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?



Fragmenting a homeomorphism of a compact manifold
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](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=101825&vfpref=html&r=37&mx-pid=283802). (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>).


