# Diffeotopy groups of non-compact 4-manifolds

Isacco Nonino  University of Bonn, Bonn, Germany<EMAIL_ADDRESS>
###### Abstract.

We provide information on diffeotopy groups of exotic smoothings of punctured
4-manifolds, extending previous results on diffeotopy groups of exotic
$\mathbb{R}^{4}$’s. In particular, we prove that for a smoothable 4-manifold
$M$ and for a non-empty, discrete set of points $S\subsetneq\mathring{M}$,
there are uncountably many distinct smoothings of $M{\smallsetminus}S$ whose
diffeotopy groups are uncountable.

We then prove that for a smoothable 4-manifold $M$ and for a non-empty,
discrete set of points $S\subsetneq\mathring{M}$, there exists a smoothing of
$M{\smallsetminus}S$ whose diffeotopy groups have similar properties as
$\mathcal{R}_{U}$, Freedman and Taylor’s universal $\mathbb{R}^{4}$.

Moreover, we prove that if $M$ is non-smoothable, both results still hold
under the assumption that $|S|\geq 2$.

##  1\. Introduction

One of the most striking results in 4-dimensional topology is the existence of
uncountably many different smooth structures on $\mathbb{R}^{4}$ (see for
example [Tau87, Gom93, GS99]). This wild behavior was later also detected in
several other classes of non-compact topological 4-manifolds (see for example
[BE98, GS99]). Up to now, it is still an open question whether _all_ non-
compact topological 4-manifolds admit uncountably many distinct smoothings. In
particular, punctured 4-manifolds (i.e, manifolds with points removed) behave
as wildly as $\mathbb{R}^{4}$. This is not surprising: one way to view
$\mathbb{R}^{4}$ is as the 4-sphere without a point.

Although during the years a lot was proved about the exotic structures on non-
compact 4-manifolds, little was known until recently on how these structures
behaved in terms of self-diffeomorphism groups.

Diffeomorphism groups are normally hard to compute. A useful approximation is
given by the _mapping class groups_. They consist of diffeomorphisms of a
manifold $M$ to itself modulo isotopy. In this paper we will refer to them as
_diffeotopy groups_ , following the terminology used in [Gom18]. Given a
smooth manifold $M$, we denote the diffeotopy group by $\mathcal{D}(M)$. In
other words, elements of $\mathcal{D}(M)$ are smooth isotopy classes of self-
diffeomorphisms of $M$. If $M$ is non-compact, we will also consider _self-
diffeomorphisms at infinity_. Roughly speaking they are defined in terms of
smooth, proper embeddings of codimension-0 submanifolds whose complement in
$M$ has compact closure. They are _isotopic_ if they are properly isotopic as
proper embeddings. We denote the _diffeotopy group at infinity_ of $M$ by
$\mathcal{D}^{\infty}(M)$.

Each self-diffeomorphim $f$ of $M$ defines a diffeomorphism at infinity simply
by restriction. We obtain a homomorphism
$r\colon\mathcal{D}(M)\to\mathcal{D}^{\infty}(M)$ by sending diffeomorphisms
to their corresponding equivalence classes at infinity. If a diffeomorphism at
infinity lies in the image of $r$ we say that it _extends_ over $M$.

Even if mapping class groups are simpler to analyze than diffeomorphism
groups, they still are quite obscure objects.

As an example, we do not know how to compute
$\mathcal{D}^{\infty}(\mathbb{R}^{4})$ yet (this is in fact related, as
explained in [Gom18], to the _smooth Schoenflies conjecture_ in dimension 4,
which is notoriously still open).

In a recently published paper, Gompf approached the question, proving that
many (in fact, uncountably many) smoothings of $\mathbb{R}^{4}$ have an
uncountable diffeotopy group $\mathcal{D}(\mathcal{R})$ [Gom18, Theorem 4.4].
The same holds for their diffeotopy group at infinity
$\mathcal{D}^{\infty}(\mathcal{R})$.

He also proved an interesting result for $\mathcal{R}_{U}$, Freedman and
Taylor’s universal smoothing of $\mathbb{R}^{4}$ [FT86]. Though nothing is
known about $\mathcal{D}(\mathcal{R}_{U})$ and
$\mathcal{D}^{\infty}(\mathcal{R}_{U})$ by themselves, we now know that the
group homomorphism
$r\colon\mathcal{D}(\mathcal{R}_{U})\to\mathcal{D}^{\infty}(\mathcal{R}_{U})$
is surjective [Gom93, Theorem 5.1]. Equivalently, every diffeomorphism at
infinity of $\mathcal{R}_{U}$ extends over $\mathcal{R}_{U}$. We will refer to
this extension property as _universal behavior_.111 Some care is required: the
precise meaning of _universal behavior_ depends on the complexity at infinity
of the non-compact 4-manifold we are working with. We will discuss this in
Section 4.

As the early results on exotic $\mathbb{R}^{4}$’s were used to distinguish
between smooth structures on other non-compact 4-manifolds, we will use
Gompf’s work as the starting point for a general discussion on diffeotopy
groups of non-compact 4-manifolds. This possibility is mentioned in [Gom18,
Section 4] and carried out in [Gom18, Theorem 4.10] for the interior of
manifolds having a specific embedding property. However in this paper we
follow a different strategy.

We will focus our attention on punctured 4-manifolds. In Section 3 and Section
4 we will prove the following two theorems:

######  Theorem 3.3.

Let $M$ be a topological 4-manifold and $S\subsetneq\mathring{M}$ a non-empty
discrete set of points. Then $M{\smallsetminus}S$ admits uncountably many
smoothings whose diffeotopy groups $\mathcal{D}^{\infty}(M{\smallsetminus}S)$
and $\mathcal{D}(M{\smallsetminus}S)$ are uncountable if:

  1. (1)

$M$ is smoothable or

  2. (2)

$M$ is non-smoothable and $|S|\geq 2$.

######  Theorem 4.4.

Let $M$ be a topological 4-manifold and $S\subsetneq\mathring{M}$ a non-empty
discrete set of points. Then $M{\smallsetminus}S$ admits a smoothing with
_universal behavior_ if:

  1. (1)

$M$ is smoothable or

  2. (2)

$M$ is non-smoothable and $|S|\geq 2$.

Here $\mathring{M}$ indicates the _manifold_ interior, i.e
$M{\smallsetminus}\partial M$. The meaning of _universal behavior_ will be
explained more in detail in Section 4.

Note that in both Theorem 3.3 and Theorem 4.4 we did not address the case $M$
a topological _non-smoothable_ 4-manifold with only one puncture. The
following natural question arises:

######  Question.

Do Theorems 3.3 and 4.4 apply also for $M{\smallsetminus}p$, where $M$ is a
topological _non-smoothable_ 4-manifold and $p\in\mathring{M}$?

The proofs of the two theorems were inspired by [FO93] and [Gom93]. In the
referenced papers uncountability of smooth structures was extended from
$\mathbb{R}^{4}$ to punctured 4-manifolds exploiting the homeomorphism
$M{\smallsetminus}p\approx M\\#\mathbb{R}^{4}$ for $p\in\mathring{M}$. We will
do the same here. This is rather intuitive: in [Gom18, Theorem 4.4], the
uncountability of $\mathcal{D}(\mathcal{R})$ is proven by exhibiting a group
injection $G\hookrightarrow\mathcal{D}^{\infty}(\mathcal{R})$, where $G$ is an
uncountable group with the discrete topology. The behavior at infinity was
enough to detect non-isotopic diffeomorphisms of $\mathcal{R}$. It is quite
natural then to try puncturing a 4-manifold $M$ in such a way to have an end
diffeomorphic to that of a previously studied $\mathbb{R}^{4}$ homeomorph. We
will then need to show how the exotic behavior at infinity can distinguish
isotopy classes of self-diffeomorphisms of the punctured manifold.

Our starting point will be the following lemma. This is an easy case of
Theorem 3.3.

######  Lemma 3.1.

Let $M$ be a closed, smooth 4-manifold and $p\in M$. Then there exists a
smoothing of $M{\smallsetminus}p$ whose diffeotopy groups
$\mathcal{D}^{\infty}(M{\smallsetminus}p)$ and
$\mathcal{D}(M{\smallsetminus}p)$ are uncountable.

The proof of this result will showcase the techniques used throughout the
paper. The main idea is the following: take an uncountable group
$G\in\mathcal{G}^{*}$ [Gom18]. $\mathcal{G}^{*}$ is the collection of all
groups with discrete topology that have an action via diffeomorphisms on
$\mathbb{R}^{4}$ that respects certain properties. Intuitively, the
diffeomorphisms defined by elements of $G\in\mathcal{G}^{*}$ interact
equivariantly with a collection of rays properly embedded in $\mathbb{R}^{4}$
and keep fixed pointwise a region disjoint from these rays. The precise
definition of $\mathcal{G}^{*}$ is given in Section 2. We construct a
$G$-action via diffeomorphisms on the punctured manifold $M{\smallsetminus}p$.
To do so, we will use $M\\#\mathbb{R}^{4}\approx M{\smallsetminus}p$. Once the
action is constructed, we exploit Gompf’s result to derive a contradiction. We
will so prove that $G$ injects into $\mathcal{D}^{\infty}(M{\smallsetminus}p)$
as well as $\mathcal{D}(M{\smallsetminus}p)$.

It is worthwhile to emphasize that diffeotopy groups of _compact_ 4-manifolds
are mostly shrouded in mystery as well. This showcases once again the
discrepancies between dimension 4 and other dimensions. The case of
$\mathbb{S}^{4}$ is for example far from understood. There are however recent
breakthroughs by Watanabe, Gabai and Budney-Gabai [Wat18, BG19, Gab20] that
sparked new life in this area of research, which still has a lot to offer in
terms of open questions.

