# Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins

David T Gay

###### Abstract

One way to better understand the smooth mapping class group of the $4$–sphere
would be to give a list of generators in the form of explicit diffeomorphisms
supported in neighborhoods of submanifolds, in analogy with Dehn twists on
surfaces. As a step in this direction, we describe a surjective homomorphism
from a group associated to loops of $2$–spheres in $S^{2}\times S^{2}$’s onto
this smooth mapping class group, discuss two natural and in some sense
complementary subgroups of the domain of this homomomorphism, show that one is
in the kernel, and give generators as above for the image of the other. These
generators are described as twists along Montesinos twins, i.e. pairs of
embedded $2$–spheres in $S^{4}$ intersecting transversely at two points.

##  1 Introduction

Given a smooth oriented manifold $X$, let $\mathop{Diff^{+}}(X)$ be the space
of orientation preserving diffeomorphisms of $X$ (fixed on a collar
neighborhood of $\partial X$ if $\partial X\neq\emptyset$). Here, inspired
heavily by Watanabe’s work [10] on homotopy groups of
$\mathop{Diff^{+}}(B^{4})$ and Budney and Gabai’s work [2] on knotted
$3$–balls in $S^{4}$, we initiate a study of
$\pi_{0}(\mathop{Diff^{+}}(S^{4}))$, i.e. the smooth mapping class group of
the $4$–sphere. We know very little about this group except that it is abelian
and that every orientation preserving diffeomorphism of $S^{4}$ is
pseudoisotopic to the identity; the group could very well be trivial, like the
topological mapping class group. Ideally we would like to find a generating
set for this mapping class group defined explicitly and geometrically, for
example as explicit diffeomorphisms supported in neighborhoods of explicit
submanifolds of $S^{4}$, in analogy with Dehn twists as generators of the
mapping class groups of surfaces. In this paper we construct a surjective
homomorphism from a limit of fundamental groups of certain embedding spaces of
$2$–spheres in $4$–manifolds onto $\pi_{0}(\mathop{Diff^{+}}(S^{4}))$, we
describe one geometrically natural subgroup of the domain of this homomorphism
which we show to be in its kernel, and we describe a “complementary”
geometrically natural subgroup and give an explicit list of generators as
above for its image in $\pi_{0}(\mathop{Diff^{+}}(S^{4}))$.

Given smooth manifolds $X$ and $Y$, let $\mathop{Emb}(X,Y)$ denote the space
of embeddings of $X$ into $Y$. Let $\\#^{n}(S^{2}\times S^{2})^{\dagger}$
refer to a punctured $\\#^{n}(S^{2}\times S^{2})$. (The puncture is not
important until the next paragraph.) Let
$\mathcal{S}_{n}=\mathop{Emb}(\amalg^{n}S^{2},\\#^{n}(S^{2}\times
S^{2})^{\dagger})$. This $\mathcal{S}$ stands for “spheres”, as in “space of
embeddings of collections of spheres”. Fix a point $p\in S^{2}$ and let
$\amalg^{n}(S^{2}\times\\{p\\})\subset\\#^{n}(S^{2}\times S^{2})^{\dagger}$
denote the union of one copy of $S^{2}\times\\{p\\}$ in each $S^{2}\times
S^{2}$ summand of $\\#^{n}(S^{2}\times S^{2})^{\dagger}$. This will be our
basepoint in $\mathcal{S}_{n}$, which we will often suppress from our
notation, with the understanding that $\mathcal{S}_{n}$ is a pointed space. We
will also be interested in two subspaces of $\mathcal{S}_{n}$: Let
$\mathcal{S}^{0}_{n}$ denote the subspace of embeddings with the property that
for each $i$ and $j$ the $i$’th component of $\amalg^{n}S^{2}$ intersects the
$\\{p\\}\times S^{2}$ in the $j$’th summand of $\\#^{n}(S^{2}\times
S^{2})^{\dagger}$ transversely at $\delta_{ij}$ points. Let
$\widehat{\mathcal{S}}_{n}$ denote the subspace of embeddings with the
property that the image of $\amalg^{n}S^{2}$ is disjoint from
$\amalg^{n}(S^{2}\times\\{p^{\prime}\\})$ for some fixed $p^{\prime}\neq p\in
S^{2}$. Note that our basepoint lies in both of these subspaces.

There is a natural homomorphism
$\mathcal{H}_{n}:\pi_{1}(\mathcal{S}_{n})\to\pi_{0}(\mathop{Diff^{+}}(S^{4}))$
defined as follows, and with more care in the next section: Given a loop of
embeddings $\alpha_{t}:\amalg^{n}S^{2}\hookrightarrow\\#^{n}(S^{2}\times
S^{2})^{\dagger}\subset\\#^{n}(S^{2}\times S^{2})$ representing some
$a\in\pi_{1}(\mathcal{S}_{n})$, for each $t$ build a $5$–dimensional cobordism
by attaching first $n$ $5$–dimensional $2$–handles to $[0,1]\times S^{4}$ with
some standard fixed attaching data in $\\{1\\}\times S^{4}$, so that the top
boundary is canonically identified with $\\#^{n}(S^{2}\times S^{2})$, and then
attach $n$ $5$–dimensional $3$–handles using $\alpha_{t}$ as the attaching
map. Varying $t$, we get a bundle over $S^{1}$ of $5$–dimensional cobordisms;
since attaching $3$–handles along the basepoint embedding
$\amalg^{n}(S^{2}\times\\{p\\})$ canonically cancels the $2$–handles, the
bundle of cobordisms is a cobordism from $S^{1}\times S^{4}$ to an interesting
$S^{4}$–bundle over $S^{1}$. We define $\mathcal{H}_{n}(a)$ to be the
monodromy of this bundle.

Thanks to the puncture, we have a natural basepoint-preserving inclusion
$\jmath:\mathcal{S}_{n}\hookrightarrow\mathcal{S}_{n+1}$, respecting the
inclusions of $\mathcal{S}^{0}_{n}$ and $\widehat{\mathcal{S}}_{n}$, and thus
inclusion-induced homomorphisms
$\jmath_{*}:\pi_{1}(\mathcal{S}_{n})\to\pi_{1}(\mathcal{S}_{n+1})$ which
commute with $\mathcal{H}_{n}$ and $\mathcal{H}_{n+1}$ and with the inclusion-
induced homomorphisms
$\imath_{*}:\pi_{1}(\mathcal{S}^{0}_{n})\to\pi_{1}(\mathcal{S}_{n})$ and
$\imath_{*}:\pi_{1}(\widehat{\mathcal{S}}_{n})\to\pi_{1}(\mathcal{S}_{n})$. As
a consequence, we have limit groups which we will denote
$\pi_{1}(\mathcal{S}_{\infty})$, $\pi_{1}(\mathcal{S}^{0}_{\infty})$,
$\pi_{1}(\widehat{\mathcal{S}}_{\infty})$ (it is not important for us to think
about the limiting spaces, just the groups, but this notation is convenient),
limiting inclusion-induced homomorphisms between them, and a limit
homomorphism
$\mathcal{H}_{\infty}:\pi_{1}(\mathcal{S}_{\infty})\to\pi_{0}(\mathop{Diff^{+}}(S^{4}))$.
Our first result is:

######  Theorem 1.

The homomorphism
$\mathcal{H}_{\infty}:\pi_{1}(\mathcal{S}_{\infty})\to\pi_{0}(\mathop{Diff^{+}}(S^{4}))$
is surjective and the kernel of $\mathcal{H}_{\infty}$ contains
$\imath_{*}(\pi_{1}(\mathcal{S}^{0}_{\infty}))$.

Our second result characterizes
$\mathcal{H}_{\infty}(\imath_{*}(\pi_{1}(\widehat{\mathcal{S}}_{\infty})))$ in
terms of a countable list of explicit generators which we will now describe.
This could in principle be all of $\pi_{0}(\mathop{Diff^{+}}(S^{4}))$,
although we have no evidence for or against that possibility.

A Montesinos twin in $S^{4}$ is a pair $W=(R,S)$ of embedded $2$–spheres
$R,S\subset S^{4}$ intersecting transversely at two points. For us, the
$2$–spheres are both oriented. Montesinos shows [8] that the boundary of a
neighborhood $\nu(W)$ of $R\cup S$ is diffeomorphic to $S^{1}\times
S^{1}\times S^{1}$ and that in fact there is a canonical parametrization
$S^{1}_{l}\times S^{1}_{R}\times S^{1}_{S}\cong\partial\nu(W)$, canonical up
to postcomposing with diffeomorphisms of $\partial\nu(W)$ which are isotopic
to the identity and precomposing with independent diffeomorphisms of
$S^{1}_{l}$, $S^{1}_{R}$ and $S^{1}_{S}$. This parametrization is
characterized by $S^{1}_{l}\times\\{b\\}\times\\{c\\}$ being homologically
trivial in $H_{1}(S^{4}\setminus(R\cup S))$, i.e. a “longitude”,
$\\{a\\}\times S^{1}_{R}\times\\{c\\}$ being a meridian for $R$, and
$\\{a\\}\times\\{b\\}\times S^{1}_{S}$ being a meridian for $S$. This then
parametrizes a neighborhood of $\partial\nu(W)$ as $[-1,1]\times
S^{1}_{l}\times S^{1}_{R}\times S^{1}_{S}$. We adopt the orientation
conventions that $S^{1}_{R}$ and $S^{1}_{S}$ have the standard meridian
orientations coming from the orientations of $R$ and $S$, that $[-1,1]$ is
oriented in the outward direction from $\nu(W)$, and that $S^{1}_{l}$ is
oriented so that the orientation of $[-1,1]\times S^{1}_{l}\times
S^{1}_{R}\times S^{1}_{S}$ agrees with the standard orientation of $S^{4}$.

######  Definition 2.

Given a Montesinos twin $W$ in $S^{4}$, parametrize a neighborhood of
$\partial\nu(W)$ as $[-1,1]\times S^{1}_{l}\times S^{1}_{R}\times S^{1}_{S}$
as above. Let $\tau_{l}:[-1,1]\times S^{1}_{l}\to[-1,1]\times S^{1}_{l}$
denote a right-handed Dehn twist. The twin twist along $W$, denoted
$\tau_{W}$, is the diffeomorphism of $S^{4}$ which is the identity outside
this neighborhood of $\partial\nu(W)$ and is equal to
$\tau_{l}\times\mathop{\rm id}\nolimits_{S^{1}_{R}}\times\mathop{\rm
id}\nolimits_{S^{1}_{S}}$ inside this neighborhood.

