# Smooth homotopy 4-sphere

Akio KAWAUCHI
Osaka City University Advanced Mathematical Institute
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
<EMAIL_ADDRESS>
( )

###### ABSTRACT

Every smooth homotopy 4-sphere is diffeomorphic to the 4-sphere.

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Keywords: 4-sphere, Smooth homotopy 4-sphere, Trivial surface-knot, O2-handle
pair, O2-sphere pair.

Mathematics Subject Classification 2010: Primary 57N13; Secondary 57Q45

1\. Introduction

As general conventions throughout this paper, a compact connected oriented
smooth $r$-dimensional manifold for $r\geq 2$ is called an $r$-manifold and a
smooth embedding and a smooth isotopy from an $r$-manifold into an
$r^{\prime}$-manifold are called an embedding and an isotopy, respectively,
unless otherwise stated. An $m$-punctured manifold of an $r$-manifold $X$ is
an $r$-manifold $X^{m(0)}$ obtained from $X$ by removing the interiors of $m$
mutually disjoint $r$-balls in the interior of $X$. The $r$-manifold
$X^{1(0)}$ is denoted by $X^{(0)}$.

By this convention, a homotopy 4-sphere is a 4-manifold $M$ homotopy
equivalent to the 4-sphere $S^{4}$, and a homotopy 4-ball is a 1-punctured
manifold $M^{(0)}$ of a homotopy 4-sphere $M$.

The main purpose of this paper is to show that every homotopy 4-sphere is
diffeomorphic to the 4-sphere $S^{4}$, so that every homotopy 4-ball is
diffeomorphic to the 4-ball $D^{4}$. For a positive integer $n$, the stable
4-sphere of genus $n$ is the connected sum 4-manifold

$\Sigma=\Sigma(n)=S^{4}\\#n(S^{2}\times S^{2})=S^{4}\\#_{i=1}^{n}S^{2}\times
S^{2}_{i},$

which is the union of an $n$-punctured manifold $(S^{4})^{n(0)}$ of the
4-sphere $S^{4}$ and $1$-punctured manifolds $(S^{2}\times
S^{2}_{i})^{(0)}\,(i=1,2,\dots,n)$ of the 2-sphere products $S^{2}\times
S^{2}_{i}\,(i=1,2,\dots,n)$ pasting the boundary 2-spheres of $(S^{4})^{n(0)}$
to the other boundary 2-spheres, where choices of the 4-balls are independent
of the diffeomorphism type of $\Sigma$.

In this paper, an argument on a trivial surface-knot in the 4-space in [11] is
used by observing that the stable 4-sphere $\Sigma$ of genus $n$ is the double
branched covering space $S^{4}(F)_{2}$ of the 4-sphere $S^{4}$ branched along
a trivial surface $F$ of genus $n$.

An orthogonal 2-sphere pair or simply an O2-sphere pair of the stable 4-sphere
$\Sigma$ is a pair $(S,S^{\prime})$ of 2-spheres $S$ and $S^{\prime}$ embedded
in $\Sigma$ meeting transversely at a point with the intersection numbers
$\mbox{Int}(S,S)=\mbox{Int}(S^{\prime},S^{\prime})=0$ and
$\mbox{Int}(S,S^{\prime})=+1$.

A pseudo-O2-sphere basis of the stable 4-sphere $\Sigma$ of genus $n$ is the
system $(S_{*},S^{\prime}_{*})$ of $n$ mutually disjoint O2-sphere pairs
$(S_{i},S_{i}^{\prime})\,(i=1,2,\dots,n)$ in $\Sigma$. Let
$N(S_{i},S_{i}^{\prime})$ be a regular neighborhood of the union $S_{i}\cup
S_{i}^{\prime}$ of the O2-sphere pair $(S_{i},S_{i}^{\prime})$ in $\Sigma$
such that $N(S_{i},S_{i}^{\prime})\,(i=1,2,\dots,n)$ are mutually disjoint.
The region of a pseudo-O2-sphere basis $(S_{*},S^{\prime}_{*})$ in $\Sigma$ of
genus $n$ is a 4-manifold $\Omega(S_{*},S^{\prime}_{*})$ in $\Sigma$ obtained
from the 4-manifolds $N(S_{i},S_{i}^{\prime})\,(i=1,2,\dots,n)$ by connecting
them by mutually disjoint 1-handles $h^{1}_{j}\,(j=1,2,\dots,n-1)$ in
$\Sigma$. Since $\Sigma$ is a simply connected 4-manifold, the region
$\Omega(S_{*},S^{\prime}_{*})$ in $\Sigma$ does not depend on any choices of
the 1-handles $h^{1}_{j}\,(j=1,2,\dots,n-1)$ and is uniquely determined by the
pseudo-O2-sphere basis $(S_{*},S^{\prime}_{*})$ up to isotopies of $\Sigma$.
The residual region

${\Omega}^{c}(S_{*},S^{\prime}_{*})=\mbox{cl}(\Sigma\setminus\Omega(S_{*},S^{\prime}_{*}))$

of the region $\Omega(S_{*},S^{\prime}_{*})$ in $\Sigma$ is always a homotopy
4-ball, which is shown by van Kampen theorem and a homological argument. An
O2-sphere basis of the stable 4-sphere $\Sigma$ of genus $n$ is a
pseudo-O2-sphere basis $(S_{*},S^{\prime}_{*})$ of $\Sigma$ such that the
residual region ${\Omega}^{c}(S_{*},S^{\prime}_{*})$ is diffeomorphic to the
4-ball. The following result is basically the main result of this paper.

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Theorem 1.1. For any two pseudo-O2-sphere bases $(R_{*},R^{\prime}_{*})$ and
$(S_{*},S^{\prime}_{*})$ of the stable 4-sphere $\Sigma$ of any genus $n\geq
1$, there is an orientation-preserving diffeomorphism $h:\Sigma\to\Sigma$
sending $(R_{i},R^{\prime}_{i})$ to $(S_{i},S^{\prime}_{i})$ for all
$i\,(i=1,2,\dots,n)$.

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The stable 4-sphere $\Sigma$ of genus $n$ admits an O2-sphere basis. If
$(R_{*},R^{\prime}_{*})$ is an O2-sphere basis of $\Sigma$ and
$(S_{*},S^{\prime}_{*})$ is the image of $(R_{*},R^{\prime}_{*})$ by an
orientation-preserving diffeomorphism $f:\Sigma\to\Sigma$, then
$(S_{*},S^{\prime}_{*})$ is also an O2-sphere basis. Thus, the following
corollary is directly obtained from Theorem 1.1.

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Corollary 1.2. Every pseudo-O2-sphere basis of the stable 4-sphere $\Sigma$ of
any genus $n\geq 1$ is an O2-sphere basis of $\Sigma$.

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In this paper, an O2-handle pair $(D\times I,D^{\prime}\times I)$ on a trivial
surface-knot $F$ in $S^{4}$ in [11] is discussed (see Section 2 for an
explanation). A system $(D_{*}\times I,D^{\prime}_{*}\times I)$ of $n$
mutually disjoint O2-handle pairs $(D_{i}\times I,D^{\prime}_{i}\times
I)\,(i=1,2,\dots,n)$ on $F$ of genus $n$ in $S^{4}$ is called an O2-handle
basis of $F$.

It is shown in Lemma 2.3 that the lift $(S(D_{*}),S(D^{\prime}_{*}))$ of the
core system $(D_{*},D^{\prime}_{*})$ of any O2-handle basis $(D_{*}\times
I,D^{\prime}_{*}\times I)$ of $F$ to $S^{4}(F)_{2}=\Sigma$ is an O2-sphere
basis of $\Sigma$. Also, in Corollary 5.3, it is shown that every O2-sphere
basis of $\Sigma$ is isotopic to such an O2-sphere basis
$(S(D_{*}),S(D^{\prime}_{*}))$ in $\Sigma$.

The following result which is called 4D Smooth Poincaré Conjecture is a direct
consequence of Corollary 1.2.

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Corollary 1.3. Any homotopy 4-sphere $M$ is diffeomorphic to the 4-sphere
$S^{4}$.

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Proof of Corollary 1.3. It is known that there is an orientation-preserving
diffeomorphism

$\kappa:M\\#\Sigma\to\Sigma$

from the connected sum $M\\#\Sigma$ onto $\Sigma$ for a positive integer $n$
(see Wall [15]). Let the connected sum $M\\#\Sigma$ be the union
$M^{(0)}\cup\Sigma^{(0)}$. Corollary 1.2 means that the image
$\kappa(\Sigma^{(0)})$ is the region $\Omega(S_{*},S^{\prime}_{*})$ of an
O2-sphere basis $(S_{*},S^{\prime}_{*})$ of $\Sigma$, so that the residual
region ${\Omega}^{c}(S_{*},S^{\prime}_{*})=\kappa(M^{0})$ is a 4-ball and
hence $M^{(0)}$ is diffeomorphic to the 4-ball $D^{4}$. By $\Gamma_{4}=0$ in
Cerf [2] or $\pi_{0}(\mbox{Diff}^{+}(S^{3}))=0$ by Hatcher [6], the
diffeomorphism $M^{(0)}\to D^{4}$ extends to a diffeomorphism $M\to S^{4}$.
$\square$

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In the topological category, it is well-known by Freedman [3] (see also [4])
that the corresponding result of Corollary 1.3 holds (i.e., every topological
4-manifold homotopy equivalent to the 4-sphere is homeomorphic to the
4-sphere). In the piecewise-linear category, it can be shown by using the
piecewise-linear versions of the techniques used in this paper (see Hudson
[10], Rourke-Sanderson [14]) that the corresponding result of Corollary 1.3
holds (i.e., every piecewise-linear 4-manifold homotopy equivalent to the
4-sphere is piecewise-linearly homeomorphic to the 4-sphere).

It is known by Wall in [15] that for every closed simply connected signature-
zero spin 4-manifold $M$ with the second Betti number
$\beta_{2}(M;{\mathbf{Z}})=2m>0$, there is a diffeomorphism

$\kappa:M\\#\Sigma(n)\to\Sigma(m+n)$

for a positive integer $n$ and by Freedman [3] (see also [4]) that there is a
homeomorphism from $W$ to $\Sigma(m)$. However, a technique used for the proof
of Theorem 1.1 cannot be generalized to this case. In fact, it is known by
Akhmedov-Park in [1] that there is a closed simply connected signature-zero
spin 4-manifold $M$ with a large second Betti number
$\beta_{2}(M;{\mathbf{Z}})=2m$ such that $M$ is not diffeomorphic to
$\Sigma(m)$. What can be said in this paper is the following corollary.

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Corollary 1.4. Let $M$ and $M^{\prime}$ be any closed (not necessarily simply
connected) 4-manifolds with the same second Betti number
$\beta_{2}(M;{\mathbf{Z}})=\beta_{2}(M^{\prime};{\mathbf{Z}})$. Then an
embedding $u:M^{(0)}\to\bar{M}^{\prime}$ extends to a diffeomorphism
$u^{+}:M\to M^{\prime}$ if and only if the embedding $u:M^{(0)}\to M^{\prime}$
induces a fundamental group isomorphism

$u_{\\#}:\pi_{1}(M^{(0)},x)\to\pi_{1}(M^{\prime},u(x)).$

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Proof of Corollary 1.4. Since the proof of the “only if”part is obvious, it
suffices to prove the “if”part. This proof is obtained by noting that the
closed complement $\tilde{D}^{4}_{0}=\mbox{cl}(M^{\prime}\setminus
u(M^{(0)}))$ is a homotopy 4-ball with 3-sphere boundary which is confirmed by
the van Kampen theorem and a homological argument. By the proof of Corollary
1.3. the homotopy 4-ball $\tilde{D}^{4}_{0}$ is diffeomorphic to the 4-ball
$D^{4}$ and the embedding $u:M^{(0)}\to M^{\prime}$ extends to a
diffeomorphism $u^{+}:M\to M^{\prime}$. $\square$

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The following corollary is obtained by combining Corollary 1.3 with the result
on a trivial $S^{2}$-link in $S^{4}$ in [11, 12].

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Corollary 1.5. Every closed 4-manifold $M$ such that the fundamental group
$\pi_{1}(M,x)$ is a free group of rank $n$ and $H_{2}(M;{\mathbf{Z}})=0$ is
diffeomorphic to the connected sum 4-manifold

$S^{4}\\#n(S^{1}\times S^{3})=S^{4}\\#_{i=1}^{n}S^{1}\times S^{3}_{i}.$

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Proof of Corollary 1.5. Let $\ell_{i}\,(i=1,2,\dots,n)$ be a system of $n$
mutually disjoint simple loops in $M$ which is homotopic to a system of loops
generation the free group $\pi_{1}(M,x)$, and $N(\ell_{i})=S^{1}\times
D^{3}_{i}\,(i=1,2,\dots,n)$ a system of mutually disjoint regular
neighborhoods of $\ell_{i}\,(i=1,2,\dots,n)$ in $M$. The 4-manifold $X$
obtained from $M$ by replacing $S^{1}\times D^{3}_{i}$ with $D^{2}\times
S^{2}_{i}$ for every $i$ is a homotopy 4-sphere by van Kampen theorem and
$H_{2}(M;{\mathbf{Z}})=0$, and hence $X$ is diffeomorphic to $S^{4}$ by
Corollary 1.3. The core 2-sphere $K_{i}=S^{2}_{i}$ of $D^{2}\times S^{2}_{i}$
forms a 2-link $L=\cup_{i=1}^{n}S^{2}_{i}$ in $X=S^{4}$ such that the
fundamental group $\pi_{1}(S^{4}\setminus L,x)(\cong\pi_{1}(M,x))$ is a
meridian-based free group of rank $n$. By [12], the 2-link $L$ is a trivial
2-link in $S^{4}$ and hence bounds mutually disjoint 3-balls in $S^{4}$. By
returning $D^{2}\times S^{2}_{i}$ to $S^{1}\times D^{3}_{i}$ for every $i$, it
is concluded that the 4-manifold $M$ must be diffeomorphic to the desired
connected sum 4-manifold. $\square$

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The following corollary which is called 4D Smooth Schoenflies Conjecture is
also obtained.

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Corollary 1.6. Any embedded 3-sphere $S^{3}$ in the 4-sphere $S^{4}$ splits
$S^{4}$ into two components of 4-manifolds which are both diffeomorphic to the
4-ball.