###  1.1. Outline

In Section 2 we will recall all the definitions and results we need from
Gompf’s paper. Section 3 and 4 will be devoted to the proofs of Theorem 3.3
and Theorem 4.4 respectively.

###  1.2. Acknowledgements

The work presented in this paper is part of my master’s thesis _Smooth
structures on non-compact 4-manifolds_ at the University of Bonn, written
under the supervision of Dr. Arunima Ray. I thank her for being extremely
supportive and for all the useful discussions we had. I also thank Prof.
Robert E. Gompf, whose outstanding work inspired my research and whose kind
answers helped me a lot during the last months.

##  2\. Group actions and Exotic $\mathbb{R}^{4}$’s

We recall the main definitions and results from [Gom18].

###  2.1. Diffeotopy groups at infinity and diffeotopy groups of the ends

Given a smooth manifold $V$, we study its diffeomorphism group modulo isotopy.

######  Definition 2.1.

Let $V$ be a smooth manifold. The group $\mathcal{D}(V)$ of isotopy classes of
self-diffeomorphisms of $V$ is called the _diffeotopy group of $V$_.

For a non-compact manifold $V$, we also have an analogous notion in terms of
diffeomorphisms at infinity. We now give a precise definition of such objects.

######  Definition 2.2.

Given a non-compact manifold $V$, a _closed neighborhood of infinity_ is a
codimension-0 submanifold $U\subseteq V$ that is a closed subset whose
complement has compact closure (see Figure 1).

######  Definition 2.3.

A _compact exhaustion_ of a connected topological manifold $M$ is a sequence
$\\{C_{i}\\}_{i=1}^{\infty}$ of compact, connected, codimension-0 submanifolds
with $C_{i}\subseteq Int_{M}(C_{i+1})$ (topological interior) and
$\bigcup_{i}C_{i}=M$. By letting $N_{i}:=\overline{M{\smallsetminus}C_{i}}$,
we obtain a corresponding _cofinal sequence of closed neighborhoods of
infinity_. Each such $N_{i}$ has connected components
$\\{N_{i}^{j}\\}_{j=1}^{k_{i}}$, where $k_{i}=\infty$ is also allowed. We can
arrange that each $N_{i}^{j}$ is noncompact by enlarging $C_{i}$ to include
all the compact components of $N_{i}$.

In the above arrangement, an _end_ $\epsilon$ of $M$ is determined by a nested
sequence $(N_{i}^{h_{i}})_{i=1}^{\infty}$ of components of the $N_{i}$; each
component is called a closed _neighborhood_ of $\epsilon$ (see Figure 1).

######  Remark 2.4.

Changing the sequence $\\{C_{i}\\}_{i}$ does not change the homeomorphism type
of space of the ends of $M$. More information about ends of spaces can be
found in [Fre31, Fre42, Pes90].

######  Example 2.5.

The Euclidean space $\mathbb{R}^{n}$ for $n\geq 2$ has one end $\epsilon$.
$\epsilon$ has a closed neighborhood that is homeomorphic to
$\mathbb{S}^{n-1}\times[0,\infty)$. The space $\mathbb{S}^{1}\times\mathbb{R}$
has two ends $\epsilon_{1}$ and $\epsilon_{2}$. They have closed neighborhoods
$N_{1}$ and $N_{2}$ respectively which are both homeomorphic to
$\mathbb{S}^{1}\times[0,\infty)$. A common example of an infinite-ended space
is the universal cover of $\mathbb{S}^{1}\vee\mathbb{S}^{1}$.

$N_{1}$$N_{2}$$U$$U$$\mathbb{S}^{1}\times\mathbb{R}$$\epsilon_{1}$$\epsilon_{2}$
Figure 1. On the top, $U$ is a closed neighborhood of infinity of
$\mathbb{S}^{1}\times\mathbb{R}$; $U$ is homeomorphic to the disjoint union of
two copies of $\mathbb{S}^{1}\times[0,\infty)$. On the bottom, two closed
neighborhoods $N_{1}$ and $N_{2}$, both homeomorphic to
$\mathbb{S}^{1}\times[0,\infty)$, of the ends $\epsilon_{1}$ and
$\epsilon_{2}$ respectively.

######  Definition 2.6.

Given non-compact smooth manifolds $V$ and $V^{\prime}$, suppose $f_{i}\colon
Y_{i}\to Y_{i}^{\prime},i=1,2$, are diffeomorphisms between closed
neighborhoods of infinity $Y_{i}\subseteq V$ and $Y_{i}^{\prime}\subseteq
V^{\prime}$. We refer to $f_{1}$ and $f_{2}$ as _equivalent_ if they agree
outside some compact subset of $V$ containing the complements of $Y_{1}$ and
$Y_{2}$. A _diffeomorphism at infinity_ from $V$ to $V^{\prime}$ is an
equivalence class of such diffeomorphisms.

If $V$ has a single end, this is referred to as a _diffeomorphism of the end_
of $V$ and $V^{\prime}$. In order to simplify the notation we write $f$
instead of $[f]$ for the equivalence class.

We say that a diffeomorphism at infinity $f$ _extends over_ $V$ if the
equivalence class $f$ contains a diffeomorphism $V\to V^{\prime}$. We can
compose such diffeomorphisms at infinity, making the set of self-
diffeomorphism of $V$ at infinity into a group. Given a group $G$, we define a
_$G$ -action at infinity_ on $V$ to be a homomorphism of $G$ into this group.

######  Definition 2.7.

Let $V$ be a non-compact smooth manifold. Two diffeomorphisms at infinity of
$V$ will be called _isotopic_ if they have representatives that are properly
isotopic as proper embeddings into $V^{\prime}$. Recall a _proper_ map has
preimage of each compact subset compact.

The group $\mathcal{D}^{\infty}(V)$ of isotopy classes of self-diffeomorphisms
of $V$ at infinity is called the _diffeotopy group of $V$ at infinity_.

We obtain a homomorphism $r\colon\mathcal{D}(V)\to\mathcal{D}^{\infty}(V)$ by
sending diffeomorphisms to their corresponding equivalence classes at
infinity.

Focusing on closed neighborhoods of the ends instead of closed neighborhoods
of infinity, we obtain an analogue of Definition 2.6.

######  Definition 2.8.

Let $V$ and $V^{\prime}$ be non-compact smooth manifolds and $\epsilon$ and
$\epsilon^{\prime}$ ends of $V$ and $V^{\prime}$ respectively. Suppose
$f_{i}\colon N_{i}\to N_{i}^{\prime},i=1,2$, are diffeomorphisms between
closed neighborhoods of $\epsilon$ and $\epsilon^{\prime}$. Assume $N_{i}$ and
$N_{i}^{\prime}$ are components of $U_{i}$ and $U_{i}^{\prime}$, which are
closed neighborhoods of infinity of $V$ and $V^{\prime}$, respectively. We
refer to $f_{1}$ and $f_{2}$ as _equivalent_ if they agree outside some
suitably large compact subset of $V$ containing the complements of $U_{1}$ and
$U_{2}$. A _diffeomorphism of the end_ $\epsilon$ is an equivalence class of
such diffeomorphisms.

We say that a diffeomorphism $f$ of $\epsilon$ _extends over_ $V$ if the
equivalence class $f$ contains a diffeomorphism $V\to V^{\prime}$. We can
compose such diffeomorphism of the end, making the set of self-diffeomorphism
of $\epsilon$ into a group. Given a group $G$, we define a _$G$ -action at the
end_ to be a homomorphism of $G$ into this group.

As before, we will denote the equivalence class with $f$ instead of $[f]$.
Note that Definition 2.8 fits with Definition 2.6 in the case that $V$ has a
single end.

######  Definition 2.9.

Let $V$ be a non-compact smooth manifold. Two diffeomorphisms of an end
$\epsilon$ of $V$ will be called _isotopic_ if they have representatives that
are properly isotopic as proper embeddings in $V^{\prime}$.

The group $\mathcal{D}^{\epsilon}(V)$ of isotopy classes of self-
diffeomorphisms of $\epsilon$ is called the _diffeotopy_ group of $\epsilon$.

For each end of the manifold $V$, we also obtain an homomorphism
$r_{\epsilon}:\mathcal{D}(V)\to\mathcal{D}^{\epsilon}(V)$ by sending
diffeomorphisms to their corresponding equivalence class with respect to the
end $\epsilon$.

######  Remark 2.10.

We will later refer to $\mathcal{D}^{\epsilon}(V)$ as a subgroup of
$\mathcal{D}^{\infty}(V)$. Strictly speaking, this is not quite correct. A
diffeomorphism $\phi$ of $\epsilon$ embeds just a closed neighborhood $U$ of
the specific end $\epsilon$. However, $\phi$ can be turned into a
diffeomorphism at infinity $\overline{\phi}$ by extending it via the identity.
Let $U$ be a component of $W$, a closed neighborhood of infinity of $V$. Let
$\overline{\phi}$ be the proper embedding of $W$ in $V$ defined by
$\overline{\phi}|_{U}=\phi$ and $\overline{\phi}=$ identity on the other
components of $W$. When we refer to $\phi$ as a diffeomorphism at infinity we
mean $\overline{\phi}$. Under this identification we can interpret
$\mathcal{D}^{\epsilon}(V)$ as a subgroup of $\mathcal{D}^{\infty}(V)$.