By the canonicity of our parametrization, $\tau_{W}$ is well-defined up to
isotopy, i.e. $[\tau_{W}]$ is a well-defined class in
$\pi_{0}(\mathop{Diff^{+}}(S^{4}))$. Incidentally, we have the following as a
consequence of our orientation conventions:

######  Lemma 3.

If $W=(R,S)$ is a Montesinos twin, then
$[\tau_{W}]^{-1}=[\tau_{(S,R)}]=[\tau_{(\overline{R},S)}]=[\tau_{(R,\overline{S})}]$.

###### Proof.

Either switching the spheres $S$ and $R$, or reversing the orientation of one
of them, reverses the orientation of $S^{1}_{R}\times S^{1}_{S}$, which then
forces the reversal of the orientation of $[-1,1]\times S^{1}_{l}$, changing a
positive Dehn twist to a negative Dehn twist. ∎

We now describe a family of twins $W(i)=(R(i),S)$, for
$i\in\mathbb{N}\cup\\{0\\}$. Figure 1 illustrates $W(3)$; $W(i)$ is the same
but with $i$ turns in the spiral rather than $3$ turns. Figure 2 and 3 give
two alternate descriptions of this twin. Orientations are not made explicit
here since our main claim is simply that the twists invoved generate a certain
group, and the inverse of a generator is still a generator. Note that both
$R(i)$ and $S$ are individually unknotted $2$–spheres, and that the twin
$W(0)$ is the trivial “unknotted twin”.

2pt $R(3)$ [r] at 2 125 $S$ [l] at 260 106

Figure 1: An illustration of $W(3)=(R(3),S)$. The picture mostly happens in
the slice
$\\{t=0\\}\subset\mathbb{R}^{4}=\\{(x,y,z,t)\\}\subset\mathbb{R}^{4}\cup\\{\infty\\}=S^{4}$.
The ring labelled $S$ is a slice through $S$, which shrinks to a point as we
move forwards and backwards in the “time” coordinate $t$. The “snake whose
tail passes through his head” is $R(3)$, which is projected onto $\\{t=0\\}$,
intersecting itself along one circle in the middle of the red disk (the
“snake’s left ear hole”) and along another circle in the middle of the blue
disk (the “right ear hole”). Blue and red indicate that these disks are pushed
slightly forwards (blue) and backwards (red) in time to resolve these
intersections; otherwise $R(3)$ lies in the slice $\\{t=0\\}$. The bright pink
dot where the ring pierces the tail is the positive point of intersection of
$R(3)$ and $S$; the faint green dot on the back side of the tail is the
negative intersection point. Figure 2: Another illustration of
$W(3)=(R(3),S)$. Here we have drawn an immersed pair of disks, one green and
one pink, with mostly ribbon intersections except for one nonribbon arc.
Pushing these two disks into $\mathbb{R}^{4}_{+}$ or $\mathbb{R}^{4}_{-}$ and
resolving ribbon intersections in the usual way gives two embedded disks
intersecting each other transversely once, and then taking one copy in
$\mathbb{R}^{4}_{+}$ and one in $\mathbb{R}^{4}_{-}$ glued along their common
boundary, i.e. doubling the ribbon disks, gives $R(3)$ (green) and $S$ (pink)
in $\mathbb{R}^{4}\subset S^{4}$. Figure 3: Yet another illustration of
$W(3)$. Here we have drawn two disjoint, embedded $2$–spheres in $S^{4}$ (the
two thick circles, becoming $2$–spheres when shrunk to points forwards and
backwards in time) and an arc connecting them. Push a finger from one of the
spheres out along this arc and then do a finger move when you encounter the
other sphere, creating a pair of transverse intersections, and the result is
$W(3)$.

Our second result is:

######  Theorem 4.

The subgroup
$\mathcal{H}_{\infty}(\imath_{*}(\pi_{1}(\widehat{\mathcal{S}}_{\infty})))$ of
$\pi_{0}(\mathop{Diff^{+}}(S^{4}))$ is generated by the twin twists
$\\{\tau_{W(i)}\mid i\in\mathbb{N}\\}$.

(In fact these automorphisms $\tau_{W(i)}$ are also examples of the “barbell
maps” discussed in [2]; readers familiar with barbell maps should be able to
use the description of $W(i)$ in Figure 3 to see the connection.)

######  Question 5.

Is
$[\tau_{W}]\in\mathcal{H}_{\infty}(\imath_{*}(\pi_{1}(\widehat{\mathcal{S}}_{\infty})))$
for an arbitrary Montesinos twin? More generally, given any embedding of
$S^{1}\times\Sigma\hookrightarrow S^{4}$, for closed surface $\Sigma$, a
tubular neighborhood gives an embedding of $[-1,1]\times
S^{1}\times\Sigma\hookrightarrow S^{4}$ which gives a diffeomorphism
$\tau\times\mathop{\rm id}\nolimits_{\Sigma}$, where $\tau$ is the Dehn twist
on $[-1,1]\times S^{1}$. Are such diffeomorphisms in
$\mathcal{H}_{\infty}(\imath_{*}(\pi_{1}(\widehat{\mathcal{S}}_{\infty})))$?

One could try to answer these questions either through Cerf theory, by
explicitly identifying a pseudoisotopy from a given diffeomorphism of $S^{4}$
to the identity, and then extracting a loop of attaching spheres for
$5$–dimensional $2$–handles, or one could try to work explicitly with the
diffeomorphisms in $S^{4}$ and try to find relationships amongst such twists,
to relate them to twists along our standard Montesinos twins $W(i)$.

The bigger questions are the following, with affirmative answers to both
showing that the smooth mapping class group of $S^{4}$ is trivial:

######  Question 6.

Is $\mathcal{H}_{\infty}(\imath_{*}(\pi_{1}(\widehat{\mathcal{S}}_{\infty})))$
trivial?

Theorem 4 could help prove this if one can exhibit explicit isotopies from
$\tau_{W(i)}$ to $\mathop{\rm id}\nolimits_{S^{4}}$.

######  Question 7.

Is
$\mathcal{H}_{\infty}(\imath_{*}(\pi_{1}(\widehat{\mathcal{S}}_{\infty})))=\pi_{0}(\mathop{Diff^{+}}(S^{4}))$?

Since $\imath_{*}(\pi_{1}(\mathcal{S}^{0}_{\infty}))$ is in the kernel of
$\mathcal{H}_{\infty}$, we know that $\mathcal{H}_{\infty}$ factors through
the quotient map
$\pi:\pi_{1}(\mathcal{S}_{\infty})\to\pi_{1}(\mathcal{S}_{\infty})/\langle\imath_{*}(\pi_{1}(\mathcal{S}^{0}_{\infty}))\rangle$,
where $\langle H\rangle$ denotes the normal closure of a subgroup $H$. Thus
one way to show that
$\mathcal{H}_{\infty}(\imath_{*}(\pi_{1}(\widehat{\mathcal{S}}_{\infty})))=\pi_{0}(\mathop{Diff^{+}}(S^{4}))$
would be to show that
$\pi\circ\imath:\pi_{1}(\widehat{\mathcal{S}}_{\infty})\to\pi_{1}(\mathcal{S}_{\infty})/\langle\imath_{*}(\pi_{1}(\mathcal{S}^{0}_{\infty}))\rangle$
is surjective. On the other hand this does not need to be true for the answer
to this question to be “yes”, since the kernel of $\mathcal{H}_{\infty}$ could
presumably be much larger than
$\langle\imath_{*}(\pi_{1}(\mathcal{S}^{0}_{\infty}))\rangle$.

In the next section we elaborate on the connection between loops of embeddings
of certain spheres and self-diffeomorphisms of other spheres, setting up the
general theory in various dimensions and codimensions. After that, we devote
one section to the proof of Theorem 1, and we break the proof of Theorem 4
into the three remaining sections.

The author would like to thank Bruce Bartlett, Sarah Blackwell, Mike Freedman,
Robert Gompf, Jason Joseph, Danica Kosanovic, Peter Lambert-Cole, Gordana
Matic, Benjamin Ruppik, Rob Schneiderman, Peter Teichner and Jeremy Van Horn-
Morris for helpful conversations along the way, and most especially Hannah
Schwartz for pointing out Montesinos’s work and Dave Gabai for initial
inspiration and for pointing out the mistake in the first version that claimed
far too much, and both Hannah and Dave for many clarifying conversations about
loops of spheres.

Much of this work was carried out during the author’s year at the Max Planck
Institute for Mathematics in Bonn, and thus the author gratefully acknowledges
the institute’s support in the form of the 2019-2020 Hirzebruch Research
Chair. This work was also supported by individual NSF grant DMS-2005554 and
NSF Focused Research Group grant DMS-1664567.

##  2 From loops of spheres to diffeomorphisms

Here we describe the homomorphism
$\mathcal{H}_{\infty}:\pi_{1}(\mathcal{S}_{\infty})\to\pi_{0}(\mathop{Diff^{+}}(S^{4}))$
as a special case of a more general family of homomorphisms turning loops of
framed embedded spheres in various dimensions into bundles of cobordisms and
hence into self-diffeomorphisms of smooth manifolds. This should mostly be
standard “Cerf theory”. In the introduction above, we had $2$–spheres embedded
in $4$–manifolds, but we did not mention framings of these $2$–spheres. Below,
we will work with framed spheres and then later when we relate this back to
the terminology of the introduction, we will see where the framings come from.

Fix an $m$–manifold $X$, for some $m\geq 2$, fix integers $0<k<m$ and $n\geq
0$, and let $\\#^{n}(S^{k}\times S^{m-k})^{\dagger}$ denote a punctured
$\\#^{n}(S^{k}\times S^{m-k})$. As in the introduction, the puncture is needed
so that we can view $\\#^{n}(S^{k}\times S^{m-k})^{\dagger}$ as a subspace of
$\\#^{n+1}(S^{k}\times S^{m-k})^{\dagger}$. Now consider the following space
of embeddings of collections of framed spheres:

$\mathcal{FS}_{n}(X,k)=\mathop{Emb}(\amalg^{n}(S^{k}\times
B^{m-k}),X\\#(\\#^{n}(S^{k}\times S^{m-k})^{\dagger}))$

Picking a fixed point $p\in S^{m-k}$ and a disk neighborhood $U$ of $p$
parametrized as $B^{m-k}$, we get a natural basepoint $\amalg^{n}(S^{k}\times
U)\subset X\\#(\\#^{n}(S^{k}\times S^{m-k})^{\dagger})$ for
$\mathcal{FS}_{n}(X,k)$, which we will again generally suppress in our
notation, understanding that $\mathcal{FS}_{n}(X,k)$ is a pointed space.

