbb{Z}$. However
this invariant is not a group homomorphism and behaves in a complicated manner
with respect to composition (see [26, Theorem 3.4]). For this reason, we find
it more useful to work with the invariants $SW_{\mathcal{O}}$.

Let $\mathfrak{s}\in\mathcal{S}(X)$ be a spinc-structure and let $f\in M(X)$.
Suppose that $f$ preserves $\mathfrak{s}$. If $sgn_{+}(f)=1$, then the
families Seiberg—Witten moduli space for the mapping cylinder of $f$ with
spinc-structure $\mathfrak{s}$ is oriented, and we obtain an integer families
Seiberg–Witten invariant $SW_{\mathfrak{s}}(f)\in\mathbb{Z}$. It is given by
$SW_{\mathfrak{s}}(f)=SW_{\mathfrak{s}}(h,f(h))$, for any $h\in\Pi^{reg}$. If
$sgn_{+}(f)=-1$ then there is no natural choice of orientation on the families
moduli space, hence we only get a mod $2$ invariant
$SW_{\mathfrak{s}}(f)\in\mathbb{Z}_{2}$ which is given by
$SW_{\mathfrak{s}}(f)=SW_{\mathfrak{s}}(h,f(h))\;({\rm mod}\;2)$, for any
$h\in\Pi^{reg}$ (the value of $SW_{\mathfrak{s}}(h,f(h))$ depends on $h$, but
its mod $2$ reduction does not).

We will make use of the following special case of the gluing formula of [3]:

######  Proposition 2.4.

Suppose that $X=X^{\prime}\\#(S^{2}\times S^{2})$, where
$b_{+}(X^{\prime})>1$. Let $\mathfrak{s}^{\prime}$ be a spinc-structure on
$X^{\prime}$ with $d(\mathfrak{s}^{\prime})=0$ and let $\mathfrak{s}_{0}$
denote the spin structure on $S^{2}\times S^{2}$. Let
$\mathfrak{s}=\mathfrak{s}^{\prime}\\#\mathfrak{s}_{0}$. Let $f^{\prime}$ be a
diffeomorphism on $X^{\prime}$ that preserves $\mathfrak{s}^{\prime}$ and
$\rho$ a diffeomorphism of $S^{2}\times S^{2}$. Suppose that $f^{\prime}$ is
trivial in a neighbourhood of a point $x_{1}\in X^{\prime}$ and that $\rho$ is
trivial in a neighbourhood of a point $x_{2}\in S^{2}\times S^{2}$. Set
$f=f^{\prime}\\#\rho$, where the connected sum is performed by removing balls
around $x_{1}$ and $x_{2}$ and identifying boundaries. Then:

  * (1)

If $sgn_{+}(\rho)=1$, then $SW_{\mathfrak{s}}(f)=0\;({\rm mod}\;2)$.

  * (2)

If $sgn_{+}(\rho)=-1$, then
$SW_{\mathfrak{s}}(f)=SW(X^{\prime},\mathfrak{s}^{\prime})\;({\rm mod}\;2)$,
where $SW(X^{\prime},\mathfrak{s}^{\prime})$ denotes the ordinary
Seiberg–Witten invariant of $(X^{\prime},\mathfrak{s}^{\prime})$.

##  3\. Non-finitely generated mapping class groups

In this section we prove that $M(X)$ is not finitely generated for certain
$4$-manifolds.

######  Theorem 3.1.

Let $X=2n\mathbb{CP}^{2}\\#10n\overline{\mathbb{CP}^{2}}$, where $n\geq 3$ is
odd. Then $M(X)$ is not finitely generated. More precisely, the following
holds:

  * (1)

There is a surjective homomorphism $\Phi:M_{+}(X)\to\mathbb{Z}^{\infty}$ from
$M_{+}(X)$ to $\mathbb{Z}^{\infty}$, where $\mathbb{Z}^{\infty}$ denotes a
free abelian group of countably infinite rank.

  * (2)

The mod $2$ reduction of $\Phi$ extends to a homomorphism
$\Phi:M(X)\to\mathbb{Z}_{2}^{\infty}$.

  * (3)

$M_{+}(X)$ has index $2$ in $M(X)$.

  * (4)

The short exact sequence $1\to M_{+}(X)\to M(X)\to\mathbb{Z}_{2}\to 0$ splits.

###### Proof.

First note that $X=X^{\prime}\\#(S^{2}\times S^{2})$, where
$X^{\prime}=(2n-1)\mathbb{CP}^{2}\\#(10n-1)\overline{\mathbb{CP}^{2}}$. It
follows from [31] that $\Gamma(X)=Aut(H^{2}(X;\mathbb{Z}))$. Observe that
$d(\mathfrak{s})=(c(\mathfrak{s})^{2}+8n)/4-2n-1=c(\mathfrak{s})^{2}/4-1$.
Hence $d(\mathfrak{s})=-1$ if and only if $c(\mathfrak{s})^{2}=0$. For each
$k\geq 1$, let $\mathcal{O}_{k}\subset\mathcal{S}(X)$ denote the set of spinc-
structures whose characteristic $c$ satisfies, $c\neq 0$, $c^{2}=0$, and $c$
is $k$ times a primitive element. We will show that the homomorphism

$\Phi=\bigoplus_{q=1}^{\infty}SW_{\mathcal{O}_{nq-q-1}}:M_{+}(X)\to\mathbb{Z}^{\infty}$

surjects to a subgroup of $\mathbb{Z}^{\infty}$ of countably infinite rank.

The decomposition $X=X^{\prime}\\#(S^{2}\times S^{2})$ yields an orthogonal
decomposition $L=L^{\prime}\oplus H$, where $L,L^{\prime}$ are the
intersection forms of $X,X^{\prime}$ and $H=H^{2}(S^{2}\times
S^{2};\mathbb{Z})$ is the hyperbolic lattice. Any characteristic $c\in L$
decomposes as $c=(c_{1},c_{2})$, where $c_{1}\in L^{\prime},c_{2}\in H$ are
characteristic. The intersection form $L^{\prime}$ is odd, hence $c_{1}\neq
0$.

We will partition $\mathcal{O}_{k}$ into two types of subsets:

  * (1)

Subsets $\\{\mathfrak{s},\bar{\mathfrak{s}}\\}$, where
$\mathfrak{s}=\mathfrak{s}^{\prime}\\#\mathfrak{s}_{0}$ and
$c(\mathfrak{s}_{0})=0$.

  * (2)

Subsets
$\\{\mathfrak{s}_{1},\mathfrak{s}_{2},\bar{\mathfrak{s}}_{1},\bar{\mathfrak{s}}_{2}\\}$,
where $\mathfrak{s}_{1}=\mathfrak{s}^{\prime}\\#\mathfrak{s}^{\prime\prime}$,
$\mathfrak{s}_{2}=\mathfrak{s}^{\prime}\\#\bar{\mathfrak{s}}^{\prime\prime}$
and where $c(\mathfrak{s}^{\prime\prime})\neq 0$.

Since $b_{+}(X)=2n=2\;({\rm mod}\;4)$, we have
$SW_{\mathfrak{s}}(t)=SW_{\bar{\mathfrak{s}}}(t)$ for all $t\in T(X)$. Hence a
subset of type of $\mathcal{O}_{k}$ of type (1) will contribute
$2SW_{\mathfrak{s}}(t)$ to $SW_{\mathcal{O}_{k}}(t)$ and a subset of type (2)
will contribute $2(SW_{\mathfrak{s}_{1}}(t)+SW_{\mathfrak{s}_{2}}(t))$.

Let $E(n)_{q}$ be the elliptic surface obtained from $E(n)$ by performing a
logarithmic transform of multiplicity $q\geq 1$. Since $n$ is odd, $E(n)_{q}$
is not spin and its intersection form is diagonal of signature $(2n-1,10n-1)$.
Hence $E(n)_{q}$ has the same intersection lattice as $X^{\prime}$.
Furthermore, we have that $E(n)_{q}\\#(S^{2}\times S^{2})$ is diffeomorphic to
$2n\mathbb{CP}^{2}\\#10n\overline{\mathbb{CP}^{2}}=X=X^{\prime}\\#(S^{2}\times
S^{2})$ [12, Page 320]. So there is an orientation preserving diffeomorphism
$\psi_{q}:E(n)_{q}\\#(S^{2}\times S^{2})\to X^{\prime}\\#(S^{2}\times S^{2})$.
We can choose $\psi_{q}$ so that it respects the decomposition
$H^{2}(E(n)_{q};\mathbb{Z})\oplus H^{2}(S^{2}\times S^{2};\mathbb{Z})\to
H^{2}(X^{\prime};\mathbb{Z})\oplus H^{2}(S^{2}\times S^{2};\mathbb{Z})$. To
see this, first let $\psi^{\prime}_{q}:E(n)_{q}\\#(S^{2}\times S^{2})\to
X^{\prime}\\#(S^{2}\times S^{2})$ be any diffeomorphism. Then by [31], every
isometry of $H^{2}(X^{\prime}\\#(S^{2}\times S^{2});\mathbb{Z})$ can be
realised by a diffeomorphism. Hence, composing $\psi^{\prime}_{q}$ with a
suitable diffeomorphism of $X$, we obtain the desired diffeomorphism
$\psi_{q}$.

