# Pseudo-isotopies of simply connected 4-manifolds

David Gabai  Department of Mathematics, Princeton University, NJ 08540, USA
<EMAIL_ADDRESS>,  David T. Gay  Department of Mathematics,
University of Georgia, Athens, GA 30602, USA<EMAIL_ADDRESS>,  Daniel Hartman
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
<EMAIL_ADDRESS>,  Vyacheslav Krushkal  Department of Mathematics, University
of Virginia, Charlottesville, VA 22903, USA<EMAIL_ADDRESS>and  Mark
Powell  School of Mathematics and Statistics, University of Glasgow, United
Kingdom<EMAIL_ADDRESS>
###### Abstract.

Perron and Quinn gave independent proofs in 1986 that every topological
pseudo-isotopy of a simply-connected, compact topological 4-manifold is
isotopic to the identity. Another result of Quinn is that every smooth pseudo-
isotopy of a simply-connected, compact, smooth 4-manifold is smoothly stably
isotopic to the identity. From this he deduced that
$\pi_{4}(\operatorname{TOP}(4)/\operatorname{O}(4))=0$. A replacement
criterion is used at a key juncture in Quinn’s proofs, but the justification
given for it is incorrect. We provide different arguments that bypass the
replacement criterion, thus completing Quinn’s proofs of both the topological
and the stable smooth pseudo-isotopy theorems. We discuss the replacement
criterion and state it as an open problem.

###### Key words and phrases:

4-manifolds, pseudo-isotopy, isotopy

###### 1991 Mathematics Subject Classification:

57K40, 57N35.

##  1\. Introduction

Let $M$ be a compact $d$-manifold. Let $\sqsubset\,\subseteq M\times[0,1]$
denote $(M\times\\{0\\})\cup(\partial M\times[0,1])$. A _pseudo-isotopy of
$M$_ is a homeomorphism $F\colon M\times[0,1]\to M\times[0,1]$ such that
$F|_{\sqsubset}=\operatorname{Id}_{\sqsubset}$. The homeomorphism $f\colon
M\to M$ obtained from restricting $F$ to $M\times\\{1\\}$ is said to be
_pseudo-isotopic to the identity_. If $F$ is level-preserving, i.e. if
$F|_{M\times\\{t\\}}$ is a homeomorphism from $M\times\\{t\\}$ to itself for
all $t\in[0,1]$, then $F$ is the trace of an isotopy; for brevity in this case
we refer to $F$ simply as an isotopy. Observe that if $F$ is an isotopy then
$F$ is isotopic rel. $\sqsubset$ to the identity map of $M\times[0,1]$, and in
particular $f$ is isotopic to the identity map of $M$. There are directly
analogous definitions in the smooth and $\operatorname{PL}$ categories.

For $M$ smooth and simply-connected, $F$ a diffeomorphism, and $d\geq 5$, Cerf
[Cerf] proved that $F$ is smoothly isotopic to the identity, which in
particular implies that $f$ is smoothly isotopic to the identity. For $d\geq
6$, Cerf’s method was extended by Hatcher-Wagoner [HW] and Igusa [Igusa-what-
happens] to a two-stage obstruction theory deciding in the non-simply-
connected setting whether $F$ is smoothly isotopic to the identity. Cerf and
Hatcher-Wagoner’s results were extended to the topological and
$\operatorname{PL}$ categories by Pedersen [BLR]*Appendix 2, making use of
work of Hudson [Hudson] and [Pedersen-top-conc]. The case $d=5$ in the non-
simply connected case is open at the time of writing, in all categories.

This article concerns the extension of Cerf’s theorem to dimension four, in
the topological category and in the smooth category after stabilising
$M\times[0,1]$ with copies of $S^{2}\times S^{2}\times[0,1]$. From now on we
set $d=4$.

###  1.1. Stable isotopy

First we discuss the smooth stable version. We recall the definition of a
stable isotopy. Assume that $M$ is a compact, smooth, simply-connected
4-manifold, and that $F\colon M\times[0,1]\to M\times[0,1]$ is a smooth
pseudo-isotopy. After an isotopy, we may assume that there is a 4-ball
$D^{4}\subseteq M$ such that $F|_{D^{4}\times[0,1]}$ is the identity. We can
then connect sum $M\times[0,1]$ with $(\\#^{k}S^{2}\times S^{2})\times[0,1]$
along $D^{4}\times[0,1]$; let $N_{k}$ denote the result. Consider $F_{k}\colon
N_{k}\rightarrow N_{k}$, extending
$F|_{(M\times[0,1])\setminus(\mathring{D}^{4}\times[0,1])}$ by the identity on
$((\\#^{k}S^{2}\times
S^{2})\times[0,1])\setminus(\mathring{D}^{4}\times[0,1])$. If there exists a
$k$ such that $F_{k}$ is smoothly isotopic rel. $\sqsubset$ to the identity,
then we say that the pseudo-isotopy $F$ is _stably isotopic to the identity_.

Here is Quinn’s $4$-dimensional smooth stable pseudo-isotopy theorem
[Quinn:isotopy]*Theorem 1.4.

######  Theorem 1.1.

Let $M$ be a compact, smooth, simply-connected 4-manifold and let $F\colon
M\times[0,1]\to M\times[0,1]$ be a smooth pseudo-isotopy. Then $F$ is stably
isotopic to the identity.

In Section 7 we show that Quinn’s proof of one of the steps in his argument,
the _Replacement Criterion_ in [Quinn:isotopy]*Section 4.5, does not work. We
do not know whether the replacement criterion holds, and this is an
interesting open question; we state it in 7.1.

The first main goal of this article is to fix Quinn’s proof of Theorem 1.1. We
will use the majority of Quinn’s argument, but we modify it to replace his use
of the replacement criterion. Our modification uses a method called
_factorisation_ , which first appeared in [Gabai-22]*Lemma 3.15, to split a
pseudo-isotopy in two; see below Remark 4.3. One of the two resulting pseudo-
isotopies can be stably isotoped to the identity using Quinn’s methods. We
present a new argument, a novel application of Quinn’s sum square move, to
resolve the other pseudo-isotopy. We exploit extra $S^{2}\times S^{2}$
summands to find geometrically dual, framed embedded spheres to certain
surfaces, when they are required.

######  Remark 1.2.

Stabilisation is necessary in Theorem 1.1. In dimension 4, smooth pseudo-
isotopy does not imply smooth isotopy for 1-connected 4-manifolds, as shown
first by Ruberman [ruberman-isotopy], with later examples constructed by
Kronheimer-Mrowka [Kronheimer-Mrowka-K3], Baraglia-Konno [Baraglia-Konno], Lin
[Lin-dehn-twist-stabn], and Konno-Mallick-Taniguchi [Konno-Mallick-Taniguchi],
among others.

Quinn deduced the following result from Theorem 1.1, stated as
[Quinn:isotopy]*Theorem 1.2. Here $\operatorname{TOP}(4)$ denotes the
topological group of homeomorphisms of $\mathbb{R}^{4}$ that fix the origin,
and $\operatorname{TOP}(4)/\operatorname{O}(4)$ denotes the homotopy fibre of
the canonical map $\operatorname{BO}(4)\to\operatorname{BTOP}(4)$.

######  Theorem 1.3.

$\pi_{4}(\operatorname{TOP}(4)/\operatorname{O}(4))=0$.

The space $\operatorname{TOP}(4)/\operatorname{O}(4)$ is an important
universal space in smoothing theory. For example, Theorem 1.3 was the final
step in showing [FQ]*Theorem 8.7A, that
$\operatorname{TOP}(4)/\operatorname{O}(4)\to\operatorname{TOP}/\operatorname{O}$
is 5-connected. In combination with Lees and Lashof’s immersion theory [Lees,
Lashof-70-I, Lashof-70-II, Lashof-71], this implies [FQ]*Theorem 8.7B, which
states that for $M$ a noncompact, connected 4-manifold, concordance classes of
smooth structures correspond bijectively with $H^{3}(M,\partial
M;\mathbb{Z}/2)$.

######  Remark 1.4.

There is now an alternative proof due to Gabai [Gabai-22]*Theorem 2.5, of the
fact that smoothly pseudo-isotopic diffeomorphisms of a compact 1-connected
4-manifold are stably isotopic. However Gabai’s proof does not trivialise the
given pseudo-isotopy, so does not recover Theorem 1.1. Thus Gabai’s theorem
[Gabai-22]*Theorem 2.5 cannot be applied to prove Theorem 1.3.

We note that Perron’s article [Perron], which we will discuss more below, did
not address Theorem 1.1, and it is not clear how to approach it using Perron’s
method, due to his use of the Alexander trick. Thus as far as we know Theorem
1.3 cannot be deduced from Perron’s work.

The only proof known to us of Theorem 1.3 makes use of our corrected proof of
Theorem 1.1.

###  1.2. Topological isotopy

Now we discuss the topological 4-dimensional analogue of Cerf’s theorem.

######  Theorem 1.5.

Let $M$ be a compact, topological, 1-connected 4-manifold and let $F\colon
M\times[0,1]\to M\times[0,1]$ be a pseudo-isotopy. Then $F$ is topologically
isotopic rel. $\sqsubset$ to the identity $\operatorname{Id}_{M\times[0,1]}$.

The first proof of Theorem 1.5, for a class of $4$-manifolds discussed next,
was given by Perron [Perron]. Perron’s main result states the theorem for
smooth, compact 4-manifolds without 1-handles. In [Perron]*Section 7, Perron
deduced the statement for closed, 1-connected, topological 4-manifolds, using
an argument he attributes to Siebenmann. In Section 5 we will explain how this
deduction can be extended further to compact manifolds with boundary using
work of Boyer [Boyer86, Boyer93] to deduce all cases of Theorem 1.5. This
extension is necessary because Casson [Kirby-problems]*Remarks following
Problem 4.18 showed there are compact, contractible, smooth 4-manifolds with
boundary that do not admit a handle decomposition with no 1-handles.

Quinn [Quinn:isotopy]*Theorem 1.4 gave an independent proof of Theorem 1.5,
for all compact, topological, 1-connected 4-manifolds. Quinn’s work may be
thought of as a natural extension of Cerf’s approach to dimension $4$. A
(presently unknown) extension to non-simply connected $4$-manifolds would
likely involve a combination of the higher-dimensional Hatcher-Wagoner theory
for non-trivial fundamental groups and a suitable extension of Quinn’s
approach.

The second main goal of this article is to fix Quinn’s proof of Theorem 1.5,
which also relied on his replacement criterion [Quinn:isotopy]*Section 4.5.
Thus, given the extension of Perron’s proof to manifolds with boundary,
discussed above, and our completion of Quinn’s argument, there are two
independent proofs of Theorem 1.5 in full generality. As with the fix of
Theorem 1.1, we will retain the majority of Quinn’s argument, however we will
again split the given pseudo-isotopy into two using factorisation. One of the
resulting pseudo-isotopies can be resolved using Quinn’s argument from
[Quinn:isotopy]*Section 4.6. For the second we arrange for a judicious
application of the Alexander trick.

######  Remark 1.6.

As discussed above, there is a well established obstruction theory for pseudo-
isotopies of non-simply-connected manifolds in dimensions $\geq 6$, in both
topological and smooth categories. An analogue of such a theory is not
presently known for non-simply-connected $4$-manifolds. However, it follows
from the work of Budney-Gabai [Budney-Gabai-2] that the simply-connected
hypothesis cannot be removed in Theorem 1.5.

###  1.3. Classification of homeomorphisms up to isotopy

An important application of Theorem 1.5 is to classify homeomorphisms of
1-connected 4-manifolds up to isotopy. To achieve this one also needs a
classification of homeomorphisms up to pseudo-isotopy.