Then we get a homomorphism

$\mathcal{FH}_{n}:\pi_{1}(\mathcal{FS}_{n}(X,k))\to\pi_{0}(\mathop{Diff^{+}}(X))$

defined as follows. Represent an element of $\pi_{1}(\mathcal{FS}_{n}(X,k))$
by a $1$–parameter family of embeddings $\beta_{t}:\amalg^{n}(S^{k}\times
B^{m-k})\hookrightarrow X\\#(\\#^{n}(S^{k}\times S^{m-k})^{\dagger})$ with
$\beta_{0}=\beta_{1}=\amalg^{n}(S^{k}\times U)$. We will use this to build a
$(m+2)$–manifold $Z_{\beta}$ which is a cobordism from $S^{1}\times X$ to some
$X$ bundle over $S^{1}$, and the monodromy of this bundle will be
$\mathcal{H}_{n}([\beta_{t}])$. To build $Z_{\beta}$, first let $Y$ be
$[0,1]\times X$ with $n$ $(m+1)$–dimensional $k$–handles attached along $n$
unlinked $0$–framed unknotted $S^{k-1}$’s in a ball in $X$. Thus $Y$ is a
cobordism from $X$ to $X\\#(\\#^{n}(S^{k}\times S^{m-k}))$. Now consider
$S^{1}\times Y$, which is a cobordism from $S^{1}\times X$ to $S^{1}\times
X\\#(\\#^{n}(S^{k}\times S^{m-k}))$. Identifying $S^{1}$ with $[0,1]/1\sim 0$
and using $t$ as the $S^{1}$–coordinate, we can now use each $\beta_{t}$ as
the attaching map for $n$ $(m+1)$–dimensional $(k+1)$–handles attached to
$\\{t\\}\times X\\#(\\#^{n}(S^{k}\times S^{m-k}))\subset\\{t\\}\times Y$; the
result is our construction of $Z_{\beta}$. Thus $Z_{\beta}$ is a bundle over
$S^{1}$, each fiber of which is a $(m+1)$–dimensional cobordism from $X$ to an
$m$–manifold $X_{t}$, built with $n$ $k$–handles and $n$ $(k+1)$–handles. Note
that the $(k+1)$–handle attached along $\beta_{0}=\beta_{1}$ is the canonical
cancelling handle for the corresponding $k$–handle, and thus $X_{0}$ is
canonically diffeomorphic to $X$. Therefore the full $(m+2)$–dimensional
cobordism $Z_{\beta}$ is a cobordism from $S^{1}\times X$ to some $X$ bundle
over $S^{1}$ with a monodromy which is well-defined by the homotopy class of
the loop of embeddings $\beta_{t}$. This defines $\mathcal{FH}_{n}$ and it is
straightforward to see that $\mathcal{FH}_{n}$ is a group homomorphism.

Thanks to the punctures, we have basepoint preserving inclusions

$\ldots\subset\mathcal{FS}_{n}(X,k)\subset\mathcal{FS}_{n+1}(X,k)\subset\ldots$

and thus induced maps on $\pi_{1}$ and thus a direct limit

$\ldots\to\pi_{1}(\mathcal{FS}_{n}(X,k))\to\pi_{1}(\mathcal{FS}_{n+1}(X,k))\to\ldots\to\pi_{1}(\mathcal{FS}_{\infty}(X,k))$

Again, we do not really care about the limiting spaces, just the groups. Thus
one should think of an element of $\pi_{1}(\mathcal{FS}_{\infty}(X,k))$ as an
equivalence class of loops in some $\mathcal{FS}_{n}(X,k)$, where two such
loops are equivalent if they become homotopic after including into some
$\mathcal{FS}_{N}(X,k)$ for some $N\geq n$. It is not hard to see that these
induced maps on $\pi_{1}$ commute with the $\mathcal{FH}_{n}$ homomorphisms,
so that finally we get the limit homomorphism

$\mathcal{FH}_{\infty}:\pi_{1}(\mathcal{FS}_{\infty}(X,k))\to\pi_{0}(\mathop{Diff^{+}}(X))$

As in the unframed setting of the introduction, we have two natural subspaces
of $\mathcal{FS}_{n}(X,k)$: Let $\mathcal{FS}^{0}_{n}(X,k)$ denote those
embeddings of $\amalg^{n}(S^{k}\times B^{m-k}$) into $X\\#(\\#^{n}(S^{k}\times
S^{m-k}))$ for which the $S^{k}\times\\{0\\}$ in the $i$’th $S^{k}\times
B^{m-k}$ transversely intersects the $\\{p\\}\times S^{m-k}$ in the $j$’th
$S^{k}\times S^{m-k}$ summand transversely at $\delta_{ij}$ points. Let
$\widehat{\mathcal{FS}}_{n}$ denote the subspace of embeddings with the
property that the image of $\amalg^{n}(S^{k}\times B^{m-k})$ is disjoint from
$\amalg^{n}(S^{k}\times\\{p^{\prime}\\})$ for some fixed $p^{\prime}\in
S^{m-k}\setminus U$. Note that our basepoint lies in both of these subspaces.

It is standard that $\imath_{*}(\pi_{1}(\mathcal{FS}^{0}_{n}(X,k)))$ is in the
kernel of $\mathcal{FH}_{n}$, because when the $S^{k}\times\\{0\\}$’s are dual
to the $\\{p\\}\times S^{m-k}$’s for all $t$ in a loop of embeddings
$\alpha_{t}$, then for all $t$ the $k$–handles and $(k+1)$–handles cancel
“uniquely” (this is Cerf’s l’unicité de mort [3]). Thus the cobordism
$Z_{\alpha}$ becomes a bundle over $S^{1}$ with each fiber supporting a Morse
function without critical points, so that the monodromy at the top is
$\mathop{\rm id}\nolimits_{X}$.

Now we discuss the relationship to spaces of spheres without framings. Let

$\mathcal{S}_{n}(X,k)=\mathop{Emb}(\amalg^{n}S^{k},X\\#(\\#^{n}(S^{k}\times
S^{m-k})^{\dagger}))$

There is an obvious “framing forgetting” map of pointed spaces

$\mathcal{F}:\mathcal{FS}_{n}(X,k)\to\mathcal{S}_{n}(X,k)$

given by restricting an embedding of $S^{k}\times B^{m-k}$ to
$S^{k}=S^{k}\times\\{0\\}$. Palais [9] shows that such maps are locally
trivial and thus satisfy the homotopy lifting property. The fiber of
$\mathcal{F}$ is the space of framings of a fixed $S^{k}$, i.e. (up to
homotopy) maps from $S^{k}$ to $SO(m-k)$. Note that the fiber over the
basepoint is actually a subspace of $\mathcal{FS}^{0}_{n}(X,k)$ and thus
$\pi_{1}$ of the fiber lands in the kernel of $\mathcal{FH}_{n}$. As a
consequence, even though

$\mathcal{F}_{*}:\pi_{1}(\mathcal{FS}_{n}(X,k))\to\pi_{1}(\mathcal{S}_{n}(X,k))$

may not be injective, if the fiber is not simply connected, we still have that
$\mathcal{FH}_{n}$ induces a well-defined homomorphism $\mathcal{H}_{n}$ from
the image $\mathcal{F}_{*}(\pi_{1}(\mathcal{FS}_{n}(X,k)))$ of
$\mathcal{F}_{*}$ in $\pi_{1}(\mathcal{S}_{n}(X,k))$ to
$\pi_{0}(\mathop{Diff^{+}}(S^{m})$. All of this also commutes with the
inclusion maps from $n$ to $n+1$ giving $\mathcal{H}_{\infty}$.

When $m=4$ and $k=2$, the fibers of $\mathcal{F}$ are connected, i.e. a sphere
in a $4$–manifold with self-intersection $0$ has only one framing up to
isotopy (because $\pi_{2}(SO(2))=0$). Therefore

$\mathcal{F}_{*}:\pi_{1}(\mathcal{FS}_{n}(S^{4},2))\to\pi_{1}(\mathcal{S}_{n}(S^{4},2)=\mathcal{S})$

is surjective and so $\mathcal{FH}_{n}$ and $\mathcal{FH}_{\infty}$ induce
homomorphisms $\mathcal{H}_{n}$ and

$\mathcal{H}_{\infty}:\pi_{1}(\mathcal{S}_{\infty})\to\pi_{0}(\mathop{Diff^{+}}(S^{m})$

##  3 Surjectivity of $\mathcal{H}_{\infty}$

In this section we will prove Theorem 1, i.e. that
$\mathcal{H}_{\infty}:\pi_{1}(\mathcal{S}_{\infty})\to\pi_{0}(\mathop{Diff^{+}}(S^{4}))$
is surjective. (We have already seen in the previous section that the kernel
of $\mathcal{H}_{\infty}$ contains
$\imath_{*}(\pi_{1}(\mathcal{S}^{0}_{\infty}))$.) Since $\mathcal{H}_{\infty}$
is a limit of maps $\mathcal{H}_{n}$, what we are really trying to prove is
the following:

######  Theorem 8.

For any orientation preserving diffeomorphism $\phi:S^{4}\to S^{4}$, there is
some $n\in\mathbb{N}$ and some loop $\alpha_{t}$ in $\mathcal{S}_{n}$ based at
the basepoint $\amalg^{n}(S^{2}\times\\{p\\})$, such that
$\mathcal{H}_{n}([\alpha_{t}])=[\phi]\in\pi_{0}(\mathop{Diff^{+}}(S^{4})$.

In other words, once we prove this theorem then we have:

###### Proof of Theorem 1.

This is an immediate corollary to Theorem 8 and the discussion in the previous
section. ∎

In fact, to make things more concrete, we will prove this:

######  Theorem 9.

For any orientation preserving diffeomorphism $\phi:S^{4}\to S^{4}$, there is
a $6$–manifold $Z$ with the following properties:

  1. 1.

$Z$ is a bundle over $S^{1}$ and is a cobordism from $S^{1}\times S^{4}$ to
$S^{1}\times_{\phi}S^{4}=([0,1]\times S^{4})/(1,p)\sim(0,\phi(p))$, with the
three fibrations over $S^{1}$, of $Z$, $S^{1}\times S^{4}$ and
$S^{1}\times_{\phi}S^{4}$, being compatible.

  2. 2.

For each $t\in S^{1}$, the fiber $Y_{t}$ over $t$ in $Z$, being a
$5$–dimensional cobordism from $S^{4}$ to a $4$–manifold $X_{t}$ which is
diffeomorphic to $S^{4}$, comes with a handle decomposition with $n$
$2$–handles and $n$ $3$–handles.