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Proof of Corollary 1.6. The splitting components are both homotopy 4-balls by
van Kampen theorem and a homological argument, which are diffeomorphic to the
4-ball by the proof of Corollary 1.3. $\square$

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The paper is organized as follows: In Section 2, the stable 4-sphere $\Sigma$
is identified with the double branched covering space $S^{4}(F)_{2}$ of
$S^{4}$ branched along a trivial surface-knot $F$. In Section 3, a homological
version of Theorem 1.1 is shown. Throughout Section 4, the proof of Theorem
1.1 is done. In Section 5, an isotopic deformation of an orientation-
preserving diffeomorphism of the stable 4-sphere $\Sigma$ is studied by
combining the proof of Theorem 1.1 with Gabai’s 4D light-bulb theorem in [5].
In fact, in Theorem 5.1, it is shown that every orientation-preserving
diffeomorphism of $\Sigma$ is isotopic to the lift of an equivalence of a
trivial surface-knot $F$ in $S^{4}$ to $\Sigma$ modulo a diffeomorphism of
$\Sigma$ with a support of a 4-ball disjoint from the lift of $F$. Corollary
5.3 is also shown there.

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2\. The stable 4-sphere as the double branched covering space of the 4-sphere
branched along a trivial surface-knot

A surface-knot of genus $n$ in the 4-sphere $S^{4}$ is a closed surface $F$ of
genus $n$ embedded in $S^{4}$. It is also called a 2-knot if $n=0$, i.e., $F$
is the 2-sphere $S^{2}$. Two surface-knots $F$ and $F^{\prime}$ in $S^{4}$ are
equivalent by an equivalence $f$ if $F$ is sent to $F^{\prime}$ orientation-
preservingly by an orientation-preserving diffeomorphism $f:S^{4}\to S^{4}$.

A trivial  surface-knot of genus $n$ in $S^{4}$ is a surface-knot $F$ of genus
$n$ which is the boundary of a handlebody of genus $n$ embedded in $S^{4}$,
where a handlebody of genus $n$ means a 3-manifold which is a 3-ball for
$n=0$, a solid torus for $n=1$ or a boundary-disk sum of $n$ solid tori. A
surface-link in $S^{4}$ is a union of disjoint surface knots in $S^{4}$, and a
trivial surface-link is a surface-link bounding disjoint handlebodies in
$S^{4}$. A trivial surface-link in $S^{4}$ is determined regardless of the
embeddings and unique up to isotopies (see [9]).

A symplectic basis of a closed surface $F$ of genus $n$ is a system
$(x_{*},x^{\prime}_{*})$ of element pairs
$(x_{j},x^{\prime}_{j})\,(j=1,2,\dots,n)$ of $H_{1}(F;{\mathbf{Z}})$ with the
intersection numbers
$\mbox{Int}(x_{j},x_{j^{\prime}})=\mbox{Int}(x^{\prime}_{j},x^{\prime}_{j^{\prime}})=\mbox{Int}(x_{j},x^{\prime}_{j^{\prime}})=0$
for all distinct $j,j^{\prime}$ and $\mbox{Int}(x_{j},x^{\prime}_{j})=+1$ for
all $j$. By an argument on the intersection form

$\mbox{Int}:H_{1}(F;{\mathbf{Z}})\times H_{1}(F;{\mathbf{Z}})\to{\mathbf{Z}},$

any pair $(x_{1},x^{\prime}_{1})$ with $\mbox{Int}(x_{1},x^{\prime}_{1})=+1$
is extended to a symplextic basis $(x_{*},x^{\prime}_{*})$ of $F$. It is well-
known that every symplectic basis
$(x_{*},x^{\prime}_{*})=\\{(x_{j},x^{\prime}_{j})|\,j=1,2,\dots,n\\}$ is
realized by a system of oriented simple loop pairs
$(e_{*},e^{\prime}_{*})=\\{(e_{j},e^{\prime}_{j})|\,j=1,2,\dots,n\\}$ of $F$
with the geometric intersections $e_{j}\cap e_{j^{\prime}}=e^{\prime}_{j}\cap
e^{\prime}_{j^{\prime}}=e_{j}\cap e^{\prime}_{j^{\prime}}=\emptyset$ for all
distinct $j,j^{\prime}$ and $e_{j}\cap e^{\prime}_{j}$= a point for all $j$
which is called a loop basis of $F$.

For a surface-knot $F$ in $S^{4}$, an element $x\in H_{1}(F;{\mathbf{Z}})$ is
said to be spin if the ${\mathbf{Z}}_{2}$-reduction $[x]_{2}\in
H_{1}(F;{\mathbf{Z}}_{2})$ of $x$ has $\eta([x]_{2})=0$ for the
${\mathbf{Z}}_{2}$-quadratic function

$\eta:H_{1}(F;{\mathbf{Z}}_{2})\to{\mathbf{Z}}_{2}$

associated with a surface-knot $F$ in $S^{4}$. For a loop $\ell$ in $F$
bounding a surface $D_{\ell}$ in $S^{4}$ with $D_{\ell}\cap F=\ell$, the
${\mathbf{Z}}_{2}$-self-intersection number
$\mbox{Int}(D_{\ell},D_{\ell})\pmod{2}$ with respect to the $F$-framing is
defined as the value $\eta([\ell]_{2})$.

For every surface-knot $F$ in $S^{4}$, there is a spin basis of $F$ (see [7]).
This means that any spin pair $(x_{1},x^{\prime}_{1})$ with
$\mbox{Int}(x_{1},x^{\prime}_{1})=+1$ is extended to a spin symplectic basis
$(x_{*},x^{\prime}_{*})$ of $F$ by an argument of Arf invarinat of the
${\mathbf{Z}}_{2}$-quadratic function
$\eta:H_{1}(F;{\mathbf{Z}}_{2})\to{\mathbf{Z}}_{2}$. In particular, any spin
pair $(x_{1},x^{\prime}_{1})$ is realized by a spin pair
$(\ell_{1},\ell^{\prime}_{1})$ of $F$ which is extended to a spin loop basis
of $F$.

A 2-handle on a surface-knot $F$ in $S^{4}$ is a 2-handle $D\times I$ on $F$
embedded in $S^{4}$ such that $(D\times I)\cap F=(\partial D)\times I$, where
$I$ denotes a closed interval with $0$ as the center and $D\times 0$ is called
the core of the 2-handle $D\times I$ and identified with $D$. For a 2-handle
$D\times I$ on $F$ in $S^{4}$, the loop $\partial D$ of the core disk $D$ is a
spin loop in $F$ since $\eta([\partial D]_{2})=0$.

To save notation, if an embedding $h:D\times I\cup F\to X$ is given from a
2-handle $D\times I$ on a surface $F$ to a 4-manifold $X$, then the 2-handle
image $h(D\times I)$ and the core image $h(D)$ on $h(F)$ are denoted by
$hD\times I$ and $hD$, respectively.

An orthogonal 2-handle pair or simply an O2-handle pair on a surface-knot $F$
in $S^{4}$ is a pair $(D\times I,D^{\prime}\times I)$ of 2-handles $D\times I$
and $D^{\prime}\times I$ on $F$ which meet orthogonally on $F$, that is, which
meet $F$ only at the attaching annuli $(\partial D)\times I$ and $(\partial
D^{\prime})\times I$ so that the loops $\partial D$ and $\partial D^{\prime}$
meet transversely at just one point $q$ and the intersection $(\partial
D)\times I\cap(\partial D^{\prime})\times I$ is diffeomorphic to the square
$Q=\\{q\\}\times I\times I$ (see [11]).

An O2-handle basis of a trivial surface-knot $F$ of genus $n$ in $S^{4}$ is a
system $(D_{*}\times I,D^{\prime}_{*}\times I)$ of mutually disjoint O2-handle
pairs $(D_{i}\times I,D_{i}^{\prime}\times I)\,(i=1,2,\dots,n)$ on $F$ in
$S^{4}$ such that the loop system $(\partial D_{*},\partial
D^{\prime}_{*})=\\{(\partial D_{i},\partial
D_{i}^{\prime})|\,i=1,2,\dots,n\\}$ forms a spin loop basis of $F$.

Every trivial surface $F$ in $S^{4}$ is taken as the boundary of a standard
handlebody in the equatorial 3-sphere $S^{3}$ of the 4-sphere $S^{4}$. A
standard O2-handle basis of $F$ is an O2-handle basis of $F$ which is taken in
$S^{3}$ and a standard loop basis of $F$ is a loop basis of $F$ determined by
the attaching part of a standard O2-handle basis of $F$.

For any given spin loop basis of a trivial surface-knot $F$ of genus $n$ in
$S^{4}$, there is an O2-handle basis $(D_{*}\times I,D^{\prime}_{*}\times I)$
of a trivial surface-knot $F$ in $S^{4}$ such that the loop basis $(\partial
D_{*},\partial D^{\prime}_{*})$ coincides with the given spin loop basis of
$F$. This is because there is an equivalence $f:(S^{4},F)\to(S^{4},F)$ sending
the standard spin loop basis to the given spin loop basis of $F$ by [8, 11]
and hence there is an O2-handle basis of $F$ which is the image of the
standard O2-handle basis of $F$.

Let $p:S^{4}(F)_{2}\to S^{4}$ be the double branched covering projection
branched along $F$. The non-trivial covering involution of the double branched
covering space $S^{4}(F)_{2}$ is denoted by $\alpha$. The preimage $p^{-1}(F)$
in $\Sigma$ of $F$ which is the fixed point set of $\alpha$ and diffeomorphic
to $F$ is also written by the same notation as $F$. The following result is a
standard result.

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Lemma 2.1. For a standard O2-handle basis $(D_{*}\times I,D^{\prime}_{*}\times
I)$ of a trivial surface-knot $F$ of genus $n$ in $S^{4}$, there is an
orientation-preserving diffeomorphism

$f:S^{4}(F)_{2}\to\Sigma$

sending the 2-sphere pair system

$(S(D_{*}),S(D^{\prime}_{*}))=\\{(S(D_{i}),S(D^{\prime}_{i}))|\,i=1,2,\dots,n\\}$

to the standard O2-sphere basis $(S^{2}\times 1_{*},1\times S^{2}_{*})$ of the
stable 4-sphere $\Sigma$ of genus $n$. In particular, the 2-sphere pair system
$(S(D_{*}),S(D^{\prime}_{*}))$ is an O2-sphere basis of $\Sigma$.

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Proof of Lemma 2.1. Let $A_{i}\,(i=1,2,\dots,n)$ be mutually disjoint regular
neighborhoods of the 3-balls

$D_{i}\times I\cup D_{i}^{\prime}\times I\quad(i=1,2,\dots,n)$

in $S^{4}$. Since $A_{i}$ is a 4-ball, the closed complement
$(S^{4})^{n(0)}=\mbox{cl}(S^{4}\setminus\cup_{i=1}^{n}A_{i})$ is an
$n$-punctured manifold of $S^{4}$.

Let $P=F\cap(S^{4})^{n(0)}$ be a proper surface in $(S^{4})^{n(0)}$. Since $P$
is considered as an $n$-punctured 2-sphere, the pair $((S^{4})^{n(0)},P)$ is
considered as an $n$-punctured pair of a trivial 2-knot space $(S^{4},S^{2})$.

By using that the double branched covering space $S^{4}(S^{2})_{2}$ is
diffeomorphic to $S^{4}$, there is a diffeomorphism

$h^{\prime}:(S^{4})^{n(0)}(P)_{2}\to(S^{4})^{n(0)}.$

On the other hand, for the proper surface $P_{i}=F\cap A_{i}$ in the 4-ball
$A_{i}$, the pair $(A_{i},P_{i})$ is considered as a 1-punctured pair of a
trivial torus-knot space $(S^{4},T)$, so that the double branched covering
space $A_{i}(P_{i})_{2}$ is diffeomorphic to the 1-punctured manifold of the
double branched covering space $S^{4}(T)_{2}$.

The trivial torus-knot space $(S^{4},T)$ is the double of the product pair
$(A_{1},o)\times I=(A_{1}\times I,o\times I)$ for a trivial loop $o$ in the
interior of a 3-ball $A_{1}$ and an interval $I$, namely, $(S^{4},T)$ is
diffeomorphic to the boundary pair

$\partial((B,o)\times I^{2})=(\partial(B\times I^{2}),\partial(o\times
I^{2})),$

where $I^{m}$ denotes the $m$-fold product of $I$ for any $m\geq 2$. Thus, the
double branched covering space $S^{4}(T)_{2}$ is diffeomorphic to the boundary
$\partial(B(o)_{2}\times I^{2})$, where $B(o)_{2}$ is the double branched
covering space over $B$ branched along $o$ which is diffeomorphic to the
product $S^{2}\times I$. This means that the 5-manifold $B(o)_{2}\times I^{2}$
is the product $S^{2}\times I^{3}$, and the 4-manifold $S^{4}(T)_{2}$ is
diffeomorphic to $S^{2}\times S^{2}$, so that $A_{i}(P_{i})_{2}$ is
diffeomorphic to $(S^{2}\times S^{2})^{(0)}$. From this construction, we see
that there is an orientation-preserving diffeomorphism

$h_{i}:A_{i}(P_{i})_{2}\to(S^{2}\times S^{2})^{(0)}_{i}$

sending the O2-sphere pair $(S(D_{i}),S(D^{\prime}_{i}))$ to the standard
O2-sphere pair $(S^{2}\times 1_{i},1\times S^{2}_{i})$ of the $i$th copy
$(S^{2}\times S^{2})^{(0)}_{i}$ of $(S^{2}\times S^{2})^{(0)}$ for all $i$. By
a boundary collar concordance using $\Gamma_{4}=0$ in Cerf [2] or a boundary
collar isotopy using $\pi_{0}(\mbox{Diff}^{+}(S^{3}))=0$ by Hatcher [6], the
diffeomorphisms $h^{\prime}$ and $h_{i}\,(i=1,2,\dots,n)$ extend to a desired
orientation-preserving diffeomorphism $h:S^{4}(F)_{2}\to\Sigma$. $\square$

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The identification of $S^{4}(F)_{2}=\Sigma$ is fixed by an orientation-
preserving diffeomorphism $f:S^{4}(F)_{2}\to\Sigma$ given in Lemma 2.1.
Combining a result of [11], we have the following corollary.

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Corollary 2.2. For any two O2-handle bases $(D_{*}\times
I,D^{\prime}_{*}\times I)$ and $(E_{*}\times I,E^{\prime}_{*}\times I)$ of a
trivial surface-knot $F$ of genus $n$ in $S^{4}$, there is an orientation-
preserving $\alpha$-equivariant diffeomorphisn $\tilde{f}$ of $\Sigma$ sending
the 2-sphere pair system $(S(D_{*}),S(D^{\prime}_{*}))$ to the 2-sphere pair
system $(S(E_{*}),S(E^{\prime}_{*}))$. In particular, the 2-sphere pair system
$(S(D_{*}),S(D^{\prime}_{*}))$ for every O2-handle bases $(D_{*}\times
I,D^{\prime}_{*}\times I)$ is an O2-sphere basis of $\Sigma$.