The classification of diffeomorphisms up to smooth pseudo-isotopy was done
using modified surgery by Kreck [Kreck-isotopy-classes] for closed, smooth,
1-connected 4-manifolds. Quinn gave the classification for homeomorphisms of
closed, topological, 1-connected 4-manifolds in [Quinn:isotopy], with a
correction of the analysis of the normal invariants given by Cochran-Habegger
[Cochran-Habegger]. It is also straightforward to adapt Kreck’s argument to
the topological category.

The outcome [Quinn:isotopy]*Theorem 1.1 of the classification of
homeomorphisms up to pseudo-isotopy, combined with Theorem 1.5, is that the
natural map

$\pi_{0}\operatorname{Homeo}^{+}(M)\xrightarrow{\cong}\operatorname{Aut}(H_{2}(M),\lambda_{M}),$

sending an isotopy classes of orientation-preserving homeomorphisms of $M$ to
the induced isometry of the intersection pairing $\lambda_{M}\colon
H_{2}(M)\times H_{2}(M)\to\mathbb{Z}$, is an isomorphism. Surjectivity is due
to Freedman [F].

For homeomorphisms of compact, 1-connected, topological manifolds, with
possibly nonempty boundary, the classification up to pseudo-isotopy, and hence
up to isotopy by Theorem 1.5, was completed by Orson-Powell [Orson-Powell].
The corresponding surjectivity result in the case of nonempty boundary is due
to Boyer [Boyer86].

### Organisation

We begin by explaining our remedy for the proof of Theorem 1.1. In Section 2
we recall the start of Quinn’s proof in the smooth setting, and state the key
propositions from his paper that we will use. Section 3 recalls two key
geometric constructions: Whitney spheres and the sum square move. Section 4
contains the details of our fix for the proof of Theorem 1.1.

Section 5 proves a key lemma on a decomposition of compact 1-connected
topological 4-manifolds. It has two aims. It gives a key input for Quinn’s
topological approach, and it enables us to explain how Perron’s method can be
applied to all compact 1-connected 4-manifolds, and not just those that are
either closed or admit a handle decomposition without 1-handles.

In Section 6 we adapt our fix from Section 4 to the topological case. Finally,
in Section 7 we give more details on the problem with Quinn’s proof of the
Replacement Criterion, and pose the question of whether the replacement lemma
holds.

### Acknowledgements

We are grateful to Frank Quinn for graciously discussing his work with us.

David Gabai was partially supported by NSF grant DMS-2304841. David T. Gay and
Daniel Hartman were partially supported by NSF grant DMS-2005554. David T. Gay
was also supported in part by Simons Foundation grant MP-TSM-00002714.
Vyacheslav Krushkal was supported in part by NSF Grant DMS-2105467. Mark
Powell was partially supported by EPSRC New Investigator grant EP/T028335/2
and EPSRC New Horizons grant EP/V04821X/2.

##  2\. Morse functions, nested eye Cerf graphics, and the start of Quinn’s
proof

In this section we summarise the start of Quinn’s proof assuming the input of
a smooth pseudo-isotopy. We explain the position one is in prior to Quinn’s
use of the replacement criterion, and to set up notation. After the
replacement criterion, Quinn’s argument for completing the stable smooth proof
is also valid, and we make use of this too, in the form of Theorem 2.2 below.

Throughout this section and Section 4 we work in the smooth category. In
Sections 5 and 6 we will explain how to adapt the arguments to the topological
category, unstably.

A smooth pseudo-isotopy $F\colon M\times[0,1]\to M\times[0,1]$ of a smooth
compact 1-connected 4-manifold can be translated into a 1-parameter family of
generalised Morse functions $G_{t}\colon M\times[0,1]\to[0,1]$, where
$G_{0}=\operatorname{pr}_{2}$ and $G_{1}=\operatorname{pr}_{2}\circ F$. Here
$\operatorname{pr}_{2}$ denotes projection onto the second factor, namely
$[0,1]$. Both $G_{0}$ and $G_{1}$ have no critical points. The family $G_{t}$
is also accompanied by a 1-parameter family of gradient-like vector fields
$\xi_{t}$ subordinate to $G_{t}$.

Given a pair $(G_{t},\xi_{t})$, one can in turn recover a pseudo-isotopy
$F\colon M\times[0,1]\to M\times[0,1]$. The proof of the fact that pseudo-
isotopy implies isotopy consists of _deformations_ of these 1-parameter
families, i.e. a 1-parameter family of 1-parameter families. A deformation
corresponds to an isotopy of the pseudo-isotopy $F$. The aim is to deform
$(G_{t},\xi_{t})$ until there are no critical points of $G_{t}$, for all
$t\in[0,1]$. Such a 1-parameter family corresponds to a pseudo-isotopy $F$
that is an isotopy.

Using $\operatorname{Wh}_{2}(\\{1\\})=\\{1\\}$, by the work of Hatcher-Wagoner
[HW]*Chapter VI, Theorem 2 one can arrange by a deformation of the family that
the Cerf graphic associated to $G_{t}$ consists of finitely many nested eyes –
pairs of index 2 and 3 critical points that are born near time $t_{b}$, and
die at time $t_{d}>t_{b}$, with no rearrangements. More precisely, we let
$t_{b}$ be the time of the final birth, and let $t_{d}$ be the time of the
first death. See Figure 1. According to Hatcher-Wagoner [HW]*Chapter VI,
Theorem 2, we may assume that there are no handle slides, no critical value
crossings, and that the births and deaths are independent. In particular,
there are no trajectories between any pair of index 3 critical points, and
similarly for any pair of index 2 critical points. Moreover at the moment of
each birth and shortly thereafter, the index $2$ and $3$ critical points that
appear do not have trajectories to any other critical points, other than the
unique trajectory between them. The same holds at each death, and shortly
before.

$t_{b}$$t_{f}$$t_{w}$$t_{d}$middle middle level

Figure 1. A family of nested eyes.

If we can ‘close’ the innermost eye in the family of nested eyes, one at a
time, then we complete the proof of the smooth stable isotopy theorem. We
formalise this in the following inductive hypothesis.

######  Inductive Hypothesis 2.1 ($\mathbf{E(n)}$).

Let $M$ be a smooth 1-connected 4-manifold and let $(G_{t},\xi_{t})$ be as
above, a 1-parameter family of generalised Morse functions and gradient like
vector fields on $M\times[0,1]$. Suppose that the associated Cerf graphic
consists of $n$ nested eyes with cancelling pairs of index 2 and 3 critical
points, with no handle slides. Then after stabilising $M\times[0,1]$ with
$(\\#^{p}S^{2}\times S^{2})\times[0,1]$, for some $p$, there is a deformation
of $(G_{t},\xi_{t})$ to a 1-parameter family without critical points.

We will prove the base case $E(1)$ and we will deduce the case $E(n)$ from
$E(n-1)$. The core of Quinn’s proof starts with a nested eye family, and works
on the innermost eye. We set up notation and recall the salient points of
Quinn’s argument for removing the innermost eye.

Let us suppose we have a family of $n$ nested eyes, for some $n\geq 1$. In the
level sets $G_{t}^{-1}(1/2)$, for $t_{b}<t<t_{d}$, we have a copy of
$M\\#_{i=1}^{n}S^{2}\times S^{2}$. For each $t$ we have $n$ spheres
$\\{A_{i}^{t}\\}_{i=1}^{n}$, which are the ascending spheres of the index 2
critical points, and $n$ spheres $\\{B_{i}^{t}\\}_{i=1}^{n}$, which are the
descending spheres of the index 3 critical points. Shortly after the last
birth time $t_{b}$, the sphere $A_{i}^{t}$ is the sphere
$S^{2}\times\\{\operatorname{pt}\\}$ in the $i$th $S^{2}\times S^{2}$ summand.
Similarly the sphere $B_{i}^{t}$ is the sphere $\\{\operatorname{pt}\\}\times
S^{2}$ in the $i$th $S^{2}\times S^{2}$ summand. The $A_{i}^{t}$ and the
$B_{i}^{t}$ are enumerated so that $A_{1}$ and $B_{1}$ correspond to the
innermost eye, and then moving outwards, as shown in e.g. Figure 6. Let
$A^{t}:=\cup_{i=1}^{n}A_{i}^{t}$ and let $B^{t}:=\cup_{i=1}^{n}B_{i}^{t}$.

We may consider that $A^{t}$ stays fixed as $t$ varies in $t_{b}<t<t_{d}$,
$B^{t}$ moves by an isotopy, and $A^{t}\cup B^{t}$ undergoes a regular
homotopy. We may also assume that there are times $t_{f}$ and $t_{w}$, with
$t_{b}<t_{f}<1/2<t_{w}<t_{d}$, such that at time $t_{f}$ a collection of
finger moves occur, and at time $t_{w}$ a collection of Whitney moves occur.
During these moves, intersections of $B_{j}^{t}$ with $A_{i}^{t}$ are created
or removed, respectively. We consider the _middle-middle level_ ,
$G_{1/2}^{-1}(1/2)$. We can let the finger and Whitney discs evolve so that we
see both simultaneously in the middle-middle level. We write
$A_{i}:=A_{i}^{1/2}$ and $B_{j}:=B^{1/2}_{j}$; that is, in the middle-middle
level we drop the $t$ from the notation.

In the middle-middle level, we call the finger move discs for $A_{i}$, $B_{j}$
intersections $\\{V^{ij}_{k}\\}$, and write $V^{ij}:=\cup_{k}V^{ij}_{k}$.
Analogously we call the Whitney discs $\\{W^{ij}_{\ell}\\}$, and write
$W^{ij}:=\cup_{\ell}W^{ij}_{\ell}$. We also write $V:=\cup_{i,j}V^{ij}$ and
$W:=\cup_{i,j}W^{ij}$.

Working on the innermost eye, Quinn showed in [Quinn:isotopy]*Sections 4.2–4.4
that we can assume, after a deformation, that both

$\big{(}\bigcup_{k}\partial V^{11}_{k}\cup\partial W^{11}_{k}\big{)}\cap
A_{1}$

and

$\big{(}\bigcup_{k}\partial V^{11}_{k}\cup\partial W^{11}_{k}\big{)}\cap
B_{1}$

are arcs in $A_{1}$ and $B_{1}$ respectively. We will refer to this as
_Quinn’s arc condition._ See Figure 2.

$A_{1}$$B_{1}$$W^{11}_{1}$$W^{11}_{2}$$V^{11}_{1}$$V^{11}_{2}$$S_{V^{11}_{2}}$$S_{V^{11}_{1}}$

Figure 2. Finger and Whitney discs with boundaries forming an arc in $A_{1}$
and in $B_{1}$. The Whitney spheres from Section 3.1 corresponding to the
finger move discs are also shown.

For a disc $D$, let $\mathring{D}$ denote its interior. The following
statement summarizes the result of [Quinn:isotopy]*Section 4.6.

######  Theorem 2.2 (Quinn).

Suppose that $M$ is smooth, compact, and 1-connected, and suppose that
$F\colon M\times[0,1]\xrightarrow{\cong}M\times[0,1]$ is a smooth pseudo-
isotopy. Consider an associated 1-parameter family $(G_{t},\xi_{t})$, and
suppose that the associated Cerf graphic consists of $n$ nested eyes with
cancelling pairs of index 2 and 3 critical points, and no handle slides.
Consider the data of spheres and finger/Whitney discs $(A,B,V,W)$ in the
middle-middle level.

If Quinn’s arc condition is satisfied for $(A,B,V,W)$, and if the algebraic
intersection numbers $\mathring{V}^{11}_{k}\cdot\mathring{W}^{11}_{\ell}$
vanish for all $k,\ell$, then after stabilisation of the pseudo-isotopy with
$(\\#^{p}S^{2}\times S^{2})\times[0,1]$, for some $p$, there is a smooth
deformation of the family to one with $(n-1)$ nested eyes.