Let $\rho$ be a diffeomorphism of $S^{2}\times S^{2}$ which acts as $-1$ on
$H^{2}(S^{2}\times S^{2};\mathbb{Z})$ and is trivial in a neighbourhood of
some point. Such diffeomorphisms can easily be constructed, for example, take
the product $r\times r$ of two copies of a reflection on $S^{2}$ and then
isotopy it to act trivially in a neighbourhood of a point. Define a
diffeomorphism $f_{0}\in M(X)$ by setting $f_{0}=id_{X^{\prime}}\\#\rho$,
where the connected sum is performed by removing a ball of $(S^{2}\times
S^{2})$ on which $\rho$ acts trivially. For each $q\geq 1$, define a
diffeomorphism $f_{q}\in M(X)$ by setting
$f_{q}=\psi_{q}\circ(id_{E(n)_{q}}\\#\rho)\circ\psi_{q}^{-1}$. Note that
$sgn_{+}(f_{0})=sgn_{+}(f_{q})=-1$. Also $t_{q}=f_{q}\circ f_{0}\in T(X)$.

We claim that $SW_{\mathcal{O}_{nq-q-1}}(t_{q})=2\;({\rm mod}\;4)$ and that
$SW_{\mathcal{O}_{nq^{\prime}-q^{\prime}-1}}(t_{q})=0\;({\rm mod}\;4)$ for all
$q^{\prime}>q$. This implies that the elements $\\{\Phi(t_{q})\\}_{q\geq 1}$
are linearly independent. To see this, first note that $\Phi(t_{q})\in
2\mathbb{Z}^{\infty}$ by charge conjugation symmetry and that the image of
$\\{\Phi(t_{q})/2\\}_{q\geq 1}$ is a basis for $\mathbb{Z}_{2}^{\infty}$, by
the above claim. Now suppose
$n_{1}\Phi(t_{1})+n_{2}\Phi(t_{2})+\cdots+n_{r}\Phi(t_{r})=0$ for some
$n_{1},\dots,n_{r}\in\mathbb{Z}$, not all zero. Without loss of generality we
can assume that $gcd(n_{1},\dots,n_{r})=1$. Then
$n_{1}\Phi(t_{1})/2+\cdots+n_{r}\Phi(t_{r})/2=0$. But
$\\{\Phi(t_{q})/2\\}_{q_{\geq}1}$ are linearly independent in
$\mathbb{Z}_{2}^{\infty}$, so $n_{1},\dots,n_{r}$ are all even, a
contradiction.

Now we prove the claim. Let $q,q^{\prime}\geq 1$ and set
$k=nq^{\prime}-q^{\prime}-1$. By partitioning $\mathcal{O}_{k}$ into subsets
of type (1) and (2) as described above, we can then write
$SW_{\mathcal{O}_{k}}(t_{q})$ as a sum of contributions from sets of type (1)
and (2). Consider a contribution from a subset
$\\{\mathfrak{s},\bar{\mathfrak{s}}\\}$ of type (1). So
$\mathfrak{s}=\mathfrak{s}^{\prime}\\#\mathfrak{s}_{0}$. The contribution is
$2SW_{\mathfrak{s}}(t_{q})$. Since $f_{q}$ and $f_{0}$ both preserve
$\mathfrak{s}$, we have that

$\displaystyle SW_{\mathfrak{s}}(t_{q})$
$\displaystyle=SW_{\mathfrak{s}}(f_{q}\circ f_{0})$
$\displaystyle=SW_{\mathfrak{s}}(f_{q})+SW_{\mathfrak{s}}(f_{0})\;({\rm
mod}\;2)$ $\displaystyle=SW(E(n)_{q},\mathfrak{s}^{\prime})\;({\rm mod}\;2)$

where the last equality is due to Proposition 2.4. Let $f\in
H^{2}(E(n)_{q};\mathbb{Z})$ denote the class of the multiple fibre. Then $f$
is non-zero, primitive and $f^{2}=0$. From the well-known calculation of the
Seiberg–Witten invariants for elliptic surfaces [21, Chapter 3], we have that
$SW(E(n)_{q},\mathfrak{s}^{\prime})=0$ unless $c(\mathfrak{s}^{\prime})$ is a
multiple of $f$. More precisely, $SW(E(n)_{q},\mathfrak{s}^{\prime})$ is zero
unless $c(\mathfrak{s}^{\prime})=2(qk+a)f-(nq-q-1)f$, where $0\leq k\leq n-2$
and $0\leq a<q$. In such a case
$SW(E(n)_{q},\mathfrak{s}^{\prime})=(-1)^{k}\binom{n-2}{k}$. In particular,
$SW(E(n)_{q},\mathfrak{s}^{\prime})=\pm 1$ if
$c(\mathfrak{s}^{\prime})=(nq-q-1)f$ and
$SW(E(n)_{q},\mathfrak{s}^{\prime})=0$ if $c(\mathfrak{s})=uf$, $u>nq-q-1$.
Now suppose that $q^{\prime}\geq q$. We have that
$SW(E(n)_{q},\mathfrak{s}^{\prime})=0$ unless $c(\mathfrak{s})$ is a multiple
of $f$. But since
$\mathfrak{s}=\mathfrak{s}^{\prime}\\#\mathfrak{s}_{0}\in\mathcal{O}_{k}$,
this can only happen if $c(\mathfrak{s}^{\prime})=\pm kf$ (recall that
$\mathcal{O}_{k}$ is the set of spinc-structures whose characteristic $c$
satisfies, $c\neq 0$, $c^{2}=0$, and $c$ is $k$ times a primitive element).
Hence if $q^{\prime}>q$, then every pair
$\\{\mathfrak{s},\bar{\mathfrak{s}}\\}$ of type (1) contributes $0$ mod $4$ to
$SW_{\mathcal{O}_{k}}(t_{q})$. If $q^{\prime}=q$, then there is exactly one
pair $\\{\mathfrak{s},\bar{\mathfrak{s}}\\}$ of type (1) which contributes $2$
mod $4$ to $SW_{\mathcal{O}_{k}}(t_{q})$ and all other pairs are $0$ mod $4$.

Now consider the contribution from a set
$\\{\mathfrak{s}_{1},\mathfrak{s}_{2},\bar{\mathfrak{s}}_{1},\bar{\mathfrak{s}}_{2}\\}$
of type (2), where
$\mathfrak{s}_{1}=\mathfrak{s}^{\prime}\\#\mathfrak{s}^{\prime\prime}$,
$\mathfrak{s}_{2}=\mathfrak{s}^{\prime}\\#\bar{\mathfrak{s}}^{\prime\prime}$
and where $c(\mathfrak{s}^{\prime\prime})\neq 0$. As seen above, the
contribution is
$2(SW_{\mathfrak{s}_{1}}(t_{q})+SW_{\mathfrak{s}_{2}}(t_{q}))$. We will show
that $SW_{\mathfrak{s}_{1}}(t_{q})+SW_{\mathfrak{s}_{2}}(t_{q})=0\;({\rm
mod}\;2)$, hence all subsets of type (2) contribute $0$ mod $4$ and this will
prove the claim.

Observe that
$\mathfrak{s}_{2}=f_{q}(\mathfrak{s}_{1})=f_{0}(\mathfrak{s}_{1})$. Let
$h\in\Pi^{reg}$. Then

$\displaystyle SW_{\mathfrak{s}_{1}}(t_{q})$
$\displaystyle=SW_{\mathfrak{s}_{1}}(h,t_{q}(h))$
$\displaystyle=SW_{\mathfrak{s}_{1}}(h,f_{q}f_{0}(h))$
$\displaystyle=SW_{\mathfrak{s}_{1}}(h,f_{q}(h))+SW_{\mathfrak{s}_{1}}(f_{q}(h),f_{q}f_{0}(h))$
$\displaystyle=SW_{\mathfrak{s}_{1}}(h,f_{q}(h))-SW_{\mathfrak{s}_{2}}(h,f_{0}(h)).$

Similarly,
$SW_{\mathfrak{s}_{2}}(t_{q})=SW_{\mathfrak{s}_{2}}(h,f_{q}(h))-SW_{\mathfrak{s}_{1}}(h,f_{0}(h))$.
Hence

$\displaystyle SW_{\mathfrak{s}_{1}}(t_{q})+SW_{\mathfrak{s}_{2}}(t_{q})$
$\displaystyle=\left(SW_{\mathfrak{s}_{1}}(h,f_{q}(h))+SW_{\mathfrak{s}_{2}}(h,f_{q}(h))\right)$
$\displaystyle\quad\quad\quad-\left(SW_{\mathfrak{s}_{1}}(h,f_{0}(h))+SW_{\mathfrak{s}_{2}}(h,f_{0}(h))\right)$
$\displaystyle=\left(SW_{\mathfrak{s}_{1}}(h,f_{q}(h))-SW_{\mathfrak{s}_{1}}(f_{q}(h),f_{q}^{2}(h))\right)$
$\displaystyle\quad\quad\quad-\left(SW_{\mathfrak{s}_{1}}(h,f_{0}(h))-SW_{\mathfrak{s}_{1}}(f_{0}(h),f_{0}^{2}(h))\right)$
$\displaystyle=SW_{\mathfrak{s}_{1}}(h,f_{q}^{2}(h))-SW_{\mathfrak{s}_{1}}(h,f_{0}^{2}(h))$
$\displaystyle=SW_{\mathfrak{s}_{1}}(f_{q}^{2})-SW_{\mathfrak{s}_{1}}(f_{0}^{2})$
$\displaystyle=0\;({\rm mod}\;2)$

where the last line follows from Proposition 2.4. This proves the claim. Hence
we have proven that there exists a surjective homomorphism
$M_{+}(X)\to\mathbb{Z}^{\infty}$.

The fact that the mod $2$ reduction of $\Phi$ extends to $M(X)$ follows by
noting that $\frac{1}{2}SW_{\mathcal{O}_{k}}\in H^{1}(M_{+}(X);\mathbb{Z})$
extends to $H^{1}(M(X);\widehat{\mathbb{Z}})$ and then applying the mod $2$
reduction map $\widehat{\mathbb{Z}}\to\mathbb{Z}_{2}$.