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Proof of Corollary 2.2. By uniqueness of an O2-handle pair in [11, Theorem
3.2], there is an equivalence $f:(S^{4},F)\to(S^{4},F)$ keeping $F$ fixed such
that the O2-handle basis $(D_{*}\times I,D^{\prime}_{*}\times I)$ of $F$ is
transformed into the O2-handle basis $(E_{*}\times I,E^{\prime}_{*}\times I)$
of $F$. The lifting diffeomorphism $\tilde{f}:S^{4}(F)_{2}\to S^{4}(F)_{2}$
combined with Lemma 2.1 implies the first result. By taking $(E_{*}\times
I,E^{\prime}_{*}\times I)$ a standard O2-handle basis, the second result is
obtained. $\square$

x

An $n$-rooted disk family is the triplet $(d,d_{*},b_{*})$ where $d$ is a
disk, $d_{*}$ is a system of $n$ mutually disjoint disks
$d_{i}\,(i=1,2,\dots,n)$ in the interior of $d$ and $b_{*}$ is a system of $n$
mutually disjoint bands $b_{i}\,(i=1,2,\dots,n)$ in the $n$-punctured disk
$\mbox{cl}(d\setminus d_{*})$ such that $b_{i}$ spans an arc in the loop
$\partial d_{i}$ and an arc in the loop $\partial d$. Let $\hat{b}_{*}$ denote
the centerline system of the band system $b_{*}$.

In the following lemma, it is shown that there is a canonical $n$-rooted disk
family $(d,d_{*},b_{*})$ associated with an O2-handle basis $(D_{*}\times
I,D^{\prime}_{*}\times I)$ of a trivial surface-knot $F$ of genus $n$ in
$S^{4}$.

x

Lemma 2.3. Let $(D_{*}\times I,D^{\prime}_{*}\times I)$ be an O2-handle basis
of a trivial surface-knot $F$ of genus $n$ in $S^{4}$, and $(d,d_{*},b_{*})$
an $n$-rooted disk family. Then there is an embedding

$\varphi:(d,d_{*},b_{*})\times I\to(S^{4},D_{*}\times I,D^{\prime}_{*}\times
I)$

such that

(1) the surface $F$ is the boundary of the handlebody $V$ of genus $n$ given
by

$V=\varphi(\mbox{cl}(d\setminus d_{*})\times I),$

(2) there is an identification

$\varphi(d_{*}\times I,d_{*})=(\varphi(d_{*})\times
I,\varphi(d_{*}))=(D_{*}\times I,D_{*})$

as 2-handle systems on $F$ and

(3) there is an identification

$\varphi(b_{*}\times I,\hat{b}_{*}\times I)=(D^{\prime}_{*}\times
I,D^{\prime}_{*})$

as 2-handle systems on $F$.

x

In other words, Lemma 2.3 says that the 2-handle system $D_{*}\times I$
attaches to the handlebody $V$ along a longitude system of $V$ and the
2-handle system $D^{\prime}_{*}\times I$ attaches to $V$ along a meridian
system of $V$.

x

Proof of Lemma 2.3. If $(E_{*}\times I,E^{\prime}_{*}\times I)$ is a standard
O2-handle basis of $F$, then it is easy to construct such an embedding
$\varphi^{\prime}:(d,d_{*},b_{*})\times I\to(S^{4},E_{*}\times
I,E^{\prime}_{*}\times I)$. In general, by uniqueness of an O2-handle pair in
[11, Theorem 3.2], there is an equivalence $f:(S^{4},F)\to(S^{4},F)$ keeping
$F$ fixed and sending the standard O2-handle basis $(E_{*}\times
I,E^{\prime}_{*}\times I)$ of $F$ into the O2-handle basis $(D_{*}\times
I,D^{\prime}_{*}\times I)$ of $F$. The composite embedding
$\varphi=f\varphi^{\prime}:(d,d_{*},b_{*})\times I\to(S^{4},D_{*}\times
I,D^{\prime}_{*}\times I)$ is a desired embedding. $\square$

x

The embedding $\varphi$ in Lemma 2.3 is called a bump embedding. The 3-ball
$B=\varphi(D\times I)$, the handlebody $V$ in Lemma 2.3 and the pair $(B,V)$
are respectively called a bump 3-ball, a bump handlebody and a bump pair of
$F$ in $S^{4}$.

For a bump embedding

$\varphi:(d,d_{*},b_{*})\times I\to(S^{4},D_{*}\times I,D^{\prime}_{*}\times
I),$

there is an embedding $\tilde{\varphi}^{\prime}:d\times I\to S^{4}(F)_{2}$
with $p\tilde{\varphi}^{\prime}=\varphi$. The images
$\tilde{\varphi}^{\prime}(d_{*}\times I)$ and
$\tilde{\varphi}^{\prime}(b_{*}\times I)$ are respectively considered as
2-handle systems $\tilde{D}_{*}\times I$ and $\tilde{D}^{\prime}_{*}\times I$
on $F$ in $S^{4}(F)_{2}$ by the rules of Lemma 2.3 (1)-(3), so that
$(\tilde{D}_{*}\times I,\tilde{D}^{\prime}_{*}\times I)$ is an O2-handle basis
of $F$ in $(S^{4})(F)_{2}$ with $p(\tilde{D}_{*}\times
I,\tilde{D}^{\prime}_{*}\times I)=(D_{*}\times I,D^{\prime}_{*}\times I)$. The
embedding

$\tilde{\varphi}:(d,d_{*},b_{*})\times I\to(S^{4}(F)_{2},\tilde{D}_{*}\times
I,\tilde{D}^{\prime}_{*}\times I)$

has $p\tilde{\varphi}=\varphi$ and is called a lifting bump embedding of the
bump embedding $\varphi$. The bump 3-ball $\tilde{\varphi}(d\times I)$ and the
bump handlebody $\tilde{\varphi}(\mbox{cl}(d\setminus d_{*})\times I)$ are
respectively denoted by $B$ and $V$ in $S^{4}(F)_{2}$ unless confusion might
occur.

The composite embedding

$\alpha\tilde{\varphi}:(d,d_{*},b_{*})\times
I\to(S^{4}(F)_{2},\alpha\tilde{D}_{*}\times
I,\alpha\tilde{D}^{\prime}_{*}\times I)$

is another lifting bump embedding of the bump embedding $\varphi$. The bump
3-ball $\alpha\tilde{\varphi}(d\times I)=\alpha(B)$ and the bump handlebody
$\alpha\varphi(\mbox{cl}(d\setminus d_{*})\times I)=\alpha(V)$ are
respectively denoted by $\bar{B}$ and $\bar{V}$ in $S^{4}(F)_{2}$. Then we
have

$V\cap\bar{V}=B\cap\bar{B}=F\quad\mbox{in}\quad S^{4}(F)_{2}.$

For an O2-handle basis $(\tilde{D}_{*}\times I,\tilde{D}^{\prime}_{*}\times
I)$ of a trivial surface-knot $F$ in $S^{4}$, the O2-handle bases
$(\tilde{D}_{*}\times I,\tilde{D}^{\prime}_{*}\times I)$ and
$(\alpha\tilde{D}_{*}\times I,\alpha\tilde{D}^{\prime}_{*}\times I)$ of $F$ in
$S^{4}(F)_{2}$ are respectively denoted by

$(D_{*}\times I,D^{\prime}_{*}\times I),\quad\mbox{and}\quad(\bar{D}_{*}\times
I,\bar{D}^{\prime}_{*}\times I).$

Note that the unions $S(D_{i})=D_{i}\cup\bar{D}_{i}$ and
$S(D^{\prime}_{i})=D^{\prime}_{i}\cup\bar{D}^{\prime}_{i}$ are 2-spheres in
$S^{4}(F)_{2}$ such that $(S(D_{i}),S(D^{\prime}_{i}))$ is an O2-sphere pair
in $S^{4}(F)_{2}$.

For a lifting bump embedding, we have the following lemma.

x

Lemma 2.4. Let $\tilde{\varphi}:(d,d_{*},b_{*})\times I\to(\Sigma,D_{*}\times
I,D^{\prime}_{*}\times I)$ be a lifting bump embedding. Let
$u:\Sigma^{(0)}\to\Sigma$ be an embedding. Assume that the image
$\tilde{\varphi}(d\times I)$ is in the interior of $\Sigma^{(0)}$ to define
the composite embedding $u\tilde{\varphi}:(d,d_{*},b_{*})\times
I\to(\Sigma,uD_{*}\times I,uD^{\prime}_{*}\times I)$. Then there is a
diffeomorphism $g:\Sigma\to\Sigma$ which is isotopic to the identity such that
the composite embedding $gu\tilde{\varphi}:(d,d_{*},b_{*})\times
I\to(\Sigma,guD_{*}\times I,guD^{\prime}_{*}\times I)$ is identical to the
lifting bump embedding $\tilde{\varphi}:(d,d_{*},b_{*})\times
I\to(\Sigma,D_{*}\times I,D^{\prime}_{*}\times I)$.

x

Proof of Lemma 2.4. The 0-section $(d,d_{*},b_{*})\times 0$ of the line bundle
$(d,d_{*},b_{*})\times I$ of the $n$-rooted disk family $(d,d_{*},b_{*})$ is
identified with $(d,d_{*},b_{*})$. Then the disk $u(d)$ moves into the disk
$d$ in $\Sigma$ and then move the disk system $u(d_{*})$ and the band system
$u(b_{*})$ into the disk system $d_{*}$ and the band system $b_{*}$ in the
disk $d$, respectively. The result of these deformations is attained by a
diffeomorphism $g^{\prime}$ of $\Sigma$ which is isotopic to the identity.
Thus, we have

$g^{\prime}u\tilde{\varphi}(d,d_{*},b_{*})=\tilde{\varphi}(d,d_{*},b_{*}).$

Then there is a diffeomorphism $g^{\prime\prime}$ of $\Sigma$ which is
isotopic to the identity such that the composite embedding
$g^{\prime\prime}g^{\prime}u:\Sigma^{(0)}\to\Sigma$ preserves the normal line
bundle of the disk $d$ in $\Sigma^{(0)}$ and $\Sigma$. Thus, we have

$gu\tilde{\varphi}(d,d_{*},b_{*})\times I=\tilde{\varphi}(d,d_{*},b_{*})\times
I$

for the diffeomorphism $g=g^{\prime\prime}g^{\prime}$ of $\Sigma$ which is
isotopic to the identity.

$\square$

x

In Lemma 2.4, note that any disk interior of the disk systems $gu\bar{D}_{*}$
and $gu\bar{D}^{\prime}_{*}$ does not meet the bump 3-ball
$B=\tilde{\varphi}(d\times I)$ in $\Sigma$.

In fact, since $gu$ defines an embedding from $B\cup\bar{B}$ with
$B\cap\bar{B}=F$ into $\Sigma$ and we have $gu(B,F)=(B,F)$, the complement
$gu(\bar{B})\setminus F$ of $F$ in the 3-ball $gu(\bar{B})$ does not meet the
bump 3-ball $B$, which means that any disk interior of the disk systems
$gu\bar{D}_{*}$ and $gu\bar{D}^{\prime}_{*}$ does not meet the bump 3-ball
$B$.

Unless $\Sigma^{(0)}$ and $\Sigma$ have the same genus $n$, this property
cannot be guaranteed.

x

3\. Proof of a homological version of Theorem 1.1

The following lemma is related to the intersection numbers of the lifting
O2-sphere bases between two O2-handle bases of $F$ in $S^{4}$.

x

Lemma 3.1. Let $(D_{*}\times I,D^{\prime}_{*}\times I)$ be an O2-handle basis
of a trivial surface-knot $F$ of genus $n$ in $S^{4}$, and
$(\ell_{*},\ell^{\prime}_{*})=(\partial D_{*},\partial D^{\prime}_{*})$ a spin
loop basis of $F$. For a 2-handle $E\times I$ on $F$ in $S^{4}$, assume that
the homology class $[e]\in H_{1}(F;{\mathbf{Z}})$ of the loop $e=\partial E$
is written as

$[e]=\sum_{j=1}^{n}k_{j}[\ell_{j}]+\sum_{j=1}^{n}s_{j}[\ell_{j}^{\prime}]$

in $H_{1}(F;{\mathbf{Z}})$ for some integers $k_{j},s_{j}\,(j=1,2,\dots,n)$.
Then the homology class $[S(E)]\in H_{2}(\Sigma;{\mathbf{Z}})$ is written as

$[S(E)]=\sum_{j=1}^{n}k_{j}[S(D_{j})]+\sum_{j=1}^{n}s_{j}[S(D^{\prime}_{j})].$

x

Proof of Lemma 3.1. Let $N(F)^{c}=\mbox{cl}(S^{4}\setminus N(F))$ for a
regular neighborhood $N(F)$ of $F$ in $S^{4}$. We may consider that the disk
$\overset{\frown}{E}=E\cap N(F)^{c}$ meets transversely the disks
$\overset{\frown}{D}_{j}=D_{j}\cap N(F)^{c}$,
$\overset{\frown}{D^{\prime}}_{j}=D^{\prime}_{j}\cap
N(F)^{c}\,(j=1,2,\dots,n)$ with a finite number of points in the interiors.
Since the disks $\overset{\frown}{E}$, $\overset{\frown}{D}_{j}$ and
$\overset{\frown}{D^{\prime}}_{j}$ are lifted to the double disks with
opposite signs, the intersections
$\overset{\frown}{E}\cap\overset{\frown}{D}_{j}$ and
$\overset{\frown}{E}\cap\overset{\frown}{D^{\prime}}_{j}$ do not contribute
the following intersection numbers

$\mbox{Int}([S(E)],[S(D_{j})]),\quad\mbox{Int}([S(E)],[S(D^{\prime}_{j})])\quad(j=1,2,\dots,n)$

in $\Sigma$. Examining the geometric intersections between the lift of $E\cap
N(F)$ to $S(F)_{2}$ and the lifts of $D_{j}\cap N(F),D^{\prime}_{j}\cap
N(F)\,(j=1,2,\dots,n)$ to $S(F)_{2}$, we have that

$\mbox{Int}([S(E)],[S(D_{j})])=s_{j}\quad\mbox{and}\quad\mbox{Int}([S(E)],[S(D^{\prime}_{j})])=k_{j}$

for all $j$. Since

$\mbox{Int}([S(D_{i})],[S(D_{j})])=\mbox{Int}([S(D^{\prime}_{i})],[S(D^{\prime}_{j})])=0$

and

$\mbox{Int}([S(D_{i})],[S(D^{\prime}_{j}])=\delta_{ij}$

for all $i,j$, the desired identity is obtained. $\square$

x

The following lemma is a homological version of Theorem 1.1.

x

Lemma 3.2. For any pseudo-O2-sphere bases $(R_{*},R^{\prime}_{*})$ and
$(S_{*},S^{\prime}_{*})$ of the stable 4-sphere $\Sigma$ of genus $n$, there
is an $\alpha$-invariant orientation-preserving diffeomorphism
$\tilde{f}:\Sigma\to\Sigma$ which induces an isomorphism

$\tilde{f}_{*}:H_{2}(\Sigma;{\mathbf{Z}})\to H_{2}(\Sigma;{\mathbf{Z}})$

such that

$[\tilde{f}R_{i}]=[S_{i}]\quad\mbox{and}\quad[\tilde{f}R_{i}^{\prime}]=[S^{\prime}_{i}]$

for all $i$.

x

Proof of Lemma 3.2. It suffices to show this lemma when
$(R_{*},R^{\prime}_{*})$ is a particular O2-sphere basis of $\Sigma$, because
the composite diffeomorphism of two $\alpha$-invariant orientation-preserving
diffeomorphisms of $\Sigma$ is also a $\alpha$-invariant orientation-
preserving diffeomorphism of $\Sigma$. By Lemma 2.3, assume that
$(R_{*},R^{\prime}_{*})=(S(D_{*}),S(D^{\prime}_{*}))$ for an O2-handle basis
$(D_{*}\times I,D^{\prime}_{*}\times I)$ of a trivial surface-knot $F$ of
genus $n$ in $S^{4}$.