Quinn used the problematic replacement criterion to arrange for the algebraic
intersection condition in Theorem 2.2 to hold. Since we cannot appeal to the
replacement criterion, we must provide an alternative argument.

##  3\. Whitney spheres and the sum square move

Here we take a short digression to recall two important constructions that we
shall need in Section 4. In Section 3.1 we recall the construction of Whitney
spheres. In Section 3.2 we recall Quinn’s sum square move.

###  3.1. Whitney spheres

We recall a construction of the _Whitney sphere_ $S_{V_{k}^{ij}}$ associated
with a finger move disc $V_{k}^{ij}$. These spheres have appeared in different
guises in the literature, cf. [Quinn:isotopy]*Section 4.3, [FQ]*Section 3.1,
Ex. (2), [COP20]*Section 4.2, [ST-2019]*Section 2. We use the terminology and
the description from [ST-2019].

The description is given in ${\mathbb{R}}^{3}\times{\mathbb{R}}$ where $A_{i}$
and the finger disc $V_{k}^{ij}$ are in ${\mathbb{R}}^{3}\times\\{0\\}$, and
$B_{j}$ is represented as (arc $\subseteq{\mathbb{R}}^{3}$)$\times[-1,1]$,
Figure 3. The Whitney sphere is drawn red, and it consists of two discs,
$D_{i}\subseteq{\mathbb{R}}^{3}\times\\{i\\}$, $i=-1,1$, joined by an annulus
(circle $\subseteq{\mathbb{R}}^{3}$)$\times[-1,1]$. Each $D_{i}$ is
constructed using two copies of the finger move disc $V_{k}^{ij}$, so overall
the Whitney sphere contains four pushed-off copies of $V_{k}^{ij}$.

The Whitney sphere is framed and embedded and can be assumed to lie in an
arbitrarily small neighbourhood of $V_{k}^{ij}$.

$V_{k}^{ij}$$A_{i}$$B_{j}$$t=-1$$-1<t<0$$t=0$$0<t<1$$t=1$$D_{-1}$$D_{1}$

Figure 3. A description of the Whitney sphere $S_{V_{k}^{ij}}$ in
$\mathbb{R}^{3}\times\mathbb{R}$.

###  3.2. The sum square move

We recall Quinn’s sum square move [Quinn:isotopy]*Section 4.2, which can be
used to modify finger or Whitney disc configurations by a deformation of the
pseudo-isotopy. The data for the move is a framed embedded square $S$ in the
middle-middle level, with interior disjoint from the spheres $A$ and $B$, and
from the discs $V$. The square has two edges on the $V$ discs, denoted $V_{0}$
and $V_{i}$ (anticipating the proof below) in Figure 4, one edge on $A$, and
one on $B$. New $V$ discs are obtained by cutting $V_{0},V_{i}$ along the
boundary edges of the sum square $S$, and gluing in two parallel copies of
$S$. The effect of the move on the boundaries of the discs, on $A$ and $B$
spheres, is illustrated in Figure 7.

$B$$A$$V_{0}$$V_{i}$$S$

Figure 4. The sum square move

Figure 4 is a $3$-dimensional model for the sum square. Here $A,V_{0},V_{i}$,
and a neighbourhood of the arc of $\partial S$ in $B$ are pictured in the
“present”,
${\mathbb{R}}^{3}\times\\{0\\}\subseteq{\mathbb{R}}^{3}\times{\mathbb{R}}$.
The rest of $B$ extends into the past and the future. The framing of $S$ along
its boundary is determined in the $3$-dimensional model by a non-vanishing
vector field on $\partial S$ which is normal to $S$ and tangent to $A,B$ and
the $V$ discs; this framing has to admit an extension over $S$ for the move to
give rise to embedded $V$ discs.

A justification is given in [Quinn:isotopy]*Section 4.2 for why the sum square
move gives a deformation of the pseudo-isotopy. As usual in the subject, given
the boundary data $\partial S$, the challenge is to find a framed, embedded
$S$ disjoint from other given surfaces. The proof in Section 4 below explains
how to achieve this in relevant situations in the stable setting. Note that if
$S\cap W=\emptyset$, then the $V$ discs do not acquire new intersections with
$W$ as a result of this move.

##  4\. The proof of the inductive step

We continue with the notation and setup from Section 2, and describe our
correction to Quinn’s proof of Theorem 1.1.

Suppose that there are $m$ finger discs $\\{V^{11}_{k}\\}_{k=1}^{m}$. Then
there are also $m$ Whitney discs $\\{W^{11}_{\ell}\\}_{\ell=1}^{m}$ for
$A_{1}$–$B_{1}$ intersections. We fix an orientation on each of these discs.
Define

$T_{\ell}:=\\#_{k=\ell}^{m}S_{V^{11}_{k}},$

where the connected sum is formed by tubing the Whitney spheres
$S_{V^{11}_{k}}$ together along arcs in the $W^{11}_{k}$ that are parallel to
one of the boundary arcs of $W^{11}_{k}$, for $k>\ell$. (The tubing could be
done along any arcs, not necessarily arcs in the Whitney discs; generally this
would result in algebraically cancelling pairs of intersections which are fine
for the applications below.)

######  Lemma 4.1.

The spheres $\\{T_{\ell}\\}_{\ell=1}^{m}$ are mutually disjoint, framed, and
embedded in the complement of $A\cup B\cup V$. For some choice of orientations
on the $T_{\ell}$ we have algebraic intersection numbers $T_{\ell}\cdot
W^{11}_{k}=\delta_{k\ell}$. That is, the collection $\\{T_{\ell}\\}$ forms a
collection of algebraically dual spheres to the Whitney discs
$\\{W^{11}_{k}\\}$.

###### Proof.

The $\delta_{k\ell}$ terms can be seen in Figure 2. Each intersection of
$\mathring{V}^{11}_{\ell}$ with $\mathring{W}^{11}_{k}$ gives rise to four
intersections between $S_{{V}^{11}_{\ell}}$ and $\mathring{W}^{11}_{k}$. These
come in algebraically cancelling pairs, so do not contribute to the
intersection numbers in the statement of the lemma. ∎

We write

$J_{kp}:=-\mathring{V}^{11}_{k}\cdot\mathring{W}^{11}_{p}\in\mathbb{Z}$

and let

$\Sigma_{kp}:=\\#^{J_{kp}}T_{p}$

denote $J_{kp}$ mutually disjoint parallel, oriented copies of $T_{p}$, tubed
together by annuli contained in a regular neighbourhood of $T_{p}$ that are
disjoint from $A$, $B$, $V$, and $W$. The spheres $\Sigma_{kp}$ are mutually
disjoint, framed, and embedded in the complement of $A\cup B\cup V$. Note that

$\mathring{W}_{\ell}^{11}\cdot\Sigma_{kp}=\mathring{W}_{\ell}^{11}\cdot\\#^{J_{kp}}T_{p}=J_{kp}\delta_{p\ell}=-(\mathring{V}^{11}_{k}\cdot\mathring{W}^{11}_{p})\delta_{p\ell}.$

Now we want to create a new family of discs whose interiors have trivial
algebraic intersection numbers with the interior of every disc
$W^{11}_{\ell}$. We define

$\widetilde{V}^{11}_{k}:=V^{11}_{k}\\#_{p=1}^{m}\Sigma_{kp}.$

Note that after a small isotopy, the interiors of discs
$\\{\widetilde{V}^{11}_{k}\\}$ may be assumed to be disjoint from the
interiors of $\\{{V}^{11}_{k}\\}$. We compute

$\displaystyle\begin{split}\widetilde{V}^{11}_{k}\cdot\mathring{W}^{11}_{\ell}&=\mathring{V}^{11}_{k}\cdot\mathring{W}^{11}_{\ell}+\sum_{p=1}^{m}\Sigma_{kp}\cdot\mathring{W}^{11}_{\ell}\\\
&=\mathring{V}^{11}_{k}\cdot\mathring{W}^{11}_{\ell}-\sum_{p=1}^{m}(\mathring{V}^{11}_{k}\cdot\mathring{W}^{11}_{p})\delta_{p\ell}\\\
&=\mathring{V}^{11}_{k}\cdot\mathring{W}^{11}_{\ell}-\mathring{V}^{11}_{k}\cdot\mathring{W}^{11}_{\ell}=0.\end{split}$
(1)

Finally, for each $k$ we define

$\widehat{V}^{ij}_{k}:=\begin{cases}\widetilde{V}^{11}_{k}&(i,j)=(1,1)\\\
V^{ij}_{k}&(i,j)\neq(1,1).\end{cases}$

That is, we replace the discs associated with the innermost eye, but leave all
other discs in $\\{V^{ij}_{k}\\}$ unchanged. As we explain below, the new
family of discs $\\{\widehat{V}^{ij}_{k}\\}$ will be used to factorize the
given pseudo-isotopy into two, which we can then trivialize separately.

######  Lemma 4.2.

The discs $\widehat{V}^{ij}_{k}$ are pairwise disjoint, framed, and embedded.

###### Proof.

Each of the $\widetilde{V}^{11}_{k}$ discs lives in a regular neighbourhood of
the discs $V^{11}$, together with arcs on $A_{1}$, say, given by $\partial
W^{11}\cap A_{1}$. Since the interiors of the boundary arcs of $V^{11}$ and of
$W^{11}$ are disjoint by Quinn’s arc condition, and since the $V^{ij}$ discs
can be assumed to miss a regular neighbourhood of $V^{11}$, we do not create
extra intersections. Also note that the spheres $\Sigma_{kp}$ are framed,
mutually disjoint, disjoint from all $\\{{V}^{ij}_{k}\\}$, and embedded. Hence
tubing into them again yields framed and embedded finger discs. ∎

Next we will use the freedom to introduce additional $S^{2}\times S^{2}$
summands by stabilising the pseudo-isotopy with $(\\#^{m}S^{2}\times
S^{2})\times[0,1]$. In particular such an operation introduces $m$ additional
$S^{2}\times S^{2}$ summands into the middle-middle level. This paragraph is
specific to the stable proof. The crucial feature of $\widehat{V}^{11}$ is
that the algebraic intersection numbers between the components of
$\widehat{V}^{11}$ and the components of $W^{11}$ are all zero. We are free to
modify $\widehat{V}^{11}$ as long as they continue satisfying this condition
and Lemma 4.2 continues to be satisfied. We take $\widehat{V}^{11}$ as defined
above, and tube each component into $S^{2}\times\\{\operatorname{pt}\\}$ in
its own newly added $S^{2}\times S^{2}$ summand. Since $\widehat{V}^{11}$
consists of $m$ discs, we add $m$ $S^{2}\times S^{2}$ summands. Now, in the
stabilised middle-middle level, the entire collection of discs
$\widehat{V}^{11}$ has a geometrically dual collection of framed embedded
spheres that are disjoint from $A\cup B\cup W\cup(V\setminus V^{11})$.

######  Remark 4.3.

We have structured the proof so that the steps that use stabilisation by
copies of $(S^{2}\times S^{2})\times[0,1]$ are separated from the rest of the
proof. This is a device to avoid having to repeat text verbatim during the
topological proof in Section 6.

Now we perform _factorisation_ [Gabai-22]*Lemma 3.15. That is, we introduce
_ex nihilo_ a cancelling finger-Whitney pair. To explain this we introduce
some notation. If we have a finger disc $V_{k}$, and we use it as a Whitney
disc, then we write it as $\underline{V_{k}}$, to indicate the same disc in
the middle-middle level, but with its _modus operandi_ altered. Analogously if
we have a Whitney disc $W_{\ell}$ that we wish to use as a finger disc, we
write it as $\underline{W_{\ell}}$.