The fact that $M_{+}(X)$ has index $2$ in $M(X)$ follows immediately from
$sgn_{+}(f_{0})=-1$. Furthermore, it is easy to see that $f_{0}^{2}$ is
isotopic to a Dehn twist on the neck of the connected sum
$X^{\prime}\\#(S^{2}\times S^{2})$. By using the circle action on $S^{2}\times
S^{2}$, it follows that this is isotopic to the identity. So $f_{0}$ defines a
splitting $\mathbb{Z}_{2}\to M(X)$ of the sequence $1\to M_{+}(X)\to
M(X)\to\mathbb{Z}_{2}\to 0$. ∎

##  4\. Split extensions

Let $L=\mathbb{Z}^{n}$ denote the standard diagonal lattice of rank $n$ with
orthonormal basis $e_{1},\dots,e_{n}$. The isometry group of $L$ is the
hyperoctahedral group $H_{n}$, which is also the Coxeter group of type
$BC_{n}$. This group is easily seen to be generated by permutations of
$e_{1},\dots,e_{n}$ and the reflections $r_{1},\dots,r_{n}$ in the hyperplanes
orthogonal to $e_{1},\dots,e_{n}$. The reflections generate a normal subgroup
isomorphic to $\mathbb{Z}_{2}^{n}$ and $H_{n}$ is the semidirect product
$H_{n}=S_{n}\ltimes\mathbb{Z}_{2}^{n}$.

Let $X=n\mathbb{CP}^{2}$ be the connected sum of $n$ copies of
$\mathbb{CP}^{2}$. Then $H^{2}(X;\mathbb{Z})$ is isomorphic to $L$ with
$e_{1},\dots,e_{n}$ corresponding to generators of
$H^{2}(\mathbb{CP}^{2};\mathbb{Z})$ for the $n$ summands of $X$. It is not
hard to see that $\Gamma(X)$ is equal to the full isometry group of $L$. This
will also follow from the construction given below.

######  Theorem 4.1.

For $X=n\mathbb{CP}^{2}$, there is a splitting $\Gamma(X)\to M(X)$.

###### Proof.

We will construct a smooth fibre bundle $\pi:E\to B$ with fibres diffeomorphic
to $X$ and such that the monodromy of the local system
$R^{2}\pi_{*}\mathbb{Z}$ yields an isomorphism $\rho:\pi_{1}(B)\to Aut(L)$.
The geometric monodromy of the family defines a lift
$\widetilde{\rho}:\pi_{1}(B)\to\pi_{0}(Diff(X))=M(X)$ of $\rho$ to $M(X)$.
Then $\widetilde{\rho}\circ\rho^{-1}:Aut(L)\to M(X)$ is the desired splitting
(this also proves that $\Gamma(X)\cong Aut(L)$).

Let $C_{m}$ be the space of $m$-tuples of distinct points on $S^{4}$. Clearly
$C_{1}$ is diffeomorphic to $S^{4}$. For $m>1$ there is a natural map
$C_{m}\to C_{m-1}$ given by forgetting the $m$-th point. This map gives
$C_{m}$ the structure of a fibre bundle over $C_{m-1}$ with fibre $F_{m-1}$
the $4$-sphere with $m-1$ points removed. Since
$\pi_{1}(F_{m-1})=\pi_{0}(F_{m-1})=1$, the long exact sequence in homotopy
yields an isomorphism $\pi_{1}(C_{m})\cong\pi_{1}(C_{m})$. Since
$\pi_{1}(C_{1})=\pi_{1}(S^{4})=1$, it follows by induction that
$\pi_{1}(C_{n})=1$ for all $n$.

Fix an orientation on $S^{4}$. Let $\widetilde{C}_{n}$ denote the space
consisting of an $n$-tuple $(x_{1},\dots,x_{n})$ of distinct points of $S^{4}$
together with an $n$-tuple $(I_{1},\dots,I_{n})$, where $I_{j}$ is a complex
structure on $T_{x_{j}}S^{4}$ which induces the given orientation. The
forgetful map $\widetilde{C}_{n}\to C_{n}$ which forgets the complex
structures $I_{1},\dots,I_{n}$ gives $\widetilde{C}_{n}$ the structure of a
fibre bundle over $C_{n}$. Since the space of complex structures on
$\mathbb{R}^{4}$ compatible with a given orientation is isomorphic to
$SO(4)/U(2)\cong S^{2}$, it follows that the fibres of $\widetilde{C}_{n}\to
C_{n}$ are isomorphic to $(S^{2})^{n}$. The long exact sequence in homotopy
implies that $\pi_{1}(\widetilde{C}_{n})=1$.

Consider the trivial family $\widetilde{E}_{0}=\widetilde{C}_{n}\times
S^{4}\to\widetilde{C}_{n}$. This family is equipped with $n$ sections
$s_{1},\dots,s_{n}$, where
$s_{j}((x_{1},\dots,x_{n}),(I_{1},\dots,I_{n}))=x_{j}$. The normal bundle of
$s_{j}$ is $N_{j}=T_{x_{j}}S^{4}$. The complex structure $I_{j}$ gives $N_{j}$
the structure of a complex rank $2$ vector bundle. Therefore, we can form a
family $\widetilde{E}_{n}$ by blowing up $\widetilde{E}_{0}$ along the
sections $s_{1},\dots,s_{n}$. More precisely, consider the fibre bundle over
$\widetilde{C}_{n}$ with fibre $\mathbb{CP}^{2}$ given by the projective
bundle $\mathbb{P}(\mathbb{C}\oplus N_{j})$. This bundle has a natural section
$t_{j}$ corresponding to the $1$-dimensional subbundle
$\mathbb{C}\subset\mathbb{C}\oplus N_{j}$. The normal bundle of $t_{j}$ is
isomorphic to $N_{j}$. Since $s_{j}$ and $t_{j}$ have isomorphic normal
bundles, we can attach $\mathbb{P}(\mathbb{C}\oplus N_{j})$ to
$\widetilde{E}_{0}$ by removing tubular neighbourhoods of $s_{j}$ and $t_{j}$
and identifying the boundaries.

The hyperoctahedral group $H_{n}=S_{n}\ltimes\mathbb{Z}_{2}^{n}$ acts on
$\widetilde{C}_{n}$ as follows. The permutation group $S_{n}$ acts by
permuting the points $x_{1},\dots,x_{n}$ as well as the corresponding complex
structures $I_{1},\dots,I_{n}$. The group $\mathbb{Z}_{2}^{n}$ is generated by
reflections $r_{1},\dots,r_{n}$. We let $r_{j}$ act by fixing
$x_{1},\dots,x_{n}$, sending $I_{j}$ to $-I_{j}$ and fixing the remaining
complex structures. The action of $H_{n}$ is free and we let
$B=\widetilde{C}_{n}/H_{n}$ be the quotient. It follows that $\pi_{1}(B)\cong
H_{n}$. The action of $H_{n}$ on $\widetilde{C}_{n}$ lifts to an action on
$\widetilde{E}_{0}=\widetilde{C}_{n}\times S^{4}$ which acts trivially on the
$S^{4}$ factor. It is not hard to see that $\widetilde{E}_{n}$ can be
constructed in such a way that the action of $H_{n}$ extends to it. Now let
$E=\widetilde{E}_{n}/H_{n}$. This is a family $\pi:E\to B$ over $B$ with
fibres diffeomorphic to $n\mathbb{CP}^{2}$. The monodromy of the local system
$R^{2}\pi_{*}\mathbb{Z}$ is easily seen to yield an isomorphism
$\rho:\pi_{1}(B)\to Aut(L)$. As explained above, this yields a splitting
$\Gamma(X)\to M(X)$. ∎

##  5\. Non-split extensions

Let $X$ be a compact, simply connected, smooth $4$-manifold with intersection
form $L=H^{2}(X;\mathbb{Z})$. Let $\Sigma$ be a compact surface (orientable or
non-orientable). Suppose that $\rho:\pi_{1}(\Sigma)\to\Gamma(X)$ is a
homomorphism. Letting $\Gamma(X)$ act on
$L_{\mathbb{R}}=\mathbb{R}\otimes_{\mathbb{Z}}L$, we obtain a flat vector
bundle $H_{\rho}\to\Sigma$ which has a covariantly constant bilinear form of
signature $(b_{+}(X),b_{-}(X))$. Let $H^{+}_{\rho}$ denote a maximal positive
definite subbundle of $H_{\rho}$. The choice of subbundle $H^{+}_{\rho}$ is
not unique, but all such subbundles are isomorphic. In particular the
Stiefel–Whitney classes $w_{j}(H^{+}_{\rho})\in H^{j}(\Sigma;\mathbb{Z}_{2})$
depend only on $\rho$.

######  Theorem 5.1.

Let $X$ be a compact, simply connected, smooth $4$-manifold with $b_{+}(X)=2$
and let $\rho:\pi_{1}(\Sigma)\to\Gamma(X)$ be a homomorphism. Suppose that
$w_{2}(H^{+}_{\rho})\neq 0$ and suppose there exists a characteristic $c\in L$
which is $\rho$-invariant and satisfies $c^{2}>\sigma(X)$. Then $\rho$ does
not lift to a homomorphism $\widetilde{\rho}:\Gamma(X)\to M(X)$.