######  Remark 2.11.

If $V$ has ends $\epsilon_{i}$ for $i\in\Lambda$, then the direct product
$\prod_{i\in\Lambda}\mathcal{D}^{\epsilon_{i}}(V)$ can be interpreted as a
subgroup of $\mathcal{D}^{\infty}(V)$ as follows. Take a product
$(\phi_{i})_{i\in\Lambda}$, where $\phi_{i}$ embeds a closed neighborhood
$U_{i}$ of the end $\epsilon_{i}$ inside $V$ for each $i$. Then
$(\phi_{i})_{i\in\Lambda}$ defines a diffeomorphism $\phi$ at infinity of $V$:
take a closed neighborhood of infinity $U$, where $U_{i}$’s are components of
$U$, and define $\phi$ to embed $U_{i}$ via $\phi_{i}$ and the remaining
components via the identity. Identifying $(\phi_{i})_{i\in\Lambda}$ with
$\phi$ we turn $\prod_{i\in\Lambda}\mathcal{D}^{\epsilon_{i}}(V)$ into a
subgroup of $\mathcal{D}^{\infty}(V)$.

###  2.2. Group actions

We follow [Gom18] for the terminology.

Let $\mathbb{R}^{4}$ denote the Euclidean space with its standard smoothing.
Let $\Sigma$ be a countable, discrete set and let
$\Gamma\colon\Sigma\times[0,\infty)\hookrightarrow\mathbb{R}^{4}$ be an
injection such that $\gamma_{s}\colon=\Gamma|_{\\{s\\}\times[0,\infty)}$ is a
ray for each $s\in\Sigma$. Let $G$ be a group with the discrete topology.

######  Definition 2.12.

[Gom18, Definition 4.1] Fix $\Gamma$ and $G$ as above. A $G$-action via
diffeomorphisms on $\mathbb{R}^{4}$ will be called _$\Gamma$ -compatible_ if
$G$ acts effectively on $\Sigma$ so that for each $g\in G$ and
$(s,t)\in\Sigma\times[0,\infty)$ we have $g\circ\Gamma(s,t)=\Gamma(g(s),t)$
and so that the stabilizer of each $s$ fixes pointwise a neighborhood of
$\gamma_{s}([0,\infty))$.

Let $\mathcal{G}$ be the set of all groups $G$ so that, after some choice of
$\Gamma$, $G$ acts $\Gamma$-compatibly on $\mathbb{R}^{4}$. $\mathcal{G}^{+}$
denotes the subset for which the action can be chosen to preserve orientation,
and $\mathcal{G}^{*}\subseteq\mathcal{G}^{+}$ is the subset for which there is
such a $\Gamma$-compatible action fixing pointwise a neighborhood of some ray
$\gamma$ in $\mathbb{R}^{4}$ whose image is disjoint from that of $\Gamma$. We
will mainly be interested in the class $\mathcal{G}^{*}\subsetneq\mathcal{G}$.

######  Example 2.13.

  1. (1)

$\mathbb{Q}^{3}$ lies in $\mathcal{G}^{*}$. Take $\Gamma$ to be the inclusion
$\mathbb{Q}^{3}\times[0,\infty)\subset\mathbb{R}^{3}\times\mathbb{R}=\mathbb{R}^{4}$
where $\mathbb{Q}^{3}$ is given the discrete topology. Take the obvious action
on $\mathbb{R}^{3}\times[0,\infty)$, tapered to be trivial on
$\mathbb{R}^{3}\times(-\infty,-1]$. This defines a $\Gamma$-compatible action
of $\mathbb{Q}^{3}$ on $\mathbb{R}^{4}$. By [Gom18, Proposition 4.2], many
uncountable groups such as $\mathbb{Q}^{\omega}$, where $\omega$ denotes the
first infinite cardinal number, lie in
$\mathcal{G}^{*}\subsetneq\mathcal{G}^{+}\subsetneq\mathcal{G}$.

  2. (2)

Every countable group $G$ of Euclidean or hyperbolic isometries of
$\mathbb{R}^{3}$ lies in $\mathcal{G}$, and if it preserves orientation, in
$\mathcal{G}^{+}$. Choose a point $p\in\mathbb{R}^{3}$ with trivial stabilizer
under the elements $g\in G$. Define then
$\Gamma:G\times[0,\infty)\to\mathbb{R}^{3}\times\mathbb{R}=\mathbb{R}^{4}$ by
setting $\Gamma(g,t)=(g(p),t)$.

###  2.3. Main results from [Gom18]

We next state the results of [Gom18] that we will generalize.

######  Lemma 2.14 ([Gom18, Theorem 4.4, Case (1)]).

There are uncountably many diffeomorphism types of $\mathbb{R}^{4}$
homeomorphs $\mathcal{R}$222An $\mathbb{R}^{4}$ homeomorph is a manifold
homeomorphic to $\mathbb{R}^{4}$ but not necessarily diffeomorphic to it. We
follow[Gom18] for the terminology. satisfying the following characterization.
$\mathcal{R}$ embeds in $\mathbb{R}^{4}$ and every $G\in\mathcal{G}$ has an
action on $\mathcal{R}$ that injects into the diffeotopy groups
$\mathcal{D}(\mathcal{R})$ and $\mathcal{D}^{\infty}(\mathcal{R})$. The
elements of $G$ define orientation preserving diffeomorphisms of $\mathcal{R}$
if and only if the defining $G$-action on $\mathbb{R}^{4}$ preserves
orientation.

We briefly discuss the construction in the proof of [Gom18, Theorem 4.4]; this
will be useful in order to better understand our own construction in Section
3.

######  Construction 2.15.

This is the construction of an $\mathbb{R}^{4}$ homeomorph $\mathcal{R}$
satisfying the characterization in Lemma 2.14. We do not provide the proof for
the injection property; it can be found in [Gom18].

We will build $\mathcal{R}$ as an end sum.333This is the non-compact analogue
of the boundary sum. Given two smoothings $\mathcal{R}_{1},\mathcal{R}_{2}$ of
$\mathbb{R}^{4}$, the end sum $\mathcal{R}_{1}\natural\mathcal{R}_{2}$ is
defined as follows: let $\gamma_{1}:[0,\infty)\to\mathcal{R}_{1}$ and
$\gamma_{2}:[0,\infty)\to\mathcal{R}_{2}$ be two smooth properly embedded rays
with tubular neighborhoods $\nu_{1},\nu_{2}$ respectively. Then
$\mathcal{R}_{1}\natural\mathcal{R}_{2}:=\mathcal{R}_{1}\cup_{\phi_{1}}([0,1]\times\mathbb{R}^{3})\cup_{\phi_{2}}\mathcal{R}_{2}$
, where $\phi_{1}:[0,\frac{1}{2})\times\mathbb{R}^{3}\to\nu_{1}$ and
$\phi_{2}:(\frac{1}{2},1]\times\mathbb{R}^{3}\to\nu_{2}$ are orientation
preserving diffeomorphisms which respect the $\mathbb{R}^{3}$ bundle
structures. For a more comprehensive treatment of the end sum operation, see
[CG19]. For a fixed $G\in\mathcal{G}$, $\Sigma$ and $\Gamma$ (as in Definition
2.12), where $G$ acts $\Gamma$-compatibly on $\mathbb{R}^{4}$, identify
$\Sigma$ with $\mathbb{Z}^{+}$. We will find an increasing sequence $(t_{n})$
such that, when each ray $\gamma_{n}:=\Gamma|_{n}$ is restricted to the
interval $[t_{n},\infty)$, we get a multiray $\overline{\Gamma}$ with an
equivariant444By equivariant we mean $g\circ\varphi_{n}(x)=\varphi_{g(n)}(x)$.
tubular neighborhood map, whose restriction over each truncated ray we denote
by $\varphi_{n}\colon[t_{n},\infty)\times\mathbb{R}^{3}\to\mathbb{R}^{4}$.

We truncate the rays $\gamma_{n}$ for each $n$ so that they can be tubed
without getting undesired intersections.

To do so, construct the sequence by induction on $n$, starting with $t_{1}=0$.
Given $n\geq 1$, suppose that for each $i\leq n-1$ we have already
equivariantly defined each $\varphi_{i}|_{[t_{i},t_{n}]\times\mathbb{R}^{3}}$
so that the image lies in a compact set $K_{in}$ disjoint from each other
$K_{jn}$ and from each $\gamma_{j}((t_{n},\infty))$ with $j\leq n$. Choose
$t_{n+1}\geq t_{n}+1$ so that $\gamma_{n+1}([t_{n+1},\infty))$ avoids each
$K_{in}$ and the ball of radius $n$. Then define
$\varphi_{i}|_{[t_{n},t_{n+1}]\times\mathbb{R}^{3}}$ for each $i\leq n$ so
that the induction hypotheses are extended to $n+1$.

If we end sum $\mathbb{R}^{4}$ with copies of a fixed $\mathbb{R}^{4}$
homeomorph along each $\varphi_{n}$, the resulting $\mathbb{R}^{4}$
homeomorph, call it $\mathcal{R}$, inherits a $G$-action.

######  Remark 2.16.

Start with $G\in\mathcal{G}^{*}$ with a $\Gamma$-compatible action on
$\mathbb{R}^{4}$ for some multiray $\Gamma$. By definition, there is a ray
$\gamma$ in $\mathbb{R}^{4}$, whose image is disjoint from $\Gamma$, with a
$G$-fixed neighborhood. Since in Construction 2.15 we used end-sums along the
(truncated) rays $\gamma_{n}$ to construct $\mathcal{R}$, $\gamma$ defines a
proper embedding of $[0,\infty)$ in $\mathcal{R}$ as well. Then the $G$-action
inherited by $\mathcal{R}$ fixes pointwise $\gamma$ and its neighborhood.