Another way to think about this is to consider two families of discs in the
middle level at time $1/2$. Whitney moves describe the motion of $B$ forward
in time, and finger discs describe the motion of $B$ backwards in time.
Underlining a disc indicates reversing the time direction associated with that
disc, i.e. reversing whether it is considered as a finger or Whitney disc.

As indicated above, the plan is to introduce a trivial family in the middle of
the family. We have a family where finger moves corresponding to the discs $V$
occur at $t_{f}<1/2$, and then Whitney moves occur using the discs $W$ at time
$t_{w}>1/2$. By a deformation, we may and shall assume that the spheres
$A^{t}$ and $B^{t}$ are constant during the time interval
$[t_{f},t_{f}+4\varepsilon]$, where $\varepsilon\ll 1/2-t_{f}$.

Consider a new family, which proceeds as before until $t_{f}$. Then shortly
after $t_{f}$, at time $t_{f}+\varepsilon$ say, Whitney moves are performed to
remove all $A$, $B$ intersections using Whitney discs
$\underline{\widehat{V}}$. Then at time $t_{f}+3\varepsilon$, these Whitney
moves are reversed by finger moves corresponding to the discs $\widehat{V}$.
Then for time $t>1/2$ the family is again as before: at the time $t_{w}$ the
Whitney moves guided by the discs $W$ are performed, and from time $t_{d}$
onward the critical points are cancelled. This new family is related to the
original family by a deformation, in which the Whitney moves, and their
subsequent undoing, are performed progressively less and less.

We have now accomplished factorisation. We pass from the finger-Whitney
configuration $V\cdot W$ to the configuration
$V\cdot\underline{\widehat{V}}\cdot\widehat{V}\cdot W$.

Note that at times $t_{f}+\varepsilon<t<t_{f}+3\varepsilon$ the spheres
$A_{i},B_{j}$ intersect geometrically in $\delta_{ij}$ points. Now, deform the
family so that all the index 2 and 3 critical points are cancelled at time
$t_{f}+2\varepsilon$.

The outcome is two nested eye families. First we consider the left hand family
of nested eyes. If $(i,j)\neq(1,1)$, then all finger moves involving $A_{i}$
and $B_{j}$ are undone with exactly the same disc, acting as a Whitney disc.
Thus we can deform the family to eradicate all of these extra intersections
from ever occurring.

######  Lemma 4.4.

In a family of nested eyes, if there are no handle slides, i.e. no $2/2$ nor
$3/3$ trajectories, then we can perform a deformation via bigon moves, beak
moves, and their inverses, to move the innermost eye outermost.

###### Proof.

This is a standard fact in Cerf theory. It follows from the sequence of beak
moves, bigon moves, and their inverses shown in Figure 5, in the case of two
eyes. We illustrated the exchange move in Lemma 4.4 in the case of two eyes,
but it can also be applied to move the innermost eye past an arbitrary number
of eyes, to make it outermost. There are no obstructions to performing these
moves because by hypothesis there are no trajectories between critical points
in different eyes of the same index.

beak movebeak movebigon movebigon move

Figure 5. A sequence of Cerf moves that switch the order of nesting of two
concentric eyes.

∎

Applying Lemma 4.4, we now remove all eyes apart from the previous innermost
eye. We move the innermost eye outermost, and then all remaining eyes have a
unique intersection for all time, i.e. $|A^{t}_{i}\pitchfork B^{t}_{i}|=1$ for
all $t\in[t_{b},t_{d}]$. Hence they can be removed by the case $m=k=1$ of
[HW]*Chapter V, Proposition 1.1.

In the right hand nested eye family, by equation (1) the finger and Whitney
discs for the innermost eye satisfy that the interiors of
$\widetilde{V}^{11}_{k}$ and $W^{11}_{\ell}$ intersect algebraically
trivially, for all $k$, $\ell$. Therefore the innermost eye of the right hand
nested eyes can be removed by Theorem 2.2. We are left with the configuration
shown in Figure 6. The right hand nested eye family now has $(n-1)$ nested
eyes, and so by 2.1, all of these $(n-1)$ eyes can be removed.

$A_{1}$$B_{1}$$A_{1}$$B_{1}$$A_{2}$$B_{2}$$A_{n}$$B_{n}$$A_{2}$$B_{2}$$A_{n}$$B_{n}$

Figure 6. A deformation of the family leads to a modification of the Cerf
graphic as shown. The finger/Whitney discs change via factorisation from
$V\cdot W$ to $V\cdot\underline{\widehat{V}}\cdot\widehat{V}\cdot W$.

######  Remark 4.5.

The remainder of the proof is specific to the smooth stable case.

It remains to consider the innermost eye in the left hand family. This is
characterised by the collection of (finger, Whitney) discs
$(V^{11},\underline{\widehat{V}}^{11})$. The unions of the boundaries form
circles rather than arcs (after a small push of the boundary of
$\widehat{V}^{11}$ to make the boundaries disjoint).

We would like to convert $(V^{11},\underline{\widehat{V}}^{11})$ into finger,
Whitney discs satisfying Quinn’s arc condition. We will achieve this using the
sum square move (Section 3.2). First introduce a trivial arc-pair $V_{0}$,
$W_{0}$ and rename
$(V_{i},W_{i}):=(V_{i}^{11},\underline{\widehat{V}_{i}}^{11})$, for $i\geq 1$.
Abusing notation, again denote $V:=\cup_{i}V_{i},W:=\cup_{i}W_{i}$. We will
use sum squares $S_{i}$ with boundaries on $V_{0}$ and $V_{i}$, for each
$i\geq 1$, as indicated in Figure 7.

$+$$-$$+$$-$$+$$\partial V_{0}$$\partial S$$\partial W_{0}$$\partial
V_{i}$$\partial W_{i}$

Figure 7. Rearranging the boundaries of the finger and Whitney discs, on $A$
and on $B$, using the sum square move.

Following Quinn [Quinn:isotopy]*Sections 4.2 and 4.3, make the sum square
$S_{i}$ framed and disjoint from $A$, $B$, and $V$, using spheres dual to $A$
and $B$, that are disjoint from $V$ but intersect $W$. We add more copies of
$S^{2}\times S^{2}$, one for each $i$, and pipe each $S_{i}$ into its own
$S^{2}\times\\{\operatorname{pt}\\}$.

Now the $S_{i}$ and the $W_{j}=\underline{\widehat{V}_{j}}^{11}$ have a
collection of framed embedded geometrically dual spheres, disjoint from
everything else and each other, arising from the new $S^{2}\times S^{2}$
summands. (See the paragraph following the proof of Lemma 4.2 for the
discussion of dual spheres to $W_{j}$.) Resolve all $S_{i}$, $W_{j}$
intersections by tubing $S_{i}$ into parallel copies of the dual spheres to
the $W_{j}$. Resolve any self-intersections of $S_{i}$ by tubing $S_{i}$ into
parallel copies of the dual spheres to $S_{i}$. This produces embedded squares
$S_{i}$ with interiors disjoint from $A\cup B\cup V\cup W$, and correctly
framed. Performing the sum square moves creates a family satisfying Quinn’s
arc condition. Since $V$ and $\widehat{V}$ had disjoint interiors, by the
sentence preceding equation (1), the discs in the new family after the sum
square move also all have pairwise disjoint interiors.

This allows us to stably, smoothly trivialize the pseudo-isotopy given by
$(V,\underline{\widehat{V}})$. Thus the remaining eye, the formerly innermost
eye in the left hand family, can also be removed. This completes the proof of
the inductive step, and hence completes the proof of Theorem 1.1. ∎

##  5\. A decomposition theorem for topological 4-manifolds

We prove a decomposition lemma (Lemma 5.4) for compact 1-connected topological
4-manifolds. The proof is an adaptation of the proofs in [Perron]*Section 7
which Perron attributes to Siebenmann. The argument given there is for closed
1-connected 4-manifolds, and relies on the Freedman-Quinn classification of
1-connected closed topological 4-manifolds. We explain how this argument can
be adapted to compact 1-connected 4-manifolds, with nonempty boundary, using
Boyer’s classification [Boyer86, Boyer93] of compact 1-connected topological
4-manifolds. This has two aims.

  1. (i)

Theorem 1.5 stated by Quinn is more general than the result of Perron, because
it encompasses all compact 1-connected topological 4-manifolds with nonempty
boundary. Perron’s result in the case of nonempty boundary has the assumption
that the 4-manifold is smooth and has a handle decomposition with no
1-handles. We clarify that Theorem 1.5 can be deduced from Perron’s result,
using the decomposition lemma proven in this section.

  2. (ii)

The first step of Quinn’s proof of Theorem 1.5 uses a decomposition result,
[Quinn:isotopy]*Proposition 1.5. The proof given in this section, following
Siebenmann’s argument, seems to us to be easier than Quinn’s proof in
[Quinn:isotopy]*Section 6 of his Proposition 1.5, so this represents a
simplification of Quinn’s proof.

######  Remark 5.1.

Both Perron and Quinn remark that Theorem 1.5 might follow from
[Perron]*Theorem 0 and [Quinn:isotopy]*Proposition 1.5, via the argument of
[Perron]*Section 7. However, [Quinn:isotopy]*Proposition 1.5 allows 1-handles,
and so we do not believe this is the case. Boyer’s classification [Boyer86,
Boyer93], which was completed after [Quinn:isotopy], avoids 1-handles, as
stated in Theorem 5.2 below.

We will make use of the following result of Boyer [Boyer86, Boyer93]; see in
particular [Boyer93]*p. 35.

######  Theorem 5.2 (Boyer).

Let $M$ be a 1-connected 4-manifold with connected nonempty boundary $Y$. Then
$M$ is homeomorphic to a manifold of the form $Y\times[0,1]\cup H\cup C$,
where $H$ consists of finitely many 2-handles attached to $Y\times\\{1\\}$,
such that $\partial(Y\times[0,1]\cup H)=Y\times\\{0\\}\sqcup\Sigma$, where
$\Sigma$ is a $\mathbb{Z}$-homology 3-sphere, and $C$ is a compact,
contractible 4-manifold with $\partial C\cong\Sigma$.

Since Boyer’s result requires that $\partial M=Y$ is connected, first we
dispense with the case that $\partial M$ is disconnected.

######  Lemma 5.3.

Suppose that every pseudo-isotopy is isotopic rel. $\sqsubset$ to the identity
for 1-connected compact 4-manifolds with connected boundary. Then every
pseudo-isotopy is isotopic rel. $\sqsubset$ to the identity for all
1-connected compact 4-manifolds.

###### Proof.

Let $M$ be a 1-connected compact 4-manifold with boundary components
$Y_{1},\dots,Y_{n}$. Let $F\colon M\times[0,1]\to M\times[0,1]$ be a pseudo-
isotopy and let $f:=F|_{M\times\\{1\\}}$. Choose pairwise disjoint locally
flat embedded arcs $\gamma_{2},\dots,\gamma_{n}$ in $M$, such that for each
$\gamma_{i}$, one endpoint is on $Y_{1}$ and the other endpoint is on $Y_{i}$.
The image $F(\gamma_{i}\times[0,1])$ is a concordance from $\gamma_{i}$ to
$f(\gamma_{i})$. Observe that (forgetting the $[0,1]$ coordinate in
$M\times[0,1]$) concordance implies homotopy, and homotopy implies isotopy for
arcs in a 4-manifold, so $F$ is isotopic to a homeomorphism, again denoted
$F$, such that $F(\gamma_{i}\times[0,1])$ is the trace of an isotopy, for each
$i$. By a further isotopy we can assume $F$ fixes each $\gamma_{i}\times[0,1]$
pointwise, and in fact fixes an open neighbourhood of each
$\gamma_{i}\times[0,1]$ pointwise, $i=2,\dots,n$. Removing these
neighbourhoods and restricting $F$ to their complement yields a pseudo-isotopy
$F|$ of a 1-connected 4-manifold with connected boundary. By hypothesis $F|$
is isotopic rel. $\sqsubset$ to the identity. Extending this isotopy over the
removed neighbourhoods with the identity map yields an isotopy of the original
$F$ to the identity of $M\times[0,1]$, rel. $\sqsubset$. ∎

From now on we assume that $\partial M=Y$ is connected. The following
Decomposition Lemma and its proof are modelled on [Perron]*Lemma 7.2. We check
that by invoking Boyer’s classification the proof extends to the case of
nonempty boundary, providing some necessary details. The statement is closely
related to [Quinn:isotopy]*Proposition 1.5.