###### Proof.

Consider first the case that $\Sigma$ is orientable of genus $g$. Recall that
$\pi_{1}(\Sigma)$ admits a presention

$\pi_{1}(\Sigma)=\langle
a_{1},b_{1},\dots,a_{g},b_{g}\;|\;[a_{1},b_{1}]\cdots[a_{g},b_{g}]\rangle$

Suppose that $\rho$ admits a lift $\widetilde{\rho}:\pi_{1}(\Sigma)\to M(X)$.
Let $\alpha_{j}$ be a diffeomorphism of $X$ whose isotopy class is
$\widetilde{\rho}(a_{j})$ and let $\beta_{j}$ be a diffeomorphism of $X$ whose
isotopy class is $\widetilde{\rho}(b_{j})$. Then
$[\alpha_{1},\beta_{1}]\cdots[\alpha_{g},\beta_{g}]$ is isotopic to the
identity. The surface $\Sigma$ can be constructed from a wedge of $2g$ circles
by attaching a $2$-cell whose attaching map represents
$[a_{1},b_{1}]\cdots[a_{g},b_{g}]$ in $\pi_{1}(\vee_{i=1}^{2g}S^{1})$. We will
construct a smooth family $\pi:E\to\Sigma$ whose fibres are diffeomorphic to
$X$ as follows. Over the $1$-skeleton $\vee_{i=1}^{2g}S^{1}$, we take the
wedge sum of mapping cylinders associated to the diffeomorphisms
$\alpha_{1},\beta_{1},\dots,\alpha_{g},\beta_{g}$. A choice of isotopy from
$[\alpha_{1},\beta_{1}]\cdots[\alpha_{g},\beta_{g}]$ to the identity allows us
to extend this family over the $2$-cell and in this way we obtain the family
$\pi:E\to\Sigma$. By construction, the local system $R^{2}\pi_{*}\mathbb{Z}$
has monodromy $\rho$. Now suppose that $w_{2}(H^{+}_{\rho})\neq 0$ and that
there exists a characteristic $c\in L$ which is $\rho$-invariant and satisfies
$c^{2}>\sigma(X)$. This contradicts [2, Theorem 1.1], hence $\rho$ does not
lift to $M(X)$.

The case that $\Sigma$ is non-orientable is similar. Recall that
$\pi_{1}(\Sigma)$ admits a presentation

$\pi_{1}(\Sigma)=\langle a_{1},\dots,a_{k}\;|\;a_{1}^{2}\cdots
a_{k}^{2}\rangle$

where $\Sigma$ has Euler characteristic $1-k$. If $\rho$ lifts to a
homomorphism $\widetilde{\rho}:\pi_{1}(\Sigma)\to M(X)$, then we choose
diffeomorphisms $\alpha_{1},\dots,\alpha_{k}$ where the isotopy class of
$\alpha_{j}$ is $\widetilde{\rho}(a_{j})$. Then
$\alpha_{1}^{2}\cdots\alpha_{k}^{2}$ is isotopic to the identity. A choice of
such an isotopy allows us to construct a smooth family $\pi:E\to\Sigma$ with
fibres diffeomorphic to $X$ and such that the monodromy of
$R^{2}\pi_{*}\mathbb{Z}$ is $\rho$. As before, this contradicts [2, Theorem
1.1], hence $\rho$ does not lift to $M(X)$. ∎

######  Remark 5.2.

A similar argument was used in [17] to prove the non-triviality of $T(X)$ for
$X=2\mathbb{CP}\\#n\overline{\mathbb{CP}^{2}}$, $n\geq 11$.

######  Corollary 5.3.

Let $X=(S^{2}\times S^{2})\\#X^{\prime}$, where $b_{+}(X^{\prime})=1$,
$b_{-}(X^{\prime})\geq 10$. Then there does not exist a splitting
$\Gamma(X)\to M(X)$.

###### Proof.

Let $L=H^{2}(X;\mathbb{Z})$, $L^{\prime}=H^{2}(X^{\prime};\mathbb{Z})$ denote
the intersection lattices of $X$ and $X^{\prime}$ and let $H=H^{2}(S^{2}\times
S^{2}\;\mathbb{Z})$. So $L\cong H\oplus L^{\prime}$. Since $X=(S^{2}\times
S^{2})\\#X^{\prime}$, we have that $\Gamma(X)=Aut(H^{2}(X;\mathbb{Z}))$ by
[31]. Let $x,y\in H$ be a basis with $x^{2}=y^{2}=0$, $\langle x,y\rangle=1$.
Since $b_{+}(X^{\prime})=1$ and $\sigma(X^{\prime})<0$, it follows that
$X^{\prime}$ is not spin and it follows that $L^{\prime}\cong H^{\prime}\oplus
E_{8}\oplus L^{\prime\prime}$, where $H^{\prime}$ has basis
$x^{\prime},y^{\prime}$, $(x^{\prime})^{2}=(y^{\prime})^{2}=0$, $\langle
x^{\prime},y^{\prime}\rangle=1$, $E_{8}$ is the negative definite $E_{8}$
lattice and $L^{\prime\prime}$ is a diagonal lattice with basis
$e_{1},\dots,e_{m}$, where $m=b_{-}(X^{\prime})-9$, with $e_{i}^{2}=-1$ for
all $i$, $\langle e_{i},e_{j}\rangle=0$ for $i\neq j$.

Let $u=x+y$, $v=x^{\prime}+y^{\prime}$. Then $u^{2}=v^{2}=2$, $\langle
u,v\rangle=0$. Let $r_{u}$ be the reflection $r_{u}(x)=x+\langle x,u\rangle u$
and define $r_{v}$ similarly. Consider the isometry $f(x)=r_{u}r_{v}(x)$. Then
$f\in\Gamma(X)$ and $f^{2}=1$. Hence we obtain a homomorphism
$\rho:\pi_{1}(\mathbb{RP}^{2})\to\Gamma(X)$ which sends the generator of
$\pi_{1}(\mathbb{RP}^{2})$ to $f$. Since $f$ acts as $-1$ on the maximal
positive definite subspace of $H^{2}(X;\mathbb{R})$ spanned by $u$ and $v$, we
have that $w_{2}(H^{+}_{\rho})\neq 0$. Let $c=e_{1}+\dots+e_{m}$. Then $c$ is
a characteristic that $c^{2}>\sigma(X)$ and $\langle c,u\rangle=\langle
c,v\rangle=0$. Then $r_{u}(c)=r_{v}(c)=c$ and hence $f(c)=c$. Then Theorem 5.1
implies that $\rho$ does not lift to $M(X)$. Hence the subgroup $\langle
f\rangle\subseteq\Gamma(X)$ does not lift to $M(X)$, in particular, there does
not exist a splitting $\Gamma(X)\to M(X)$. ∎

######  Remark 5.4.

Note that in the above proof $u\in L$ can be realised by an embedded
$2$-sphere in $X$, namely the diagonal $S^{2}\subset S^{2}\times S^{2}$. By a
result of Seidel [27], it follows that $r_{u}$ can be lifted to an element
$\hat{r}_{u}\in M(X)$ of order $2$. Since $\Gamma(X)=Aut(L)$, it follows that
there is a diffeomorphism of $X$ sending $u$ to $v$. It follows that $v$ can
also be realised by an embedded $2$-sphere and hence $r_{v}$ can be lifted to
an element $\hat{r}_{v}\in M(X)$ of order $2$. Then $\hat{r}_{u}\hat{r}_{v}$
is a lift of $f$ to $M(X)$. If $u,v$ could be represented by disjoint embedded
$2$-spheres, then $\hat{r}_{u},\hat{r}_{v}$ commute (since
$\hat{r}_{u},\hat{r}_{v}$ can be constructed to have disjoint supports) and
then $\hat{r}_{u}\hat{r}_{v}$ would be an involutive lift of $f$,
contradicting the corollary above. We deduce that $u,v$ can be represented by
embedded spheres, but they can not be represented by disjoint embedded spheres
even though $\langle u,v\rangle=0$.

##  6\. Nielsen realisation

As explained in the introduction, the following result shows that the Nielsen
realisation problem fails for
$X=X^{\prime}\\#p\mathbb{CP}^{2}\\#q\overline{\mathbb{CP}^{2}}$ whenever
$p+q\geq 4$.

######  Theorem 6.1.

Let $X=X^{\prime}\\#p\mathbb{CP}^{2}\\#q\overline{\mathbb{CP}^{2}}$ where
$X^{\prime}$ is a compact, smooth, simply-connected $4$-manifold and $p+q\geq
4$. Then $M(X)$ contains a subgroup isomorphic to $\mathbb{Z}_{2}^{4}$ which
can not be lifted to $Diff(X)$.

###### Proof.

To each summand of $\mathbb{CP}^{2}$ or $\overline{\mathbb{CP}^{2}}$ in $X$,
there is a corresponding embedded $2$-sphere of self-intersection $\pm 1$. Let
$E_{1},\dots,E_{4}$ be any four of them. Let $t_{1},\dots,t_{4}\in M(X)$ be
the corresponding Dehn twists around these spheres. Then $t_{1},\dots,t_{4}$
are involutons [27] and they commute since $E_{1},\dots,E_{4}$ are disjoint.
Hence the group $G\subseteq M(X)$ generated by $t_{1},\dots,t_{4}$ is
isomorphic to $\mathbb{Z}_{2}^{4}$. Now suppose that $G$ can be lifted to
$Diff(X)$. Hence we can find commuting diffeomorphisms
$\sigma_{1},\dots,\sigma_{4}$ such that the isotopy class of $\sigma_{i}$ is
$t_{i}$.