Let $(\ell_{*},\ell^{\prime}_{*})=(\partial D_{*},\partial D^{\prime}_{*})$,
which is a spin loop basis of $F$. By a base change of
$H_{2}(\Sigma;{\mathbf{Z}})$, we can write

$[S_{i}]=\sum_{j=1}^{n}k_{ij}[R_{j}]+\sum_{j=1}^{n}s_{ij}[R^{\prime}_{j}],\quad[S^{\prime}_{i}]=\sum_{j=1}^{n}k^{\prime}_{ij}[R_{j}]+\sum_{j=1}^{n}s^{\prime}_{ij}[R^{\prime}_{j}]$

for some integers
$k_{ij},s_{ij},k^{\prime}_{ij},s^{\prime}_{ij}\,(i,j=1,2,\dots,n)$. By
definition, we have the intersection numbers

$\displaystyle\mbox{Int}([R_{i}],[R_{j}])$ $\displaystyle=$
$\displaystyle\mbox{Int}([S_{i}],[S_{j}])=0,$
$\displaystyle\mbox{Int}([R_{i}^{\prime}],[R_{j}^{\prime}])$ $\displaystyle=$
$\displaystyle\mbox{Int}([S_{i}^{\prime}],[S_{j}^{\prime}])=0,$
$\displaystyle\mbox{Int}([R_{i}],[R_{j}^{\prime}])$ $\displaystyle=$
$\displaystyle\mbox{Int}([S_{i}],[S_{j}^{\prime}])=\delta_{ij}$

for all $i,j$. By counting the identity

$\mbox{Int}([S_{1}],[S^{\prime}_{1}])=\sum_{j=1}^{n}k_{1j}s^{\prime}_{1j}+\sum_{j=1}^{n}s_{1j}k^{\prime}_{1j}=1,$

the homology classes $x_{1}$ and $x^{\prime}_{1}$ in $H_{1}(F;{\mathbf{Z}})$
defined by

$x_{1}=\sum_{j=1}^{n}k_{1j}[\ell_{j}]+\sum_{j=1}^{n}s_{1j}[\ell_{j}^{\prime}]\quad\mbox{and}\quad
x^{\prime}_{1}=-\sum_{j=1}^{n}k_{1j}^{\prime}[\ell_{j}]+\sum_{j=1}^{n}s_{1j}^{\prime}[\ell_{j}^{\prime}]$

have the intersection number $\mbox{Int}(x_{1},x^{\prime})=1$ in $F$. Then the
pair $(x_{1},x^{\prime}_{1})$ is realized by a simple loop pair
$(e_{1},e^{\prime}_{1})$ meeting at a single point in $F$. To show that the
simple loops $e_{1}$ and $e^{\prime}_{1}$ are spin simple loops, note that

$\mbox{Int}([S_{1}],[S_{1}])=\sum_{j=1}^{n}2k_{1j}s_{1j}=0\quad\mbox{and}\quad\mbox{Int}([S^{\prime}_{1}],[S^{\prime}_{1}])=\sum_{j=1}^{n}2k^{\prime}_{1j}s^{\prime}_{1j}=0,$

so that

$\sum_{j=1}^{n}k_{1j}s_{1j}=0\quad\mbox{and}\quad\sum_{j=1}^{n}k^{\prime}_{1j}s^{\prime}_{1j}=0.$

The ${\mathbf{Z}}_{2}$-quadratic function
$\eta:H_{1}(F;{\mathbf{Z}}_{2})\to{\mathbf{Z}}_{2}$ associated with $F$ in
$S^{4}$ is given by the identity

$\eta(x+y)=\eta(x)+\eta(y)+\mbox{Int}(x,y)_{2}$

for all ${\mathbf{Z}}_{2}$-homology classes $x,y\in H_{1}(F;{\mathbf{Z}}_{2})$
and the ${\mathbf{Z}}_{2}$-intersection number $\mbox{Int}(x,y)_{2}$ in $F$.
Hence the values of the ${\mathbf{Z}}_{2}$-homology classes
$[e_{1}]_{2},[e^{\prime}_{1}]_{2}\in H_{1}(F;{\mathbf{Z}}_{2})$ by the
${\mathbf{Z}}_{2}$-quadratic function $\eta$ are calculated to be

$\eta([e_{1}]_{2})=\sum_{j=1}^{n}k_{1j}s_{1j}=0\pmod{2},\quad\eta([e^{\prime}_{1}]_{2})=-\sum_{j=1}^{n}k^{\prime}_{1j}s^{\prime}_{1j}=0\pmod{2},$

by using $\eta([\ell_{j}]_{2})=\eta([\ell^{\prime}_{j^{\prime}}]_{2})=0$ and
$\mbox{Int}([\ell_{j}]_{2},[\ell_{j^{\prime}}]_{2})_{2}=\delta_{jj^{\prime}}$
for all $j,j^{\prime}$. Thus, the simple loops $e_{1}$ and $e^{\prime}_{1}$
are spin simple loops. Then the spin loop pair $(e_{1},e^{\prime}_{1})$
extends to a spin loop basis $(e_{i},e^{\prime}_{i})\,(i=1,2,\dots,n)$ of $F$
in $S^{4}$.

By [8, 11], there is an equivalence $f:(S^{4},F)\to(S^{4},F)$ sending the spin
loop basis $(\ell_{*},\ell^{\prime}_{*})$ to the spin loop basis
$(e_{*},e^{\prime}_{*})$. Apply Lemma 3.1 to the 2-handle $f(D_{1})\times I$
under the O2-handle basis $(D_{*}\times I,D^{\prime}_{*}\times I)$ of $F$.
Since the loop $\partial f(D_{1})=e_{1}$ represents the homology class

$\sum_{j=1}^{n}k_{1j}[\ell_{j}]+\sum_{j=1}^{n}s_{1j}[\ell_{j}^{\prime}],$

in $H_{1}(F;{\mathbf{Z}})$, the homology class $[S(fD_{1})]\in
H_{2}(\Sigma;{\mathbf{Z}})$ is equal to

$\sum_{j=1}^{n}k_{1j}[R_{j}]+\sum_{j=1}^{n}s_{1j}[R^{\prime}_{j}]=[S_{1}].$

Replace
$(\tilde{f}R_{*},\tilde{f}R^{\prime}_{*})=(S(fD_{*}),S(fD^{\prime}_{*}))$ with
$(R_{*},R^{\prime}_{*})=(S(D_{*}),S(D^{\prime}_{*}))$. Then we can write

$[S_{1}]=[R_{1}]\quad\mbox{and}\quad[S_{1}^{\prime}]=\sum_{j=1}^{n}k_{1j}^{\prime}[R_{j}]+\sum_{j=1}^{n}s_{1j}^{\prime}[R_{j}^{\prime}]$

with $s^{\prime}_{11}=1$. Let $(e_{i},e^{\prime}_{i})\,(i=1,2,\dots,n)$ be a
spin loop basis of $F$ in $S^{4}$ with

$[e_{1}]=[\ell_{1}]\quad\mbox{and}\quad[e_{1}^{\prime}]=\sum_{j=1}^{n}k_{1j}^{\prime}[\ell_{j}]+\sum_{j=1}^{n}s_{1j}^{\prime}[\ell_{j}^{\prime}].$

By [11], there is an equivalence $f:(S^{4},F)\to(S^{4},F)$ sending
$(\ell_{*},\ell^{\prime}_{*})$ to $(e_{*},e^{\prime}_{*})$. Apply Lemma 3.1 to
the 2-handles $fD_{1}\times I$ and $fD^{\prime}_{1}\times I$ under the
O2-handle basis $(D_{*}\times I,D^{\prime}_{*}\times I)$ of $F$. Then the
lifting diffeomorphism $\tilde{f}:\Sigma\to\Sigma$ sends the homology classes
$[R_{1}]=[S(D_{1})]$ and $[R^{\prime}_{1}]=[S(D^{\prime}_{1})]$ to the
homology classes $[S(fD_{1})]=[S_{1}]$ and
$[S(fD^{\prime}_{1})]=[S_{1}^{\prime}]$, respectively. By continuing this
process, there is an equivalence $f:(S^{4},F)\to(S^{4},F)$ such that the
lifting diffeomorphism $\tilde{f}:\Sigma\to\Sigma$ sends the homology classes
$[R_{i}]=[S(D_{i})]$ and $[R^{\prime}_{i}]=[S(D_{i}^{\prime})]$ to the
homology classes $[S(fD_{i})]=[S_{i}]$ and
$[S(fD_{i}^{\prime})]=[S_{i}^{\prime}]$ for all $i$, respectively. $\square$

x

4\. Proof of Theorem 1.1

Throughout this section, the proof of Theorem 1.1 is done. As well as the
proof of Lemma 3.2, it suffices to show this theorem when
$(R_{*},R^{\prime}_{*})$ is an O2-sphere basis of $\Sigma$ with
$(R_{*},R^{\prime}_{*})=(S(D_{*}),S(D^{\prime}_{*}))$ for an O2-handle basis
$(D_{*}\times I,D^{\prime}_{*}\times I)$ of a trivial surface-knot $F$ of
genus $n$ in $S^{4}$.

Let $\Omega(S_{*},S^{\prime}_{*})$ be the region of the pseudo-O2-sphere basis
$(S_{*},S^{\prime}_{*})$ of $\Sigma$. The 4-manifold obtained from
$\Omega(S_{*},S^{\prime}_{*})$ by adding a 4-ball $D^{4}$ in place of the
residual region ${\Omega}^{c}(S_{*},S^{\prime}_{*})$ is diffeomorphic to
$\Sigma$. This means that there is an orientation-preserving embedding

$u:\Sigma^{(0)}\to\Sigma$

such that

$(uS(D_{*}),uS(D^{\prime}_{*}))=(S_{*},S^{\prime}_{*}).$

By Lemma 3.2, after applying an $\alpha$-invariant orientation-preserving
diffeomorphism $\tilde{f}:\Sigma\to\Sigma$, we assume that the homology
classes $[uS(D_{i})]=[S_{i}]$ and $[uS(D^{\prime}_{i})]=[S^{\prime}_{i}]$ are
identical to the homology classes $[R_{i}]=[S(D_{i})$ and
$[R^{\prime}_{i}]=[S(D^{\prime}_{i})]$ for all $i$, respectively. Let $(B,V)$
be a bump pair of the O2-handle basis $(D_{*}\times I,D^{\prime}_{*}\times I)$
of $F$ in $S^{4}$ defined soon after Lemma 2.3. Recall that the two lifts of
$(B,V)$ to $\Sigma$ under the double branched covering projection
$p:S^{4}(F)_{2}\to S^{4}$ are denoted by $(B,V)$ and $(\bar{B},\bar{V})$.

For the proof of Theorem 1.1, we provide with three lemmas. The first lemma is
as follows.

x

Lemma 4.1. There is a diffeomorphism $g$ of $\Sigma$ which is isotopic to the
identity such that the composite embedding

$gu:\Sigma^{(0)}\to\Sigma$

preserves the bump pair $(B,V)$ in $\Sigma$ identically and has the property
that every disk interior in the disk systems $gu\bar{D}_{*}$ and
$gu\bar{D}^{\prime}_{*}$ meets every disk in the disk systems $\bar{D}_{*}$
and $\bar{D}^{\prime}_{*}$ only with the intersection number $0$.

x

Proof of Lemma 4.1. By Lemma 2.4, there is a diffeomorphism
$g^{\prime}:\Sigma\to\Sigma$ which is isotopic to the identity such that the
composite embedding $g^{\prime}u:\Sigma^{(0)}\to\Sigma$ preserves the bump
pair $(B,V)$ in $\Sigma$ identically and has the property that any disk
interior of the disk system $u\bar{D}_{*}$ and $u\bar{D}^{\prime}_{*}$ does
not meet the O2-handle basis $(D_{*}\times I,D^{\prime}_{*}\times I)$ in
$\Sigma$ and every disk interior of the disk system $u\bar{D}_{*}$ and
$u\bar{D}^{\prime}_{*}$ transversely meets the disk system $\bar{D}_{*}$ and
$\bar{D}^{\prime}_{*}$.

The overlapping parts $g^{\prime}uD_{i}=D_{i}$ and
$g^{\prime}uD^{\prime}_{i}=D^{\prime}_{i}$ do not contribute to the
intersection numbers on the homology classes
$[g^{\prime}uS(D_{i})]=[S(D_{i})]$ and
$[g^{\prime}uS(D^{\prime}_{j})]=[S(D^{\prime}_{j})]$ in $\Sigma$ for all $i$,
$j$, which are given by

$\mbox{Int}([S(D_{i})],[S(D_{j})])=\mbox{Int}([S(D^{\prime}_{i})],[S(D^{\prime}_{j})])=0$

and

$\mbox{Int}([S(D_{i})],[S(D^{\prime}_{j}])=\delta_{ij}$

for all $i,j$. Thus, after a slight isotopic deformation $g$ of $g^{\prime}$
keeping $(D_{*}\times I,D^{\prime}_{*}\times I)$ fixed, the composite
embedding $gu:\Sigma^{(0)}\to\Sigma$ preserves the bump pair $(B,V)$ in
$\Sigma$ identically and has the property that any disk interior of the disk
system $u\bar{D}_{*}$ and $u\bar{D}^{\prime}_{*}$ does not meet the O2-handle
basis $(D_{*}\times I,D^{\prime}_{*}\times I)$ in $\Sigma$ and every disk
interior of the disk system $u\bar{D}_{*}$ and $u\bar{D}^{\prime}_{*}$
transversely meets the disk system $\bar{D}_{*}$ and $\bar{D}^{\prime}_{*}$
only with the intersection number $0$. This completes the proof of Lemma 4.1.
$\square$

x

By Lemma 4.1, we can assume that the orientation-preserving embedding

$u:\Sigma^{(0)}\to\Sigma$

sends the bump pair $(B,V)$ to itself identically and has the property that
every disk interior in the disk systems $u\bar{D}_{*}$ and
$u\bar{D}^{\prime}_{*}$ meets every disk in the disk systems $\bar{D}_{*}$ and
$\bar{D}^{\prime}_{*}$ only with intersection number $0$. Then we have the
following lemma:

x

Lemma 4.2. There is a diffeomorphism $g$ of $\Sigma$ which is isotopic to the
identity such that the composite embedding

$gu:\Sigma^{(0)}\to\Sigma$

sends the disk systems $D_{*}$ and $D^{\prime}_{*}$ identically and and the
disk interiors of the disk systems $gu\bar{D}_{*}$, $gu\bar{D}^{\prime}_{*}$
to be disjoint from the disk systems $\bar{D}_{*}$ and $\bar{D}^{\prime}_{*}$
in $\Sigma$.

x

Proof of Lemma 4.2. Between the open disks $u\mbox{Int}(D_{i})$,
$u\mbox{Int}(D^{\prime}_{i^{\prime}})$ for all $i,i^{\prime}$ and the open
disks $\mbox{Int}(D_{j})$, $\mbox{Int}(D^{\prime}_{j^{\prime}})$ for all
$j,j^{\prime}$, suppose that an open disk, say $u\mbox{Int}(D_{i})$, meets an
open disk, say $\mbox{Int}(D_{j})$ with a pair of points with opposite signs.