  3. 3.

The $n$ $2$–handles for each $Y_{t}$ are attached along a fixed standard
collection of $n$ standardly framed $S^{1}$’s in $S^{4}$, so that the
$4$–manifold immediately above these $2$–handles is canonically identified
with $\\#^{n}(S^{2}\times S^{2})$

  4. 4.

The $n$ $3$–handles for each $Y_{t}$ are attached along a moving family of $n$
framed $S^{2}$’s in $\\#^{n}(S^{2}\times S^{2})$ such that, ignoring the
framings, these $S^{2}$’s are given by a loop of embeddings of
$\amalg^{n}S^{2}$ into $\\#^{n}(S^{2}\times S^{2})$ all missing a single fixed
point and starting and ending at the standard embedding
$\amalg^{n}(S^{2}\times\\{p\\})$, i.e. a based loop $\alpha_{t}$ in
$\mathcal{S}_{n}$.

Then:

###### Proof of Theorem 8.

This is an immediate corollary of Theorem 9, since the homomorphism
$\mathcal{H}_{n}$ comes precisely from constructions of cobordisms as in
Theorem 9. ∎

We will use Cerf theoretic techniques, beginning with a pseudoisotopy.

######  Lemma 10.

Every orientation preserving self-diffeomorphism of $S^{4}$ is pseudoisotopic
to the identity.

###### Proof.

Consider an orientation preserving diffeomorphism $\phi:S^{4}\to S^{4}$. Let
$f:S^{5}\to\mathbb{R}$ be projection onto the last coordinate in
$\mathbb{R}^{6}$, and for any interval $I\subset\mathbb{R}$, let
$S^{5}_{I}=f^{-1}(I)$. Let $V$ be a smooth vector field on
$S^{5}\setminus\\{(0,0,0,0,0,\pm 1)\\}$ which is orthogonal to all level sets
of $f$ and scaled so that $df(V)=1$. Let
$X=S^{5}_{[-1,0]}\cup_{\phi}S^{5}_{[0,1]}$, where $\phi:\partial
S^{5}_{[0,1]}=S^{4}\to-S^{4}=\partial S^{5}_{[-1,0]}$ is now seen as an
orientation reversing gluing diffeomorphism. Arrange the gluing (i.e. the
smooth structure on $X$) so that the vector field $V$ on the two halves of $X$
is still a smooth vector field on $X$, which we call $V_{X}$. Note that $X$
also inherits the Morse function $f$, which we label $f_{X}:X\to\mathbb{R}$,
and we can use the same notation $X_{I}=f_{X}^{-1}(I)\subset X$. The point is
that if $I\subset(-\infty,0]$ or if $I\subset[0,\infty)$ then
$X_{I}=S^{5}_{I}$, i.e. they are actually equal sets, not just diffeomorphic
manifolds.

Now note that $X$ is homotopy equivalent to $S^{5}$ and therefore [6, 7]
diffeomorphic to $S^{5}$. For some small $\epsilon>0$ we can assume that the
diffemorphism $\Phi:S^{5}\to X$ is the identity on
$S^{5}_{[-1,-1+\epsilon]}=X_{[-1,-1+\epsilon]}$ and on
$S^{5}_{[1-\epsilon,1]}=X_{[1-\epsilon,1]}$. Using flow along $V$ and $V_{X}$,
respectively, and the standard identification of $\partial
S^{5}_{[-1,-1+\epsilon]}$ with $S^{4}$, we can parametrize both
$S^{5}_{[-1+\epsilon,1-\epsilon]}$ and $X_{[-1+\epsilon,1-\epsilon]}$ as
$[-1+\epsilon,1-\epsilon]\times S^{4}$ and then $\Phi$ restricts to give a
diffeomorphism from $[-1+\epsilon,1-\epsilon]\times S^{4}$ to itself.
Furthermore, this map is the identity on $\\{-1+\epsilon\\}\times S^{4}$ and
by continuity must equal $\phi$ on $\\{1-\epsilon\\}\times S^{4}$. After
reparametrizing $[-1+\epsilon,1-\epsilon]$ as $[0,1]$ we get the desired
pseudoisotopy.

∎

Now we begin the Cerf theory.

###### Proof of Theorem 9.

Given the orientation preserving diffeomorphism $\phi:S^{4}\to S^{4}$, let
$\Phi:[0,1]\times S^{4}\to[0,1]\times S^{4}$ be a pseudoisotopy from the
identity to $S^{4}$, i.e. $\Phi(0,p)=(0,p)$ and $\Phi(1,p)=(1,\phi(p))$. Let
$f_{0}:[0,1]\times S^{4}\to[0,1]$ be projection onto the first factor, let
$V_{0}$ be the unit vector field on $[0,1]\times S^{4}$ in the $[0,1]$
direction, let
$(f_{1},V_{1})=\Phi^{*}(f_{0},V_{0})=(f_{0}\circ\Phi,D\Phi^{-1}(V_{0}))$, and
let $(f_{t},V_{t})$ be a generic homotopy of Morse functions with gradient-
like vector fields from $(f_{0},V_{0})$ to $(f_{1},V_{1})$. Hatcher and
Wagoner (Chapter VI, Proposition 3, page 214 of [5]) show that this family of
functions with gradient-like vector fields can be homotoped rel
$t\in\\{0,1\\}$ so as to arrange the following properties (see Figure 4):

2pt $2$ [tr] at 45 20 $2$ [tr] at 49 24 $2$ [tr] at 56 31 $3$ [br] at 45 67
$3$ [br] at 49 63 $3$ [br] at 56 55

Figure 4: A nice Cerf graphic with only critical points of index $2$ and $3$.

  * •

The only Morse critical points of any $f_{t}$ are critical points of index $2$
and $3$.

  * •

None of the functions $f_{t}$ have two critical points mapping to the same
critical value.

  * •

All critical points of index $2$ stay below all critical points of index $3$.
In other words, for every $t$, if $p$ is a critical point of index $2$ for
$f_{t}$ and $q$ is a critical point of index $3$ for $f_{t}$ then
$f_{t}(p)<f_{t}(q)$.

  * •

There are no handle slides. In other words, none of the vector fields $V_{t}$
have flow lines between critical points of the same index.

  * •

All births of cancelling pairs of critical points happen before all deaths of
cancelling pairs, and no two births or deaths happen at the same time.

  * •

At the moments of birth and death, the handle pair dying or being born does
not run over any other handles. In other words, there are no $V_{t}$ flow
lines between a non-Morse birth/death critical point and any other critical
point.

Now consider the $6$–manifold $Z=[0,1]\times[0,1]\times
S^{4}/(1,s,p)\sim(0,\Phi(s,p))$. We will show that this has the properties
advertised in the statement of Theorem 9. Currently we have the first
property, namely that $Z$ is a bundle over $S^{1}$ and is a cobordism from
$S^{1}\times S^{4}$ to $S^{1}\times_{\phi}S^{4}=([0,1]\times
S^{4})/(1,p)\sim(0,\phi(p))$, with the three fibrations over $S^{1}$, of $Z$,
$S^{1}\times S^{4}$ and $S^{1}\times_{\phi}S^{4}$, being compatible. Each
fiber $Y_{t}$ of $Z\to S^{1}$ gets a handle decomposition from $(f_{t},V_{t})$
(allowing for birth and death handle decompositions) but it will take some
work to arrange that these satisfy the remaining properties as stated in
Theorem 9. First let us characterize each fiber $Y_{t}$, with its given handle
decomposition, as much as possible in terms of handle attaching data in
$\\{t\\}\times S^{4}\subset S^{1}\times S^{4}$.

Suppose that the births in our Morse functions $f_{t}$ occur at times
$0<t_{1}<t_{2}<\ldots<t_{n}<1/4$ and that the deaths occur at times
$3/4<t^{\prime}_{n}<\ldots<t^{\prime}_{2}<t^{\prime}_{1}<1$. We will show how
each cobordism $Y_{t}$ can be described as built from
$\\{t\\}\times[0,1]\times S^{4}$ according to handle attaching data, and to do
this we establish a few conventions. First, all of our embeddings of spheres
and disks of various dimensions are framed embeddings, but to keep the
terminology minimal we will sometimes suppress mention of framings. Later we
will make sure to carefully relate this back to the case of loops of unframed
spheres. Second, given a framed embedding $\epsilon$ of a sphere in a manifold
$X$, let $X(\epsilon)$ denote the result of surgering $X$ along $\epsilon$.
Third, a framed embedding $\delta$ of a disk $D^{k}$ into an $n$–manifold $X$
gives us several auxiliary pieces of information: We get a framed embedding
$\alpha$ of $S^{k-1}=\partial D^{k}$ into $X$. We get the surgered manifold
$X(\alpha)$. And, in $X(\alpha)$, there is a natural framed embedding $\beta$
of $S^{k}$ into $X(\alpha)$ which coincides with $\delta$ on the part of
$X(\alpha)$ away from the surgery, and which coincides with the co-core of the
surgery inside the surgered region. This is the basic model for the result of
attaching a $(n+1)$–dimensional cancelling $k$–$(k+1)$ handle pair to
$[0,1]\times X$, with the attaching data for the pair being described entirely
in $X$ by the framed disk $\delta$. Thus we also see that $X(\alpha)(\beta)$
is canonically identified with $X$. Here we will work with the case $k=2$ and
$k+1=3$, but later in the paper we will be interested in the case $k=1$ and
$k+1=2$.

The notation introduced above also makes sense when the $\delta$’s, $\alpha$’s
or $\beta$’s are embeddings of disjoint unions of disks or spheres. Using
this, we now describe the form of the explicit $t$–varying handle attaching
data that gives the fiberwise construction of $Z$, i.e. the data that shows
how to construct each $Y_{t}$ starting with $\\{t\\}\times[0,1]\times S^{4}$
and attaching various $5$–dimensional handles.

  * •

For $0\leq t<t_{1}$, no handles are attached, i.e.
$Y_{t}=\\{t\\}\times[0,1]\times S^{4}$.

  * •

For $t=t_{1}$, there is a framed embedding $\delta_{1}$ of a disk $D^{2}$ into
$S^{4}$ such that $Y_{t_{1}}$ is built from $\\{t_{1}\\}\times[0,1]\times
S^{4}$ by attaching a cancelling pair of a $5$–dimensional $2$–handle and
$3$–handle, with the $2$–handle attached along the framed embedding
$\alpha^{1}_{t_{1}}=\delta_{1}|_{S^{1}}$, and with the $3$–handle attached
along the resulting framed embedding $\beta^{1}_{t_{1}}$ of $S^{2}$ into
$S^{4}(\alpha^{1}_{t_{1}})$.