######  Lemma 5.4 (Decomposition Lemma).

Assume that $M^{4}=Y\times[0,1]\cup H\cup C$ is as in Theorem 5.2 and that
$\partial M=Y$ is connected. The contractible manifold $C$ is contained in the
interior of a topologically embedded 4-ball $D^{4}\subseteq M$. It follows
that we can write $M$ as $M=Y\times[0,1]\cup H\cup D^{4}$, where $H$ consists
of finitely many 2-handles attached to $Y\times\\{1\\}$.

###### Proof.

Let $C_{1}$ and $C_{1}^{\prime}$ be two copies of $C$. The union
$C_{1}\cup_{\partial}(-C_{1}^{\prime})$ is a homotopy sphere and so is
homeomorphic to $S^{4}$ [F]. It follows that $C_{1}^{\prime}\subseteq S^{4}$,
and (removing a $4$-ball from $S^{4}$) $C_{1}^{\prime}\subseteq D^{4}$.
Considering a $4$-ball $D^{4}$ in the interior of $C$, we have inclusions

$C_{1}^{\prime}\subseteq D^{4}\subseteq C\subseteq M.$ (2)

Let $W:=M\setminus\mathring{C}$ and let
$W^{\prime}:=M\setminus\mathring{C}_{1}^{\prime}$. We claim there exists a
homeomorphism rel. boundary $f\colon W\xrightarrow{\cong}W^{\prime}$. Assuming
the claim, since every homeomorphism of a homology 3-sphere $\Sigma$ extends
over any given contractible 4-manifold111This follows from Boyer’s
classification [Boyer86], but can also be proved more directly. Let
$f\colon\Sigma\xrightarrow{\cong}\Sigma$, and fix $C^{4}\simeq\\{\ast\\}$ with
$\partial C=\Sigma$. Then $D:=C\cup_{f}-C$ is a homotopy 4-sphere, so by [F]
$D\cong S^{4}=\partial D^{5}$. Consider $D^{5}$ as a rel. boundary
$h$-cobordism from $C$ to $C$ rel. $f$, and apply the $h$-cobordism theorem
[FQ]*Theorem 7.1A. with boundary $\Sigma$, this extends to a homeomorphism
$\overline{f}\colon M=W^{\prime}\cup C_{1}^{\prime}\xrightarrow{\cong}W\cup
C=M$ that sends $C_{1}^{\prime}$ to $C$. For the copy of $D^{4}$ as in (2),
the image $\overline{f}(D^{4})$ is a 4-ball in $M$ whose interior contains
$C$.

It remains to prove the claim: for a fixed 1-connected, compact $M$, removing
two different copies of the same contractible 4-manifold, $C$ and
$C_{1}^{\prime}$, from $M$, yields homeomorphic 4-manifolds. To prove it, we
apply Boyer’s classification from [Boyer86]. First note we can apply the
tubing trick from the proof of Lemma 5.3, together with homotopy implies
isotopy for arcs, to reduce to the case of connected boundary. We assert that
Boyer’s classifying invariants coincide for a pair of 4-manifolds if and only
if they coincide after gluing a contractible 4-manifold to both along
homeomorphic homology 3-discs in the boundaries. Boyer’s invariants coincide
after gluing a contractible 4-manifold because both manifolds $W\cup C$ and
$W^{\prime}\cup C_{1}^{\prime}$ are copies of $M$.

It remains to see the assertion that Boyer’s classifying invariants are not
affected by gluing contractible manifolds along homology 3-discs. Boyer’s main
invariant is the intersection form of the 4-manifold $M$, together with an
isomorphism of its algebraic boundary with the linking form on the torsion
submodule of $H_{1}(Y)$. His secondary invariant only applies for a pair of
spin 4-manifolds, and (after fixing a spin structure on $Y$) lies in a
subgroup of $H^{1}(Y;\mathbb{Z}/2)$. It records the spin structure induced on
$Y$ by the 4-manifold. These homological invariants are not affected by gluing
on a contractible manifold, and so the assertion holds. ∎

As promised in (i) at the start of the section, we deduce Theorem 1.5 using
[Perron] and the Decomposition Lemma 5.4.

###### Proof of Theorem 1.5 using [Perron].

By Lemma 5.3 we assume without loss of generality that $\partial M$ is
connected. By the Decomposition Lemma 5.4, we can write $M$ as $M=Y\times
I\cup H\cup D^{4}$, where $I=[0,1]$ denotes the collar coordinate.

Let $F\colon M\times[0,1]\to M\times[0,1]$ be a pseudo-isotopy. Recall that
$F|_{\sqsubset}=\operatorname{Id}_{\sqsubset}$. Isotope $F$, using uniqueness
of collars, to arrange that $F$ restricted to a collar on $\sqsubset$ is the
identity, and in particular is the identity on $(M\times\\{0\\})\cup(Y\times
I\times[0,1])$.

Now we consider the cores
$V:=\sqcup_{i}(D^{2}\times\\{0\\})_{i}\subseteq\sqcup_{i}(D^{2}\times
D^{2})_{i}=H$ of the 2-handles $H$. If we were to turn the 2-handles upside
down, then these would be the cocores of the upside-down 2-handles.

The main step of Perron’s proof, which culminates in the statement of
[Perron]*Lemme 5.2 is to take a tubular neighbourhood $\mathcal{N}$ of the
union $V$ of the cocores of his 2-handles (the cores of our 2-handles $H$),
and perform an isotopy of the pseudo-isotopy $F$, rel. the collar on
$\sqsubset$, to arrange that its restriction to this neighbourhood is the
inclusion $\mathcal{N}\times[0,1]\to M\times[0,1]$. Possibly after a further
isotopy, we may take $\mathcal{N}=H$, and hence as a result we have a pseudo-
isotopy $F^{\prime}\colon M\times[0,1]\to M\times[0,1]$ that is the identity
away from $\mathring{D}^{4}\times[0,1]\subseteq M\times[0,1]$, and is the
identity on $D^{4}\times\\{0\\}$. By Alexander’s coning trick, with cone point
$(0,1)\subseteq\mathring{D}^{4}\times[0,1]$, we can further isotope
$F^{\prime}$ rel. $(M\times[0,1])\setminus(\mathring{D}^{4}\times[0,1])$ to
the identity. ∎

We go on to explain our fix of Quinn’s proof of Theorem 1.5, which, as stated
in (ii), starts with an application of Lemma 5.4.

##  6\. The topological pseudo-isotopy theorem

We start by recalling Quinn’s strategy for the proof of Theorem 1.5, which
begins in [Quinn:isotopy]*Section 5. Our fix starts after the statement of
Theorem 6.3.

Let $M$ be a compact, 1-connected topological 4-manifold and let $F\colon
M\times[0,1]\to M\times[0,1]$ be a topological pseudo-isotopy, i.e. a
homeomorphism that is the identity on $\sqsubset$. By Lemma 5.3 we may and
shall assume that $\partial M=:Y$ is connected. By the Decomposition Lemma
5.4, $M=Y\times[0,1]\cup H\cup D^{4}$, where $H$ consists of 2-handles added
to $Y\times\\{1\\}$. Define

$X:=Y\times[0,1]\cup H\subseteq M,$

and note that $\partial M=Y\times\\{0\\}\subseteq X$. The goal, as on
[Quinn:isotopy]*p. 363, is to prove the following.

######  Proposition 6.1.

The pseudo-isotopy $F$ is isotopic rel. $\partial M=Y\times\\{0\\}$ to a
pseudo-isotopy $F^{\prime}$ that is the identity on $X$.

Since $M\setminus X\subseteq D^{4}\subseteq M$, we have that $(M\setminus
X)\times[0,1]\subseteq D^{4}\times[0,1]\subseteq M\times[0,1]$, we can apply
the Alexander trick to $F^{\prime}$, pushing towards $(0,1)\in
D^{4}\times[0,1]\subseteq M\times[0,1]$ as the cone point, to isotope
$F^{\prime}$ to the identity. Hence it suffices to prove Proposition 6.1.

Next, Quinn assumes, after an isotopy, that $F$ is the identity on
$B\times[0,1]$, where $B\subseteq M\setminus X$ is a 4-ball. Write $M_{0}$ for
$M$ with the origin of $B$ removed. High dimensional smoothing theory applies
since $M_{0}$ is connected and non-compact, by immersion theory [Lees,
Lashof-70-I, Lashof-70-II, Lashof-71], and Quinn’s prior result [Quinn-
annulus]*Corollary 2.2.3 that
$\operatorname{TOP}(4)/\operatorname{O}(4)\to\operatorname{TOP}/\operatorname{O}$
is 3-connected. We obtain a smooth structure on $M_{0}$, hence on
$M_{0}\times[0,1]$, and then seek to smooth $F$. The unique obstruction
[Kirby-Siebenmann:1977-1] to smoothing $F$ lives in $H^{3}(M_{0},\partial
M_{0};\mathbb{Z}/2)=0$. Hence there is an isotopy of $F|_{M_{0}\times[0,1]}$
to a smooth pseudo-isotopy of $M_{0}$. This step is common to both Perron’s
and Quinn’s proofs, and it is well explained by Perron [Perron]*Corollaire
1.3.

The resulting smooth pseudo-isotopy $F|_{M_{0}\times[0,1]}$ has a
corresponding 1-parameter family of Morse functions and gradient-like vector
fields $(G_{t},\xi_{t})$. Since we are in the non-compact setting there can be
infinitely many critical points. However only finitely many of them intersect
$X$ for some $t$, because $X$ and $[0,1]$ are compact. Quinn defines a control
function $\gamma\colon M_{0}\to[0,\infty)$ to be $0$ on $M\setminus B$ and
$(1/d(x,0))-1$ on $B$, where $d$ denotes the standard Euclidean metric on the
ball $B$. There is then a controlled Reduction to Eyes Lemma
[Quinn:isotopy]*Lemma 5.1, proven in [Quinn:isotopy]*Section 5.3. The outcome
is a one parameter family of Morse data $(G_{t},\xi_{t})$ on
$M_{0}\times[0,1]$, related to the original data by a deformation of the
family, such that the associated Cerf graphic consists of nested eyes with
critical points of index 2 and 3, there are no handle slides, no
rearrangements, and the birth and deaths are independent. There is also an
associated 1-parameter family of handle structures where the handles have
controlled diameter, i.e. their image under $\gamma$ has diameter less than
some given constant.

As mentioned above, there are finitely many critical points, $n$ say, that
intersect $X$ for some $t\in[0,1]$. Since there are no handle slides and the
births and deaths are independent, we can apply the argument in Lemma 4.4 to
perform a deformation of the family such that the $n$ eyes intersecting $X$
are the $n$ innermost eyes in the nesting.

Again the strategy is to remove one of these $n$ eyes at a time, starting with
the innermost eye. The proof is by induction on $n$, using the following
inductive hypothesis, which is the topological, non-compact version of 2.1.