Let $a$ be a simple arc in the open disk $\mbox{Int}(D_{j})$ joins the pair of
points whose interior does not meet the intersection points
$\mbox{Int}(D_{j})\cap u\mbox{Int}(D_{*})$.

Let $T(a)$ be the torus obtained from the 2-sphere $uS(D_{i})$ by a surgery
along a 1-handle $h^{1}(a)$ on $uS(D_{i})$ whose core is the arc $a$.

Slide the arc $a$ along the open disk $\mbox{Int}(D_{j})$ without moving the
endpoints so that

(i) the 1-handle $h^{1}(a)$ passes once time through a thickening
$S^{\prime}_{j}\times I$ of a 2-sphere $S^{\prime}_{j}$ parallel to the
2-sphere $S(D^{\prime}_{j})$ (not meeting $S(D^{\prime}_{j})$), and

(ii) the intersection $h^{1}(a)\cap(S^{\prime}_{j}\times I)$ is a 2-handle
$d_{j}\times I$ on the torus $T(a)$ which is a strong deformation retract of
the 2-handle $h^{1}(a)$ on $T(a)$.

x

Let $\tilde{u}S(D_{i})$ be the 2-sphere obtained from $T(a)$ by the surgery
along the 2-handle $d_{j}\times I$. This 2-sphere $\tilde{u}S(D_{i})$ is
obtained from the 2-sphere $uS(D_{i})$ as $\tilde{u}S(D_{i})=\gamma uS(D_{i})$
by the composite embedding $\gamma u:\Sigma^{(0)}\to\Sigma$ for a
diffeomorphism $\gamma:\Sigma\to\Sigma$ which is isotopic to the identity.
This isotopy of $\gamma$ keeps the outside of a regular neighborhood of
$h^{1}(a)$ in the image of $\gamma$ fixed.

Regard the 2-handle $d_{j}\times I$ on $T(a)$ as a 1-handle of the 2-sphere
$\tilde{u}S(D_{i})$. Let $\tilde{\tilde{u}}S(D_{i})$ be the 2-sphere obtained
from $T(a)$ by the surgery along the 2-handle
$\mbox{cl}(S^{\prime}_{j}\setminus d_{j})\times I$ and regard this 2-handle as
a 1-handle on the 2-sphere $\tilde{\tilde{u}}S(D_{i})$. Then by a homotopy
deformation of a 1-handle (see [9, Lemma 1.4]), the 2-sphere
$\tilde{u}S(D_{i})$ is isotopically deformed into the 2-sphere
$\tilde{\tilde{u}}S(D_{i})$. Thus, there is a diffeomorphism
$\gamma^{\prime}:\Sigma\to\Sigma$ which is isotopic to the identity such that

$\tilde{\tilde{u}}S(D_{i})=\gamma^{\prime}\tilde{u}S(D_{i})=\gamma^{\prime}\gamma
uS(D_{i}).$

This isotopy of $\gamma^{\prime}$ keeps the outside of a regular neighborhood
of $S^{\prime}_{j}\times I$ fixed.

By these operations, the total geometric intersection number between the open
disks $\gamma^{\prime}\gamma u\mbox{Int}(D_{i})$, $\gamma^{\prime}\gamma
u\mbox{Int}(D^{\prime}_{i^{\prime}})$ for all $i,i^{\prime}$ and the open
disks $\mbox{Int}(D_{j})$, $\mbox{Int}(D^{\prime}_{j^{\prime}})$ for all
$j,j^{\prime}$ is reduced by 2.

By continuing this process, we have a diffeomorphism $g:\Sigma\to\Sigma$ which
is isotopic to the identity such that the composite embedding
$gu:\Sigma^{(0)}\to\Sigma$ sends the disk systems $D_{*}$ and $D^{\prime}_{*}$
identically and the open disks $ug\mbox{Int}(D_{i})$,
$ug\mbox{Int}(D^{\prime}_{i^{\prime}})$ are disjoint from the open disks
$\mbox{Int}(D_{j})$, $\mbox{Int}(D^{\prime}_{j^{\prime}})$ for all
$i,i^{\prime},j,j^{\prime}$. This completes the proof of Lemma 4.2. $\square$

x

For the O2-sphere basis $(S(D_{*}),S(D^{\prime}_{*}))$ of $\Sigma$, let
$q_{*}=\\{q_{i}=S(D_{i})\cap S(D^{\prime}_{i})|\,i=1,2,\dots,n\\}$ be the
transverse intersection point system between $S(D_{*})$ and
$S(D^{\prime}_{*})$.

The diffeomorphism $g$ of $\Sigma$ in Lemma 4.2 is deformed so that the disks
$guD_{i}$ and $D_{i}$ are separated, and then the disks $guD^{\prime}_{i}$ and
$D^{\prime}_{i}$ are separated while leaving the transverse intersection point
$q_{i}$. By this deformation, we obtain a pseudo-O2-sphere basis
$(guS(D_{*}),guS(D^{\prime}_{*}))$ of $\Sigma$ which meets the O2-sphere basis
$(S(D_{*}),S(D^{\prime}_{*}))$ at just the transverse intersection point
system $q_{*}$.

Next, the diffeomorphism $g$ of $\Sigma$ is deformed so that a disk
neighborhood system of $q_{*}$ in $guS(D_{*})$ and a disk neighborhood system
of $q_{*}$ in $S(D_{*})$ are matched, and then a disk neighborhood system of
$q_{*}$ in $guS(D^{\prime}_{*})$ and a disk neighborhood system of $q_{*}$ in
$S(D^{\prime}_{*})$ are matched.

Thus, there is a diffeomorphism $g$ of $\Sigma$ which is isotopic to the
identity such that the meeting part of the pseudo-O2-sphere basis
$(guS(D_{*}),guS(D^{\prime}_{*}))$ and the O2-sphere basis
$(S(D_{*}),S(D^{\prime}_{*}))$ is just a disk neighborhood pair system
$(d_{*},d^{\prime}_{*})$ around the transverse intersection point system
$q_{*}$.

Now, assume that for an embedding $u:\Sigma^{(0)}\to\Sigma$, the meeting part
of the pseudo-O2-sphere basis $(uS(D_{*}),uS(D^{\prime}_{*}))$ and the
O2-sphere basis $(S(D_{*}),S(D^{\prime}_{*}))$ is just a disk neighborhood
pair system $(d_{*},d^{\prime}_{*})$ of $q_{*}$. Then we have the following
lemma:

x

Lemma 4.3. There is an orientation-preserving diffeomorphism $h$ of $\Sigma$
such that the composite embedding

$hu:\Sigma^{(0)}\to\Sigma$

preserve the O2-sphere basis $(S(D_{*}),S(D^{\prime}_{*}))$ identically.

x

For the proof of Lemma 4.3, Lemma 4.4, called Framed Light-bulb Diffeomorphism
Lemma is provided which is easily proved in comparison with the isotopy
version of this lemma using Gabai’s 4D light-bulb theorem in [5] established
in Section 6.

A 4D solid torus is a 4-manifold $Y$ in $S^{4}$ which is diffeomorphic to
$S^{1}\times D^{3}$ and $Y^{c}=\mbox{cl}(S^{4}\setminus Y)$. Let $Y_{*}$ be a
system of mutually disjoint 4D solid tori $Y_{i},\,(i=1,2,\dots,n)$ in
$S^{4}$, and $Y^{c}_{*}$ the system of the 4-manifolds
$Y^{c}_{i},\,(i=1,2,\dots,n)$. Let

$\cap Y^{c}_{*}=\cap_{i=1}^{n}Y^{c}_{i}.$

Then Lemma 4.4 is stated as follows.

x

Lemma 4.4 (Framed Light-bulb Diffeomorphism Lemma). Let $D_{*}\times I$ be a
system of mutually disjoint 2-handles $D_{i}\times I\,(i=1,2,\dots,n)$ in
$\cap Y^{c}_{*}$ such that $(\partial D_{i})\times I$ attaches to $\partial
Y^{c}_{i}$ for all $i$. If $E_{*}\times I$ is any system of mutually disjoint
2-handles $E_{i}\times I\,(i=1,2,\dots,n)$ in $\cap Y^{c}_{*}$ such that

$(\partial E_{i})\times I=(\partial D_{i})\times I$

for all $i$, then there is an orientation-preserving diffeomorphism
$h:S^{4}\to S^{4}$ sending $Y_{*}$ identically such that

$h(D_{*}\times I,D_{*})=(E_{*}\times I,E_{*}).$

x

Proof of Lemma 4.4. Let $\ell_{*}$ be the system of the loops
$\ell_{i}=\partial D_{i}=\partial E_{i}\,(i=1,2,\dots,n)$. Let
$c:\ell_{*}\times[0,1]\to D_{*}$ be a boundary collar function of $D_{*}$ with
$c(x,0)=x$ for all $x\in\ell_{*}$, and $c^{\prime}:\ell_{*}\times[0,1]\to
E_{*}$ a boundary collar function of $E_{*}$ with $c^{\prime}(x,0)=x$ for all
$x\in\partial\ell_{*}$. Assume that

$c(x,t)\times I=c^{\prime}(x,t)\times I$

for all $x\in\ell_{*}$ and $t\in[0,1]$. Let $\nu(\partial
D_{*})=c(\ell_{*}\times[0,1])$ and
$\hat{D}_{*}=\mbox{cl}(D_{*}\setminus\nu(\partial D_{*}))$. Similarly, let
$\nu(\partial E_{*})=c^{\prime}(\ell_{*}\times[0,1])$ and
$\hat{E}_{*}=\mbox{cl}(E_{*}\setminus\nu(\partial E_{*}))$. Consider
$D_{*}\times I$ and $E_{*}\times I$ in $S^{4}$. Let $\beta_{*}$ be the system
of arcs $\beta_{i}\,(i=1,2,\dots,n)$ such that $\beta_{i}$ is an arc in
$\ell_{i}$, and $\beta_{*}^{c}$ the system of the arcs
$\beta^{c}_{i}=\mbox{cl}(\ell_{i}\setminus\beta_{i})\,(i=1,2,\dots,n)$.

By 3-cell moves within a regular neighborhood of
$c(\beta_{*}\times[0,1])\times I\cup\hat{D}_{*}\times I$ in $S^{4}$, there is
an orientation-preserving diffeomorphism $h^{\prime}$ of $S^{4}$ such that

$h^{\prime}D_{*}\times I)=c(\beta^{c}_{*}\times[0,1])\times I).$

Since $\nu(\partial D_{*})$ is identical to $\nu(\partial E_{*})$ and hence
$c(\beta^{c}_{*}\times[0,1]$ is identical to
$c^{\prime}(\beta^{c}_{*}\times[0,1]$, by 3-cell moves within a regular
neighborhood of $c(\beta_{*}\times[0,1])\times I\cup\hat{E}_{*}\times I$ in
$S^{4}$, there is an orientation-preserving diffeomorphism $h^{\prime\prime}$
of $S^{4}$ such that

$h^{\prime\prime}(c^{\prime}(\beta^{c}_{*}\times[0,1])\times I)=E_{*}\times
I.$

The composite diffeomorphism $h^{\prime\prime}h^{\prime}$ is an orientation-
preserving diffeomorphism of $S^{4}$ sending $c(\ell_{*}\times[0,1])$ (and
hence identically and $D_{*}\times I$ to $E_{*}\times I$. Let $N(\ell_{*})$ be
a regular neighborhood system of the loop system $\ell_{*}$ in $S^{4}$ meeting
$c(\ell_{*}\times[0,1])$ regularly which is a system of $n$ mutually disjoint
4D tori. Since the diffeomorphism $h^{\prime\prime}h^{\prime}$ is deformed
into an orientation-preserving diffeomorphism $h$ of $S^{4}$ which sends
$N(\ell_{*})$ identically such that $h(D_{*}\times I,D_{*})=(E_{*}\times
I,E_{*})$. Here, the 4D torus system $N(\ell_{*})$ can be replaced with any
given $Y_{*}$ because $N(\ell_{*})$ is isotopic to $Y_{*}$. This completes the
proof of Lemma 4.4. $\square$

x

The proof of Lemma 4.3 is done as follows.

x

Proof of Lemma 4.3. Consider the 4-manifold $X$ obtained from $\Sigma$ by
replacing a regular neighborhood system $N(S(D^{\prime}_{*}))=S^{2}\times
D^{2}_{*}$ of the 2-sphere system $S(D^{\prime}_{*})$ in $\Sigma$ with the 4D
solid torus system $Y_{*}=D^{3}\times S^{1}_{*}$. The 4-manifold $X$ is
diffeomorphic to the 4-sphere $S^{4}$. Let $\tilde{E}_{*}\times I$ and
$E_{*}\times I$ be the 2-handle systems in $X=S^{4}$ attached to the 4D solid
torus system $Y_{*}$ which are obtained from the thickening 2-sphere systems
$uS(D_{*})\times I$ and $S(D_{*})\times I$ in $\Sigma$, respectively. Lemma
4.4 can be used for the 2-handle systems $\tilde{E}_{*}\times I$ and
$E_{*}\times I$ attached to $Y_{*}$. Thus, there is an orientation-preserving
diffeomorphism $\rho:S^{4}\to S^{4}$ sending $Y_{*}$ identically such that

$\rho u(\tilde{E}_{*}\times I,\tilde{E}_{*})=(E_{*}\times I,E_{*}).$

By returning the 4D solid torus system $Y_{*}$ in $X$ to the regular
neighborhood system $N(S(D^{\prime}_{*}))$ of the 2-sphere system
$S(D^{\prime}_{*})$ in $\Sigma$, there is an orientation-preserving
diffeomorphism $\tilde{\rho}:\Sigma\to\Sigma$ sending $N(S(D^{\prime}_{*}))$
identically such that

$\tilde{\rho}uS(D_{*})=S(D_{*}).$

Next, for the pseudo-O2-sphere basis
$(S(D_{*}),\tilde{\rho}uS(D^{\prime}_{*}))$ and the O2-sphere basis
$(S(D_{*}),S(D^{\prime}_{*}))$ in $\Sigma$, consider the 4-manifold
$X^{\prime}$ obtained from $\Sigma$ by replacing a regular neighborhood system
$N(S(D_{*}))=S^{2}\times D^{2}_{*}$ of the 2-sphere system $S(D_{*})$ in
$\Sigma$ with the 4D solid torus system $Y^{\prime}_{*}=D^{3}\times
S^{1}_{*}$. The 4-manifold $X^{\prime}$ is diffeomorphic to the 4-sphere
$S^{4}$. Let $\tilde{E}^{\prime}_{*}\times I$ and $E^{\prime}_{*}\times I$ be
the 2-handle systems in $X=S^{4}$ attached to the 4D solid torus system
$Y^{\prime}_{*}$ which are obtained from the thickening 2-sphere systems
$\tilde{\rho}uS(D^{\prime}_{*})\times I$ and $S(D^{\prime}_{*})\times I$ in
$\Sigma$, respectively.