##  3\. Proof of Theorem 3.3

The goal of this section is to extend Lemma 2.14; how far can we extend this
result to general punctured 4-manifolds? The starting point will be the case
of a once punctured _smooth, closed_ 4-manifold $M$ (Lemma 3.1). This first
step will give us the construction we will use to prove Theorem 3.3 in full
generality.

######  Lemma 3.1.

Let $M$ be a closed smooth 4-manifold and $p\in M$. Then there exists a
smoothing of $M{\smallsetminus}p$ whose diffeotopy groups
$\mathcal{D}^{\infty}(M{\smallsetminus}p)$ and
$\mathcal{D}(M{\smallsetminus}p)$ are uncountable.

###### Proof.

We start by choosing an $\mathbb{R}^{4}$ homeomorph $\mathcal{R}$. For every
uncountable group $G\in\mathcal{G}^{*}$ there exists (Lemma 2.14) an
$\mathbb{R}^{4}$ homeomeorph, call it $\mathcal{R}$, with a $G$-action fixing
pointwise a neighborhood of some ray $\gamma$ (see Remark 2.16). $G$ injects
in the diffeotopy groups $\mathcal{D}(\mathcal{R})$ and
$\mathcal{D}^{\infty}(\mathcal{R})$ by Lemma 2.14.

$M\\#\mathbb{R}^{4}$ is homeomorphic to $M{\smallsetminus}p$; smooth it as the
smooth connected sum between $M$ and $\mathcal{R}$. This gives a smoothing of
$M{\smallsetminus}p$.

As stated before $G$ acts on $\mathcal{D}(\mathcal{R})$ and
$\mathcal{D}^{\infty}(\mathcal{R})$. Call the group homomorphism $\theta$, i.e
each $g\in G$ defines a self-diffeomorphism $\theta(g)$ of $\mathcal{R}$.555In
Definition 2.6 we defined a $G$-action at infinity as a homomorphism of $G$
into the diffeomorphism group at infinity. Composing this with the quotient
map, we obtain a homomorphism into the diffeotopy group at infinity. In order
to simplify the notation, we will omit the quotient map. When we say that a
$G$-action at infinity defines a diffeomorphism at infinity we usually mean an
isotopy class of diffeomorphisms at infinity. We do the same for $G$-actions
via diffeomorphisms. We extend the $G$-action to an action on
$\mathcal{D}(M{\smallsetminus}p)$ and
$\mathcal{D}^{\infty}(M{\smallsetminus}p)$. Each $g$ will define a self-
diffeomorphism of the punctured manifold. To do so, take a disk $D$ inside the
$G$-fixed neighborhood of $\gamma$; use this disk to perform the smooth
connected sum with $M$. Let $g\in G$. Since the $G$-action is trivial in $D$,
the diffeomorphism $\theta(g)$ is the identity on $D$. We can then extend it
via the identity to $M^{4}{\smallsetminus}\mathring{D^{\prime}}$, where
$D^{\prime}$ is a disk in $M$ around the point $p$. Hence for each $g\in G$ we
get a diffeomorphism of $M{\smallsetminus}p$ which is the identity on
$M{\smallsetminus}\mathring{D^{\prime}}$.

This defines a $G$-action on $\mathcal{D}(M{\smallsetminus}p)$ (and, by post-
composing it with $r$, also on $\mathcal{D}^{\infty}(M{\smallsetminus}p)$).
Call this action $\widetilde{\theta}$; the diffeomorphism defined by $g\in G$
is then $\widetilde{\theta}(g)$.

$M{\smallsetminus}\mathring{D^{\prime}}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\gamma}$$\mathcal{R}{\smallsetminus}\mathring{D}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}id}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}id}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}id}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\theta(g)}$$p$$D^{\prime}$$D$
Figure 2. The construction of the diffeomorphism $\widetilde{\theta}(g)$. The
$G$-fixed ray $\gamma$ going to infinity is in violet. In light gray, the two
4-disks $\mathring{D^{\prime}}$ and $\mathring{D}$ used to perform the
connected sum. In blue we indicate the definition of $\widetilde{\theta}(g)$
on $M\\#\mathcal{R}$: $\theta(g)$ is extended via the identity on the
boundaries of the disks.

We give the above construction, which assigns for each $\theta(g)$
diffeomorphism of $\mathcal{R}$ the diffeomorphism $\widetilde{\theta}(g)$ of
$M{\smallsetminus}p$, the name $\\#$, reminding us that the connected sum is
used to define it. So $\\#(\theta(g))=\widetilde{\theta}(g)$ is defined as
$\widetilde{\theta}(g)|_{\mathcal{R}{\smallsetminus}D}=\theta(g)|_{\mathcal{R}{\smallsetminus}D}$
and $\widetilde{\theta}(g)|_{M{\smallsetminus}D^{\prime}}=id$. See Figure 2
for a visual representation of $\\#$.

Define $\phi$ (resp. $\widetilde{\phi}$) to be the composition $r|\circ\theta$
(resp. $r|\circ\widetilde{\theta}$) where
$r\colon\mathcal{D}(\mathcal{R})\to\mathcal{D}^{\infty}(\mathcal{R})$ is the
homomorphism we defined in Section 2.666By $r|$ we mean the restriction of $r$
on $\theta(G)$ (resp. $\widetilde{\theta}(G)$).

Finally define $\\#^{\infty}$ to be the homomorphism sending an equivalence
class $\phi(g)\in\phi(G)\subseteq\mathcal{D}^{\infty}(\mathcal{R})$ to the
corresponding equivalence class $\widetilde{\phi}(g)$ in
$\widetilde{\phi}(G)\subseteq\mathcal{D}^{\infty}(M{\smallsetminus}p)$. A
representative of $\phi(g)$ is given by the embedding of some closed
neighborhood of infinity $U$ defined as $\theta(g)|_{U}$. Shrink $U$ if
necessary to make it disjoint from $D$. Then $\theta(g)|_{U}$ embeds $U$ in
$M{\smallsetminus}p$ as well. In particular, the restriction of $\theta(g)$ on
$U$ is the same as the restriction of $\widetilde{\theta}(g)$ on $U$ by our
connected sum construction. So $\\#^{\infty}(\phi(g))$ is the equivalence
class of the diffeomorphism at infinity given by the embedding
$\widetilde{\theta}(g)|_{U}$ of $U$ in $M{\smallsetminus}p$, i.e the
equivalence class $\widetilde{\phi}(g)$.

We summarize everything we have obtained so far in the following diagram:

${\mathcal{D}(\mathcal{R})}$${\mathcal{D}^{\infty}(\mathcal{R})}$${G}$${\theta(G)}$${\phi(G)}$${\widetilde{\theta}(G)}$${\widetilde{\phi}(G)}$${\mathcal{D}(M{\smallsetminus}p)}$${\mathcal{D}^{\infty}(M{\smallsetminus}p)}$$\scriptstyle{r}$$\scriptstyle{\widetilde{\theta}}$$\scriptstyle{\theta}$$\scriptstyle{\\#}$$\scriptstyle{r|}$$\subseteq$$\scriptstyle{\\#^{\infty}}$$\subseteq$$\scriptstyle{r|}$$\subseteq$$\subseteq$$\scriptstyle{r}$

Gompf proved [Gom18, Theorem 4.4] that the map $\phi$ (hence also $\theta$) in
the upper row is injective, i.e the $G$-action defines an injection of $G$ in
$\mathcal{D}^{\infty}(\mathcal{R})$ (hence and also in
$\mathcal{D}(\mathcal{R})$) (see Lemma 2.14 above). By our definitions of
$\widetilde{\theta},\widetilde{\phi},\\#$ and $\\#^{\infty}$, both the left
triangle and the right square in the diagram commute.

We want to prove that $\widetilde{\phi}=r\circ\widetilde{\theta}$ is an
injection. To do so, it suffices to show that $\\#^{\infty}$ is injective,
since by commutativity of the diagram
$\widetilde{\phi}=\\#^{\infty}\circ\phi$, and we already know that $\phi$ is
injective.

######  Claim.

The group homomorphism $\\#^{\infty}$ constructed as above is injective

###### Proof of the claim.

For ease of notation, throughout the proof we call $\phi(g)=:g$, meaning that
this is the class of diffeomorphisms of infinity that behave like the
diffeomorphism $\theta(g)$ on some closed neighborhood of infinity $U$.

We prove the claim by contradiction. Suppose it is not true. Then there exists
an element $g\neq id\in\mathcal{D}^{\infty}(\mathcal{R})$ (i.e, $g\neq e\in
G$) which is sent to $id\in\mathcal{D}^{\infty}(M{\smallsetminus}p)$. We will
show that if $\\#^{\infty}(g)=id$, then
$g=id\in\mathcal{D}^{\infty}(\mathcal{R})$ to begin with.

If $\\#^{\infty}(g)=id\in\mathcal{D}^{\infty}(M{\smallsetminus}p)$, then there
are two representatives of their equivalence classes that are properly
isotopic as embeddings of $U$, a closed neighborhood of infinity in
$M{\smallsetminus}p$. By our definition of $\\#^{\infty}(g)$, a representative
is given by the restricion of $\widetilde{\theta}(g)$ on some closed
neighborhood of infinity of $M\\#\mathcal{R}$ entirely contained in
$\mathcal{R}{\smallsetminus}\mathring{D}$. Then shrinking $U$ if necessary, we
can assume that the two representatives of $\widetilde{\phi}(g)$ and $id$
which are _properly isotopic_ embed $U$ in $M{\smallsetminus}p$ as
$\theta(g)|_{U}$ and as $id$. So we have $H:U\times[0,1]\to
M{\smallsetminus}p$ a proper isotopy with $H_{0}=\theta(g)|_{U}$ and
$H_{1}=id$.