Consider the fixed point set $F$ of $\sigma_{1}$. Since $\sigma_{1}$ acts on
$H^{2}(X;\mathbb{Z})$ as a reflection in a $\pm 1$ sphere, it follows from [8,
Proposition 2.4] that $F$ consists of a single copy of $\mathbb{RP}^{2}$,
together with some isolated points and some $2$-spheres. Let $G_{0}$ be the
subgroup of $G$ generated by $\sigma_{2},\sigma_{3},\sigma_{4}$. Since
$\sigma_{2},\sigma_{3},\sigma_{4}$ commute with $\sigma_{1}$, they act on $F$
and in particular must send the copy of $\mathbb{RP}^{2}$ to itself. Hence
$G_{0}$ acts on $\mathbb{RP}^{2}$. We claim that the action is effective. To
see this, suppose $f\in G_{0}$ fixes $\mathbb{RP}^{2}$ pointwise. Since $f$ is
an orientation preserving involution, it must act on the normal bundle of
$\mathbb{RP}^{2}$ in $X$ as either the identity or multiplication by $-1$.
Hence either $f$ or $\sigma_{1}f$ fixes $\mathbb{RP}^{2}$ pointwise and acts
trivially on the normal bundle. For a diffeomorphism of finite order, this can
only happen if the diffeomorphism is the identity. Hence $f$ or $\sigma_{1}f$
is the identity, but $f\in G_{0}$, so $f\neq\sigma_{1}$ and it must be that
$f$ is the identity.

A finite group action on $\mathbb{RP}^{2}$ by diffeomorphisms is conjugate to
a subgroup of $PO(3)\cong SO(3)$. Since $G_{0}$ is abelian, its action on the
standard representation of $SO(3)$ can be simultaneously diagonalised, so
$G_{0}$ is isomorphic to a subgroup of
$\\{diag(\epsilon_{1},\epsilon_{2},\epsilon_{3})\in
SO(3)\\}\cong\mathbb{Z}_{2}^{2}$, which is impossible since $|G_{0}|=8$. So
$G$ does not lift to $Diff(X)$. ∎

##  7\. Boundary Dehn twists

Let $X^{(n)}$ be obtained from $X$ by removing $n$ disjoint open balls. So
$X^{(n)}$ is a compact $4$-manifold with boundary consisting of $n$ copies of
$S^{3}$. Let $Diff(X^{(n)},\partial X^{(n)})$ denote the group of
diffeomorphisms of $X^{(n)}$ which are the identity in a neighbourhood of the
boundary. Let $M_{n}(X)=\pi_{0}(Diff(X^{(n)},\partial X^{(n)})$ denote the
group of components of $Diff(X^{(n)},\partial X^{(n)})$. It is known that the
map $M_{n}(X)\to M(X)$ is surjective and that the kernel is generated by Dehn
twists on the boundary components [11]. More precisely, if $S^{3}\to X^{(n)}$
is a boundary component, then $X^{(n)}$ has a tubular neighbourhood
$[0,1]\times S^{3}\to X$. The Dehn twist on this boundary component is defined
by taking a non-trivial loop $\alpha_{t}:[0,1]\to SO(4)$ and defining
$\phi:[0,1]\times S^{3}\to[0,1]\times S^{3}$ by $\phi(t,x)=(t,\alpha_{t}(x))$,
where $SO(4)$ acts on $S^{3}$ in the standard way. We assume that $\alpha_{t}$
is smooth and equals the identity in a neighbourhood of $\\{0,1\\}$, hence
$\phi$ can be extended to an element of $Diff(X^{(n)},\partial X^{(n)})$ by
taking it to be the identity outside of the tubular neighbourhood.

Let $K_{n}(X)$ denote the kernel of $M_{n}(X)\to M(X)$, so we have an short
exact sequence

$1\to K_{n}(X)\to M_{n}(X)\to M(X)\to 1.$

Furthermore, we have a surjection $\mathbb{Z}_{2}^{n}\to K_{n}(X)$ given by
Dehn twists on the boundary components ([11, Proposition 3.1]).

######  Proposition 7.1.

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

  * (1)

If $X$ is spin, then $K_{n}(X)$ is either $\mathbb{Z}_{2}^{n}$ or
$\mathbb{Z}_{2}^{n}/\Delta\mathbb{Z}_{2}$, for all $n$, where
$\Delta\mathbb{Z}_{2}$ is the diagonal copy of $\mathbb{Z}_{2}$.

  * (2)

If $X$ is not spin, then $K_{n}(X)=0$ for all $n$, hence $M_{n}(X)\cong M(X)$.

###### Proof.

Part (1) is given by [11, Corollary 2.5] and part (2) by [22, Corollary A.5].
∎

In light of Proposition 7.1, boundary Dehn twists are only interesting when
$X$ is spin. In this case, we either have $K_{n}(X)\cong\mathbb{Z}_{2}^{n}$ or
$K_{n}(X)\cong\mathbb{Z}_{2}^{n}/\Delta\mathbb{Z}_{2}$. Which of these two
cases occurs is completely determined by the $n=1$ case. We consider this case
in more detail. There is a Serre fibration

(7.1) $Diff(X^{(1)},\partial X^{(1)})\to Diff(X)\to Emb\left(D^{4},X\right)$

where $Emb\left(D^{4},X\right)$ is the space of embeddings of a disc in $X$
which can be extended to a diffeomorphism. Furthermore, there is a homotopy
equivalence $Emb\left(D^{4},X\right)\cong F(X)$, where $F(X)$ is the oriented
frame bundle of $X$ [11]. Since $X$ is simply-connected and spin,
$\pi_{1}(F(X))\cong\mathbb{Z}_{2}$. Then the fibration (7.1) induces an exact
sequence

$\pi_{1}(Diff(X))\buildrel\phi\over{\longrightarrow}\mathbb{Z}_{2}\to
M_{1}(X)\to M(X)\to 1.$

In the absence of a metric we can define the spin bundle of $X$ to be the
universal cover $\widetilde{F}(X)\to F(X)$ of $F(X)$. Since
$\pi_{1}(F(X))\cong\mathbb{Z}_{2}$, $\widetilde{F}(X)\to F(X)$ is a double
cover. Since $Emb\left(D^{4},X\right)\cong F(X)$, it follows that $\phi$ is
the map that measures whether or not a loop of diffeomorphisms of $X$ lifts to
a loop in the spin bundle of $X$. This leads to an alternative description of
the group $M_{1}(X)$ when $X$ is spin. Let $SpinDiff(X)$ be the group whose
elements consist of a diffeomorphism $f\in Diff(X)$ and a choice of lift of
$f_{*}:F(X)\to F(X)$ to $\widetilde{F}(X)$. We have a short exact sequence
$1\to\mathbb{Z}_{2}\to SpinDiff(X)\to Diff(X)\to 1$ and the connecting
homomorphism $\pi_{1}(Diff(X))\to\mathbb{Z}_{2}$ is precisely $\phi$. The map
$Diff(X^{(1)},\partial X^{(1)})\to Diff(X)$ admits a lift
$Diff(X^{(1)},\partial X^{(1)})\to SpinDiff(X)$ by taking the unique lift
which is the identity over $\partial X^{(1)}$. We then have a commutative
diagram

$\textstyle{\pi_{1}(Diff(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\phi}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(X)}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{0}(SpinDiff(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

from which it follows that $M_{1}(X)\to\pi_{0}(SpinDiff(X))$ is an
isomorphism. If $\phi$ is non-trivial, then $K_{1}(X)=0$ and $M_{1}(X)\to
M(X)$ is an isomorphism. This happens for $S^{2}\times S^{2}$, as seen by
taking a loop of diffeomorphisms given by a circle action which rotates one of
the spheres. Similarly, $\phi$ is non-trivial for $X=S^{4}$ or for a connected
sum of copies of $S^{2}\times S^{2}$. If $\phi$ is trivial, then
$K_{1}(X)\cong\mathbb{Z}_{2}$ and $M_{1}(X)\to M(X)$ is an extension of $M(X)$
by $\mathbb{Z}_{2}$, hence corresponds to a class $\xi_{X}\in
H^{2}(M(X);\mathbb{Z}_{2})$. It is natural to ask what this class is and in
particular, whether or not it is trivial. First, we need some examples of spin
$4$-manifolds where $\phi=0$.

######  Theorem 7.2.

Let $X$ be a compact, smooth, simply-connected $4$-manifold. If $X$ is
homeomorphic to $K3$ then $\phi=0$. Similarly, if $X=X^{\prime}\\#(S^{2}\times
S^{2})$, where $X^{\prime}$ is homeomorphic to $K3$, then $\phi=0$.

###### Proof.

In [4], it is proven that if $E\to S^{2}$ is a smooth family of $K3$ surfaces
over $S^{2}$, then $w_{2}(TE)=0$. As explained in [18], this implies that the
homomorphism $\phi$ is zero. The same argument works for any $X$ that is
homeomorphic to $K3$, since by [20], the Seiberg–Witten invariant of the spin
structure of $X$ is odd.

Next, suppose $X=X^{\prime}\\#(S^{2}\times S^{2})$, where $X^{\prime}$ is
homeomorphic to $K3$. Suppose that $\phi$ is non-zero. This means that the
boundary Dehn twist $\tau\in M_{1}(X)$ is trivial. But this would imply that
the Dehn twist on the neck of $K3\\#X^{\prime}$ becomes trivial upon connected
sum with $S^{2}\times S^{2}$. However this contradicts [19] (in [19] the
theorem is stated only for $X^{\prime}=K3$, but the exact same proof works for
any smooth $4$-manifold homeomorphic to $K3$). ∎

Recall that an involution $f$ on a simply-connected spin $4$-manifold $X$ is
called even or odd according to whether or not $f$ lifts to an involution on
the spin bundle of $X$.

######  Proposition 7.3.

Suppose $X$ is spin and that $\phi=0$, so that the extension class $\xi_{X}\in
H^{2}(M(X);\mathbb{Z}_{2})$ is defined. Suppose that $f$ is an odd involution.
Then $\xi_{X}(f)\neq 0$. In particular, the extension $\mathbb{Z}_{2}\to
M_{1}(X)\to M(X)$ is non-trivial.