Lemma 4.4 can be used for the 2-handle systems $\tilde{E}^{\prime}_{*}\times
I$ and $E^{\prime}_{*}\times I$ attached to $Y^{\prime}_{*}$. Thus, there is
an orientation-preserving diffeomorphism $\rho^{\prime}:S^{4}\to S^{4}$
sending $Y^{\prime}$ identically such that

$\rho^{\prime}u(\tilde{E}^{\prime}_{*}\times
I,\tilde{E}^{\prime}_{*})=(E^{\prime}_{*}\times I,E^{\prime}_{*}).$

By returning the 4D solid torus system $Y^{\prime}_{*}$ in $X^{\prime}$ to the
regular neighborhood system $N(S(D_{*}))$ of the 2-sphere system $S(D_{*})$ in
$\Sigma$, there is an orientation-preserving diffeomorphism
$\tilde{\rho}^{\prime}:\Sigma\to\Sigma$ sending $N(S(D_{*}))$ identically such
that

$\tilde{\rho}^{\prime}\tilde{\rho}uS(D^{\prime}_{*})=S(D^{\prime}_{*}).$

The composite diffeomorphism
$h=\tilde{\rho}^{\prime}\tilde{\rho}:\Sigma\to\Sigma$ preserve the O2-sphere
basis $(S(D_{*}),S(D^{\prime}_{*}))$ identically. This completes the proof of
Lemma 4.3. $\square$

x

In Lemma 4.3, since $(uS(D_{*}),uS(D^{\prime}_{*}))=(S_{*},S^{\prime}_{*})$
and $(R_{*},R^{\prime}_{*})=(S(D_{*}),S(D^{\prime}_{*}))$, the proof of
Theorem 1.1 is completed. $\square$

x

5\. A classification of orientation-preserving diffeomorphisms of the stable
4-sphere

Let $\mbox{Diff}^{+}(D^{4},\mbox{rel }\partial)$ be the orientation-preserving
diffeomorphism group of the 4-ball $D^{4}$ keeping the boundary $\partial
D^{4}$ by the identity. An identity-shift of a 4-manifold $\Sigma$ is a
diffeomorphism $\iota:\Sigma\to\Sigma$ obtained from the identity map
$1:\Sigma\to\Sigma$ by replacing the identity on a 4-ball in $\Sigma$ disjoint
from $F$ with an element of $\mbox{Diff}^{+}(D^{4},\mbox{rel }\partial)$.

The following result is obtained by using Framed Light-bulb Isotopy Lemma
(explained soon) based on Gabai’s 4D light-bulb theorem in [5] instead of
Framed Light-bulb Diffeomorphism Lemma in the proof of Theorem 1.1.

x

Theorem 5.1. Let $\Sigma$ be the stable 4-sphere $\Sigma$ of any genus $n\geq
1$, and $h:\Sigma\to\Sigma$ any orientation-preserving diffeomorphism. Then
there is an $\alpha$-equivariant diffeomorphism $\tilde{f}:\Sigma\to\Sigma$
such that the composite diffeomorphism $\tilde{f}h:\Sigma\to\Sigma$ induces
the identity isomorphism

$(\tilde{f}h)_{*}=1:H_{2}(\Sigma;{\mathbf{Z}})\to H_{2}(\Sigma;{\mathbf{Z}}).$

Further for any such $\alpha$-equivariant diffeomorphism
$\tilde{f}:\Sigma\to\Sigma$, the composite diffeomorphism
$\tilde{f}h:\Sigma\to\Sigma$ is isotopic to an identity-shift $\iota$ of
$\Sigma$.

x

To prove Theorem 5.1 in the process of the proof of Theorem 1.1, we need to
show that the diffeomorphism $h$ of $\Sigma$ in Lemma 4.3 can be replaced by a
diffeomorphism $h$ of $\Sigma$ which is isotopic to the identity. For this
purpose, we need the following lemma, coming from Gabai’s 4D light-bulb
theorem in [5, Theorem 10.4]. Note that the assumption of Framed Light-bulb
Isotopy Lemma adds an additional condition to that assumption of Framed Light-
bulb Diffeomorphism Lemma (Lemma 4.4).

x

Lemma 5.2 (Framed Light-bulb Isotopy Lemma). Let $Y_{*}$ be a system of
mutually disjoint 4D solid tori $Y_{i}\,(i=1,2,\dots,n)$ in $S^{4}$. Let
$D_{*}\times I$ be a system of mutually disjoint 2-handles $D_{i}\times
I\,(i=1,2,\dots,n)$ in $\cap Y^{c}_{*}$ such that $(\partial D_{i})\times I$
attaches to $\partial Y^{c}_{i}$ for all $i$. If $E_{*}\times I$ is any system
of mutually disjoint 2-handles $E_{i}\times I\,(i=1,2,\dots,n)$ in $\cap
Y^{c}_{*}$ such that

$(\partial E_{i})\times I=(\partial D_{i})\times I$

for all $i$ and the unions $D_{i}\cup E_{i}\,(i=1,2,\dots,n)$ are mutually
disjoint, then there is a diffeomorphism $h:S^{4}\to S^{4}$ which is
$Y_{*}$-relatively isotopic to the identity such that

$h(D_{*}\times I,D_{*})=(E_{*}\times I,E_{*}).$

x

Proof of Lemma 5.2. As in the proof of Lemma 4.4, assume that a boundary
collar system of $D_{*}$ coincides with a boundary collar system of $E_{*}$.

First, we show the assertion of the special case $n=1$. Let $Y=S^{1}\times
D^{3}$. Let $D$ and $E$ be proper disks in $Y^{c}\cong D^{2}\times S^{2}$
admitting trivial line bundles $D\times I,E\times I$ such that

$(\partial D)\times I=(\partial E)=\ell\times I\subset\partial Y^{c}.$

If the 2-sphere $D\cup(-E)$ is not null-homologous in $Y^{c}$, the disk $D$ is
$\partial$-relatively homologous to the 2-cycle $E+mS$ in $(Y^{c},\partial
Y^{c})$ for a 2-sphere generator $[S]$ of
$H_{2}(Y^{c};{\mathbf{Z}})\cong{\mathbf{Z}}$ and some non-zero integer $m$.
Then the self intersection $\mbox{Int}([D],[D])$ in $Y^{c}$ with the given
framing of $\partial D$ in $\partial Y^{c}$ must be $2m\neq 0$. In fact, since
$\mbox{Int}([E],[E])=0$ in $Y^{c}$ with the given framing of $\partial E$ in
$\partial Y^{c}$ and $\mbox{Int}([S],[S])=0$, we have
$\mbox{Int}([D],[D])=2\mbox{Int}([S],[E])=2m$. This contradicts
$\mbox{Int}([D],[D])=0$. Thus, the 2-sphere $D\cup(-E)$ is null-homologous in
$Y^{c}$. By Gabai’s 4D light-bulb theorem in [5, Theorem 10.4], there is a
diffeomorphism $\lambda:Y^{c}\to Y^{c}$ which is $\partial$-relatively
isotopic to the identity such that $\lambda D=E$, which extends to a
diffeomorphism $h=\lambda^{+}:S^{4}\to S^{4}$ which is $Y^{c}$-relatively
isotopic to the identity such that $hD=E$. A diffeomorphism of $S^{4}$
preserves trivial line bundles on disks if the line bundles on the boundary
circles are preserved. This is because a sole obstruction that a disk admits a
trivial line bundle extending a given line bundle on the boundary circle in
$S^{4}$ is that the self intersection number of the disk with the given
boundary framing by the line bundle is $0$. Thus, the diffeomorphism $h$ of
$S^{4}$ has $h(D\times I,D)=(E\times I,E)$ and the assertion of the special
case $n=1$ is shown.

For the proof in general case, let $K=K(\ell_{*})$ be a connected graph in
$S^{4}$ obtained from the loop system $\ell_{*}$ of the loops
$\ell_{i}=\partial D_{i}=\partial E_{i},\,(i=1,2,\dots,n)$ by adding mutually
disjoint $n-1$ simple arcs not meeting any interior disks of the disk systems
$D_{*}$ and $E_{*}$. For every $k$ with $1\leq k\leq n$, let $Y_{k}$ be a
regular neighborhood of the disk-arc union

$\ell_{k}\cup_{1\leq i\leq n,i\neq k}E_{i}\cup_{j=1}^{n}a_{j}$

in $S^{4}$, which is a 4D solid torus in $S^{4}$.

By the proof of the special case $n=1$, there is a diffeomorphism $h_{1}$ of
$S^{4}$ which is $Y_{1}$-relatively isotopic to the identity such that
$h_{1}D_{1}=E_{1}$. Next, for the disk systems $h_{1}(D_{*})$ and $E_{*}$,
there is a diffeomorphism $h_{2}$ of $S^{4}$ which is $Y_{2}$-relatively
isotopic to the identity such that $h_{2}h_{1}D_{1}=E_{1}$ and
$h_{2}h_{1}D_{2}=E_{2}$. Continuing this process, there is a diffeomorphism
$h=h_{n}\dots h_{2}h_{1}$ of $S^{4}$ which is $N(K)$-relatively isotopic to
the identity for a regular neighborhood $N(K)$ of $K$ such that
$hD_{i}=E_{i}\,(i=1,2,\dots,n)$. Thus, for a regular neighborhood
$N(\ell_{*})$ of the loop system $\ell_{*}$ in $N(K)$, this diffeomorphism $h$
of $S^{4}$ is $N(\ell_{*})$-relatively isotopic to the identity and has
$hD_{i}=E_{i}\,(i=1,2,\dots,n)$. The regular neighborhood system $N(\ell_{*})$
can be regarded as the 4D solid torus system $Y_{*}$ as in the proof of Lemma
4.4. This completes the proof of Lemma 5.2. $\square$

x

The proof of Theorem 5.1 is done as follows.

x

Proof of Theorem 5.1. Let $\Sigma=S^{4}(F)_{2}$ by a standard O2-handle basis
$(D_{*}\times I,D^{\prime}_{*}\times I)$ of $F$. On the lemmas used for the
proof of Theorem 1.1, Lemma 3.2 says that there is an $\alpha$-invariant
orientation-preserving diffeomorphism $\tilde{f}$ of $\Sigma$ such that the
composite $\tilde{f}h:\Sigma\to\Sigma$ induces the identity
$(\tilde{f}h)_{*}=1:H_{2}(\Sigma;{\mathbf{Z}})\to H_{2}(\Sigma;{\mathbf{Z}})$.
For any such $\tilde{f}$, Lemmas 4.1 and 4.2 say that there is a
diffeomorphism $g_{1}$ of $\Sigma$ which is isotopic to the identity such that
the composite embedding

$g_{1}\tilde{f}h:\Sigma\to\Sigma$

sends the bump pair $(B,V)$ in $\Sigma$ identically and has the property that
the disk interiors of the disk systems $g_{1}\tilde{f}h\bar{D}_{*}$ and
$g_{1}\tilde{f}h\bar{D}^{\prime}_{*}$ are disjoint from the disk systems
$\bar{D}_{*}$ and $\bar{D}^{\prime}_{*}$.

By the proof of Lemma 4.3 and the use of Lemma 5.2 instead of Lemma 4.4, there
is a diffeomorphism $g_{2}:\Sigma\to\Sigma$ which is isotopic to the identity
such that the composite diffeomorphism $g_{2}g_{1}\tilde{f}h$ of $\Sigma$
preserve the O2-sphere basis $(S(D_{*}),S(D^{\prime}_{*}))$ identically.