We wish to show that there is such a proper isotopy between representatives of
$g$ and $id\in\mathcal{D}^{\infty}(\mathcal{R})$, i.e a proper isotopy
$G:U^{\prime}\times[0,1]\to\mathcal{R}$ between embeddings
$\theta(g)|_{U^{\prime}}$ and $id$ of $U^{\prime}$ a closed neighborhood of
infinity of $\mathcal{R}$.

We can consider the track $H$ of the proper isotopy, i.e the (level
preserving) proper embedding
$H:U\times[0,1]\hookrightarrow(M{\smallsetminus}p)\times[0,1]$. Take
$C:=H(U\times[0,1])\cap((M{\smallsetminus}\mathring{D^{\prime}})\times[0,1])$
(see Figure 3). $U\times[0,1]$ is closed; $H$ is proper and hence a closed
map, which means $H(U\times[0,1])$ is closed as well.
$(M{\smallsetminus}\mathring{D^{\prime}})\times[0,1]$ is compact, so $C$ is
compact as it is the intersection between a closed and a compact set.

Take the preimage $H^{-1}(C)$, which is a compact subset of $U\times[0,1]$.
Exhaust $U\times[0,1]$ by compact sets $K_{n}\times[0,1]$; since $H^{-1}(C)$
is compact, it is contained in some $K_{N}\times[0,1]$, $N\in\mathbb{N}$.
Henceforth, we can find a closed neighborhood of infinity $U^{\prime}\subseteq
U$ so that $U^{\prime}\times[0,1]$ is disjoint from $H^{-1}(C)$ (see Figure
3). Then the image of $U^{\prime}\times[0,1]$ under $H$ is entirely contained
in $\mathcal{R}{\smallsetminus}\mathring{D}$, so the restriction of $H$ to
$U^{\prime}$ defines a proper isotopy between $\theta(g)|_{U^{\prime}}$ and
$id$ as embeddings of $U^{\prime}$ in $\mathcal{R}$. This means that $g=id$ as
equivalence classes in $\mathcal{D}^{\infty}(\mathcal{R})$. This is the
desired contradiction, finishing the proof of the claim.

$M{\smallsetminus}\mathring{D^{\prime}}$$\mathcal{R}{\smallsetminus}\mathring{D}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}U^{\prime}}$${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}U}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}H^{-1}(C)}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\theta(g)(U^{\prime})}$${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}\theta(g)(U)}$
Figure 3. A schematic representation of the track $H$. The image
$H(U\times[0,1])$ is in green lines. The compact set $C$ is colored in light
blue; its preimage under $H$ is bounded by blue lines. The closed neighborhood
of infinity $U^{\prime}$ is in violet; the picture shows how
$H(U^{\prime}\times[0,1])$ is entirely contained in
$\mathcal{R}{\smallsetminus}\mathring{D^{4}}$.

∎

Using the claim, we get that $\widetilde{\phi}$ is indeed an injection.
Clearly, since $\widetilde{\phi}=r\circ\widetilde{\theta}$, the same holds for
$\widetilde{\theta}$. This proves that $G$ injects in
$\mathcal{D}(M{\smallsetminus}p)$ an
$\mathcal{D}^{\infty}(M{\smallsetminus}p)$, showing that both are uncountable.
∎

We started with once punctured closed, smooth 4-manifolds. The following
corollary extends the result of Lemma 3.1 to give uncountably many smoothings
having the desired property.

######  Corollary 3.2.

Let $M$ be a closed, smooth 4-manifold and $p\in M$. Then there exist
uncountably many smoothings of $M{\smallsetminus}p$ whose diffeotopy groups
$\mathcal{D}^{\infty}(M{\smallsetminus}p)$ and
$\mathcal{D}(M{\smallsetminus}p)$ are uncountable.

###### Proof.

Given an uncountable group $G\in\mathcal{G}^{*}$ (see Example 2.13 (1)) we can
choose $\mathcal{R}$ from an _uncountable_ family of $\mathbb{R}^{4}$
homeomorphs $\\{\mathcal{R}_{t}\\}_{t}$ by equivariantly varying the model
summand in the construction (see [Gom18, Lemma 3.2]).

Consider the family of smoothings of $M{\smallsetminus}p$ given by
$\\{M\\#\mathcal{R}_{t}\\}_{t\in\mathbb{R}}$. Since only countably many
$\mathbb{R}^{4}$ homeomorphs can realize a given end [Gom93], this uncountable
family contains an uncountable subfamily of smoothings of $M{\smallsetminus}p$
with pairwise non-diffeomorphic ends. Hence, this represents an uncountable
family of distinct smoothings of $M{\smallsetminus}p$ so that each one
satisfies the properties of the above Lemma 3.1. ∎

We proved the basic case. We can now state the result in full generality; the
techniques we will use in the proof are analogous to the ones used in the
proofs of Lemma 3.1 and Corollary 3.2.

######  Theorem 3.3.

Let $M$ be a topological 4-manifold, $S\subsetneq\mathring{M}$ a non-empty
discrete set of points. Then $M{\smallsetminus}S$ admits uncountably many
smoothings whose diffeotopy groups $\mathcal{D}^{\infty}(M{\smallsetminus}S)$
and $\mathcal{D}(M{\smallsetminus}S)$ are uncountable if:

  1. (1)

$M$ is smoothable or

  2. (2)

$M$ is non-smoothable and $|S|\geq 2$.

######  Remark 3.4.

We impose the assumption $S\subsetneq\mathring{M}$ in order to avoid removing
points on the (possibly non-empty) boundary $\partial M$. Going through the
proof of Lemma 3.1, we see that we made almost no use of the empty boundary
assumption. However if $p\in\partial M$, then the initial idea, connect-
summing with $\mathcal{R}$, needs to be modified. We have to take the boundary
sum with $\mathbb{H}^{4}$, the upper half 4-space. We do not know how the
diffeotopy groups of $\mathbb{H}^{4}$ behave (note that this is
$D^{4}{\smallsetminus}pt$ where the point lies in the boundary
$\mathbb{S}^{3}$, a case we will not cover) and so no direct conclusions can
be made from this construction.

###### Proof of Theorem 3.3.

We already covered case (1) when $|S|=1$ and $M$ is closed (by Remark 3.4 also
compact with possibly non-empty boundary) in Lemma 3.1. To prove Theorem 3.3,
it suffices to prove case (1) for non-compact 4-manifolds; if $M$ is a
_smooth, compact_ 4-manifold, $|S|\geq 2$ then apply the result to
$M^{\prime}:=M{\smallsetminus}p$, in which case $M^{\prime}$ is non-compact,
and $S^{\prime}:=S{\smallsetminus}p$, where $p\in S$. Similarly, if $M$ is a
non-smoothable 4-manifold (2) follows from (1) applied to
$M^{\prime}:=M{\smallsetminus}p$ (by Quinn’s _smoothability theorem_ [Qui82]
$M{\smallsetminus}p$ has a smooth structure) and
$S^{\prime}:=S{\smallsetminus}p$, where $p\in S$.

We first prove (1) for $M$ a non-compact 4-manifold and $|S|=1$. Once we have
this, we will show how to obtain the general result.

######  Claim.

Let $M$ be a non-compact 4-manifold and $p\in\mathring{M}$. Then there exist
uncountably many smoothings of $M{\smallsetminus}p$ whose diffeotopy groups
$\mathcal{D}^{\infty}(M{\smallsetminus}p)$ and
$\mathcal{D}(M{\smallsetminus}p)$ are uncountable.

###### Proof of the claim.

The proof follows that of Lemma 3.1. $M$ is non-compact and hence smoothable
[Qui82]; fix a smoothing. Choose an uncountable $G\in\mathcal{G}^{*}$ (see
Example 2.13) and an associated $\mathbb{R}^{4}$ homeomorph $\mathcal{R}$.
Smooth $M{\smallsetminus}p$ as $M\\#\mathcal{R}$, using a disk $D$ in the
$G$-fixed neighborhood of $\gamma$ in $\mathcal{R}$ to perform the smooth
connected sum. $G$ acts on $\mathcal{D}(\mathcal{R})$ and
$\mathcal{D}^{\infty}(\mathcal{R})$. Keeping the notation consistent, we call
the group homomorphism $\theta$. We extend the $G$-action to an action on
$\mathcal{D}(M{\smallsetminus}p)$ and
$\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$, where $\epsilon_{p}$ is the
end corresponding to the puncture at $p$ (see Definitions 2.3 and 2.9). We do
so as in Lemma 3.1. We extend $\theta(g)$ via the identity over
$M{\smallsetminus}\mathring{D^{\prime}}$, where $D^{\prime}$ is a disk in $M$
around the point $p$. Hence for each $g\in G$ we get a diffeomorphism of
$M{\smallsetminus}p$ which is the identity on
$M{\smallsetminus}\mathring{D^{\prime}}$. This defines a $G$-action on
$\mathcal{D}(M{\smallsetminus}p)$ (and, by post-composing with
$r_{\epsilon_{p}}|$, also on
$\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$). As in Lemma 3.1 call this
action $\widetilde{\theta}$; the diffeomorphism defined by $g\in G$ is
$\widetilde{\theta}(g)$.