######  Inductive Hypothesis 6.2 ($\mathbf{E(n)}$).

Let $M$, $X\subseteq M$, and $M_{0}$ be as above. Let $(G_{t},\xi_{t})$ be a
1-parameter family of generalised Morse functions and gradient like vector
fields on $M_{0}\times[0,1]$. Suppose that the associated Cerf graphic
consists of nested eyes with cancelling pairs of index 2 and 3 critical
points, with no handle slides, no rearrangements, and independent births and
deaths. Suppose there are at most $n$ eyes that, for some time $t$, intersect
$X\times[0,1]$, and that these are the $n$ innermost eyes in the Cerf graphic.
Then there is a deformation of $(G_{t},\xi_{t})$ to a 1-parameter family
without any critical points that intersect $X\times[0,1]$.

We will prove the base case $n=1$ and the inductive step. We will use the
notation from Section 2, which we briefly recall. We have birth time $t_{b}$,
and a death time $t_{d}$. In the level sets $G_{t}^{-1}(1/2)$, for
$t_{b}<t<t_{d}$, we have ascending spheres $\\{A_{i}^{t}\\}$ of the index 2
critical points, and $n$ descending spheres $\\{B_{i}^{t}\\}$ of the index 3
critical points. The $A_{i}^{t}$ and the $B_{i}^{t}$ are enumerated so that
the $n$ eyes that intersect $X$ are first, with $A_{1}^{t}$ and $B_{1}^{t}$
correspond to the innermost eye among these $n$ eyes, and then moving
outwards. The spheres $A_{1}^{t},\dots,A_{n}^{t}$ and
$B_{1}^{t},\dots,B_{n}^{t}$ can intersect $X\times[0,1]$, whereas $A_{i}^{t}$
and $B_{j}^{t}$ are disjoint from $X\times[0,1]$, for all $i,j>n$ and for all
$t\in[0,1]$.

Let $A^{t}:=\cup_{i=1}^{n}A_{i}^{t}$ and $B^{t}:=\cup_{i=1}^{n}B_{i}^{t}$. We
also have a finger move time $t_{f}$ and a Whitney move time $t_{w}$, with
$t_{b}<t_{f}<1/2<t_{w}<t_{d}$. In the middle-middle level $G_{1/2}^{-1}(1/2)$
we see the spheres $A_{i}:=A_{i}^{1/2}$, $B_{j}:=B^{1/2}_{j}$, the finger move
discs $\\{V^{ij}_{k}\\}$ and the Whitney discs $\\{W^{ij}_{\ell}\\}$. We write
$V^{ij}:=\cup_{k}V^{ij}_{k}$, $W^{ij}:=\cup_{\ell}W^{ij}_{\ell}$,
$V:=\cup_{i,j}V^{ij}$, and $W:=\cup_{i,j}W^{ij}$.

The fact that $A_{i}^{t}$ and $B_{j}^{t}$ are disjoint from $X\times[0,1]$,
for all $i,j>n$ and for all $t\in[0,1]$, implies that in the middle-middle
level all the discs $V^{ij}$ and $W^{ij}$ are disjoint from $X$ for $j>n$; an
intersection of one of these discs with $X$ would entail a motion of
$B_{j}^{t}$, for some $j>n$, through $X\times[0,1]$.

The following statement summarizes the result of [Quinn:isotopy]*Sections 4.6
and 5.2. This is the topological analogue of Theorem 2.2, proved using the
disc embedding theorem.

######  Theorem 6.3 (Quinn).

In the situation described above, if Quinn’s arc condition is satisfied for
$A_{1},B_{1}$, and if the algebraic intersection numbers
$\mathring{V}^{11}_{k}\cdot\mathring{W}^{11}_{\ell}$ vanish for all $k,\ell$,
then there is a topological deformation of the family to one with $(n-1)$
nested eyes that intersect $X\times[0,1]$.

We need to arrange a situation that we can apply this result to prove the
inductive step. As in the smooth stable case discussed in Section 4, the
problem with the proof in [Quinn:isotopy] is in the use of the replacement
criterion to arrange $\mathring{V}^{11}_{k}\cdot\mathring{W}^{11}_{\ell}=0$,
to arrange that the hypotheses of Theorem 6.3 are satisfied. We give an
alternative argument below.

As a result of the use of control theory, discussed above, $V^{ij}$ and
$W^{ij}$ are disjoint from $X$ for $j>n$. Quinn’s primary concern in
[Quinn:isotopy]*Section 5.2 is to preserve this condition as an eye is
cancelled. This guarantees that the associated eyes, with enumeration $>n$,
remain disjoint from $X\times[0,1]$. During this proof, Quinn mentions the
replacement criterion in the second paragraph of [Quinn:isotopy]*p. 366. Since
we do not use the replacement criterion in our fix, this paragraph becomes
irrelevant. We observe that our new proof preserves $V^{ij}\cap X=W^{ij}\cap
X=\emptyset$ for $j>n$.

As in Section 4, we perform the factorisation to replace the family with discs
$V\cdot W$ with the concatenation of moves corresponding to discs
$V\cdot\underline{\widehat{V}}\cdot\widehat{V}\cdot W$. The discs
$\widetilde{V}^{11}$ are defined as before, by tubing into Whitney spheres.
Ignore the modification to these discs from the paragraph just above Remark
4.3; as stated in that remark, this modification was specific to the smooth
stable proof. In particular, we do not stabilise the pseudo-isotopy with
$(\\#^{m}S^{2}\times S^{2})\times[0,1]$. Then for each $k$ we define

$\widehat{V}^{ij}_{k}:=\begin{cases}\widetilde{V}^{11}_{k}&(i,j)=(1,1)\\\
V^{ij}_{k}&(i,j)\neq(1,1).\end{cases}$

We emphasise that in the second case we make this definition for all the
possibly infinitely many values of $(i,j)\neq(1,1)$. The factorisation
involves all the eyes, not just the $n$ innermost eyes that we are trying to
cancel. The only discs that are modified are the $V^{11}$, and these already
intersect $X$. So we continue to have $\widehat{V}^{ij}_{k}\cap
X=V^{ij}_{k}\cap X=\emptyset$ for $j>n$ and for all $k$.

As in Section 4, we cancel all the pairs of index $2-3$ critical points for a
small period of time in the middle, and consider again the two families of
(possibly infinitely many) nested eyes corresponding to a 1-parameter family
of Morse data on $M_{0}\times[0,1]$.

By the same argument as in Section 4, we remove all but the innermost eye in
the left hand family of nested eyes. We will return to analysing this eye
later.

In the innermost eye of the right hand family, the finger and Whitney discs
once more satisfy the algebraic intersection hypothesis, as well as Quinn’s
arc condition. Thus by Theorem 6.3 we can remove the innermost eye without
causing any new critical points to intersect $X\times[0,1]$, and maintaining
control. Then the right hand family contains $(n-1)$ nested eyes that
intersect $X\times[0,1]$. If $n=1$ there are no eyes left and so we are done
for the right hand family. Otherwise, by 6.2 we can remove $(n-1)$ nested eyes
by a deformation, without causing any other critical points to intersect
$X\times[0,1]$, in the family that is the outcome of this deformation.

Now we analyse the left hand family, consisting of one remaining eye. This has
finger discs $V^{11}$, and then Whitney discs $\underline{\widehat{V}}^{11}$
made from tubing the discs in $V^{11}$ into Whitney spheres. We do not need to
worry about control here, nor intersections with $X\times[0,1]$, because we
have reduced this family to a single eye. We just need to see that the pseudo-
isotopy corresponding to this family is topologically isotopic to the
identity.

Let $t_{b+}$ be a time shortly after the birth time $t_{b}$. Then $A^{t_{b+}}$
and $B^{t_{b+}}$ denotes the ascending and descending spheres, respectively,
of the index 2 and 3 critical points, in the middle level
$G_{t_{b+}}^{-1}(1/2)$, shortly after the birth. Fixing a regular
neighbourhood of $A^{t_{b+}}\cup B^{t_{b+}}$ determines a decomposition of
$G_{t_{b+}}^{-1}(1/2)$ as $M_{0}\\#S^{2}\times S^{2}$.

######  Lemma 6.4.

We may assume, after a deformation of the 1-parameter family, that every
finger move, when viewed in the middle-middle level, occurs entirely within
the $\\#S^{2}\times S^{2}$ summand of $M_{0}\\#S^{2}\times S^{2}$.

###### Proof.

Since $M_{0}\\#S^{2}\times S^{2}$ is simply-connected and the union of the
ascending and descending spheres of the critical points are
$\pi_{1}$-negligible, the complement $N:=\big{(}M_{0}\\#S^{2}\times
S^{2}\big{)}\setminus\big{(}A^{t_{b+}}\cup B^{t_{b+}}\big{)}$ is simply-
connected.

Consider the collection of finger move arcs, with ends joined together via
disjoint arcs on $A^{t_{b+}}$ and $B^{t_{b+}}$. By the previous paragraph
these loops are null homotopic in the complement of the ascending and
descending spheres, and hence bound immersed discs with interiors in $N$. See
the left picture in Figure 8. We can push intersections between these discs,
and self-intersections, off the boundary, over the part of the boundary of the
discs consisting of the finger move arc. We obtain a collection of mutually
disjoint embedded discs. We can use these to isotope all the finger moves into
an arbitrarily small neighbourhood of $A^{t_{b+}}\cup B^{t_{b+}}$. We already
chose such a neighbourhood in order to determine the decomposition of
$G_{t_{b+}}^{-1}(1/2)$ as $M_{0}\\#S^{2}\times S^{2}$. Hence we can shrink the
finger move by an isotopy to lie in the $\\#S^{2}\times S^{2}$ summand.

Figure 8. The finger move arc bounds an embedded disc.

This change can be realised by a deformation of the whole family, because we
can apply an ambient isotopy consistently for all times before $t_{f}$. ∎

By Lemma 6.4, we assume that in our left hand family, each finger move, with
finger disc $V^{11}_{k}$, occurs within the $S^{2}\times S^{2}$ factor in the
middle level. Also each Whitney move, with Whitney disc
$\underline{\widetilde{V}}^{11}_{k}$, occurs in a neighbourhood of $A_{1}\cup
B_{1}\cup\widetilde{V}^{11}$. We built the Whitney discs
$\underline{\widetilde{V}}^{11}_{k}$ using Whitney spheres, and these can be
assumed to lie in an arbitrary neighbourhood of $V^{11}$. Hence each Whitney
move occurs in a neighbourhood of $A_{1}\cup B_{1}\cup V^{11}$.

We assumed that $A_{1}$ stays fixed, and we have now arranged that all motions
of $B_{1}$ occur within the $S^{2}\times S^{2}$ summand. Hence all motions of
$A_{1}$ and $B_{1}$ stay within this summand. This translates to a pseudo-
isotopy supported in a copy of $D^{4}\subseteq M_{0}$. That is, we have an
isotopic pseudo-isotopy $F^{\prime}\colon M_{0}\times[0,1]\to
M_{0}\times[0,1]$, a decomposition

$M_{0}\times[0,1]\cong(M_{0}\setminus\mathring{D}^{4})\times[0,1]\cup_{\partial
D^{4}\times[0,1]}D^{4}\times[0,1],$ (3)

and a pseudo-isotopy $P\colon D^{4}\times[0,1]\to D^{4}\times[0,1]$, such that
$F^{\prime}$ has the form

$F^{\prime}=\operatorname{Id}\cup P$

with respect to the decomposition (3). By Alexander’s coning trick, again with
cone point $(0,1)\subseteq D^{4}\times[0,1]$, the pseudo-isotopy $P$ is
topologically isotopic rel. $\sqsubset$ to
$\operatorname{Id}_{D^{4}\times[0,1]}$. Hence $F^{\prime}$ is isotopic to
$\operatorname{Id}_{M_{0}\times[0,1]}$, and the pseudo-isotopy corresponding
to our left hand family is topologically isotopic to
$\operatorname{Id}_{M_{0}\times[0,1]}$. This completes the proof of the
inductive step, and hence completes the proof of Theorem 1.5. ∎

##  7\. The disc replacement criterion

We discuss the problems with the given proof of the replacement criterion in
[Quinn:isotopy]*Section 4.5, and mention a couple of related open problems.