###### Proof.

As explained above, the extension $1\to\mathbb{Z}_{2}\to M_{1}(X)\to M(X)\to
1$ is isomorphic to the extension
$1\to\mathbb{Z}_{2}\to\pi_{0}(SpinDiff(X))\to M(X)\to 1$. But $f$ defines a
class $[f]\in M(X)$ such that $[f]^{2}=1$, but any lift of $f$ to the spin
bundle is not an involution. So there is no splitting $M(X)\to M_{1}(X)$ and
more precisely, $\xi_{X}(f)\neq 0$. ∎

######  Corollary 7.4.

If $X=K3$ or $K3\\#(S^{2}\times S^{2})$, then $\xi_{X}\in
H^{2}(M(X);\mathbb{Z}_{2})$ is non-trivial.

###### Proof.

This is immediate from Theorem 7.2 and Proposition 7.3, since both $K3$ and
$K3\\#(S^{2}\times S^{2})$ admit odd involutions. ∎

In what follows we will completely determine the class $\xi_{X}\in
H^{2}(M(X);\mathbb{Z}_{2})$ when $X$ is homeomorphic to $K3$.

######  Proposition 7.5.

Let $\pi:E\to B$ be a smooth fibre bundle, where $B$ is a compact surface and
the fibres of $E$ are diffeomorphic to a compact, simply-connected, smooth
spin $4$-manifold $X$. Then

  * (1)

There exists a spinc-structure $\mathfrak{s}_{E/B}$ on the vertical tangent
bundle $TE/B=Ker(\pi_{*})$ whose restriction to each fibre is spin.

  * (2)

Let $ind(D)\in K^{0}(B)$ denote the families index of the Dirac operator $D$
with respect to the spinc-structure $\mathfrak{s}_{E/B}$. Then

$c_{1}(ind(D))=(\sigma(X)/16)w_{2}(TE/B)\;({\rm mod}\;2).$

###### Proof.

(1) follows immediately from [2, Proposition 2.1]. The Dirac operator $D$ for
the spinc-structure $\mathfrak{s}_{E/B}$ defines a family of elliptic
operators parametrised by $B$ and $ind(D)$ is the families index. Then
$c_{1}(ind(D))=c_{1}(\mathcal{L})$, where $\mathcal{L}=det(ind(D))$ is the
determinant line bundle of $D$. Suppose that the family $E$ is determined by
transition function $\psi_{ij}$ valued in $Diff(X)$. Let
$\widetilde{\psi}_{ij}$ be lifts of $\psi_{ij}$ to $SpinDiff(X)$. Then
$\widetilde{\psi}_{ij}\widetilde{\psi}_{jk}\widetilde{\psi}_{ki}=g_{ijk}$,
where $g_{ijk}$ is a $\mathbb{Z}_{2}$-valued cocycle, defining a class
$[g_{ijk}]\in H^{2}(B;\mathbb{Z}_{2})$. Clearly $w_{2}(TE/B)=[g_{ijk}]$.
Observe that $c(\mathfrak{s}_{E/B})\in H^{2}(B;\mathbb{Z})$ is a lift of
$[g_{ijk}]$ to integer coefficients. Therefore we can represent
$c(\mathfrak{s}_{E/B})$ as an integer-valued $2$-cocycle $c_{ijk}$ such that
$c_{ijk}=g_{ijk}\;({\rm mod}\;2)$. Choose real-valued smooth functions
$u_{ij}$ such that $c_{ijk}=u_{ij}+u_{jk}+u_{ki}$. Set $f_{ij}=e^{2\pi
iu_{ij}}$. Then $f_{ij}$ define transition functions for a complex line bundle
whose first Chern class is $[c_{ijk}]$. Note that $f_{ij}=h_{ij}^{2}$, where
$h_{ij}=e^{\pi iu_{ij}}$. Then $h_{ij}h_{jk}h_{ki}=(-1)^{g_{ijk}}$. Define
$Spin^{c}Diff(X)=U(1)\times_{\mathbb{Z}_{2}}SpinDiff(X)$. Then
$\varphi_{ij}=h_{ij}\widetilde{\psi}_{ij}$ is a $2$-cocycle valued in
$Spin^{c}Diff(X)$.

Consider now the transition functions for the determinant line bundle
$\mathcal{L}$. Since $\mathfrak{s}_{E/B}$ restricts to a spin structure on the
fibres, the spinor bundles have a quaternionic structure on each fibre. It
follows that $\widetilde{\psi}_{ij}$ induces a trivial action on the
determinant line. However, the $U(1)$-factor $h_{ij}$ in
$\varphi_{ij}=h_{ij}\widetilde{\psi}_{ij}$ acts on the spinor bundles as
scalar multiplication which then acts on the determinant line by $h_{ij}^{d}$,
where $d$ is the virtual rank of $ind(D)$, which is $d=-\sigma(X)/8$. Hence
$\mathcal{L}$ has transition functions
$h_{ij}^{-\sigma(X)/8}=f_{ij}^{-\sigma(X)/16}$. Recalling that $f_{ij}$ are
transition functions for a line bundle with Chern class
$c(\mathfrak{s}_{E/B})$, it follows that

$c_{1}(ind(D))=c_{1}(\mathcal{L})=-\frac{\sigma(X)}{16}c(\mathfrak{s}_{E/B})=\frac{\sigma(X)}{16}w_{2}(TE/B)\;({\rm
mod}\;2).$

∎

######  Proposition 7.6.

Let $\pi:E\to B$ be a smooth fibre bundle, where the fibres of $E$ are
homeomorphic to $K3$. Then $w_{2}(TE/B)=w_{2}(H^{+})$, where $H^{+}\to B$
denote the bundle whose fibre over $b$ is a maximal positive definite subspace
of $H^{2}(E_{b};\mathbb{R})$.

###### Proof.

Since $H^{2}(B;\mathbb{Z}_{2})$ is detected by maps of compact surfaces into
$B$, it suffices to prove the result when $B$ is a compact surface. Then by
Proposition 7.5, $w_{2}(TE/B)=c_{1}(ind(D))\;({\rm mod}\;2)$. On the other
hand, since the fibres are homeomorphic to $K3$, their Seiberg–Witten with
respect to the spin structure is odd [20]. Then by [4, Corollary 1.3],
$c_{1}(ind(D))=w_{2}(H^{+})$. ∎

Let $L$ be a lattice and $A=Aut(L)$ the group of automorphisms. Over the
classifying space $BA$ we have the tautological flat bundle $H=EA\times_{A}L$.
Let $H^{+}\to BA$ be a maximal positive subbundle. This defines a
characteristic class $w_{2}(H^{+})\in H^{2}(Aut(L);\mathbb{Z}_{2})$.

######  Theorem 7.7.

Let $X$ be a smooth $4$-manifold which is homeomorphic to $K3$. Let $L_{X}$ be
the intersection lattice of $X$. Then the extension class $\xi_{X}\in
H^{2}(M(X);\mathbb{Z}_{2})$ is the pullback of $w_{2}(H^{+})\in
H^{2}(Aut(L_{X});\mathbb{Z}_{2})$ under the map $M(X)\to Aut(L_{X})$.

###### Proof.

Let $B$ be a compact surface and consider a map $\iota:B\to BM(X)$. This is
equivalent to a homomorphism $\rho:\pi_{1}(B)\to M(X)$. We claim that $\rho$
is the geometric monodromy of a family $E\to B$. We can take $B$ to be given
by attaching a $2$-cell to a wedge of $k$ circles. Each circle defines a
generator $g_{i}\in\pi_{1}(B)$ and the $2$-cell defines a relation
$r=r(g_{1},\dots,g_{k})$, which is a word in the $g_{i}$. Choose a lift
$f_{i}\in Diff(X)$ of $\rho(g_{i})\in M(X)$. Then we can construct a family
$E_{1}$ over the $1$-skeleton on $B$ as a wedge of mapping cylinders
corresponding to the diffeomorphisms $f_{1},\dots,f_{k}$. Since
$g_{1},\dots,g_{k}$ satisfy $r$, it follows that $r(f_{1},\dots,f_{k})$ is
isotopic to the identity. Choosing such an isotopy, we can extend $E_{1}$ over
the $2$-cell, giving the desired family $E\to B$. As explained in [4, Remark
4.20], we can assume that the family $E\to B$ is smooth. Now consider the
obstruction to lifting the structure group of $E$ to $SpinDiff(X)$. This is
easily seen to coincide with the obstruction to lifting $\rho:\pi_{1}(B)\to
M(X)$ to $M_{1}(X)$, which is $\iota^{*}(\xi_{X})\in H^{2}(B;\mathbb{Z}_{2})$.
On the other hand, the obstruction to lifting the structure group of $E$ to
$SpinDiff(X)$ is $w_{2}(TE/B)$, which by Proposition 7.6 equals
$w_{2}(H^{+})$. Since $H^{2}(M(X);\mathbb{Z}_{2})$ is detected by maps of
compact surfaces $B$ into $BM(X)$, the result is proven. ∎

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11institutetext: Department of Intelligent Control and Decision Support
Systems, Digital Technologies Center, Gdańsk University of Technology, G.
Narutowicza 11/12, 80-233 Gdańsk, Poland

# Neural modelling of dynamic systems with time delays based on an adjusted
NEAT algorithm

Krzysztof Laddach 11 0000-0001-9122-2167 Rafał Łangowski Corresponding author
- e-mail address<EMAIL_ADDRESS>0000-0003-1150-9753

###### Abstract

A problem related to the development of an algorithm designed to find an
architecture of artificial neural network used for black-box modelling of
dynamic systems with time delays has been addressed in this paper. The
proposed algorithm is based on a well-known NeuroEvolution of Augmenting
Topologies (NEAT) algorithm. The NEAT algorithm has been adjusted by allowing
additional connections within an artificial neural network and developing
original specialised evolutionary operators. This resulted in a compromise
between the size of neural network and its accuracy in capturing the response
of the mathematical model under which it has been learnt. The research
involved an extended validation study based on data generated from a
mathematical model of an exemplary system as well as the fast processes
occurring in a pressurised water nuclear reactor. The obtaining simulation
results demonstrate the high effectiveness of the devised neural (black-box)
models of dynamic systems with time delays.