We give the above construction, which assigns for each $\theta(g)$
diffeomorphism of $\mathcal{R}$ the diffeomorphism $\widetilde{\theta}(g)$ of
$M{\smallsetminus}p$, the name $\\#_{p}$. Call $\phi$ (resp.
$\widetilde{\phi}$) the composition $r|\circ\theta$ (resp.
$r_{\epsilon_{p}}|\circ\widetilde{\theta}$). Finally define $\\#_{p}^{\infty}$
to be the homomorphism sending an equivalence class
$\phi(g)\in\phi(G)\subseteq\mathcal{D}^{\infty}(\mathcal{R})$ to the
corresponding equivalence class $\widetilde{\phi}(g)$ in
$\widetilde{\phi}(G)\subseteq\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$.

We can summarize the construction so far in a diagram:

${\mathcal{D}(\mathcal{R})}$${\mathcal{D}^{\infty}(\mathcal{R})}$${G}$${\theta(G)}$${\phi(G)}$${\tilde{\theta}(G)}$${\tilde{\phi}(G)}$${\mathcal{D}(M{\smallsetminus}p)}$${\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)\par\par}$$\scriptstyle{r}$$\scriptstyle{\theta}$$\scriptstyle{\tilde{\theta}}$$\scriptstyle{r|}$$\scriptstyle{\\#_{p}}$$\subseteq$$\scriptstyle{\\#_{p}^{\infty}}$$\subseteq$$\scriptstyle{r_{\epsilon_{p}}|}$$\subseteq$$\subseteq$$\scriptstyle{r_{\epsilon_{p}}}$

As in the proof of Lemma 3.1, we shall show that the homomorphism
$\\#^{\infty}_{p}$ is an injection. We prove this by contradiction. We use the
same notation conventions as in Lemma 3.1. We show that if
$\\#_{p}^{\infty}(g)=id\in\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$,
then $g=id\in\mathcal{D}^{\infty}(\mathcal{R})$ to begin with.

If $\\#_{p}^{\infty}(g)=id\in\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$,
then there are two representatives of their equivalence classes that are
properly isotopic as embeddings (in $M{\smallsetminus}p$) of $U$, a
neighborhood of the end $\epsilon_{p}$. Shrinking $U$ if necessary, we can
assume that such representatives of $\widetilde{\phi}(g)$ and $id$ embed $U$
in $M{\smallsetminus}p$ as $\theta(g)|_{U}$ and $id$. So we have
$H:U\times[0,1]\to M{\smallsetminus}p$, a proper isotopy with
$H_{0}=\theta|_{U}(g)$ and $H_{1}=id$.

We wish to show that there is such a proper isotopy between representatives of
$g,id\in\mathcal{D}^{\infty}(\mathcal{R})$, i.e a proper isotopy
$G:U^{\prime}\times I\to\mathcal{R}$ between embeddings
$\theta|_{U^{\prime}}(g)$ and $id$ of $U^{\prime}$, a closed neighborhood of
infinity of $\mathcal{R}$.

Again consider the track of the proper isotopy $H$, the (level preserving)
proper embedding
$H:U\times[0,1]\hookrightarrow(M{\smallsetminus}p)\times[0,1]$. Find a compact
set $K$ separating $U$, a closed neighborhood of $\epsilon_{p}$, from the
other components of a neighborhood of infinity $V$ of $M{\smallsetminus}p$
(see Figure 4). Take $C:=H(U\times[0,1])\cap(K\times[0,1])$ (see Figure 4). As
before, $H(U\times[0,1])$ is closed and $K\times[0,1]$ is compact, so $C$ is
compact.

The preimage $H^{-1}(C)$ under the proper map $H$ is compact in
$U\times[0,1]$. Hence it lies in some compact $K_{N}\times[0,1]$ and we can
find a closed neighborhood $U^{\prime}\subseteq U$ of the end $\epsilon_{p}$
so that $U^{\prime}\times[0,1]$ is disjoint from $H^{-1}(C)$.

We now know that the image of $U^{\prime}\times[0,1]$ under $H$ avoids the
compact region $K\times[0,1]$. Note that $U^{\prime}\times[0,1]$ is connected.
For $t=1$ we have $H_{1}=id$, so in $(M{\smallsetminus}p)\times 1$,
$U^{\prime}$ is contained in the $\mathcal{R}{\smallsetminus}\mathring{D}$
part. Since $K$ disconnects the ends of $M{\smallsetminus}p$ , and the image
under the embedding $H$ of the connected $U^{\prime}\times[0,1]$ must stay
connected, we can conclude that $H(U^{\prime}\times[0,1])$ is entirely
contained in $(\mathcal{R}{\smallsetminus}\mathring{D})\times[0,1]$ (see
Figure 4).

$M{\smallsetminus}\mathring{D^{\prime}}$$\mathcal{R}{\smallsetminus}\mathring{D}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}U^{\prime}}$${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}U}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}H^{-1}(C)}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\theta(g)(U^{\prime})}$${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}\theta(g)(U)}$$K$
Figure 4. A schematic representation of the track of $H$. The compact subset
$K$ that separates the punctured end from the other ends of
$M{\smallsetminus}p$ is colored in gray. The set $C:=H(U\times[0,1])\cap
K\times[0,1]$ is colored in light blue; the closed neighborhood of infinity
$U^{\prime}$ is in violet. As the picture shows, $H(U^{\prime}\times[0,1])$ is
entirely contained in $(\mathcal{R}{\smallsetminus}\mathring{D})\times[0,1]$.

Restricting $H$ to $U^{\prime}$ then defines a proper isotopy between
$\theta(g)|_{U^{\prime}}$ and $id$ as embeddings of $U^{\prime}$ in
$\mathcal{R}$, which is the desired contradiction.

As for the proof of Lemma 3.1, we obtain that $G$ injects via
$\widetilde{\phi}$ (resp. $\widetilde{\theta}$) in
$\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$ (resp.
$\mathcal{D}(M{\smallsetminus}p)$). This implies that
$\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$ and
$\mathcal{D}(M{\smallsetminus}p)$ are uncountable. Since
$\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$ is a subgroup of
$\mathcal{D}^{\infty}(M{\smallsetminus}p)$ (see Remark 2.10), the latter is
uncountable as well.

We need to prove there are uncountably many distinct smoothings of
$M{\smallsetminus}p$ satisfying this property. This works as in Corollary 3.2.
Consider the uncountable family of $\mathbb{R}^{4}$ homeomorphs
$\\{\mathcal{R}_{t}\\}_{t\in\mathbb{R}}$ with uncountable diffeotopy groups.
Take the uncountable subfamily $\\{\mathcal{R}_{s}\\}_{s\in\mathbb{R}}$ having
pairwise non-diffeomorphic ends. As in [Gom93], $M$ can only have countably
many $\mathbb{S}^{3}\times\mathbb{R}$ collared ends. So there is another
subfamily $\\{\mathcal{R}_{u}\\}_{u\in\mathbb{R}}$, with pairwise non-
diffeomorphic ends, so that no end has the same diffeomorphism type of an end
of $M$. Consider then the family of smoothings
$\\{M\\#\mathcal{R}_{u}\\}_{u\in\mathbb{R}}$. This realizes an uncountable
family of distinct smoothings of $M{\smallsetminus}p$ each one satisfying the
desired property for $\mathcal{D}(M{\smallsetminus}p)$ and
$\mathcal{D}^{\infty}(M{\smallsetminus}p)$. This completes the proof of the
claim. ∎

We have proved (1) when $M$ is non-compact and $|S|=1$; if $S$ is an arbitrary
discrete subset, the result follows from the claim. More precisely, let $S$ be
discrete and $M$ non-compact. Take $p\in S$, define $S^{\prime}$ to be
$S{\smallsetminus}p$ and $M^{\prime}:=M{\smallsetminus}S^{\prime}$. Then
$M^{\prime}$ is non-compact. By the claim,
$M^{\prime}{\smallsetminus}p=M{\smallsetminus}S$ has uncountable
$\mathcal{D}(M^{\prime}{\smallsetminus}p)$ and
$\mathcal{D}^{\infty}(M^{\prime}{\smallsetminus}p)$ (since the set $S$ is
discrete, we can find $K$ as in the proof of the claim separating the end
$\epsilon_{p}$ from the other punctured ends) and this completes the proof of
Theorem 3.3. ∎

######  Example 3.5.

Theorem 3.3 implies that $\mathbb{S}^{4}{\smallsetminus}\\{p,q\\}$ with $p\neq
q\in\mathbb{S}^{4}$, which is homeomorphic to
$\mathbb{S}^{3}\times\mathbb{R}$, has uncountably many smoothings whose
diffeotopy groups
$\mathcal{D}(\mathbb{S}^{4}{\smallsetminus}\\{p,q\\})$ and
$\mathcal{D}^{\infty}(\mathbb{S}^{4}{\smallsetminus}\\{p,q\\})$ are
uncountable.

In the proof of Theorem 3.3 we constructed a $G$-action on
$\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$. By carefully modifying the
construction, we obtain the following corollary.

######  Corollary 3.6.

Let $G\in\mathcal{G}^{*}$ be uncountable. Let $M$ be a topological 4-manifold,
$S\subsetneq\mathring{M}$ a non-empty discrete set of points. Call $\Lambda$
the index set of $S$. Let p1 and p2 be the following properties:

  * •

p1: _The direct product $\prod_{i\in\Lambda}G$ injects into the diffeotopy
groups $\mathcal{D}(M{\smallsetminus}S)$ and
$\mathcal{D}^{\infty}(M{\smallsetminus}S)$_.

  * •

p2: _The direct product $G\times G\cdots\times G=G^{|S|-1}$, where $S$ is
finite, injects into the diffeotopy groups $\mathcal{D}(M{\smallsetminus}S)$
and $\mathcal{D}^{\infty}(M{\smallsetminus}S)$_.