  * •

For $t_{1}\leq t\leq t_{2}$, we have a $t$–varying framed embedding
$\alpha_{t}=\alpha^{1}_{t}$ of $S^{1}$ into $S^{4}$ and a $t$–varying framed
embedding $\beta_{t}=\beta^{1}_{t}$ of $S^{2}$ into $S^{4}(\alpha_{t})$. For
$t_{1}\leq t<t_{2}$, $Y_{t}$ is built from $\\{t\\}\times[0,1]\times S^{4}$ by
attaching a $2$–handle along $\alpha_{t}$ and then a $3$–handle along
$\beta_{t}$. At $t=t_{1}$, the $\alpha_{t}$ and $\beta_{t}$ agree with the
$\alpha^{1}_{t}$ and $\beta^{1}_{t}$ from the previous point.

  * •

For $t=t_{2}$, there is a framed embedding $\delta_{2}$ of $D^{2}$ into
$S^{4}$ disjoint from the images of $\alpha^{1}_{t_{2}}$ and
$\beta^{1}_{t_{2}}$, such that $Y_{t_{2}}$ is built from
$\\{t_{2}\\}\times[0,1]\times S^{4}$ by attaching

    * -

first a $5$–dimensional $2$–handle along $\alpha^{1}_{t_{2}}$,

    * -

then a cancelling $2$–$3$–pair in which the $2$–handle is attached along
$\alpha^{2}_{t_{2}}=\delta_{2}|_{S^{1}}$ and the $3$–handle is attached along
the resulting framed $2$–sphere $\beta^{2}_{t_{2}}$ in
$S^{4}(\alpha^{1}_{t_{2}})(\alpha^{2}_{t_{2}})$, and

    * -

then a $3$–handle attached along $\beta^{1}_{t_{2}}$, which can be seen as a
framed $2$–sphere in $S^{4}(\alpha^{1}_{t_{2}})$, in
$S^{4}(\alpha^{1}_{t_{2}})(\alpha^{2}_{t_{2}})$ or in
$S^{4}(\alpha^{1}_{t_{2}})(\alpha^{2}_{t_{2}})(\beta^{2}_{t_{2}})\cong
S^{4}(\alpha^{1}_{t_{2}})$.

  * •

For $t_{2}\leq t\leq t_{3}$, we have a $t$–varying framed embedding
$\alpha_{t}=\alpha^{1}_{t}\amalg\alpha^{2}_{t}$ of $S^{1}\amalg S^{1}$ into
$S^{4}$ and a $t$–varying framed embedding
$\beta_{t}=\beta^{1}_{t}\amalg\beta^{2}_{t}$ of $S^{2}\amalg S^{2}$ into
$S^{4}(\alpha_{t})$, agreeing with the $\alpha^{1}_{t_{2}}$,
$\alpha^{2}_{t_{2}}$, $\beta^{2}_{t_{2}}$ and $\beta^{1}_{t_{2}}$ of the
preceding point when $t=t_{2}$, so that $Y_{t}$ is built from
$\\{t\\}\times[0,1]\times S^{4}$ by attaching $2$–handles along $\alpha_{t}$
and then $3$–handles along $\beta_{t}$.

  * •

This process continues with each birth at time $t_{i}$ governed by a new
framed disk $\delta_{i}$, generating a new framed $S^{1}$ $\alpha^{i}_{t_{i}}$
and a new framed $S^{2}$ $\beta^{i}_{t_{i}}$, which then join the previous
framed spheres to create
$\alpha_{t}=\alpha^{1}_{t}\amalg\ldots\amalg\alpha^{i}_{t}$ and
$\beta_{t}=\beta^{1}_{t}\amalg\ldots\amalg\beta^{i}_{t}$, which are the
attaching spheres for $2$– and $3$–handles for $t_{t}\leq t<t_{i+1}$.

  * •

Reversing time we see the deaths governed by (most likely quite different)
disks $\delta_{n}^{\prime},\ldots,\delta_{1}^{\prime}$ and the same pattern of
framed $S^{1}$’s and $S^{2}$’s in between these times.

  * •

For $t_{n}\leq t\leq t_{n}^{\prime}$, there is a $t$–parametrized family
$\alpha_{t}$ of framed embeddings of $\amalg^{n}S^{1}$ into $S^{4}$ and a
$t$–parametrized family $\beta_{t}$ of framed embeddings of $\amalg^{n}S^{2}$
into $S^{4}(\alpha_{t})$ which constitute the attaching data for the $n$
$2$–handles and $n$ $3$–handles used to construct each $Y_{t}$ in this range.

We will now improve the format of this data somewhat. First, we can arrange
that for some small $\epsilon>0$, on the time interval $t_{1}\leq t\leq
t_{1}+\epsilon$ the embeddings $\alpha_{t}$ and $\beta_{t}$ are independent of
$t$, i.e. the first framed circle and sphere do not move for a short time
after their birth. Next, since a birth happens at a point, we can make the
second birth happen at an earlier time so that in fact
$t_{1}<t_{2}<t_{1}+\epsilon$. Repeating this, and doing the same in reverse
with the deaths, we can assume that on the whole interval $t_{1}\leq t\leq
t_{n}$ and on the whole interval $t_{n}^{\prime}\leq t\leq t_{1}^{\prime}$,
the $\alpha_{t}$ and $\beta_{t}$ are independent of $t$ except for the fact
that at each $t_{i}$ a new $\alpha^{i}_{t}$ and $\beta^{i}_{t}$ is added to
the mix (and ditto for the deaths).

Thus now the governing data for the constructions of each $Y_{t}$ can be more
succinctly described by the following data:

  * •

A framed embedding $\delta=\delta_{1}\amalg\ldots\amalg\delta_{n}$ of
$\amalg^{n}D^{2}$ into $S^{4}$ (for the births), defining framed embeddings
$\alpha$ of $\amalg^{n}S^{1}$ into $S^{4}$ and $\beta$ of $\amalg^{n}S^{2}$
into $S^{4}(\alpha)$.

  * •

A framed embedding
$\delta^{\prime}=\delta_{1}^{\prime}\amalg\ldots\amalg\delta_{n}^{\prime}$ of
$\amalg^{n}D^{2}$ into $S^{4}$ (for the deaths), defining framed embeddings
$\alpha^{\prime}$ of $\amalg^{n}S^{1}$ into $S^{4}$ and $\beta^{\prime}$ of
$\amalg^{n}S^{2}$ into $S^{4}(\alpha^{\prime})$

  * •

A $t$–parameterized family, for $t\in[1/4,3/4]$, of framed embeddings
$\alpha_{t}=\alpha^{1}_{t}\amalg\ldots\amalg\alpha^{n}_{t}$ of
$\amalg^{n}S^{1}$ into $S^{4}$, with $\alpha_{1/4}=\alpha$ and
$\alpha_{3/4}=\alpha^{\prime}$.

  * •

A $t$–parameterized family, for $t\in[1/4,3/4]$, of framed embeddings
$\beta_{t}=\beta^{1}_{t}\amalg\ldots\amalg\beta^{n}_{t}$ of $\amalg^{n}S^{2}$
into $S^{4}(\alpha_{t})$, with $\beta_{1/4}=\beta$ and
$\beta_{3/4}=\beta^{\prime}$.

We would now like to modify this data (by a homotopy of the family
$(f_{t},V_{t})$) so that $\delta=\delta^{\prime}$ is a fixed standard
embedding and so that the family $\alpha_{t}$ is actually invariant with
respect to $t$, i.e. $\alpha_{t}=\alpha=\alpha^{\prime}$ for all $t$. Since
any collection of $n$ disks is isotopic to any other, it is easy to arrange
that $\delta=\delta^{\prime}$ (we tack the necessary isotopies on at the
beginning and end of the $t$–parameterized family of handle attaching data).
Thus we now assume $\delta=\delta^{\prime}$ is standard.

Next focus on $\alpha^{1}_{t}$, which is a loop of embeddings of $S^{1}$
starting and ending at a standard embedding. By an isotopy rel endpoints we
can arrange that $\alpha^{1}_{t}$ is independent of $t$ at a fixed point on
$S^{1}$ (since $\pi_{1}(S^{4})=0$), and then on a fixed interval neighborhood
of that point (since $\pi_{1}(S^{3})=0$). These are achieved by homotopies of
the family of embeddings of $S^{1}$, i.e. isotopies of isotopies, and can be
extended to all of $S^{4}$ by the parametrized isotopy extension theorem, so
that the other $\alpha^{i}_{t}$’s move out of the way when we move
$\alpha^{1}_{t}$. Also, when we move the $\alpha^{i}_{t}$’s, we can realize
this as a homotopy of $(f_{t},V_{t})$ by just modifying the gradient-like
vector field $V_{t}$ in $f_{t}^{-1}[a,b]$ for some small $0<a<b$ with $b$
below the lowest critical value of all the $f_{t}$’s. Thus we do not need to
worry about how this affects the framed $2$–spheres $\beta_{t}$, they will
still be a family as above of framed embeddings of $\amalg^{n}S^{2}$ into
$S^{4}(\alpha_{t})$. Now we use the fact that the space of embeddings of
$B^{1}$ in $B^{4}$ with fixed endpoints on $\partial B^{4}$ is simply
connected [1] to finally arrange that $\alpha^{1}_{t}$ is independent of $t$,
i.e. that $\alpha^{1}_{t}=\alpha^{1}_{1/4}=\alpha^{1}_{3/4}$.