By the proof of Lemma 2.1, a region ${\Omega}(S(D_{*}),S(D^{\prime}_{*}))$ is
taken to be the double branched covering space $A(P^{\prime})_{2}$ of a 4-ball
$A$ branched along a proper 1-punctured trivial surface $P^{\prime}$ of $F$
and the residual region ${\Omega}^{c}(S(D_{*}),S(D^{\prime}_{*}))$ is taken to
be the double branched covering space $A^{c}(d)_{2}$ of the 4-ball
$A^{c}=\mbox{cl}(S^{4}\setminus A)$ branched along the proper trivial disk
$d=\mbox{cl}(F\setminus P^{\prime})$ of $F$. Then for a diffeomorphism $g_{3}$
of $\Sigma$ which is isotopic to the identity, the composite diffeomorphism
$g_{3}g_{2}g_{1}\tilde{f}h$ of $\Sigma$ preserve a rigion
${\Omega}(S(D_{*}),S(D^{\prime}_{*}))$ identically, which defines a
$\partial$-identical diffeomorphism $v:A^{c}(d)_{2}\to A^{c}(d)_{2}$. Since
the double branched covering space $A^{c}(d)_{2}$ is a 4-ball and the lift
$\tilde{d}$ of the disk $d$ to $A^{c}(d)_{2}$ is trivial, one can find a
3-ball $\Delta$ in $A^{c}(d)_{2}$ with $\partial\Delta=d\cup d^{\prime}$ for a
disk $d^{\prime}$ in the 3-sphere $\partial A^{c}$ with $\partial
d^{\prime}=-\partial d$. Then the $\partial$-identical diffeomorphism $v$ is
$\partial$-relatively isotopic to a $\partial$-identical diffeomorphism
$\tilde{v}:A^{c}(d)_{2}\to A^{c}(d)_{2}$ which preserves a regular
neighborhood $N(\Delta)$ of the 3-ball $\Delta$ identically. Thus, there is a
diffeomorphism $g_{4}$ of $\Sigma$ which is isotopic to the identity such that
the composite diffeomorphism $g_{4}g_{3}g_{2}g_{1}\tilde{f}h$ of $\Sigma$
preserves the 4-manifold ${\Omega}(S(D_{*}),S(D^{\prime}_{*}))\cup N(\Delta)$
identically and defines a $\partial$-identical diffeomorphism
$\tilde{v}^{\prime}$ from the 4-ball

$\mbox{cl}({\Omega}^{c}(S_{*},S^{\prime}_{*})\setminus
N(\Delta))=\mbox{cl}(A^{c}(d)_{2}\setminus N(\Delta))$

to itself. In other words, we have $g_{4}g_{3}g_{2}g_{1}\tilde{f}h=\iota$ for
an identity-shift $\iota$ of $\Sigma$ and hence the composite diffeomorphism
$\tilde{f}h$ of $\Sigma$ is isotopic to the identity-shift $\iota$ of
$\Sigma$. This completes the proof of Theorem 5.1. $\square$

x

The identity-shift $\iota$ in Theorem 5.1 is needed because at present it
appears unknown whether $\pi_{0}(\mbox{Diff}^{+}(D^{4},\mbox{rel }\partial)$
is trivial or not. However, it is known that the identity-shift $\iota$ is
concordant to the identity since $\Gamma_{5}=0$ (see Kervaire [13]). In the
piecewise-linear category, this notion is not needed and we have that every
piecewise-linear orientation-preserving homeomorphism $f:\Sigma\to\Sigma$ for
the stable 4-sphere $\Sigma$ of any genus $n$ is piecewise-linearly isotopic
to an $\alpha$-equivariant orientation-preserving piecewise-linear
homeomorphism $\tilde{f}:\Sigma\to\Sigma$.

x

The following result is a consequence of Theorem 5.1.

x

Corollary 5.3. Every O2-sphere basis of $\Sigma$ is isotopic to the O2-sphere
basis $(S(D_{*}),S(D_{*}^{\prime}))$ constructed from an O2-handle basis
$(D_{*}\times I,D^{\prime}_{*}\times I)$ of a trivial surface-knot $F$ of
genus $n$ in $S^{4}$.

x

Proof of Corollary 5.3. Let $(S_{*},S^{\prime}_{*})$ be an O2-sphere basis of
$\Sigma$ such that $[S_{i}]=[S(D_{i})]$ and
$[S^{\prime}_{i}]=[S(D^{\prime}_{i})]$ in $H_{2}(\Sigma;{\mathbf{Z}})$ for all
$i\,(i=1,2,\dots,n)$. Such an O2-sphere basis $(S(D_{*}),S(D_{*}^{\prime}))$
is given by Lemma 3.2. Then we show that the O2-sphere basis
$(S_{*},S^{\prime}_{*})$ is isotopic to the O2-sphere basis
$(S(D_{*}),S(D_{*}^{\prime}))$. Construct an embedding
$u:\Sigma^{(0)}\to\Sigma$ sending the O2-sphere basis
$(S(D_{*}),S(D_{*}^{\prime}))$ to the O2-sphere basis
$(S_{*},S^{\prime}_{*})$, which extends to a diffeomorphism
$h=u^{+}:\Sigma\to\Sigma$ inducing the identity isomorphism
$h_{*}=1:H_{2}(\Sigma;{\mathbf{Z}})\to H_{2}(\Sigma;{\mathbf{Z}})$. By the
proof of Theorem 5.1, there is a diffeomorphism $g$ of $\Sigma$ isotopic to
the identity such that the composite diffeomorphism $gh$ preserves the
O2-sphere basis $(S(D_{*}),S(D^{\prime}_{*}))$ of $\Sigma$ identically, which
shows that the O2-sphere basis $(S_{*},S^{\prime}_{*})$ of $\Sigma$ is
isotopic to the O2-sphere basis $(S(D_{*}),S(D_{*}^{\prime}))$ of $\Sigma$
because

$(S(D_{*}),S(D_{*}^{\prime}))=gh(S(D_{*}),S(D_{*}^{\prime}))=g(S_{*},S^{\prime}_{*}).\quad\square$

x

Acknowledgments. This work was partly supported by Osaka City University
Advanced Mathematical Institute (MEXT Joint Usage/Research Center on
Mathematics and Theoretical Physics JPMXP0619217849). Since writing the first
draft of this paper (in an announcement form) in late November of 2019, I
stayed at Beijing Jiaotong University, China from December 18, 2019 to January
4, 2020 while improving the paper. I would like to thank Liangxia Wan and
Rixin Zhang (graduate student) for their warm hospitalities.

x

## References

  * [1] A. Akhmedov and B. D. Park, Geography of simply connected spin symplectic 4-manifolds, Math. Res. Lett., 17 (2010), 483-492.
  * [2] J. Cerf, Sur les difféomorphismes de la sphere de dimension trois ($\Gamma_{4}=0$), Springer Lecture Notes in Math. 53 (1968).
  * [3] M. Freedman, The disk theorem for four-dimensional manifolds, Proc. Internat. Congr. Math. (Warsaw, Poland) (1983), 647-663.
  * [4] M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton Univ. Press (1990).
  * [5] D. Gabai, The 4-dimensional light-bulb theorem, preprint, arXiv:1705.09989v2 (2017).
  * [6] A. Hatcher, A proof of the Smale conjecture, Diff($S^{3}$) $\simeq$ O(4), Ann. of Math., 117 (1983), 553-607.
  * [7] J. A. Hillman and A. Kawauchi, Unknotting orientable surfaces in the 4-sphere, J. Knot Theory Ramifications 4(1995), 213-224.
  * [8] S. Hirose, On diffeomorphisms over surfaces trivially embedded in the 4-sphere, Algebraic and Geometric Topology 2(2002), 791-824.
  * [9] F. Hosokawa and A. Kawauchi, Proposals for unknotted surfaces in four-space, Osaka J. Math. 16 (1979), 233-248.
  * [10] J. F. P. Hudson, Piecewise-linear topology, Benjamin (1969).
  * [11] A. Kawauchi, Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link, Topology and its Applications (to appear).
  * [12] A. Kawauchi,Triviality of a surface-link with meridian-based free fundamental group, preprint.
  * [13] M. Kervaire, Groups of homotopy spheres: I, Ann. of Math. 77 (1963) 504-537.
  * [14] C. Rourke and B. Sanderson, Introduction to piecewise-linear topology, Ergebnesse der Math., 69 (1972), Springer-Verlag.
  * [15] C. T. C. Wall, On simply-connected 4-manifolds, J. London Math. Soc., 39 (1964), 141-149.


# ZMCintegral-v5.1: Support for Multi-function Integrations on GPUs

Xiao-Yan Cao  Jun-Jie Zhang  Xi’an Research Institute of High-Tech  Northwest
Institute of Nuclear Technology

###### Abstract

In this new version of ZMCintegral, we have added the functionality of multi-
function integrations, i.e. the ability to integrate more than $10^{3}$
different functions on GPUs. The Python API remains the similar as the
previous versions. For integrands less than 5 dimensions, it usually takes
less than 10 minutes to finish the evaluation of $10^{3}$ integrations on one
Tesla v100 card. The performance scales linearly with the increasing of the
GPUs.

###### keywords:

Monte Carlo integration , multi-function integrations , Numba , Ray.

††journal: Computer Physics Communications

NEW VERSION PROGRAM SUMMARY

Program Title:  ZMCintegral
Licensing provisions:  Apache-2.0
Programming language:  Python
Journal reference of previous version:  Hong-Zhong Wu, Jun-Jie Zhang, Long-
Gang Pang, Qun Wang, Comput. Phys. Commun. 248(2020):106962 and Jun-Jie Zhang,
Hong-Zhong Wu, Comput. Phys. Commun. 251(2020):107240
Does the new version supersede the previous version?:  Yes
Reasons for the new version:  When solving the Boltzmann equation with
radiations[1], one encounters different collision integrals for different
energy beams. In the relativistic QED plasma, the collision terms involve
various Feynman graphs[2] and usually the contribution from each graph is of
great interest. In these circumstances one needs to integrate many functions
of different forms simultaneously. In our previous versions[3, 4], we focused
on single integration of high dimensions and functions with parameters.
Therefore, it is necessary to include the functionality for integrating many
functions which have different dimensions, forms and integration domains.
Summary of revisions:

  * 1.

Multi-function Integrations

Suppose we have a series of integrations defined as

$\displaystyle f_{n}(\mathbf{x})$ $\displaystyle=$ $\displaystyle
a_{n}\text{cos}(\mathbf{k}_{n}\cdot\mathbf{x})+b_{n}\text{sin}(\mathbf{k}_{n}\cdot\mathbf{x}),$
(1)

where $n=1,2,3,...,100$. The above integration series can be treated as a set
of Harmonic bases if one wishes to evaluate the contribution of each Harmonic
mode. In our previous versions, these series cannot be manipulated in a
convenient and efficient way. It is worth noting that the integration domains
or dimensions can be different, for example

$\displaystyle g_{n}(x_{1},x_{2})$ $\displaystyle=$ $\displaystyle
a_{n}|x_{1}+x_{2}|\ \text{for }0<n<50$ $\displaystyle
g_{n}(x_{1},x_{2},x_{3})$ $\displaystyle=$ $\displaystyle
b_{n}|x_{1}+x_{2}-x_{3}|\ \text{for }50\leq n\leq 100.$ (2)

The support of multi-functions gives the users full flexibility to integrate
as many different functions as possible.

  * 1.

Test on GPUs

As an illustrative example, we report the solution of Eq. (1) with
$\mathbf{x}=(x_{1},x_{2},x_{3},x_{4})$, $a_{n}=b_{n}=1$. The ranges for all
components are taken to be $[0,1]$ and
$\mathbf{k}_{n}=(\frac{n+50}{2\pi},\frac{n+50}{2\pi},\frac{n+50}{2\pi},\frac{n+50}{2\pi})$
such that the integration is highly fluctuating around the zero line. The
hardware condition in this case is taken to be: Intel(R) Xeon(R) Silver 4110
<EMAIL_ADDRESS>CPU with 10 processors + one Nvidia Tesla V100 GPU. The results
are shown in Fig. 1.

Figure 1: Integration results of Eq. (1) at various $n$. The red shaded area,
drawn from $[\overline{F_{n}}-\triangle F_{n},\overline{F_{n}}+\triangle
F_{n}]$ with $F_{n}$ denoting the result of the $n\text{-th}$ integral, is
obtained via ZMCintegral (analytical results in solid black line). Here,
$\overline{F_{n}}$ and $\triangle F_{n}$ are the average value and standard
deviation of 10 independent evaluations. It takes about one minute for each
independent calculation (of all series in Eq. (1)) on one Tesla V100 card. For
each $n$ we have taken $10^{6}$ samples.

Nature of problem: ZMCintegral is an easy to use Python package for doing high
dimensional integrations on distributed GPU clusters. Using the Python
libraries Numba[5] and Ray[6], as well as the NVIDIA CUDA[7] capability,
ZMCintegral offers a succinct Python interface to evaluate numerical
integrations for physical problems. In this updated version, we mainly focus
on the problems where the users have various functions to integrate. These
integrations can take different forms and domains.
Solution method: This new version contains three Python classes.
ZMCintegral_normal utilizes the stratified-sampling and heuristic-tree-search
techniques, while ZMCintegral_functional and ZMCintegral_multifunctions use
the direct-Monte Carlo method for each integrand which benefits mainly from
the heavily distributed GPU clusters.
Additional comments: If the integrations are high-dimensional (e.g.
dimensionality of 8-12), users are encouraged to use ZMCintegral_normal. If
the integrations are middle-dimensional (e.g. dimensionality of 1-7) but with
a large parameter space, we suggest users to try ZMCintegral_functional. If
the integrations contains many different integrands and domains (e.g. $10^{4}$
different integrations), then ZMCintegral_multifunctions is suggested. The
detailed instructions can be found here: [8].

## Acknowledgment

The authors are supported in part by the Major State Basic Research
Development Program (973 Program) in China under Grant No. 2015CB856902 and by
the National Natural Science Foundation of China (NSFC) under Grant No.
11535012. The Computations are performed at the GPU servers of department of
modern physics at USTC. We are thankful for the valuable help from Prof. Qun
Wang of department of modern physics at USTC.

## References

  * [1] J. Oxenius, Kinetic Theory of Particles and Photons, Springer Berlin Heidelberg, 1986.  doi:10.1007/978-3-642-70728-5.
  * [2] V. Morozov, G. Röpke, Kinetic theory of radiation in non-equilibrium relativistic plasmas, Annals of Physics 324 (6) (2009) 1261–1302.  doi:10.1016/j.aop.2009.02.001.
  * [3] H.-Z. Wu, J.-J. Zhang, L.-G. Pang, Q. Wang, ZMCintegral: A package for multi-dimensional monte carlo integration on multi-GPUs, Computer Physics Communications (2019) 106962doi:10.1016/j.cpc.2019.106962.
  * [4] J.-J. Zhang, H.-Z. Wu, ZMCintegral-v5: Support for integrations with the scanning of large parameter space on multi-GPUs, Computer Physics Communications 251 (2020) 107240.  doi:10.1016/j.cpc.2020.107240.
  * [5] S. Kwan Lam, A. Pitrou, S. Seibert, Numba: a llvm-based python jit compiler, 2015, pp. 1–6.  doi:10.1145/2833157.2833162.
  * [6] P. Moritz, R. Nishihara, S. Wang, A. Tumanov, R. Liaw, E. Liang, M. Elibol, Z. Yang, W. Paul, M. I. Jordan, I. Stoica, Ray: A distributed framework for emerging AI applications, in: 13th USENIX Symposium on Operating Systems Design and Implementation (OSDI 18), USENIX Association, Carlsbad, CA, 2018, pp. 561–577.
URL https://www.usenix.org/conference/osdi18/presentation/moritz

  * [7] I. Buck, T. Purcell, A Toolkit for Computation on GPUs, 2004.
  * [8] https://github.com/Letianwu/ZMCintegral.git.


# cantnlp@LT-EDI-2023: Homophobia/Transphobia Detection in Social Media
Comments using Spatio-Temporally Retrained Language Models

Sidney G.-J. Wong, Matthew Durward, Benjamin Adams  Jonathan Dunn
University of Canterbury, Christchurch, New Zealand
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

This paper describes our multiclass classification system developed as part of
the LT-EDI@RANLP-2023 shared task. We used a BERT-based language model to
detect homophobic and transphobic content in social media comments across five
language conditions: English, Spanish, Hindi, Malayalam, and Tamil. We
retrained a transformer-based cross-language pretrained language model, XLM-
RoBERTa, with spatially and temporally relevant social media language data. We
also retrained a subset of models with simulated script-mixed social media
language data with varied performance. We developed the best performing seven-
label classification system for Malayalam based on weighted macro averaged F1
score (ranked first out of six) with variable performance for other language
and class-label conditions. We found the inclusion of this spatio-temporal
data improved the classification performance for all language and task
conditions when compared with the baseline. The results suggests that
transformer-based language classification systems are sensitive to register-
specific and language-specific retraining.

