# 3 Proof of Theorem 1.2

For the rest of this section, we fix a separable topological group H, and assume that \( \phi : { \mathrm { H o m e o } } ( M ) \to H \) is a homomorphism. For simplicity, we first treat only the case where M is closed; modifications for the case where \( \partial M \ne \emptyset \) and the relative case of \( \mathrm { H o m e o } _ { 0 } ( M \, \mathrm { r e l } \, N ) \) are discussed in Section 4.1, along with a comment for noncompact M.

The proof is somewhat involved, so we have divided it into three major steps. The first is general set-up; the second a “localized” version of continuity (for homeomorphisms with support in a small ball), and the third step improves this local result to a global version by careful use of the fragmentation property. There is a delicate balancing act between steps 2 and 3; in particular, it will be necessary to construct a particular kind of efficient cover of the manifold M to use in the fragmentation argument.

# Step 1: set-up for the proof

Since \( \phi \) is a group homomorphism, it suffices to show continuity at the identity. In other words, we need to prove the following.

3.1. For any neighborhood V of the identity in H, there exists a neighborhood U of the identity in \( \operatorname { H o m e o } _ { 0 } ( M ) \) such that \( U \subset \phi ^ { - 1 } ( \breve { V } ) \).

We first use the Baire category theorem to extract an “approximate” version of condition 3.1.

Lemma 3.2. Let V be a neighborhood of the identity in H. There exists a neighborhood U of the identity in \( \operatorname { H o m e o } _ { 0 } ( M ) \) such that U is contained in the closure of \( \phi ^ { - 1 } ( V ) \).

Remark 3.3. The proof given below works generally for homomorphisms from any Polish group to a separable group; see e.g. [4] for one instance of this (where it is called “Baire category continuity”) and [14] for another in the context of groups with the “Steinhaus property”.

Proof of Lemma 3.2. Take a smaller neighborhood of the identity \( V _ { 0 } \subset V \) such that \( V _ { 0 } \) is symmetric (i.e. \( v \in V _ { 0 } \Leftrightarrow v ^ { - 1 } \in V _ { 0 } ) \), and such that \( V _ { 0 } ^ { 4 } \subset V \). Let \( \{ h _ { i } \} \) be a countable dense subset of H, so that

\[ H = \bigcup _ { i } h _ { i } V _ { 0 } . \]

Let \( W = \phi ^ { - 1 } ( V _ { 0 } ^ { 2 } ) . \). For each translate \( h _ { i } V _ { 0 } \) that intersects the image of \( \mathrm { H o m e o } _ { 0 } ( M ) \), choose an element \( \phi ( g _ { i } ) \in h _ { i } V _ { 0 } \). Then \( \phi ( g _ { i } ) = h _ { i } v _ { i } \) for some \( v _ { i } \in V \), and so

\[ h _ { i } V _ { 0 } = \phi ( g _ { i } ) v _ { i } ^ { - 1 } V _ { 0 } \subset \phi ( g _ { i } ) V _ { 0 } ^ { 2 } . \]

Thus

\[ H = \bigcup _ { i } \phi ( g _ { i } ) V _ { 0 } ^ { 2 } \]

and, taking pre-images, we have

\[ { \mathrm { H o m e o } } ( M ) = \bigcup _ { i } g _ { i } W \]

Since \( { \mathrm { H o m e o } } _ { 0 } ( M ) \) is a Baire space, it cannot be covered by countably many nowhere dense sets. Thus, W is dense in the neighborhood of some \( g \in { \mathrm { H o m e o } } _ { 0 } ( M ) \), so

\[ W W ^ { - 1 } = \phi ^ { - 1 } ( V _ { 0 } ^ { 4 } ) \subset \phi ^ { - 1 } ( V ) \]

is dense in some neighborhood of the identity in \( \mathrm { H o m e o } _ { 0 } ( M ) \). This proves the lemma.

Of course, improving “dense in a neighborhood of the identity” to “contains a neighborhood of the identity” is a nontrivial matter and the main goal of this work!

# Step 2: A localized version (after Rosendal)

As in Step 1, assume that we have fixed a homomorphism \( \phi : { \mathrm { H o m e o } } ( M ) \to H \), and a neighborhood V of the identity in H, with the aim of showing that \( \phi \) satisfies condition 3.1.

The “localized version” of Condition 3.1 that we aim to prove here states, loosely speaking, that a homeomorphism \( f \in { \mathrm { H o m e o } } _ { 0 } ( M ) \) with sufficiently small support lies in \( \phi ^ { - 1 } ( V ) \). The precise statement that we will use in the next step is given in Lemma 3.8 below. Our strategy is to build up to this statement gradually, using a series of lemmata guided by Rosendal’s work in [13].

Notation 3.4. As in Lemma 3.2, we start by fixing a smaller, symmetric neighborhood of the identity \( V _ { 0 } \subset V \) such that \( V _ { 0 } ^ { 8 } \subset V \). Let \( W = \phi ^ { - 1 } ( V _ { 0 } ) \).

The first lemma is a very rough version of our end goal. It states that in any neighborhood of any point of M, we can find an open ball so that all diffeomorphisms supported on that ball are restrictions of elements in \( \phi ^ { - 1 } ( V _ { 0 } ^ { 2 } ) \subset \phi ^ { - 1 } ( V ) \).

Lemma 3.5. Let \( B \subset M \) be an embedded ball. There exists a ball \( B ^ { \prime } \subset B \) such that for every \( f \in { \mathrm { H o m e o } } _ { 0 } ( M ) \) with \( \operatorname { s u p p } ( f ) \subset B ^ { \prime } \), there is an element \( w _ { f } \in W ^ { 2 } \) with \( \operatorname { s u p p } ( w ) \subset B \) and such that the restriction of \( w _ { f } \) to \( B ^ { \prime } \) agrees with f.

Proof. Let \( B \subset M \) be an embedded ball. The argument from the proof of Lemma 3.2 implies that there exists a countable set \( \{ g _ { i } \} \subset { \mathrm { H o m e o } } _ { 0 } ( M ) \) such that

\[ { \mathrm { H o m e o } } ( M ) = \bigcup _ { i } g _ { i } W . \]

We first prove a related claim for these translates of W.

Claim 3.6. There exists a ball \( B ^ { \prime } \subset B \), and a left translate \( g _ { i } W \) such that if \( \operatorname { s u p p } ( f ) \subset B ^ { \prime } \), then there exists \( w _ { f } \in g _ { i } W \) such that

i) \( \operatorname { s u p p } ( w _ { f } ) \subset B \), and

ii) the restriction of \( w _ { f } \) to \( B ^ { \prime } \) agrees with f.

Proof of claim. Let \( B _ { i } \), \( i = 1 , 2 , \dots \). be a sequence of disjoint balls with disjoint closures and with the closure of \( \bigcup _ { i = 1 } ^ { \infty } B _ { i } \) contained in B.

We will show that for some i, every \( f \in { \mathrm { H o m e o } } _ { 0 } ( M ) \) with \( \operatorname { s u p p } ( f ) \subset B _ { i } \) agrees with the restriction of an element of \( g _ { i } W \) supported on B.

Suppose for contradiction that this is not the case. Then there is a sequence \( f _ { i } \in \)\( \mathrm { H o m e o } _ { 0 } ( M ) \) with \( \operatorname { s u p p } ( f _ { i } ) \subset B _ { i } \) and such that \( f _ { i } \) does not agree with the restriction to