Then there are uncountably many smoothings of $M{\smallsetminus}S$ for which
property p1 holds if $M$ is a smoothable 4-manifold.

Moreover, there are uncountable many smoothings of $M{\smallsetminus}S$ for
which property p2 holds when $M$ is a non-smoothable compact 4-manifold and
$|S|\geq 2$.

###### Proof.

The cases $|S|=1$ for p1 and $|S|=2$ for p2 follow from Theorem 3.3. We just
need to prove the p1 case for $|S|\geq 2$; the p2 case descends from it.

We start with a non-compact 4-manifold $M$. Fix an uncountable
$G\in\mathcal{G}^{*}$. Let $M$ be non-compact, $S$ discrete, $|S|\geq 2$. The
index set $\Lambda$ must be countable otherwise $S$ would not be discrete, so
we can identify it with a subset of $\mathbb{Z}^{+}$. Start with $p_{1}\in S$.
In the proof of Theorem 3.3 we constructed a smoothing of
$M{\smallsetminus}p_{1}$ with an injection
$G\hookrightarrow\mathcal{D}^{\epsilon_{p_{1}}}(M{\smallsetminus}p_{1})$. Fix
this smoothing and call the resulting smooth manifold $M^{\prime}$. Now take
$p_{2}\in S$ and smooth $M^{\prime}{\smallsetminus}p_{2}$ as in the proof of
Theorem 3.3 so that $G$ injects into
$\mathcal{D}^{\epsilon_{p_{2}}}(M{\smallsetminus}\\{p_{1},p_{2}\\})$. Since
$S$ is discrete we can choose the disk $D^{\prime}$ away from $p_{1}$ to
perform the connected sum with $\mathcal{R}$. The smooth structure is not
changed near the puncture $p_{1}$. Hence $G$ injects in
$\mathcal{D}^{\epsilon_{p_{1}}}(M{\smallsetminus}\\{p_{1},p_{2}\\})$ as well.

Now continue the procedure for each index $i\in\Lambda$. In the end we get a
smoothing of $M{\smallsetminus}S$ so that $G$ injects into
$\mathcal{D}^{\epsilon_{p_{i}}}(M{\smallsetminus}S)$ for each $i\in\Lambda$.
The direct product
$\prod_{i\in\Lambda}\mathcal{D}^{\epsilon_{p_{i}}}(M{\smallsetminus}S)$ is a
subgroup of $\mathcal{D}^{\infty}(M{\smallsetminus}S)$ (see Remark 2.11).
Hence we get that the direct product $\prod_{i\in\Lambda}G$ injects into
$\mathcal{D}^{\infty}(M{\smallsetminus}S)$. If $M$ is compact and smoothable,
the proof is analogous.

As for p2, start with $M$ a non-smoothable compact 4-manifold, $|S|\geq 3$.
Take $p\in S$ and define $M^{\prime}:=M{\smallsetminus}p$,
$S^{\prime}=S{\smallsetminus}p$. Then $M^{\prime}$ is non-compact and
$|S^{\prime}|\geq 2$. Apply the case of p1 to $M^{\prime}$ and $S^{\prime}$ to
get that $G^{|S^{\prime}|}$ injects into
$\mathcal{D}(M^{\prime}{\smallsetminus}S^{\prime})$ and
$\mathcal{D}^{\infty}(M^{\prime}{\smallsetminus}S^{\prime})$. Note that
$|S^{\prime}|=|S|-1$. Then use the equality
$M^{\prime}{\smallsetminus}S^{\prime}=M{\smallsetminus}S$. To get
uncountability in both cases, choose different $\mathcal{R}$ summands in the
construction as we did in the proof of Theorem 3.3. Thus we can conclude. ∎

######  Remark 3.7.

If we take in consideration different uncountable groups
$G_{i}\in\mathcal{G}^{*}$ for $i\in\Lambda$, where $\Lambda$ is the index set
of $S$, then we can use the proof of Corollary 3.6 combining more group
actions together. This would show that for $\\{G_{i}\\}_{i}$ a family in
$\mathcal{G}^{*}$, the same results as in Corollary 3.6 apply for the product
$\prod_{i\in\Lambda}G_{i}$.

##  4\. Proof of Theorem 4.4

The goal of this section is to extend the following result to general
punctured 4-manifolds.

######  Lemma 4.1 ([Gom18, Theorem 5.1]).

Every diffeomorphism of the end of $\mathcal{R}_{U}$ extends over
$\mathcal{R}_{U}$. That is, the homomorphism
$r\colon\mathcal{D}(\mathcal{R}_{U})\to\mathcal{D}^{\infty}(\mathcal{R}_{U})$
is surjective.

We will be able to achieve a similar statement; in the case of non-compact
4-manifolds, however, we shall restrict only to diffeomorphisms of the
punctured ends since the other ends might have a wild behavior that we cannot
detect with the tools we have developed so far. As an example, there might be
an end that is not topologically collared by any $M\times\mathbb{R}$, where
$M$ is a compact 3-manifold.

As we did for Lemma 3.1, we begin with the case of smooth, compact
4-manifolds. The starting point of the proof will be again a connected sum
argument. As we previously saw (see Remark 3.4), as long as the point is
assumed to be taken out from the interior of the manifold, we can allow the
manifold $M$ to have non-empty boundary.

######  Lemma 4.2.

Let $M$ be a smooth compact 4-manifold and $p\in\mathring{M}$. Then there is a
smoothing of $M{\smallsetminus}p$ so that every diffeomorphism at infinity of
$M{\smallsetminus}p$ extends over $M{\smallsetminus}p$. That is, the
homomorphism
$r\colon\mathcal{D}(M{\smallsetminus}p)\to\mathcal{D}^{\infty}(M{\smallsetminus}p)$
is surjective.

###### Proof.

Smooth $M{\smallsetminus}p$ as the smooth connected sum $M\\#\mathcal{R}_{U}$,
where $\mathcal{R}_{U}$ is Freedman and Taylor’s universal $\mathbb{R}^{4}$
[FT86]. We want to show that
$r\colon\mathcal{D}(M{\smallsetminus}p)\to\mathcal{D}^{\infty}(M{\smallsetminus}p)$
is surjective.

Consider $\phi\in\mathcal{D}^{\infty}(M{\smallsetminus}p)$ and pick $\phi$ as
a representative of the class. Then $\phi$ is a proper embedding of $V$, a
closed neighborhood of infinity of $M{\smallsetminus}p$. By shrinking $V$ if
necessary, we can assume that it lies entirely in the
$\mathcal{R}_{U}{\smallsetminus}\mathring{D}$ part. Call
$\phi(V)=:V^{\prime}$.

Our goal is to restrict $\phi$ to a diffeomorphism at infinity of
$\mathcal{R}_{U}$. At present, $V^{\prime}$ is a closed neighborhood of
infinity of $M{\smallsetminus}p$, but it might intersect the
$M^{4}{\smallsetminus}\mathring{D^{\prime}}$ part non-trivially.

$V^{\prime}$ is a closed subset of $M{\smallsetminus}p$. The intersection with
$M{\smallsetminus}\mathring{D^{\prime}}$ is compact; call it $C$.
$\phi^{-1}(C)$ is then a compact subset of $V$ since $\phi$ is proper. Exhaust
$V$ by compact subsets $K_{n}$; then there exists an $N\in\mathbb{N}$ so that
$\phi^{-1}(C)\subseteq K_{N}$. We can find $T\subseteq V$, a closed
neighborhood of infinity of $M{\smallsetminus}p$, and, since
$V\subsetneq\mathcal{R}_{U}{\smallsetminus}\mathring{D}$, also of
$\mathcal{R}_{U}$, so that the image under $\phi$ lies entirely in
$\mathcal{R}_{U}{\smallsetminus}\mathring{D^{4}}$ (see Figure 5).

$M{\smallsetminus}\mathring{D^{\prime}}$$\mathcal{R}_{U}{\smallsetminus}\mathring{D}$$V^{\prime}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}T}$${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}V}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}C}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}\phi^{-1}(C)}$

Figure 5. Restricting $\phi$ to a diffeomorphism at infinity of
$\mathcal{R}_{U}$. The intersection $C$ between $V^{\prime}=\phi(V)$ and
$M{\smallsetminus}\mathring{D^{\prime}}$ is colored in light blue. The closed
neighborhood of infinity $T$ whose image under $\phi$ is contained in
$\mathcal{R}_{U}{\smallsetminus}\mathring{D}$ is colored in violet.

.

Here $\phi$ defines a diffeomorphism at infinity of
$\mathcal{R}_{U}{\smallsetminus}\mathring{D}$, hence also of
$\mathcal{R}_{U}$. By Lemma 4.1, $\phi$ can be extended to a diffeomorphism of
$\mathcal{R}_{U}$. For ease of notation, call it $\phi$ as well.

The map $\phi$ defines a smooth embedding of $D$, the disk used to perform the
connected sum, inside $\mathcal{R}_{U}$. By the _Cerf-Palais disk theorem_
[Cer61, Pal59], there is an ambient isotopy $H$ of $\mathcal{R}_{U}$ between
the two embeddings $\phi|_{D}:D\hookrightarrow\mathcal{R}_{U}$ and
$\nu:D\hookrightarrow\mathcal{R}_{U}$, where $\nu$ denotes the inclusion.
Moreover, $H$ can be assumed to be trivial on the complement of a compact
subset $K$ of $\mathcal{R}_{U}$.