We begin with a meta argument that there ought to be a problem with given
proof of the replacement criterion. The proof given in [Quinn:isotopy]*Section
4.5 does not use the simply-connected hypothesis, nor the hypothesis that the
boundaries of finger and Whitney discs are arranged in an arc. The same proof,
if valid, would allow one to drastically alter the Hatcher-Wagoner secondary
obstruction, in particular in the
$\operatorname{Wh}_{1}(\pi_{1}(M);\pi_{2}(M))$ summand, leading to a
contradiction with the work of Hatcher-Wagoner and Igusa [HW, Igusa-what-
happens].

###  7.1. Description of the problem

Now we highlight a specific issue with the proof given by Quinn in
[Quinn:isotopy]*Section 4.5 and also illustrate a connection between Quinn’s
idea and the method of factoring used in our corrected proof. We give a
thorough account of the approach in [Quinn:isotopy]*Section 4.5 because the
exposition in that reference is short on detail, so that pinpointing the error
is not immediate. Since the replacement criterion remains an open problem,
stated as 7.1 below, we think it is important to explain the precise problem
with the original approach.

The goal of [Quinn:isotopy]*Section 4.5 is to prove that in a single eye, with
$A$, $B$, $V$, and $W$ satisfying Quinn’s embedded arc condition, if there is
a collection of alternative Whitney discs $\widetilde{W}$ with the same
boundaries as $W$ but with interiors disjoint from $W$, then the pseudo-
isotopy can be smoothly deformed to one with the same $A$, $B$, and $V$, but
with $W$ replaced with $\widetilde{W}$. Figure 9 illustrates this setup. Quinn
initially reconfigures things slightly by introducing a new $2$–sphere
$\widetilde{A}$ that is parallel to $A$ but is not yet seen as the ascending
sphere of a $2$–handle. Then one displaces $\widetilde{W}$ slightly so that
$V$ and $\widetilde{W}$ are seen as finger and Whitney discs for the pair
$(\widetilde{A},B)$, while $V$ and $W$ remain as finger and Whitney discs for
$(A,B)$. This is illustrated in Figure 10.

2pt $A$ [c] at 205 460 $B$ [c] at 180 260 $V$ [c] at 308 205 $W$ [c] at 420
340

2pt $A$ [c] at 205 460 $B$ [c] at 180 260 $V$ [c] at 308 205 $\widetilde{W}$
[c] at 400 300

Figure 9. On the left: $A$, $B$, $V$, and $W$. On the right: the same $A$, $B$
and $V$ but now we have a different Whitney disc $\widetilde{W}$. We do not
assume $\widetilde{W}$ is isotopic to $W$ (if it were, this discussion would
be trivial), only that the interiors of $W$ and $\widetilde{W}$ are disjoint.
Note that both $W$ and $\widetilde{W}$ may intersect $V$ in their interiors,
but this is not indicated in the figure.

2pt $A$ [c] at 240 490 $B$ [c] at 100 265 $V$ [c] at 305 230 $W$ [c] at 405
350 $\widetilde{A}$ [c] at 154 490 $\widetilde{W}$ [c] at 335 340

2pt  $I\times A$ [c] at 188 420

Figure 10. On the left: $A$, $B$, $V$, and $W$ together with $\widetilde{A}$,
a parallel copy of $A$, and the new version of $\widetilde{W}$. On the right,
we highlight the fact that the apparent product $I\times W$ going from $W$ to
$\widetilde{W}$ does not necessarily exist in the ambient manifold, and we
illustrate this with a grey void where the $I\times W$ would be. However $A$
and $\widetilde{A}$ are the two ends of an embedding of $I\times A$ which is
also shown in the figure on the right. There is also no “void” on the $V$ side
of the picture because the local model is constructed so that a neighborhood
$I\times V$ of $V$ does in fact extend all the way to $\widetilde{A}$.

To describe Quinn’s first modification, we introduce the notation $B_{V}$ to
denote the result of modifying $B$ by performing a Whitney move across $V$,
and similarly $B_{W}$; $B_{W,\widetilde{W}}$ denotes the result of a
simultaneous Whitney move across $W$ and across $\widetilde{W}$. We initially
start with $A$ not moving, and with $B$ moving backwards from $t=1/2$ to
$B_{V}$ and forwards from $t=1/2$ to $B_{W}$. Although $\widetilde{A}$ is not
the ascending sphere of a $2$–handle yet, Quinn’s first modification is (a) to
make sure that the Whitney move along $V$ is wide enough so that the isotopy
from $B$ to $B_{V}$ eliminates intersections between $\widetilde{A}$ and $B$
at the same time as eliminating the corresponding intersections between $A$
and $\widetilde{A}$, and (b) to replace the Whitney move along $W$ with
simultaneous Whitney moves along $W$ and $\widetilde{W}$, so that the isotopy
from $B$ to $B_{W,\widetilde{W}}$ also eliminates intersections with
$\widetilde{A}$ as well as with $A$. This is illustrated in Figure 11. The
reason that we can make the $V$ move wide but have to do two separate narrow
moves on $W$ and $\widetilde{W}$ is that the local model is built so that the
product neighbourhood $I\times V$ does stretch from $A$ to $\widetilde{A}$,
while $W$ and $\widetilde{W}$ are not assumed to be isotopic, so that there is
not a product $I\times W$ doing the same thing on the $W$ side of the picture.

Figure 11. The spheres $A$, $\widetilde{A}$, and $B_{V}$ on the left, and the
spheres $A$, $\widetilde{A}$, and $B_{W,\widetilde{W}}$ on the right.

If we follow the $I\times S^{2}$ bounded by $A$ and $\widetilde{A}$ down to a
level below the $2$–handle corresponding to $A$, where the ambient manifold is
now $M$, the $I\times S^{2}$ is surgered to a $D^{3}$, with boundary equal to
$\widetilde{A}$. Turning things upside down, one can use this $3$–ball to
create a cancelling $3$–$4$ handle pair, where the boundary of the $3$–ball is
the attaching sphere for the $3$–handle and the $3$–ball is half of the
attaching sphere for the $4$–handle. Turning things the right way up again,
this is a new $1$–$2$ pair, and this birth can happen slightly after time
$t_{b}$, while the corresponding death happens slightly before time $t_{d}$.
This yields a Cerf graphic as in Figure 12.

2pt $B$ [c] at 199 130 $A$ [c] at 199 119 $\widetilde{A}$ [c] at 199 107
$I\times A$ [c] at 199 96

$B_{V}$ [c] at 142 130 $A$ [c] at 142 119 $\widetilde{A}$ [c] at 142 107
$I\times A$ [c] at 142 96

$B_{V}$ [c] at 58 124 $A$ [c] at 58 113 $I\times A$ [c] at 58 102

$B_{V}$ [c] at 30 119 $A$ [c] at 30 108

$B_{W,\widetilde{W}}$ [c] at 255 130 $A$ [c] at 255 119 $\widetilde{A}$ [c] at
255 107 $I\times A$ [c] at 255 96

$B_{W,\widetilde{W}}$ [c] at 340 124 $A$ [c] at 340 113 $I\times A$ [c] at 340
102

$B_{W,\widetilde{W}}$ [c] at 368 119 $A$ [c] at 368 108

Figure 12. Cerf graphic after introducing a $1$–$2$–pair. The boxes in the
interior of the main eye label the ascending and descending spheres of all the
critical points. Since the lowest critical point is index $1$, its ascending
sphere is $3$–dimensional. However, after rising up past two index $2$
critical points this $S^{3}$ is punctured twice and appears as an $I\times
S^{2}$, identified with $I\times A$, with boundary equal to
$A\amalg\widetilde{A}$. Note that ascending manifolds for birth/death points
are also shown.

Actually Quinn suggests that the new $1$–$2$ eye should start and end at the
same time as the $2$–$3$ eye, rather than being slightly shorter as indicated
in our figure. In what follows one might think that this is an important
distinction, but we will argue that in fact this does not make a difference.
For now, however, if one wishes one can move the cusps of the $1$–$2$ eye left
and right as desired. We have not done this partly just because doing so would
make it harder to keep track of ascending and descending spheres in the
“middle level”.

Looked at from the original middle level, above the original $2$–handle, the
ascending sphere for the new $2$–handle is $\widetilde{A}$ and the ascending
$3$–sphere for the new $1$–handle appears as the $I\times S^{2}$ bounded by
$A$ and $\widetilde{A}$. This means that the $A$ $2$–handle can be cancelled
with the new $1$–handle from slightly after the birth of the new $1$–$2$ pair
to slightly before the death of the new $1$–$2$ pair. Performing this
cancellation produces a Cerf graphic as in Figure 13.

2pt $B$ [c] at 199 119 $\widetilde{A}$ [c] at 199 108

$B_{V}$ [c] at 171 119 $\widetilde{A}$ [c] at 171 108

$B_{V}$ [c] at 142 124 $\widetilde{A}$ [c] at 142 113 $I\times A$ [c] at 142
102

$B_{V}$ [c] at 113 130 $\widetilde{A}$ [c] at 113 119 $A$ [c] at 113 107
$I\times A$ [c] at 113 96

$B_{V}$ [c] at 84 130 $A$ [c] at 84 119 $\widetilde{A}$ [c] at 84 107 $I\times
A$ [c] at 84 96

$B_{V}$ [c] at 58 124 $A$ [c] at 58 113 $I\times A$ [c] at 58 102

$B_{V}$ [c] at 30 119 $A$ [c] at 30 108

$B_{W,\widetilde{W}}$ [c] at 227 119 $\widetilde{A}$ [c] at 227 108

$B_{W,\widetilde{W}}$ [c] at 255 124 $\widetilde{A}$ [c] at 255 113 $I\times
A$ [c] at 255 102

$B_{W,\widetilde{W}}$ [c] at 283 130 $\widetilde{A}$ [c] at 283 119 $A$ [c] at
283 107 $I\times A$ [c] at 283 96

$B_{W,\widetilde{W}}$ [c] at 311 130 $A$ [c] at 311 119 $\widetilde{A}$ [c] at
311 107 $I\times A$ [c] at 311 96

$B_{W,\widetilde{W}}$ [c] at 340 124 $A$ [c] at 340 113 $I\times A$ [c] at 340
102

$B_{W,\widetilde{W}}$ [c] at 368 119 $A$ [c] at 368 108

Figure 13. After cancelling the $A$ $2$–handle with the $1$–handle, again
showing ascending and descending spheres in the middle level. Note now that
between the two swallowtails, the ascending sphere for the index $2$ critical
point is now $\widetilde{A}$ instead of $A$, and that the isotopy of $B$ going
from the middle-middle level one step to the right, taking $B$ to
$B_{W,\widetilde{W}}$, now removes the extra intersection points between $B$
and $\widetilde{A}$.

Note in this figure that the middle of the Cerf graphic is exactly what we
would like it to be: the ascending sphere for the $2$-handle is
$\widetilde{A}$, and $B$ moves backwards in time from $t=1/2$ by doing the
Whitney move along $V$ so that $B_{V}$ is in cancelling position with
$\widetilde{A}$ while $B$ moves forward in time from $t=1/2$ by doing the
Whitney move along $W$ and $\widetilde{W}$ so that $B_{W,\widetilde{W}}$ is in
cancelling position with $\widetilde{A}$. If that were all we saw of the Cerf
graphic, i.e. if the Cerf graphic were cut off before the swallowtails arise,
then we would have the family of handlebodies that we desire. (At the very
end, we could do less and less of the Whitney move along $W$ and since $A$
would no longer be in the picture, this would just be an isotopy of the family
that does not create extraneous intersections between spheres, and we would be
left with $B$ performing just the Whitney move along $\widetilde{W}$.)