We could proceed to do the same thing for $\alpha^{2}_{t}$ if we knew that the
parametrized isotopy extensions involved when fixing $\alpha^{2}_{t}$ (and
after that, $\alpha^{3}_{t}$ up to $\alpha^{n}_{t}$) would not mess up the
work we have already done to fix $\alpha^{1}_{t}$. Recall that $\delta_{1}$ is
a fixed disk bounded by the (now constant in $t$) circle $\alpha^{1}_{t}$; if
the other circles $\alpha^{2}_{t},\ldots,\alpha^{n}_{t}$ do not pass through
$\delta_{1}$, then we can arrange that the isotopy extensions do not need to
move $\delta_{1}$ and hence do not need to move $\alpha^{1}_{t}$. (This is
because the isotopies of isotopies of circles involved are ultimately
$3$–dimensional objects and can be arranged to miss points, and hence disks.
They cannot be assumed in general to miss circles.) To deal with this issue,
let $\delta^{1}_{t}$ be a family of framed embeddings of $D^{2}$ into $S^{4}$,
for $t\in[1/4,3/4]$, with $\partial\delta^{1}_{t}=\alpha^{1}_{t}$ and
$\delta^{1}_{1/4}=\delta_{1}$, such that $\delta^{1}_{t}$ is disjoint from
$\alpha^{2}_{t}\amalg\ldots\amalg\alpha^{n}_{t}$ for all $t\in[1/4,3/4]$. This
is also achieved by an isotopy extension, simply pushing $\delta_{1}$ out of
the way as the $S^{1}$’s $\alpha^{2}_{t}\amalg\ldots\amalg\alpha^{n}_{t}$ move
around and pass through $\delta_{1}$. Note that there is no reason to expect
that $\delta^{1}_{3/4}=\delta_{1}$. However, in the time interval
$[t^{\prime}_{2},t^{\prime}_{1}]$, after the death of all of the cancelling
handle pairs involving $\alpha^{2}_{t},\ldots,\alpha^{n}_{t}$, we can reverse
the isotopy of the $\delta^{1}_{t}$’s and get back to
$\delta^{1}_{t_{1}}=\delta_{1}$. Finally, looking at the entire family of disk
embeddings $\delta^{1}_{t}$ from $t=1/4$ to $t=t^{\prime}_{1}$, this is
homotopic rel endpoints to a constant family of embeddings
$\delta^{1}_{t}=\delta_{1}=\delta^{\prime}_{1}$, and this homotopy, i.e.
isotopy of isotopies, extends to a parametrized ambient isotopy which again
moves $\alpha^{2}_{t},\ldots,\alpha^{n}_{t}$ for $t\in[1/4,3/4]$. Note that we
can assume the disks $\delta_{t}^{2},\ldots,\delta^{n}_{t}$ are not moved by
this ambient isotopy again because they are disks and hence can be avoided by
the isotopy of the family of disks $\delta^{1}_{t}$.

Thus we get
$\alpha^{1}_{t}=\alpha^{1}_{1/4}=\alpha^{1}_{3/4}=\partial\delta_{1}$, and
that $\alpha^{2}_{t}\amalg\ldots\alpha^{n}_{t}$ is disjoint from $\delta_{1}$
for all $t$. Now we can repeat this with $\alpha^{2}_{t}$ up to
$\alpha^{n}_{t}$. This gets us to our desired situation where
$\delta=\delta^{\prime}$,
$\alpha_{t}=\alpha_{1/4}=\alpha_{3/4}=\partial\delta$, and $\delta$ is a
standard framed embedding of $\amalg^{n}D^{2}$ into $S^{4}$. With this setup,
we can now move the birth of the first $2$–$3$–pair closer to $t=0$ and the
death closer to $t=1$ until they merge in $Z=[0,1]\times[0,1]\times
S^{4}/(1,s,p)\sim(0,\Phi(s,p))$, and repeat with the second pair and so on.
After this, $Y_{0}$ is built from $[0,1]\times S^{4}$ by attaching $n$
standard cancelling $2$–$3$–pairs and $Y_{t}$ is built by attaching $n$
$2$–handles along a standardly embedded collection of $n$ framed circles
$\alpha$ and $n$ $3$–handles along a loop of framed embeddings $\beta_{t}$ of
$\amalg^{n}S^{2}$. This is precisely the conclusion of the theorem.

∎

##  4 Turning $2$–$3$–handle pairs into $1$–$2$–handle pairs

We now need to work toward the connection with Montesinos twins and the proof
of Theorem 4, that twists along Montesinos twins generate the subgroup of
$\pi_{0}(\mathop{Diff}^{+}(S^{4}))$ corresponding, via the $2$–$3$–handle pair
construction above, to loops of $2$–spheres which remain disjoint from
parallel copies of the basepoint $2$–spheres. More precisely, recall that
$\widehat{\mathcal{S}}_{n}$ is the space of embeddings of $\amalg^{n}S^{2}$
into $\\#^{n}(S^{2}\times S^{2})^{\dagger}$ which remain disjoint from
$\amalg^{n}(S^{2}\times\\{p^{\prime}\\}$ for some fixed $p^{\prime}\in S^{2}$,
with basepoint being $\amalg^{n}(S^{2}\times\\{p\\})$ for $p\neq p^{\prime}$.
We want to study the subgroup
$\mathcal{H}_{\infty}(\imath_{*}(\pi_{1}(\widehat{\mathcal{S}}_{\infty})))$.
To relate this to Montesinos twins we will first need to relate it to families
of $1$–$2$–handle pairs coming from loops of framed circles in
$\\#^{n}(S^{1}\times S^{3})$. In particular, using the notation from Section
2, in this section we will prove:

######  Theorem 11.

For any $n$,
$\mathcal{H}_{n}(\imath_{*}(\pi_{1}(\widehat{\mathcal{S}}_{n})))=\mathcal{FH}_{n}(\pi_{1}(\mathcal{FS}_{n}(S^{4},1)))$.

This means that any isotopy class of diffeomorphisms of $S^{4}$ that can be
realized by a family of cobordisms built with $n$ $2$–$3$–handle pairs
governed by a loop in $\widehat{\mathcal{S}}_{n}$ can also be realized by a
family of cobordisms built with $n$ $1$–$2$–pairs governed by a loop in
$\mathcal{FS}_{n}(S^{4},1)$, which is the space of framed embeddings of a
disjoint union of $n$ circles in $\\#^{n}(S^{1}\times S^{3})^{\dagger}$.

###### Proof.

Consider a cobordism $Z$ from $S^{1}\times S^{4}$ to $S^{1}\times_{\phi}S^{4}$
built as before as a family $Y_{t}$ of cobordisms, such that each $Y_{t}$ is
built by attaching n $2$–handles to $[0,1]\times S^{4}$ and then $n$
$3$–handles to the result. The $2$–handles are attached along a family
$\alpha_{t}$ of framed embeddings of $\amalg^{n}S^{1}$ into $S^{4}$, except
that in fact $\alpha_{t}=\alpha_{0}$ does not vary with $t$ and is a standard
embedding, so that $S^{4}(\alpha_{t})$ is canonically identified with
$\\#^{n}(S^{2}\times S^{2})$. The $3$–handles are attached along a family
$\beta_{t}$ of framed embeddings of $\amalg^{n}S^{2}$ into
$\\#^{n}(S^{2}\times S^{2})$, with
$\beta_{0}=\beta_{1}=\amalg^{n}(S^{2}\times\\{p\\})$, and with each
$\beta_{t}$ disjoint from $\amalg^{n}(S^{2}\times\\{p^{\prime}\\})$.

We will now introduce two $5$–dimensional analogues of the “dotted circle”
notation traditionally used for $4$–dimensional $1$–handles. A $5$–dimensional
$1$–handle attached to the boundary of a $5$–manifold in such a way that the
$1$–handle could be cancelled (which here means that its attaching $S^{0}$
bounds a $B^{1}$ and that the framing of the $S^{0}$ comes from a framing of
the $B^{1}$) can be represented by a “dotted unknotted $2$–sphere” together
with a choice of $3$–ball bounded by the $2$–sphere in the $4$–manifold, which
means we push the interior of the $3$–ball into the $5$–manifold and carve out
its neighborhood. The $B^{3}$ bounded by the dotted $S^{2}$ intersects the
$B^{1}$ bounded by the $1$–handle’s attaching $S^{0}$ transversely once.
Likewise, a $5$–dimensional $2$–handle attached to the boundary of a
$5$–manifold in such a way that it could be cancelled (which here means that
its attaching framed $S^{1}$ bounds a framed $B^{2}$) can be represented by a
“dotted circle” together with a choice of $2$–disk bounded by the circle in
the $4$–manifold. Here the $B^{2}$ bounded by the dotted circle intersects the
$B^{2}$ bounded by the $2$–handle’s attaching circle transversely once.

We emphasize in both these $5$–dimensional cases that, using “dotted”
notation, in principle we need to know what the disks/balls are which we are
going to push in and carve out, with the $4$–dimensional case being perhaps
exceptional because an unknot in a $S^{3}$ bounds a unique disk. Budney and
Gabai have shown [2] that unknotted $2$–spheres in $S^{4}$ can bound “knotted”
$3$–balls, and of course, although all $S^{1}$’s in $S^{4}$ are unknotted,
$2$–knots can be tied into any spanning disk for such an $S^{1}$. What really
matters is whether these spanning disks and balls are isotopic in dimension
$5$, and we leave this as an interesting question; the Budney-Gabai examples
are in fact isotopic in $B^{5}$, but there might in principle be more
complicated examples that remain nonisotopic even when pushed into $B^{5}$.
However, here we will bypass this issue by working with “dotted disks” instead
of “dotted circles” and “dotted balls” instead of “dotted spheres”.

Now, given our handle attaching data $\alpha_{t}$ and $\beta_{t}$ used to
build $Z$, since the $\alpha_{t}$’s are invariant in $t$, and
$\alpha_{t}=\alpha_{0}$ bounds a fixed collection of framed disks
$\delta_{0}$, we can instead represent the $2$–handles by a (for now,
$t$–invariant) $t$–parametrized family of $n$ dotted disks
$\alpha^{\bullet}_{t}:\amalg^{n}B^{2}\hookrightarrow S^{4}$. Note that
$\alpha^{\bullet}_{t}$ is not an extension of $\alpha_{t}$, but is rather an
embedding of dual disks to the fixed disks bounded by $\alpha_{t}$. Also, we
insist on maintaining the subscript $t$ even though these are $t$–invariant
because we will shortly modify the family so as to lose $t$–invariance. Now
the instructions for building the cobordism $Z$ are to build each $Y_{t}$ from
$[0,1]\times S^{4}$ by pushing the interior of each of these disks from
$\\{1\\}\times S^{4}$ into the interior of $[0,1]\times S^{4}$ and removing
their neighborhoods, and then attaching $3$–handles along $\beta_{t}$. Note
that, after carving out the disks but before attaching the $3$–handles, the
upper boundary of this cobordism $Y_{t}$ is the surgered $4$–manifold
$S^{4}(\partial\alpha^{\bullet}_{t})$. In other words, when looking at the
$4$–dimensional boundary, we cannot tell whether we carved out the dotted
disks or attached $2$–handles along their boundaries, because the resulting
surgeries are the same.