###### Keywords:

Neural modelling Neural network architecture search PWR black-box model.

##  1 Introduction

Nowadays, an effective handling of majority industrial plants requires
advanced algorithms based on the proper mathematical models of processes that
occur in them. These algorithms perform different tasks such as diagnostics,
monitoring, estimation, control, etc. It is known that the model makes it
possible, e.g., to predict the trajectories of selected process variables,
also in unacceptable situations in a real plant, to analyse the behaviour of a
given process, or to consider various control strategies, etc. However, the
complexity of many processes is significant, and the phenomena occurring in
them are sophisticated. Therefore, delivering the system white-box model might
become very difficult and even impossible. Thus, developing an alternative
model such as a black-box or grey-box may be justified and reasonable [16].
These models are built based on observation of behaviour of a given system.
There are different tools to create black-box models. Artificial neural
networks (ANNs) are one of the most common and have produced excellent results
in many fields of science [1]. The neural (black-box) model is based on ANN,
which according to the theorems of Kolmogorov and Cybenko has the ability to
represent any mathematical function [4, 8]. However, to achieve it, an
architecture of ANN (parameters and hyper-parameters) have to be selected for
each unique function. This problem is known in the literature, where there are
many ways to solve it, although it is still an open issue. One of the common
approach to select ANN’s architecture is to use neuroevolution [13], i.e.
genetic or evolutionary algorithms [5, 10, 17]. One of the most popular
neuroevolutionary algorithms is NeuroEvolution of Augmenting Topologies (NEAT)
[18], which has been and still is used and modified to adapt its operation to
new tasks [15].

In this paper, the authors’ neuroevolution method based on an adaptation of
the NEAT algorithm to build a black-box model of a dynamic system with time
delays is presented. The NEAT algorithm has been adjusted by allowing
additional connections within ANN and developing original specialised
evolutionary operators. An algorithm has been developed and verified by
simulation that makes it possible to find ANN architecture that represents a
compromise between its size and its accuracy in capturing the response of the
model under which it has been learnt. The derived algorithm has been marked as
dNEAT. To summarise, the main contribution of this work is to develop and
verify the dNEAT algorithm that enables the selection of ANN architecture for
black-box modelling of dynamic systems with time delays.

As applications, the single-input single-output (SISO) exemplary system and
SISO model of the fast processes in a pressurised water reactor (PWR) are
taken into account. A PWR is a non-linear, spatial and non-stationary plant
whose processes are characterised by multi-scale and complex dynamics and
involve delays associated with occurring delayed neutrons. Thus, there are
many different mathematical models of PWR that are used depending on the aim,
e.g., modelling of physical processes, diagnostics, on-line monitoring, etc.
For example, seven modelling methods of processes taking place in a PWR are
distinguished in [11]. Starting from the simplest one allows obtaining point-
parameters models of low complexity, through a one-dimensional, three-
dimensional, multi-point modelling based on fractional-order calculus, up to
building models consisting of sub-models and neural models. However, the
models built in white-box way ’pay’ for their accuracy with a significant
degree of complexity. On the other hand, the ’intelligent identification’ of
ANN-based processes in a PWR, and therefore also the building of ANN-based
black-box models, is a topic enjoying considerable activity in the scientific
research space [11]. Available in the literature neural reactor models, e.g.,
[2, 6], systems for identifying reactor parameters or states, e.g., [7, 14],
or neural controllers used in a PWR control structures, e.g., [3] confirm the
potential of ANNs in this domain. Also, the fact that the strong demand for
models of these plants results from the lack of possibility to perform
experiments on a real nuclear reactor. Thus, it seems that models of a PWR
that would allow numerous tests to be carried out in a short time and at a low
computational cost would make it possible to design control, diagnostic and
safety systems in which various scenarios of events and the controls would be
considered. From the research point of view addressed in the paper, a PWR
model has been used only to generate learning and validation data. Hence, for
this reason, a detailed description of it is not presented.

##  2 Problem statement

As it has been mentioned, the main aim of this work is to develop and verify
the authors’ algorithm - the dNEAT for black-box modelling purposes of dynamic
systems with time delays. As a type of ANN, a recurrent network (RNN) has been
selected. In RNNs, signals may flow in both directions, i.e. from input to
output and vice versa. As a result, RNNs have an internal state that depends
on the current input data and the previous network states. A recurrent network
has been selected because, similar to considered plants’ dynamics, they have
an internal feedback. An architecture of exemplary RNN is presented in Fig. 1.

Figure 1: An architecture of exemplary RNN.

Searching for a network with such architecture as in Fig. 1 is equivalent to
looking for a non-linear auto-regressive exogenous discrete model of the
process that is based on its time-delayed input and output signals:

$\begin{split}y(k)=f(y(k-1),y(k-2),y(k-3),...,y(k-dy),\\\
u(k),u(k-1),u(k-2),u(k-3),...,u(k-du)),\end{split}$ (1)

where: $y(\cdot)$, $u(\cdot)$ are the output and input at the discrete-time
instant specified by $(\cdot)$, respectively; $f(\cdot)$ denotes the function
specified by $(\cdot)$; $du$, $dy$ are the maximal levels of input and output
(recurrence) delays, respectively; $k$ is the discrete-time instant.

The following parameters and hyper-parameters (architecture of the network)
should be determined to create a black-box model using RNN: the number of
neurons in assumed single hidden and output layers (so-called hidden and
output neurons); the existence of connections between individual neurons; the
delay levels of $du$ and $dy$; the neurons’ model, i.e. type and parameters of
excitation function and activation (transfer) function; the connections’
model; and the values of weights of individual connections. Some of the above
have been restricted to predetermined values (hyper-parameters). The
excitation function is assumed, classically, as a weighted sum of inputs to a
given neuron. In turn, the activation function of neurons in the hidden layer
is assumed to a sigmoidal bipolar. This choice has been made because the
considered RNNs are intended to represent the operation of continuous plants;
thus, the activation function should be also continuous. In turn, the neuron
activation function in the output layer is assumed as linear with a
directional coefficient equal to one. It is since its task is to linearly
transform the sum of the outputs of the previous layer, i.e. the excitation
signal. The connection model has been adopted as a unit function without
delays. To fully define the architecture of RNN, it is still necessary to
select the parameters: the number of hidden neurons, the existence of
connections between individual neurons, the values of $du$ and $dy$, and the
weights of individual connections. The proposed dNEAT algorithm is used for
these purposes. It is worth adding that the NEAT algorithm (dNEAT as well)
provides parallel optimisation of an architecture of the neural network
achieved during their evolutionary development [18].

##  3 dNEAT algorithm

This section describes the dNEAT algorithm by pointing out the changes made
relative to the NEAT algorithm [18]. Hence, it focuses only on the parts that
are new concerning the NEAT algorithm. The python code of the NEAT algorithm
from [12] has been used and errors in it have been corrected, so that it works
according to the original description from [18]. The codes of the dNEAT and
mentioned the NEAT are available in [9]. The main algorithms’ parameters are:

  * •

$pop_{size}=100$ \- the target number of individuals in population;

  * •

$weight_{init_{mean}}=0$, $bias_{init_{mean}}=0$ \- the Gaussian distribution
mean values for drawing initial values of weights and biases;

  * •

$weight_{init_{stdev}}=0.5$, $bias_{init_{stdev}}=0.5$ \- the Gaussian
distribution standard deviations for drawing initial values of weights and
biases;

  * •

$du_{init-max}=20$; $dy_{init-max}=20$ \- the upper ranges of intervals for
drawing values $du$ and $dy$;

  * •

$bias_{mutate_{power}}=0.033$ \- the zero-centred Gaussian distribution
standard deviation for drawing value of bias mutation;

  * •

$bias_{mutate_{rate}}=0.2$, $bias_{replace_{rate}}=0.2$ \- the probability
that mutation will change the bias of a node by adding or assigning a random
value;

  * •

$weight_{mutate_{power}}=0.1$ \- the zero-centred Gaussian distribution
standard deviation for drawing value of weight mutation;

  * •

$weight_{mutate_{rate}}=0.6$, $weight_{replace_{rate}}=0.05$ \- the
probability that mutation will change the weight by adding or assigning a
random value;

  * •

$conn_{add_{prob}}=0.2$, $conn_{delete_{prob}}=0.2$ \- the probability that
mutation will add or delete a connection between existing neurons;

  * •

$enabled_{mutate_{rate}}=0.2$ \- the probability that mutation will change the
enabled status of a connection;

  * •

$node_{add_{prob}}=0.2$, $node_{delete_{prob}}=0.2$ \- the probability that
mutation will add a new neuron or delete an existing neuron;

  * •

$compatibility_{threshold}=2.3$ \- the individuals whose genomics distance is
less than this threshold are considered to be in the same species;

  * •

$max_{stagnation}=25$ \- the species, in which the best individual has not
shown improvement over more than this number of generations will be considered
stagnant and removed;

  * •

$species_{elitism}=3$ \- number of species that will be protected from
stagnation;

  * •

$elitism=10$ \- the number of most-fit individuals in each species that will
be preserved as-is from one generation to the next;

  * •

$survival_{threshold}=0.25$ \- the fraction for each species allowed to
reproduce each generation;

  * •

$du_{mutate_{rate}}=0.2$, $dy_{mutate_{rate}}=0.2$ \- the probability that
mutation will change the delay levels;

  * •

$du_{mutate-power}=2$, $dy_{mutate-power}=2$ \- the specifies the intervals
from which the delay mutation values are drawn.