Quinn’s goal at this point is to cancel the swallowtails while moving them to
the beginning and end of the Cerf graphic, thus widening the middle section so
that in the end $\widetilde{A}$ is the ascending sphere for the one remaining
$2$–handle for the entire time from the birth to the death. The problem is
that, at some point, the swallowtails need to shrink and that, in doing so,
the ascending $3$–sphere for the $1$–handle appears in the middle level as a
shrinking $I\times S^{2}$, in other words the $I$ factor in the $I\times
S^{2}$ needs to shrink to a point so that immediately after the swallowtail
has disappeared, the $I\times S^{2}$ becomes an $S^{2}$, the ascending sphere
for the single $2$–handle that remains. Figure 14 shows the same Cerf graphic
with smaller swallowtails, on the way towards the swallowtail cancellation,
with the emphasis on seeing how the ascending spheres change. Figure 15
accompanies Figure 14 to explain the labelling.

2pt $B$ [c] at 199 90 $\widetilde{A}$ [c] at 199 79

$B_{V}$ [c] at 171 90 $\widetilde{A}$ [c] at 171 79

$B_{V}$ [c] at 129 95 $\widetilde{A^{*}}$ [c] at 128 84 $I^{*}\\!\\!\times A$
[c] at 128 73

$B_{V}$ [c] at 99 101 $A^{*}$ [c] at 99 79 $\widetilde{A^{*}}$ [c] at 99 90
$I^{*}\\!\\!\times A$ [c] at 99 68

$B_{V}$ [c] at 72 95 $A^{*}$ [c] at 71 84 $I^{*}\\!\\!\times A$ [c] at 71 73

$B_{V}$ [c] at 30 90 $A$ [c] at 30 79

$B_{W,\widetilde{W}}$ [c] at 227 90 $\widetilde{A}$ [c] at 227 79

$B_{W,\widetilde{W}}$ [c] at 270 95 $\widetilde{A^{*}}$ [c] at 270 84
$I^{*}\\!\\!\times A$ [c] at 270 73

$B_{W,\widetilde{W}}$ [c] at 299 101 $A^{*}$ [c] at 299 79 $\widetilde{A^{*}}$
[c] at 299 90 $I^{*}\\!\\!\times A$ [c] at 299 68

$B_{W,\widetilde{W}}$ [c] at 327 95 $A^{*}$ [c] at 327 84 $I^{*}\\!\\!\times
A^{*}$ [c] at 327 73

$B_{W,\widetilde{W}}$ [c] at 368 90 $A$ [c] at 368 79

Figure 14. The problem appears here when we start to shrink the swallowtails.
If $I=[0,1]$, then $I^{*}$ is smaller, e.g. $I^{*}=[1/4,3/4]$. The new
ascending spheres $A^{*}$ and $\widetilde{A^{*}}$ are $\\{1/4\\}\times A$ and
$\\{3/4\\}\times A$, if $A$ was originally identified with $\\{0\\}\times A$
and $\widetilde{A}$ was originally identified with $\\{1\\}\times A$. These
spheres are illustrated in Figure 15. Figure 15.  Illustration to accompany
Figure 14, showing the ascending manifold $I^{*}\times A$ for the $1$–handle
and the ascending spheres $A^{*}$ and $\widetilde{A^{*}}$ for the $2$–handle.
Note that both $A^{*}$ and $\widetilde{A^{*}}$ are back to having three
intersection points with $B_{W,\widetilde{W}}$.

From these figures we can see that, in the course of the swallowtail
cancellation, the intersections between $A$ or $\widetilde{A}$ and $B$ which
were eliminated by doing Whitney moves across $W$ and $\widetilde{W}$
necessarily reappear. If, as Quinn proposes, we push the swallowtail all the
way to the end as we cancel it, the only difference will be that the interval
$I$ will shrink with one end or the other fixed, as opposed to shrinking
towards the middle as we have illustrated.

This completes our explanation of the mistake in Quinn’s proof, but now we
show that in fact the end result of Quinn’s method is nothing more than
factorization. After completely cancelling the swallowtail, we are left with
the ascending sphere for the $2$–handle at the time of the cancellation being
a sphere which we now call $\overline{A}$, equal to $\\{1/2\\}\times A$ in the
original $I\times A$. The resulting Cerf graphic with ascending and descending
spheres is shown in Figure 16. The key point to note is that now the ascending
$A$ spheres are moving in time, whereas originally they were fixed and only
the $B$ spheres moved. Furthermore, the sphere $\overline{A}$ moves forward in
time to $A$ by a Whitney move along a new Whitney disc which we call $W^{*}$,
and moves backwards in time to $\widetilde{A}$ by a Whitney move along a
Whitney disc which we call $\widetilde{A^{*}}$. These Whitney (finger) moves
are indicated by labelled arrows in Figure 16, and the discs themselves are
illustrated in Figure 17.

2pt $B$ [c] at 199 90 $\widetilde{A}$ [c] at 199 79

$B_{V}$ [c] at 143 90 $\widetilde{A}$ [c] at 143 79

$B_{V}$ [c] at 85 90 $\overline{A}$ [c] at 85 79

$V$ [c] at 171 110 $\widetilde{W},\simeq$ [c] at 228 110

$\simeq$ [c] at 59 89 $\simeq$ [c] at 113 89

$B_{V}$ [c] at 30 90 $A$ [c] at 30 79

$B_{W,\widetilde{W}}$ [c] at 255 90 $\widetilde{A}$ [c] at 255 79

$B_{W,\widetilde{W}}$ [c] at 312 90 $\overline{A}$ [c] at 312 79

$B_{W,\widetilde{W}}$ [c] at 368 90 $A$ [c] at 368 79

$\widetilde{W^{*}}$ [c] at 288 59 $W^{*}$ [c] at 340 60

Figure 16. After cancelling the swallowtails. The arrows indicate finger or
Whitney moves. The arrow labelled “$\widetilde{W},\simeq$” indicating a
Whitney move across $\widetilde{W}$ and an isotopy, the isotopy being the
Whitney move across $W$. The new finger/Whitney discs $\widetilde{W^{*}}$ and
$W^{*}$ are illustrated in the accompanying Figure 17. Figure 17. The spheres
$\overline{A}$ and $B_{W,\widetilde{W}}$, and the Whitney discs $W^{*}$ and
$\widetilde{W^{*}}$. These Whitney discs are built from a subset of the strip
labelled $I\times A$ in Figure 10.

The final observation is that the $4$–tuple
$(\overline{A},B_{W,\widetilde{W}},W^{*},\widetilde{W^{*}})$ is in fact
isotopic to the original $4$–tuple $(A,B,W,\widetilde{W})$ as seen in Figure
9. Thus after an isotopy in the middle middle level, the family illustrated in
Figure 16 becomes a family in which, starting at the middle and working to the
right, $B$ does a Whitney move across $\widetilde{W}$, then undoes that with a
finger move back across $\widetilde{W}$, and then does the original Whitney
move across $W$.

In summary, the problem in the proof given by Quinn in [Quinn:isotopy]*Section
4.5 is in the very last sentence of the proof. One can remove the right hand
swallowtail from the Cerf graphic, but the subtlety is that this necessitates
an isotopy between the spheres $A$ and $\widetilde{A}$. The resulting isotopy
intersects $B$ in general, introducing new finger and Whitney moves between
$\widetilde{A}$ and $B$ that were missed by Quinn. The outcome is a collection
of finger-Whitney moves with discs $\widetilde{W}\cdot W$. In other words, it
is exactly the same as the outcome of a factorisation inserting
$\underline{\widetilde{W}}\cdot\widetilde{W}$. But a factorisation is
technically much simpler, and one is still left with a problem to solve. So in
our proofs in the earlier sections of this paper we appeal to factorisation
instead of the method of proof of [Quinn:isotopy]*Section 4.5.

An isotopy between $A$ and $\widetilde{A}$ is destined to create intersections
with $B$ if the union of Whitney discs $\widetilde{W}\cup W$, pushed slightly
into the complement of $A$ and $B$, is a homotopically essential 2-sphere in
the complement of $A\cup B$.

This leads to the extremely interesting open question of whether the
replacement criterion holds in the smooth category, in the simply connected
case, especially if one is permitted the additional assumption that the discs
one wishes to switch are homotopic rel. boundary in the middle-middle level
$M\\#^{k}S^{2}\times S^{2}$ (but not necessarily in the complement of $A\cup
B$).

######  Problem 7.1.

Consider a pseudo-isotopy of a smooth, 1-connected 4-manifold $M$, with
associated 1-parameter family having data $(A,B,V,W)$ that satisfies Quinn’s
arc condition. Let $\widetilde{W}$ be a collection of Whitney discs in one to
one correspondence with the discs in $W$, that pairwise have the same boundary
as $W$, and induce compatible framings. Suppose that the interiors of $W$ and
$\widetilde{W}$ are disjoint, and that $W_{k}\cup\widetilde{W}_{k}$ is null
homotopic in $\pi_{2}(M\\#^{n}S^{2}\times S^{2})$, for all $k$. Is there a
deformation of the pseudo-isotopy replacing the family $W$ with the family
$\widetilde{W}$?

######  Remark 7.2.

The key inductive step in our proof of Theorem 1.1, at the end of Section 4,
involves replacing one system of finger discs $V$ with another $\widehat{V}$.
To do this we need that the replacement discs $\widehat{V}$ have framed
embedded geometrically dual spheres and then we need to stabilise the
$4$-manifold. Switching the rôles of $V$ and $W$, this can be thought of as a
stable version of the disc replacement criterion: the disc replacement
criterion is stably true provided the replacement discs have dual spheres.

###  7.2. An interesting diffeomorphism of $S^{4}$

We illustrate a potential application of Problem 7.1 in a particular example.
Start with a trivial pseudo-isotopy of $S^{4}$ whose Cerf graphic is empty.
Deform this pseudo-isotopy by creating a single $2,3$-handle eye with no
finger or Whitney moves. Deform this family of generalised Morse functions
further to one where the spheres $A,B$ undergo a single finger move and a
single Whitney move. The finger and Whitney discs are standard and satisfy
Quinn’s arc condition, as shown in the $3$-dimensional slice in Figure 18.

$A$$B$$V$$W$$S_{W}$

Figure 18. The data of finger and Whitney discs in the middle-middle level
determining a potentially nontrivial pseudo-isotopy of $S^{4}$.

Recall the construction of Whitney spheres from Section 3.1. Consider the
Whitney sphere $S_{W}$; it is disjoint from $W$ and intersects $V$ in a single
point. Consider a disc $\widetilde{W}$ whose boundary is identical to
$\partial W$ and whose interior is a slight displacement of that of $W$, tubed
into $S_{W}$. Now we consider a new pseudo-isotopy determined by the pair
$(V,\widetilde{W})$. It gives rise to a self-diffeomorphism $f$ of $S^{4}$.
Since $V$ intersects $\widetilde{W}$ in a point, there is no immediate way to
trivialise this pseudo-isotopy.

######  Conjecture 7.3.

The diffeomorphism $f\colon S^{4}\to S^{4}$ is not smoothly isotopic to the
identity.

Note that the interiors of $W$ and $\widetilde{W}$ are disjoint and
$W\cup\widetilde{W}=0\in\pi_{2}(S^{2}\times S^{2})$. Thus if the answer to 7.1
is affirmative, then $f$ would be smoothly isotopic to the identity, and 7.3
would be false.

## References