The crucial point here is that, because each $\beta_{t}$ is disjoint from
$\amalg^{n}(S^{2}\times\\{p^{\prime}\\})$, we can isotope the family
$\beta_{t}$ so that it never goes over the surgered region of
$S^{4}(\partial\alpha^{\bullet}_{t})$, and thus the entire handle attaching
data now lives in $S^{4}$. Thus we can now describe each $Y_{t}$, and thus
$Z$, via data entirely lying in $S^{4}$, i.e.
$\alpha^{\bullet}_{t}\amalg\beta_{t}:(\amalg^{n}B^{2})\amalg(\amalg^{n}S^{2})\to
S^{4}$. This is not an embedding but is an embedding when restricted to
$\amalg^{n}B^{2}$ and to $\amalg^{n}S^{2}$, and the only intersections occur
between $\amalg^{n}S^{2}$ and the interiors of the disks $\amalg^{n}B^{2}$. At
times $t=0$ and $t=1$, each $S^{2}$ intersects its corresponding $B^{2}$
transversely once and is disjoint from all the other $B^{2}$’s, i.e. the
spheres and disks are in “cancelling position”

Our goal is now to “switch the dots from the circles to the spheres”, i.e. to
think of $\beta_{t}$ as being dotted spheres, thus corresponding to
$5$–dimensional $1$–handles, and to think of $\partial\alpha^{\bullet}_{t}$ as
attaching circles for $2$–handles, rather than dotted circles describing
$2$–handles. However, as discussed above, before we can “put dots on”
$\beta_{t}$ we need to extend them to be embeddings of balls not just of
spheres. In other words, we want to extend
$\beta_{t}:\amalg^{n}S^{2}\hookrightarrow S^{4}$ to
$\beta^{\bullet}_{t}:\amalg^{n}B^{3}\hookrightarrow S^{4}$. Unfortunately, if
we could do this without moving $\beta_{t}$ and also achieving the property
that $\beta^{\bullet}_{0}=\beta^{\bullet}_{1}$, we would then have shown that
$\beta_{t}$ was trivial in $\pi_{1}(\mathop{Emb}(\amalg^{n}S^{2},S^{4}))$, and
we do not (at least this author does not) know enough about
$\pi_{1}(\mathop{Emb}(\amalg^{n}S^{2},S^{4}))$ to make such an assertion. So
we proceed carefully, and in fact we will end up modifying both
$\alpha^{\bullet}_{t}$ and $\beta_{t}$, abandoning
$\alpha^{\bullet}_{1}=\alpha^{\bullet}_{0}$ and $\beta_{1}=\beta_{0}$, but
maintaining the property that $\alpha^{\bullet}_{1}$ and $\beta_{1}$ are in
cancelling position.

Using the isotopy extension theorem we can easily extend $\beta_{t}$ to
$\beta^{\bullet}_{t}:\amalg^{n}B^{3}\hookrightarrow S^{4}$ (pick $B^{3}$’s
bounded by $\beta_{0}$ and move them around by an ambient isotopy for
$\beta_{t}$) but we should not expect that
$\beta^{\bullet}_{1}=\beta^{\bullet}_{0}$. We can, however, assume that each
component of $\partial\alpha^{\bullet}_{1}$ transversely intersects the
corresponding component of $\beta^{\bullet}_{1}$ at one fixed point in the
interior of each $B^{3}$, since $\beta_{1}$ and $\partial\alpha^{\bullet}_{1}$
are meridians of each other. Thus we know that $\partial\alpha^{\bullet}_{1}$
and $\beta^{\bullet}_{1}$ are now in cancelling position, and we still have
that $\alpha^{\bullet}_{1}$ and $\partial\beta^{\bullet}_{1}$ are in
cancelling position. Now assume that $\beta^{\bullet}_{t}=\beta^{\bullet}_{1}$
and $\alpha^{\bullet}_{t}=\alpha^{\bullet}_{1}$ for $t\in[1-\epsilon,1]$. Then
there is an isotopy from $\beta^{\bullet}_{1-\epsilon}$ back to
$\beta^{\bullet}_{0}$ (one could use the original forward isotopy in reverse,
or any other isotopy) which can be extended to an ambient isotopy and then
used on the interval $[1-\epsilon,1-\epsilon/2]$ to move both
$\beta^{\bullet}_{t}$ and $\alpha^{\bullet}_{t}$, discarding completely the
given $\beta^{\bullet}_{t}$ and $\alpha^{\bullet}_{t}$ on $[1-\epsilon,1]$.
However, although we have thrown away the original path of embeddings on this
interval, we have maintained that fact that both the pair
$\partial\beta^{\bullet}_{t}$ and $\alpha^{\bullet}_{t}$ and the pair
$\beta^{\bullet}_{t}$ and $\partial\alpha^{\bullet}_{t}$ are in cancelling
position. Now, when $t=1-\epsilon/2$, we have that
$\beta^{\bullet}_{t}=\beta^{\bullet}_{0}$ but the only thing we know about
$\partial\alpha^{\bullet}_{t}$ is that it is in cancelling position with
respect to $\beta^{\bullet}_{t}=\beta^{\bullet}_{0}$, and also that
$\partial\alpha^{\bullet}_{0}$ is also in cancelling position with respect to
$\beta^{\bullet}_{t}=\beta^{\bullet}_{0}$. But now we can use Gabai’s
$4$–dimensional lightbulb theorem to isotope
$\partial\alpha^{\bullet}_{1-\epsilon/2}$ to $\alpha^{\bullet}_{0}$ remaining
in cancelling position, and use this isotopy to get
$\partial\alpha^{\bullet}_{t}$ for $t\in[1-\epsilon/2,1$. Note that we have
lost track of, but do not really care about, the embeddings
$\alpha^{\bullet}_{t}$ for $t\in[1-\epsilon/2,1]$, but care only about their
boundaries.

To translate the above modifications into a modification of our cobordism $Z$
built as a family of cobordisms $Y_{t}$, we first cancel the $2$– and
$3$–handles for all $t\in[1-\epsilon,1]$, then switch the dots from the
$\alpha$’s to the $\beta$’s for all $t\in[0,1-\epsilon]$, thus getting a new
cobordism $Z^{\prime}$ from $S^{1}\times S^{4}$ to $S^{1}\times_{\phi}S^{4}$.
This $Z^{\prime}$ is built from cobordisms $Y^{\prime}_{t}$ which start with
$[0,1]\times S^{4}$, then experience the birth of $n$ $1$–$2$ pairs which then
evolve in $t$, with $1$–handles described by the dotted balls
$\beta^{\bullet}_{t}$ and the $2$–handles attached along the circles
$\partial\alpha^{\bullet}_{t}$, and all end up cancelling at time
$t=1-\epsilon$. Then the cancellation can be postponed until $t=1-\epsilon/2$
to give the circles $\partial\alpha^{\bullet}_{t}$ to return to their starting
positions. Finally, wrapping from $t=1-\epsilon/2$ around to $t=1\sim 0$, we
can merge the deaths and births so that in the end the cobordisms $Y_{t}$ are
described by the loop of maps
$\partial\alpha^{\bullet}_{t}\amalg\beta^{\bullet}_{t}:(\amalg^{n}S^{1})\amalg(\amalg^{n}B^{3})\to
S^{4}$, with $\beta^{\bullet}_{t}$ being dotted balls describing $1$–handles
and $\partial\alpha^{\bullet}_{t}$ being attaching circles for $2$–handles.

Finally, a further ambient isotopy can now be used to arrange that
$\beta^{\bullet}_{t}$ is independent of $t$, and thus the entire construction
is governed by the loop of framed circles $\partial\alpha^{\bullet}_{t}$.

∎

##  5 From many $1$–$2$ pairs to a single $1$–$2$ pair

We now know that diffeomorphisms of $S^{4}$ that can be realized by a family
of cobordisms built with $n$ $2$–$3$–handle pairs can also be realized by a
family of cobordisms built with $n$ $1$–$2$–handle pairs, as long as the
original governing loop of embeddings of $\amalg^{n}S^{2}$ into
$\\#^{n}(S^{2}\times S^{2})$ lies in $\widehat{\mathcal{S}_{n}}$. Before we
get to Montesinos twins, we need now to show that every diffeomorphism of
$S^{4}$ that can be a realized by a family of cobordisms built with $n$
$1$–$2$–handle pairs can be realized by family built with a single
$1$–$2$–handle pair.

######  Theorem 12.

For any $n$,
$\mathcal{FH}_{n}(\pi_{1}(\mathcal{FS}_{n}(S^{4},1)))=\mathcal{FH}_{1}(\pi_{1}(\mathcal{FS}_{1}(S^{4},1)))$.

###### Proof.

We begin again with a cobordism $Z$ from $S^{1}\times S^{4}$ to
$S^{1}\times_{\phi}S^{4}$ built as a family of cobordisms $Y_{t}$, each
$Y_{t}$ built by attaching $n$ fixed standard $1$–handles to $[0,1]\times
S^{4}$ followed by $n$ “moving” $2$–handles governed by a loop of embeddings
$\alpha_{t}:\amalg^{n}(S^{1}\times B^{3})\hookrightarrow\\#^{n}(S^{1}\times
S^{3})$. Cancelling the $1$–$2$ pairs at time $t=0\sim 1$, we revert to the
Cerf theoretic perspective to get a family $(f_{t},V_{t})$ of Morse functions
with gradient-like vector fields on $[0,1]\times S^{4}$ interpolating from
$f_{0}$, which is projection onto $[0,1]$, to $f_{1}$, which is the pullback
of $f_{0}$ via some pseudoisotopy $\Phi:[0,1]\times S^{4}\to[0,1]\times S^{4}$
from $\mathop{\rm id}\nolimits_{S}^{4}$ to $\phi$. The graphic now looks like
Figure 5, exactly as in Figure 4 except that now the critical points are of
index $1$ and $2$; there are still no handle slides.

2pt $1$ [tr] at 45 20 $1$ [tr] at 49 24 $1$ [tr] at 56 31 $2$ [br] at 45 67
$2$ [br] at 49 63 $2$ [br] at 56 55

Figure 5: A nice Cerf graphic with only critical points of index $1$ and $2$.

Theorem 2.1.1 of Chenciner’s thesis [4], restated as Hatcher and Wagoner’s
Proposition 1.4 on p.177 of [5], asserts that, given a $1$–parameter family
$f_{t}$ of Morse functions on $[0,1]\times X$ where $X$ is an $m$–manifold, if
the Cerf graphic contains a swallowtail involving critical points of index $i$
and $i+1$ as in the left of Figure 6, with $i\leq m-3$, then the swallowtail
can be cancelled to give the graphic on the right in Figure 6. This applies in
our setting because $m=4$ and $i=1=4-3$. We use this to reduce the number of
$1$–$2$ pairs using the main idea of Proposition 4 on p.217 of [5], as in the
figure on the top of p.218 of [5]. We essentially reproduce this figure here
in Figure 7 which shows how to reduce a nested pair of birth-deaths of $1$–$2$
handles to a single pair. (The other elementary moves are introducing a
swallowtail, which can always be done, and merging a death with a birth, which
can always be done if level sets are connected, which they are in our case.)