Language Condition | H | N | T | Total
---|---|---|---|---
English | 179 | 2978 | 7 | 3164
Hindi | 45 | 2423 | 92 | 2560
Malayalam | 476 | 2468 | 170 | 3114
Spanish | 200 | 450 | 200 | 850
Tamil | 453 | 2064 | 145 | 2662
Table 1:  The labelled training data broken down by language condition and class label for Task A. The class labels for Task A were homophobia (H), non-anti-LGBT+ content (N), and transphobia (T). Language Condition | CS | HT | HD | HS | NO | TT | TD | Total
---|---|---|---|---|---|---|---|---
English | 302 | 12 | 167 | 436 | 2240 | 1 | 6 | 3164
Malayalam | 152 | 57 | 419 | 69 | 2247 | 7 | 163 | 3114
Tamil | 212 | 37 | 416 | 218 | 1634 | 34 | 111 | 2662
Table 2:  The labelled training data broken down by language condition and
class label for Task B. The class labels for Task B were counter-speech (CS),
homophobic-threatening (HT), homophobic-derogation (HD), hope-speech (HS),
none-of-the-above (NO), transphobic-threatening (TT), and transphobic-
derogation (TD).

##  1 Introduction

The purpose of this shared task was to develop a classification system to
predict whether samples of social media comments contained forms of homophobia
or transphobia across different language conditions. There were no
restrictions on language models or data pre-processing methods.

The five language conditions: English, Spanish, Hindi, Malayalam, and Tamil.
In addition to the language conditions, participants were tasked with
developing a system for a three-class and seven-class classification system
defining different forms of homophobic and transphobic hate speech
Chakravarthi et al. (2021).

The main contribution of our proposed system outlined in this paper included
spatio-temporal relevant social media language data to retrain a transformer-
based language model to increase the sensitivity of the pretrained language
model (PLM). We have also created simulated samples of script-mixed social
media language data which was used as part of the retraining process.

###  1.1 Problem Description

The organisers of this shared task provided .csv files containing labelled
data of pre-processed comments of users reacting to LGBT+ videos on YouTube.
This was an expanded data set of the Homophobia/Transphobia Detection data set
(Chakravarthi et al., 2021) with the inclusion of Hindi, Malayalam, and
Spanish in addition to the pre-existing English and Tamil data.

The comments were manually annotated based on a three-class and a seven-class
classification system. The participants of the shared task were not provided
any further information on the annotation process or measures of inter-
annotator agreement. The shared task was broken down into the following tasks:

  * •

Task A involves developing a classification model for three classes across all
five language conditions as shown in Table 1.

  * •

Task B involves developing a classification model for seven classes across
three language conditions as shown in Table 2.

The organisers of this shared task provided training and validation data to
develop the system. The test data was provided once the results of the shared
task were announced. The organisers evaluated the performance of each
homophobia/transphobia detection system with weighted macro averaged F1 score.
The performance for each language and class-label condition were ranked based
on this score.

###  1.2 Related Work

Previous approaches in detecting homophobia and transphobia on social media
comments has shown varying levels of success Chakravarthi et al. (2022). In
this shared task, participants were asked to detect homophobia and transphobia
across three language conditions: English, Tamil, and an additional English-
Tamil script-mixed data set.

Participants of the shared task combined various natural language processing
methods such as statistical language models and machine learning to complete
the task. However, the performance of transformer-based language models
remained consistently high across all three language conditions.

More specifically BERT-based models with minimal fine-tuning outperformed
statistical language models using TF-IDF for feature extraction. BERT, or
Bidirectional Encoder Representations from Transformers, structures the
complex relationship between words in a language through embeddings Devlin et
al. (2019).

The best performing BERT-based system for English yielded an average weighted
macro F1 score of 0.92 compared with non-transformer-based language models
Maimaitituoheti et al. (2022). Conversely, the same BERT-based models
struggled to outperform machine learning and deep learning systems approaches
in Tamil and in the English-Tamil condition.

This suggests further work is needed to refine BERT-based to improve its
performance outside an English-context. Based on the promising results of
BERT-based language models in Chakravarthi et al. (2022), the current study
extend on this transformer-based approach to develop and refine a homophobia
and transphobia detection system across language conditions.

##  2 Methodology

In this section, we provide a system overview of our transformer-based
language model. We also provide details on our retraining and fine-tuning
procedures.

###  2.1 System Overview

Due to the number of language conditions for the current shared task, it was
unfeasible to use language-specific BERT-based models. One risk for using
independently developed language-specific BERT-based models was that there was
no control on the source data used to train the representations. For this
reason, we used a cross-lingual transformer-based language model as our
baseline language model.

XLM-RoBERTa was trained on two terabytes of CommonCrawl for 100 languages
Conneau et al. (2020). Some of these languages include English, Hindi,
Malayalam, Spanish, and Tamil. Furthermore, Romanised Hindi and Tamil have
also been included in the pretraining of this cross-lingual transformer-based
language model.

Despite these benefits, we were aware of the risk in overgeneralising the
register of language of CommonCrawl as the language used on this platform is
not reflective of the language used on social media. We could retrain PLMs for
a specific task to mitigate this issue without the need to train a PLM from
scratch.

This retraining method has shown to improve the performance of PLMs in
downstream tasks (such as label classification) for under-represented and
under-resourced languages by pretraining with additional register-specific
language data Liu et al. (2019). Therefore, we have retrained the baseline
XLM-RoBERTa PLM prior to fine-tuning the baseline XLM-RoBERTa PLM.

###  2.2 Retraining

Language | Indic | Latin
---|---|---
English | - | 50K
Spanish | - | 50K
Hindi | 50K | -
Malayalam | 50K | -
Tamil | 50K | -
SM Hindi | 37.5K | 12.5K
SM Malayalam | 37.5K | 12.5K
SM Tamil | 37.5K | 12.5K

Total
---
50K
50K
50K
50K
50K
50K
50K
50K

Table 3:  Corpus size of language samples for fine-tuning with simulated
script-mixing (SM).

Language Condition | Baseline | Retrained | Script-Mixed
---|---|---|---
English | 0.93 | 0.94 | -
Hindi | 0.93 | 0.92 | 0.97
Malayalam | 0.93 | 0.95 | 0.94
Spanish | 0.83 | (0.86) | -
Tamil | 0.70 | 0.93 | 0.93

Rank
---
7
3
4
-
3

Table 4:  Macro averaged F1 for each language condition for Task A and overall
rank for the shared task. The submitted result is in bold. Note that the
result for Spanish was invalid.

Language Condition | Baseline | Retrained | Script-Mixed
---|---|---|---
English | 0.15 | 0.54 | -
Malayalam | 0.86 | 0.86 | 0.88
Tamil | 0.77 | 0.90 | 0.80

Rank
---
6
1
4

Table 5:  Macro averaged F1 for each language condition for Task B and overall
rank for the shared task. The submitted result is in bold.

We used social media language data from the Corpus of Global Language Use
(CGLU) for retraining Dunn (2020). The CGLU is a very large digital corpora
which contains over 20 billion words associated with 10,000 point locations
across the globe.

Although the source of the CGLU social media language data comes from Twitter,
a microblogging platform, and the training data comes from YouTube, a video
sharing platform, our focus is on the written language components, and we
assume some close domain alignment. We removed hashtags and hyperlinks to
ensure the retraining data has a similar form to the training data. We also
removed multiple punctuation and blank space characters. Short tweets with
fewer than 50 characters were also systematically removed.

We controlled the spatial and temporal window of the sampled tweets by
restricting the sample of tweets to those originating in India produced
between 1 January 2019 and 31 December 2019. Once again, we wanted to closely
match retraining data with the time and geographic source of the labelled
training data Chakravarthi et al. (2021).

We used the `langdetect`111https://pypi.org/project/langdetect/ library to
detect the language condition for each tweet. For each of the five different
language conditions, we extracted a random sample of 50,000 tweets for
training. We then use the LanguageModelingModel class from the
`simpletransformers` library to retrain XLM-RoBERTa on an unlabelled corpus of
social media language data.

In addition to creating corpus training data for the five different language
conditions, we created additional corpus training data with simulated script-
mixing. A major motivation to retrain the model with the simulated script-
mixed retraining data is the lack of Romanised Malayalam in XLM-RoBERTa. We
used the transliteration.XlitEngine class from the
`ai4bharat`222https://pypi.org/project/ai4bharat-transliteration/ library to
transliterate one-fifth of the sample tweets from Indic to Latin script.

The size of our retraining corpora for each language condition is shown in
Table 3. We retrained the language model for 4 iterations and we evaluated the
training for every 500 steps. We saved the model with the best performance
determined by the loss function in our output directory.

###  2.3 Fine-tuning

Once we retrained XLM-RoBERTa with the social media language data from the
CGLU, we fine-tuned the baseline and the retrained language models with the
labelled training data.

As shown in Table 1 and Table 2, the class labels for both Task A and Task B
are highly unbalanced. We used the RandomOverSampler class from the library to
oversample the minority classes. In most cases, these minority classes related
to homophobia and transphobia.

We used the classificaton class from the `simpletransformers` library to fine-
tune the retrained PLMs with the labelled training data. We trained the
classification model for 8 iterations and we evaluated the training for every
500 steps. We also used AdamW optimization Loshchilov and Hutter (2019).

We applied the same fine-tuning strategy to Task A and Task B to maintain
consistency across the shared task. We saved the model with the best
performance determined by the loss function in our output directory.

###  2.4 Other Settings

We completed the retraining and fine-tuning in Python3 on Google Colaboratory.
We used GPU as our hardware accelerator using NVIDIA A100 Tensor Core graphics
card.

##  3 Results

The results of Task A are shown in Table 4 and the results of Task B are shown
in Table 5. Both tables compare the weighted macro averaged F1 metrics for the
classification models derived from the baseline XLM-RoBERTa and the modified
XLM-RoBERTa models produced specifically for this task. The ranking of our
models are also presented in the final column of the tables.

In Task A, English, Hindi, and Malayalam performed the best of the the
baseline classification models with a macro averaged F1 score of .93. Tamil
performed the worst of the baseline classification models. The performance of
the retrained classification models were consistently better than the baseline
classification models. Malayalam performed the best with a macro averaged F1
score of 0.95 while Spanish performed the worst with a macro averaged F1 score
of 0.86. The classification models fine-tuned on simulated script-mixed
training data did not improve the classification performance for Tamil.
Conversely, we saw a decrease in performance for Malayalam. There was a large
improvement in classification performance for Hindi.

We have highlighted the performance metric in bold in terms of the optimal
classification models submitted to the organisers for evaluation in Table 4.
Hindi and Tamil ranked third out of seven, Malayalam ranked fourth out of
seven, and English ranked seventh out of eleven. Due to issues with the
labels, the submission for the Spanish condition in Task A was invalid.
However, the macro averaged F1 metric is provided in brackets for reference.

In Task B, Malayalam performed the best of the baseline classification models
with a macro averaged F1 score of 0.86 while English performed the worst with
a macro averaged F1 score of 0.15. The performance increased once we fine-
tuned the classification model with the retrained XLM-RoBERTa with the macro
averaged F1 score for English improving from 0.15 to 0.54 and for Tamil
improving from 0.77 to 0.90. The performance remained stable between the
baseline and retrained models for Malayalam.

When we introduced the script-mixed models for Malayalam and Tamil, we saw
varying levels of performance. The macro averaged F1 score for Malayalam
increased from 0.86 to 0.88. This suggests an increase in performance
accuracy. Counterintuitively, the macro averaged score F1 for Tamil decreased
from 0.90 to 0.80 which was on par with the baseline model. This suggests a
decrease in performance accuracy.

Our Malayalam classification system fine-tuned on the script-mixed social
media language data ranked first out of six, while the Tamil classification
system fine-tuned on the script-mixed social media language data ranked fourth
out of seven despite the decrease in performance from the retrained language
model. Our English classification system fine-tuned on the retrained language
model ranked sixth out of nine.

##  4 Discussion

This paper addresses intrinsic issues related to hate speech detection in
written social media data. This was mirrored in our training, validation, and
testing data which reflects the multifaceted challenges of real-world
scenarios. Hate speech is not confined to any single language or geographic
region and its instances are often buried within the vast array of existing
textual data, particularly in the context of social media.

The employment of the XLM-RoBERTa model has demonstrated to be an effective
system in detecting homophobia and transphobia in social media comments,
particularly when the PLM has been retrained with spatio-temporal data for the
English and Tamil language conditions. These findings underline the potential
of integrating geographic language data into models as a means of enhancing
their performance not only on highly represented languages, but also on lower
underrepresented languages, thus offering a robust solution.

In the preceding sections of this paper, we have outlined and highlighted in
Table 1 and Table 2 an influencing challenge relating to the distribution of
data across the different language conditions. The imbalance observed between
language conditions as exhibited in the discrepancies in their respective
training, validation, and test data sets poses an additional obstacle when it
comes to drawing comparative inferences. However, there are opportunities that
could help balance the data and potentially improve performance.

To alleviate this issue, the utilisation of synthetic data through data
augmentation techniques could prove to be a promising approach. Data
augmentation, as a broad concept, involves expanding the existing data sets to
enhance their diversity, and therefore, the generalisability of the models
trained on them Hoffmann et al. (2022). The generation of synthetic data has
been demonstrated to be an effective mechanism in addressing biased data sets,
but it also presents a desirable practice particularly suited for hate speech
detection given the prevailing concerns over text obfuscation of such
instances Aggarwal and Zesch (2022).

To help facilitate a system that can account for these nuances, data noise
injection via character, word, or even phrasal additions could be
advantageous. In this sense, the application of synthetic data coupled with
noise injection can help address class imbalance, while also training more
robust classifiers that are less reliant on explicit instances of derogatory
terms, but are more adept at discerning underlying contextual uses of hate
speech.

There are real-world applications to our homophobia/transphobia detection
system as we can refine our model with language-specific and region-specific
information to monitor hate speech on social media directed at LGBTQ+
communities. This is particularly useful for languages that are not otherwise
as well represented in large language models such as Malayalam which saw great
improvement in performance with the addition of script-mixed retraining data.

##  5 Conclusion

We saw an improvement in performance in our retrained homophobia/transphobia
classification model when compared with our baseline model. Our unique
approach to this shared task has shown potential for retraining pretrained
language models with spatio-temporal relevant language data to improve the
performance of our homophobia/transphobia detection system.
Counterintuitively, the inclusion of script-mixed language data gave us
variable results. We will aim to refine our classification system with other
attested methods such as noise injection in order to improve the performance
of our system.

## References

  * Aggarwal and Zesch (2022) Piush Aggarwal and Torsten Zesch. 2022.  Analyzing the Real Vulnerability of Hate Speech Detection Systems against Targeted Intentional Noise.  In _Proceedings of the Eighth Workshop on Noisy User-generated Text (W-NUT 2022)_ , pages 230–242, Gyeongju, Republic of Korea. Association for Computational Linguistics.
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