Then $H_{1}:\mathcal{R}_{U}\to\mathcal{R}_{U}$ is a self-diffeomorphism of
$\mathcal{R}_{U}$ satisfying $H_{1}\circ\phi|_{D}=\nu$ and
$H_{1}|_{\mathcal{R}_{U}{\smallsetminus}K}=id$. The composition
$H_{1}\circ\phi$ defines a self-diffeomorphism of $\mathcal{R}_{U}$ which
extends the diffeomorphism at infinity $\phi$ and restricts to the identity on
the disk $D$ used to perform the connected sum. Call $H_{1}\circ\phi=:\Phi$.
By construction, $\Phi|_{\partial D}$= id; hence we can extend $\Phi$ via the
identity over $M{\smallsetminus}\mathring{D^{\prime}}$, obtaining a self-
diffeomorphism of $M{\smallsetminus}p$, call it $\Phi\\#id$, that extends
$\phi$, the diffeomorphism at infinity of $M{\smallsetminus}p$ (see Figure 6).
This concludes the proof.

$M{\smallsetminus}\mathring{D^{\prime}}$$M{\smallsetminus}\mathring{D^{\prime}}$$\psi^{-1}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}K}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}K}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\Phi|_{K}}$$id$${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}\Phi|_{D}=id}$${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\Phi|_{T}=\phi}$$\psi$${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}T}$${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\phi(T)}$$\mathcal{R}_{U}{\smallsetminus}\mathring{D}$$\mathcal{R}_{U}{\smallsetminus}\mathring{D}$
Figure 6. How the self-diffeomorphism $\Phi\\#id$ of $M\\#\mathcal{R}_{U}$ is
constructed. The map $\Phi\\#id$ embeds $T$ a closed neighborhood of infinity
as $\phi(T)$. Inside the compact region $K$ we have $\Phi=H_{1}\circ\phi$,
with the property that $\Phi|_{D}=id$. Since the composition
$\psi^{-1}\circ\Phi|_{\partial D}\circ\psi=id$, we can extend $\Phi$ via the
identity over $M{\smallsetminus}\mathring{D^{\prime}}$.

∎

Now we can prove the general version of this result. As in the proof of
Theorem 3.3, most of the techinques we will use to prove Theorem 4.4 come from
the proof of Lemma 4.2.

In the statement we use the term _universal behavior_. This indicates that the
smoothings of $M{\smallsetminus}S$ satisfy a property similar to the one of
$\mathcal{R}_{U}$ with respect to diffeotopy groups. Lemma 4.2 is analogous to
Lemma 4.1. This is because both $M{\smallsetminus}p$ and $\mathbb{R}^{4}$ have
a single end. Under the assumptions of Lemma 4.2, the diffeotopy group at
infinity $\mathcal{D}^{\infty}(M{\smallsetminus}p)$ coincides with the
diffeotopy group $\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$ of the end
at $p$. Moreover, the map $r$ coincides with $r_{\epsilon_{p}}$. When dealing
with multiple punctures, and especially non-compact manifolds, _universal
behavior_ has a slightly different meaning which we now specify.

######  Definition 4.3.

Let $M$ be a topological 4-manifold, $S\subsetneq\mathring{M}$ a non-empty
discrete set of points. A smoothing of $M{\smallsetminus}S$ has _universal
behavior_ if the the map $r_{\epsilon_{p}}$ is surjective for all777When $M$
is non-smoothable and $|S|\geq 2$ the property has to hold for a subset
$S^{\prime}\subsetneq S$, where $|S^{\prime}|=|S|-1$. points $p\in S$.

This means that we do not claim that the property holds for other ends of
$M{\smallsetminus}S$.

######  Theorem 4.4.

Let $M$ be a topological 4-manifold, $S\subsetneq\mathring{M}$ a non-empty
discrete set of points. Then $M{\smallsetminus}S$ admits a smoothing with
_universal behavior_ if:

  1. (1)

$M$ is smoothable or

  2. (2)

$M$ is non-smoothable and $|S|\geq 2$.

###### Proof.

We already covered case (1) when $|S|=1$ and $M$ is compact in Lemma 4.2. To
prove Theorem 4.4, it suffices to prove case (1) for $M$ non-compact. Case (2)
follows from (1) applied to $M^{\prime}:=M{\smallsetminus}p$ and
$S^{\prime}:=S{\smallsetminus}p$, where $p\in S$, again using Quinn’s result
[Qui82].

As in the proof of Theorem 3.3, to prove (1) it suffices to show the result is
true for a non-compact 4-manifold $M$ and $|S|=1$. Once we have this, we will
show how to obtain the general result. The compact case is analogous.

######  Claim.

Let $M$ be a non-compact 4-manifold and $p\in\mathring{M}$. Then there is a
smoothing of $M{\smallsetminus}p$ with universal behavior.

###### Proof of the claim.

We will just focus on the part of the proof of Lemma 4.2 where we used the
compact assumption. All the rest follows through, keeping in mind that the
property is valid only for the specific punctured end $\epsilon_{p}$.

We have smoothed $M{\smallsetminus}p$ as $M\\#\mathcal{R}_{U}$. Consider
$\phi\in\mathcal{D}^{\epsilon_{p}}(M{\smallsetminus}p)$ and take $\phi$ a
diffeomorphism of the end $\epsilon_{p}$ as a representative of the class. The
map $\phi$ embeds $V$, a closed neighborhood of $\epsilon_{p}$, which we can
assume lies entirely in the $\mathcal{R}_{U}{\smallsetminus}\mathring{D}$
part, in $M{\smallsetminus}p$ as $\phi(V)=V^{\prime}$. Our goal is to restrict
$\phi$ to a diffeomorphism at infinity of $\mathcal{R}_{U}$. At present,
$V^{\prime}$ is a closed neighborhood of the end $\epsilon_{p}$, but it might
intersect the $M{\smallsetminus}\mathring{D^{\prime}}$ part non-trivially.

We know $V^{\prime}$ is a closed subset of $M{\smallsetminus}p$. Consider $K$,
a compact subset separating $V$ from the other components of a closed
neighborhood of infinity of $M{\smallsetminus}p$ (see Figure 7). Since the
intersection $K\cap V^{\prime}$, call it $C$, is compact, $\phi^{-1}(C)$ is
then a compact subset of $V$ because $\phi$ is proper. Exhaust $V$ by compact
subsets $K_{n}$. Then there exists an $N\in\mathbb{N}$ so that
$\phi^{-1}(C)\subseteq K_{N}$. We can find $T\subseteq V$, a closed
neighborhood of $\epsilon_{p}$, and, since
$V\subsetneq\mathcal{R}_{U}{\smallsetminus}\mathring{D}$, also of
$\mathcal{R}_{U}$, so that the image under $\phi$ is disjoint from $K$ (see
Figure 7). Since $\phi(T)$ is connected, and does not intersect $K$, it must
lie entirely inside $\mathcal{R}_{U}{\smallsetminus}\mathring{D}$, which is
exactly what we want.

$M{\smallsetminus}\mathring{D^{\prime}}$$\mathcal{R}_{U}{\smallsetminus}\mathring{D}$$V^{\prime}$${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}T}$${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}V}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}C}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}\phi^{-1}(C)}$${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}K}$
Figure 7. Restricting $\phi$ to a diffeomorphism at infinity of
$\mathcal{R}_{U}$. The intersection $C$ between $V^{\prime}=\phi(V)$ and $K$
is colored in light blue. The closed neighborhood of infinity $T$ whose image
under $\phi$ is contained in $\mathcal{R}_{U}{\smallsetminus}\mathring{D}$ is
colored in violet.

The rest of the proof is analogous to that of Lemma 4.2. This concludes the
proof of the claim. ∎

Using the claim, we can prove (1). Suppose $S$ is discrete and let $\Lambda$
denote the index set. The set $\Lambda$ must be countable otherwise $S$ would
not be discrete, so we can identify it with a subset of $\mathbb{Z}^{+}$.

Pick $p_{1}\in S$. Using the claim, we obtain a smoothing of
$M{\smallsetminus}p_{1}$ for which the map $r_{\epsilon_{p_{1}}}$ is
surjective. Fix the smoothing and call $M^{\prime}$ the resulting smooth
manifold. Call $S^{\prime}:=S{\smallsetminus}p_{1}$. Take $p_{2}\in S$.
Applying the claim to $M^{\prime}$ and $p_{2}$, we get a smoothing of
$M^{\prime}{\smallsetminus}p_{2}$ for which the map $r_{\epsilon_{p_{2}}}$ is
surjective. Since $S$ is discrete, we can choose the disk
$\mathring{D^{\prime}}$ away from $p_{1}$ to perform the connected sum with
$U$. Then the smooth structure is not changed near the puncture $p_{1}$. Hence
$r_{\epsilon_{p_{1}}}$ remains surjective.

Repeat the procedure for each index $i\in\Lambda$. In the end we get a
smoothing of $M{\smallsetminus}S$ for which the maps $r_{\epsilon_{p_{i}}}$
are surjective for each $i\in\Lambda$. Thus, we have a smoothing of
$M{\smallsetminus}S$ with _universal behavior_. As mentioned, we can prove the
case $M$ compact, smoothable and $|S|\geq 2$ analogously.

To prove case (2), apply (1) to $M^{\prime}:=M{\smallsetminus}p$ and
$S^{\prime}:=S{\smallsetminus}p$ using Quinn’s _smoothablity theorem_ [Qui82].
Keep in mind that this step causes loss of information about the end
$\epsilon_{p}$. We obtain a smoothing of $M{\smallsetminus}S$ for which the
maps $r_{\epsilon_{p_{i}}}$ are surjective for $p_{i}\neq p$. However, we do
not know whether the same holds for the map $r_{\epsilon_{p}}$. ∎

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