2pt $i$ [t] at 46 30 $i$ [t] at 46 30 $i+1$ [t] at 12 60 $i+1$ [t] at 78 60
$i+1$ [t] at 160 60

Figure 6: Eliminating a swallowtail. Figure 7: Using a swallowtail to turn a
nested pair of “eyes” into a single eye.

Repeating this we can turn $n$ nested $1$–$2$ “eyes” into a single nested
$1$–$2$ “eye”, and then we can merge the birth again at $t=0\sim 1$. Note that
in fact we could have left this last (bottom-most) index $1$ critical point
completely unchanged in this whole process, we even did not need to cancel it
with its cancelling $2$–handle at the beginning. Thus we can easily arrange
that this last $1$–handle is still stationary, i.e. its attaching map does not
move with $t$. This shows that this cobordism can be built with a single fixed
standard $1$–handle followed by a single moving $2$–handle whose attaching map
is given by a loop of embeddings $S^{1}\times B^{3}\hookrightarrow S^{1}\times
S^{3}$. Therefore
$[\phi]\in\mathcal{FH}_{1}(\pi_{1}(\mathcal{FS}_{1}(S^{4},1)))$.

∎

######  Remark 13.

In case it is needed in another context, the general version of this theorem
is that, if $i\leq m-3$ and $X$ is an $m$–manifold, then for any $n$,
$\mathcal{FH}_{n}(\pi_{1}(\mathcal{FS}_{n}(X,i)))=\mathcal{FH}_{1}(\pi_{1}(\mathcal{FS}_{1}(X,i)))$.

##  6 Twists along Montesinos twins

As a consequence of the preceding two theorems we now know that any
diffeomorphism of $S^{4}$ arising as the monodromy of the top of a cobordism
constructed as above from a loop of $n$ $2$–spheres in $\\#^{n}(S^{2}\times
S^{2})$ which remain disjoint from a parallel copy
$\amalg^{n}(S^{2}\times\\{p^{\prime}\\})$ of the basepoint embedding
$\amalg^{n}(S^{2}\times\\{p\\})$ is isotopic to a diffeomorphism arising from
a loop of embeddings of a single circle in $S^{1}\times S^{3}$. This is
summarized as:

######  Corollary 14.

$\mathcal{H}_{\infty}(\imath_{*}(\pi_{1}(\widehat{\mathcal{S}}_{\infty})))=\mathcal{FH}_{1}(\pi_{1}(\mathcal{FS}_{1}(S^{4},1)))$

Our next goal, which will complete the proof of Theorem 4, is to show that

$\mathcal{FH}_{1}(\pi_{1}(\mathcal{FS}_{1}(S^{4},1)))$

is generated by twists along Montesinos twins as advertised.

###### Proof of Theorem 4.

Recall that $\mathcal{FS}_{1}(S^{4},1))$ is the space of framed embeddings of
$S^{1}$ in $S^{1}\times S^{3}$ while
$\mathcal{S}_{1}(S^{4},1))=\mathop{Emb}(S^{1},S^{1}\times S^{3})$ is the space
of unframed embeddings of $S^{1}$ in $S^{1}\times S^{3}$. As noted at the end
of Section 2, we might worry that the homomorphism
$\pi_{1}(\mathcal{FS}_{1}(S^{4},1)))\to\pi_{1}(\mathcal{S}_{1}(S^{4},1)))$ is
not surjective, since there are two possible framings of a circle in a
$4$–manifold. However, Budney and Gabai [2] give explicit representatives of
generators for $\pi_{1}(\mathop{Emb}(S^{1},S^{1}\times S^{3}))$ all of which
can be seen to lift to framed loops of embeddings, and thus the map is
surjective so we do not need to worry about framings anymore. (There is
probably some other more direct way to see this, the point being that there is
no loop of embeddings of $S^{1}$ in $S^{1}\times S^{3}$ which switches the two
framings of $S^{1}$.)

For the remainder of this proof, we will use the less obscure notation
$\mathop{Emb}(S^{1},S^{1}\times S^{3})$ to refer to the space
$\mathcal{S}_{1}(S^{4},1))$, the latter more complicated notation only being
helpful when placing things in the much more general context of Section 2.
Also note that in Section 2 we punctured the target space of our embeddings,
but here we drop the puncture for simplicity. The point is that, at the level
of $\pi_{1}$, the puncture is irrelevant since loops of circles are
$2$–dimensional while homotopies of loops of circles are $3$–dimensional, so
since the ambient space is $4$–dimensional everything can be assumed to miss a
point.

In fact [2] shows that every class $g\in\pi_{1}(\mathop{Emb}(S^{1},S^{1}\times
S^{3}))$ can be represented by a loop $\gamma_{t}:S^{1}\hookrightarrow
S^{1}\times S^{3}$ of embeddings such that the associated map
$\Gamma:S^{1}\times S^{1}\to S^{1}\times S^{3}$ given by
$\Gamma(t,s)=\gamma_{t}(s)$ is itself an embedding. Thus we have an embedded
torus $\Gamma:S^{1}\times S^{1}\hookrightarrow S^{1}\times S^{3}$ such that
$\Gamma(\\{0\\}\times S^{1})$ is the basepoint embedding
$C=S^{1}\times\\{p\\}$. Surgery along $C$ applied to the pair $(S^{1}\times
S^{3},\Gamma)$ yields $(S^{4},R)$ for some embedded $2$–sphere $R\subset
S^{4}$, and the $2$–sphere $S$ dual to the surgery circle is an unknotted
sphere $S\subset S^{4}$ such that $(R,S)$ forms a Montesinos twin.
Furthermore, the boundary $\partial\nu(R\cup S)$ of a neighborhood of this
twin in $S^{4}$ is the same as the boundary of a tubular neighborhood of
$\Gamma(S^{1}\times S^{1})$ in $S^{1}\times S^{3}$. When this $3$–torus is
parametrized as $S^{1}_{l}\times S^{1}_{R}\times S^{1}_{S}$ as in the
introduction, we see that the $S^{1}_{l}$ parameter corresponds to the $t$
parameter in $\Gamma(t,s)=\gamma_{t}(s)$, that the $S^{1}_{R}$ direction
corresponds to the $s$–parameter, and that the $S^{1}_{S}$ direction
corresponds to the boundary of the disk factor in the tubular neighborhood
$\nu(\Gamma(S^{1}\times S^{1}))\cong D^{2}\times S^{1}\times S^{1}$.

Because $\Gamma$ is embedded, it is relatively easy to see what
$\mathcal{H}_{1}([\gamma_{t}])$ looks like. We need an ambient isotopy
$\phi_{t}$ of $S^{1}\times S^{3}$ with $\phi_{0}=\mathop{\rm id}\nolimits$,
$\phi_{t}\circ\gamma_{0}=\gamma_{t}$ and $\phi_{1}$ equal to the identity on a
neighborhood of $C$. (This is the “circle pushing” map we get by dragging the
circle around the embedded torus and back to its starting position.) This can
happen entirely in a tubular neighborhood $D^{2}\times S^{1}\times S^{1}$ of
$\Gamma$, by spinning in the $t$ direction more and more as we move towards
the center of $D^{2}$, which we state explicitly as follows: Let $(r,\theta)$
be polar coordinates on $D^{2}$, and let $(t,s)$ be coordinates as before on
$S^{1}\times S^{1}$. Choose a smooth non-increasing function
$T:[0,1]\to[0,2\pi]$ which is $1$ on $[0,1/4]$, $0$ on $[3/4,1]$, and let
$\phi_{t}(r,\theta,t,s)=(r,\theta,t+T(r),s)$. From this it is clear that
$\phi_{1}$ is the identity on $r\in[0,1/4]$ and $r\in[3/4,1]$, and on the
intermediate $[1/4,3/4]\times S^{1}\times S^{1}\times S^{1}$ is equal to a
Dehn twist on $[1/4,3/4]\times S^{1}$ crossed with the identity in the
remaining $S^{1}\times S^{1}$ direction. Back in $S^{4}$ this is exactly the
twist $\tau_{W}$ along the twin $W=(R,S)$.

In fact [2] establishes an isomorphism

$W_{1}\times W_{2}:\pi_{1}(\mathop{Emb}(S^{1},S^{1}\times
S^{3}),C)\to\mathbb{Z}\oplus\Lambda_{3}^{1}$

where $\Lambda_{3}^{1}$ is a free abelian group on a countably infinite
generating set. The $\mathbb{Z}$ factor in $\mathbb{Z}\oplus\Lambda_{3}^{1}$
is given by the loops of $S^{1}$–reparametrizing embeddings
$\gamma_{t}(s)=\gamma_{0}(s+nt)$, and it is easy to see that $\mathcal{H}_{1}$
applied to such a loop of embeddings is isotopic to $\mathop{\rm
id}\nolimits_{S^{4}}$, i.e. this $\mathbb{Z}$ factor is in the kernel of
$\mathcal{H}_{1}$. Modulo this $\mathbb{Z}$ factor, Figure 4 in [2] gives the
first two tori $T(1)$ and $T(2)$ in an obvious family $T(i)$ of tori in
$S^{1}\times S^{3}$ which give the countably infinite generating set
corresponding to $\Lambda_{3}^{1}$. We draw $T(3)$ in Figure 8. In this
figure, the circle $C\subset S^{1}\times S^{3}$ is represented as a red
vertical line on the far right side of the torus. The torus $T(n)$ is just
like this but wraps $n$ times around the $S^{1}$ direction. Surgering along
$C$ yields our Montesinos twins $W(i)=(R(i),S)$.

2pt $C$ [r] at 230 150

Figure 8: The embedded torus $T(3)$ in $S^{1}\times S^{3}$, the obvious next
member of the family of tori described in Figure 4 of [2]. The top is glued to
the bottom, and horizontal slices are $S^{3}$’s, with the “time” coordinate
indicated in red/blue shading, as in Figure 1. Here we have exaggerated
certain features of this torus and deformed somewhat from the drawings in
Figure 4 of [2] so that the connection with the Montesinos twin
$W(3)=(R(3),S)$ in Figure 1 is visually apparent. Surgering along the red
circle $C$ collapses the vertical cylinder on the right into a ball (the tail
of the snake), with the dual sphere to $C$ becoming the tail-piercing sphere
$S$.

∎

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