###  3.1 Initialisation

Each individual is initialised as the RNN consisting of one hidden and one
output neuron. The number of output neurons reasons from the number of mapped
trajectories. During initialisation, each input neuron is connected to each
hidden neuron, and each hidden neuron is connected to each output neuron. The
initial delays $du$ and $dy$ are drawn with uniform probability distribution
from ranges of $[0,du_{init-max}]$ and $[0,dy_{init-max}]$. The initial values
of the weights and biases are drawn using Gaussian distributions with
parameters given in section 3.

###  3.2 Crossover

The crossover operator in respect of neurons and connections between them
remained the same as in the NEAT algorithm. In turn, concerning the delays
levels of $du$ and $dy$ crossover way yields:

$d_{child}=round\left(rd_{parent1}+(1-r)d_{parent2}\right),$ (2)

where: $d_{child}$ is the value of $du$ or $dy$ for offspring; $d_{parent1}$,
$d_{parent2}$ are the values of $du$ or $dy$ for parents; $r$ is a number
drawn using a uniform probability distribution from the range $[0,1]$.

###  3.3 Mutation

As for crossover, the mutation concerning neurons and connections between them
remained the same as in the NEAT algorithm. The mutation of delays $du$ and
$dy$ proceeds as follows. If draw decided that the mutation of the
corresponding delay $d(\cdot)$ should occur, the delay change $\delta
d(\cdot)$ is drawn with uniform probability distribution from the range
$[-d(\cdot)_{mutate-power},$ $d(\cdot)_{mutate-power}]$. Then $\delta
d(\cdot)$ is rounded to an integer value. If $\delta d(\cdot)$ is zero then it
is assigned a value of -1 or 1 with equal (50%) probability. The corresponding
delay is then changed according to the formula $d(\cdot)=|d(\cdot)+\delta
d(\cdot)|$. Hence, if the corresponding delay level $du$ or $dy$ is decreased,
then outgoing connections from the given input are removed. On the other hand,
if the delay is increased, outgoing connections from the new inputs are added
to all hidden neurons in the same way as during initialisation.

###  3.4 Fitness function

Since the dNEAT algorithm optimises the structure of individuals directly
during its operation, the function determining the fitness of individuals does
not need to depend on the RNN architecture. Therefore, the fitness function
yields:

$f_{i}=-\frac{1000}{N}\sum_{j=1}^{J}\left(y_{ij}-t_{j}\right)^{2},$ (3)

where: $f_{i}$ is the value of fitness function for $i$-th individual; $N$ is
the number of samples during learning phase; $y_{ij}$ are successive $j$-th
samples of the response of the $i$-th individual; $t_{j}$ are successive
$j$-th samples of the target trajectory.

It can be noticed that (3) is the mean square error (MSE) of mapping the
learning trajectory through the network response. This value is multiplied by
a relatively large negative scaling number to magnify the discrepancy in
values of the fitness function of individuals with a similar mapping
performance. It is a negative number because, by definition, the better
individual should achieve a higher fitness function value.

##  4 Applications

The developed dNEAT algorithm has been simulation-verified on two
applications. The first one is the SISO exemplary system, whereas the second
is the SISO model of fast processes in a PWR. As a result, using the dNEAT
algorithm, neural (black-box) models are created for both plants.

###  4.1 Application 1

In this application, the black-box model of an exemplary non-linear plant with
time delays described by (4) has been created:

$x(k)=-0.05x(k-1)+0.02x(k-5)+sin\left(\frac{x(k-10)}{10}\right)+u(k-15).$ (4)

The input ($u(\cdot)$) trajectory during the learning phase, i.e. in the phase
of obtaining the model by the NEAT and dNEAT algorithms (red line), and the
corresponding target (output - $x(\cdot)$) trajectory (blue line) are
presented in Fig. 2(a). In turn, the trajectories used in the verification
phase, i.e. the phase in which the responses of the obtained neural model to
the trajectories that have been not used in the learning phase are shown in
Figs. 2(b) and 2(c). In all cases, the model responses have been normalised by
dividing the plant’s response by 30. All the figures presented in this paper
are available in [9].

###  4.2 Application 2

The second application is focused on building the SISO neural model of the
fast processes in a PWR. The position of the control rods and the thermal
power, scaled by dividing by the nominal power of a PWR have been selected as
the input and output signals, respectively. The input trajectory during the
learning phase (red line) and the corresponding output trajectory (blue line)
are presented in Fig. 2(d). In turn, the trajectories used in the verification
phase are shown in Figs. 2(e) and 2(f). These trajectories are taken from
[10].

##  5 Results

In this section, the simulation results illustrating the performance of the
proposed algorithm against the results generated by the NEAT algorithm are
presented. First, the learning phase is shown. Next, the verification phase is
discussed. The stop condition has taken 2500 generations, and both algorithms
have called 10 times. All parameters present in both algorithms have had the
same values (see section 3). The average value of the fitness functions of
subsequent generations from all NEAT and dNEAT calls in both applications are
presented in Fig. 4(a). Whereas the average value of fitness functions of the
best individuals in subsequent generations from all algorithms is shown in
Fig. 4(b). The responses of the neural model obtained for the learning data
for the best and worst individuals by fitness function values, selected from
all obtained the best individuals from each algorithm call for application 1
are shown in Fig. 5(a). For simplicity, the best individuals obtained from
successive algorithms calls are called ’winners’. The responses of the best
and worst individuals from the winners for the verification phase are
presented in Figs. 5(b) and 5(c). Analogous results for application 2 are
shown in Fig. 6. The MSE values of the best and worst individuals from the
winners and the average MSE value of the winners from both phases for both
applications are given in Table 1.

(a) The learning data for application 1.

(b) The verification data for application 1 - set 1.

(c) The verification data for application 1 - set 2.

Figure 2: The learning and verification data for application 1.

(d) The learning data for application 2.

(e) The verification data for application 2 - set 1.

(f) The verification data for application 2 - set 2.

Figure 3: The learning and verification data for application 2.

(a) The average value of population fitness function.

(b) The average value of fitness function of the best individuals.

Figure 4: The trajectories of average values of the fitness functions.

(a) The responses of ’winners’ in the learning phase for application 1.

(b) The responses of ’winners’ in the verification phase - set 1 for
application 1.

(c) The responses of ’winners’ in the verification phase - set 2 for
application 1.

Figure 5: The responses of ’winners’ in both phases for application 1.

(d) The responses of ’winners’ in the learning phase for application 2.

(e) The responses of ’winners’ in the verification phase - set 1 for
application 2.

(f) The responses of ’winners’ in the verification phase - set 2 for
application 2.

Figure 6: The responses of ’winners’ in both phases for application 2.

Table 1: The values of MSE for applications 1 and 2. Phase | Algorithm | Description | Value $\left(\cdot 10^{-2}\right)$
---|---|---|---
Application 1 | Application 2
Learning | NEAT | the best of the winners | $2.85996$ | $0.01335$
the worst of the winners | $3.05342$ | $0.0279$
the average value of the winners | $2.92178$ | $0.01654$
dNEAT | the best of the winners | $0.04096$ | $0.03113$
the worst of the winners | $1.7322$ | $0.08256$
the average value of the winners | $0.43248$ | $0.05183$
Verification - set 1 | NEAT | the best of the winners | $4.27963$ | $0.02234$
the worst of the winners | $4.54949$ | $0.0326$
the average value of the winners | $4.45215$ | $0.02278$
dNEAT | the best of the winners | $0.47872$ | $0.03478$
the worst of the winners | $3.49938$ | $0.12342$
the average value of the winners | $0.98783$ | $0.08117$
Verification - set 2 | NEAT | the best of the winners | $41.99314$ | $0.00909$
the worst of the winners | $47.21822$ | $0.02924$
the average value of the winners | $42.26676$ | $0.014$
dNEAT | the best of the winners | $5.58866$ | $0.02675$
the worst of the winners | $31.34926$ | $0.09197$
the average value of the winners | $16.26694$ | $0.07378$

Analysing the results obtained, the following conclusions can be drawn. The
mean values of the fitness functions of subsequent populations indicate that
for the dNEAT algorithm they have been lower than for the NEAT. However, the
mean values of the fitness functions of the best individuals show that the
dNEAT algorithm provides a better (or comparable) performance in finding the
best individuals. Additionally, these individuals are found faster. Therefore,
the dNEAT algorithm, despite searching a larger solution space and putting
more emphasis on population diversity is able to find individuals better than
the NEAT algorithm in the task of creating neural models of dynamic systems
with time delays. The individuals found in this algorithm are more diverse
through the different levels of input and output delays. Thus, the proposed
solution makes it possible to find individuals acting as a black-box model of
a dynamic system with time delays more efficiently concerning the NEAT
algorithm.

##  6 Conclusions

In this paper, the problem of developing an algorithm of artificial neural
network architecture search for black-box modelling of dynamic systems with
time delays has been investigated. In particular, the evolutionary algorithm
dNEAT has been devised to build the required models. The specialised
evolutionary operators and additional connections within the network ensure
the proper operation of the proposed algorithm. It enabled devising the SISO
neural model of the exemplary system as well as the fast processes in a PWR.
The dNEAT algorithm has been implemented in the computational environment, and
obtained simulation results yield satisfying performance of the produced
output trajectories. Further work is needed to analyse the impact of the
algorithm parameters values on its operation and to attempt to select them
automatically, e.g., through machine learning.

####  6.0.1 Acknowledgements

Financial support of these studies from Gdańsk University of Technology by the
DEC-2/2020/IDUB/I.3.3 grant under the Argentum Triggering Research Grants -
’Excellence Initiative - Research University’ program is gratefully
acknowledged.

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