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Let $M$ be a compact manifold and let $f : M \rightarrow M$ be a homeomorphism which is isotopic to the identity. We will say that $f$ can be fragmented if it satisfies the following property. Let $\mathcal{U}$ be any open cover of $M$. There then exists homeomorphisms $f_1,\ldots,f_n$ from $M$ to itself which are isotopic to the identity and open sets $U_1,\ldots,U_n \in \mathcal{U}$ such that the support of $f_i$ is in $U_i$ and $f = f_1 \cdots f_n$.

It is known that if $M$ is a smooth manifold and $f$ is a diffeomorphism that is smoothly isotopic to the identity, then $f$ may be fragmented (and the resulting $f_i$ are also diffeomorphisms smoothly isotopic to the identity). Indeed, this is is one of the ingredients of Thurston's proof that the identity component of the group of diffeomorphisms of a compact manifold is simple. A proof can be found in chapter 2 of Banyaga's book "The structure of classical diffeomorphism groups".

However, the proof in the smooth case appears to use smoothness in a strong way.

Question : Can an arbitrary homeomorphism $f : M \rightarrow M$ which is isotopic to the identity be fragmented?

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What is the "support" of a homeomorphism $g$? Is it the closure of the set of $x\in M$ such that $g(x)\neq x$ ? –  Qfwfq Feb 10 '11 at 21:43
@unknowngoogle : Yes. –  Andy Putman Feb 10 '11 at 21:45

2 Answers 2

up vote 9 down vote accepted

If I understand you correctly, this is Corollary 1.3. of Kirby and Edwards, Deformations of spaces of imbeddings, Ann. Math. (2) 93 1971 63–88. (I couldn't find an online version of the journal article.)

For $k$-parameter families of maps, but in the smooth or PL case, see Lemma B.0.4 of my recent paper with Scott Morrison (http://arxiv.org/pdf/1009.5025).

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I am reasonably confident that this result is contained in the paper of Edwards and Kirby (Annals, 1971)

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