What is the Support for a homeomorphisms' composition? I was reading this article about Homeo group:
https://www.ams.org/journals/notices/202104/noti2252/noti2252.html?adat=April%202021&trk=2252&galt=feature&cat=feature&pdfissue=202104&pdffile=rnoti-p482.pdf
All functions are homeomorphisms from the group Homeo(M) for some non compact manifold M.
$f, g \in Homeo_c(M)$ the identity component.
$B \subset M$ an open ball.
I yellow highlighted the important assumptions and ultimately want to understand what is red highlighted. 
$i.e.$ I'd like to know how the support of the composition of this homeomorphisms work.
Rather than an explanation of this particular case (which would be highly appreciated too), I'd like to know where I can study this stuff because I'm really lost at calculating the support for the conjugate $ga^{-1}g^{-1} \;$ and so on.
Thank you in advance.

Here is some additional info, we have property 2 and want to prove property 3.
"1. Fragmentation. Let $\mathcal{O}$ be an open cover of $M$. Then $\operatorname {Diff}_c(M)$ and $\operatorname {Homeo}_c(M)$ are generated by homeomorphisms supported on elements of $\mathcal{O}$.
2. Localized perfectness. Let $B$ be an open ball in $M$. Any element of $\operatorname {Diff}_c(B)$ or $\operatorname {Homeo}_c(B)$ can be written as a product of commutators.
3. Simplicity of the commutator group. Let $g \neq id$ in $\operatorname {Diff}_c(M)$, and let $B \subset M$ be an open ball. Then any commutator in $\operatorname {Diff}_c(B)$ lies in the normal closure of $g$. The same holds replacing $\operatorname {Diff}$ with $\operatorname {Homeo}$.
While I have deliberately stated these in parallel for Diff and Homeo, the proofs of 1 and 2 are entirely different in the smooth and $C^0$ case. Fragmentation is easy for diffeomorphisms: an element of $\operatorname {Diff}_c(M)$ is always the time-one map of a compactly supported time dependent vector field, and cutting that vector field off by smooth bump functions is the only tool required for fragmentation."
This article didn't define support.
 A: Lemma 
Let $f,g \in Homeo(M)$ be self-homeomorphisms, M a Manifold. 
With supports $supp(f) = A$ and $supp(g) = B$, respectively. 
Accordingly with the definition given by @Lee Mosher. 
Let $\beta \in M - (A \cup B)$, then
$fg(\beta) = f(\beta) = \beta$ 
Therefore $supp(fg) \subset A \cup B$
If we assume $A \cap B = \emptyset$, we can guarantee $A \cup B = supp(fg)$: 
Let $\alpha \in A \cup B$ 
Case 1) $\alpha \in A$ $\subset M - B$ 
$fg(\alpha) = f(\alpha) \neq \alpha $, then $\alpha \in supp(fg)$ 
Case 2) $\alpha \in B$ $\subset M - A$ 
$fg(\alpha) = f(\alpha') \;\;\;$ for some $\alpha' \neq \alpha$ 
$\;\;\;\;\;\;\;\;\; = \alpha' \;\;\;\;\;\;\;$ Since g is injective, necessarily $\alpha' = g(\alpha) \in B$ 
then $\alpha \in supp(fg)$ 
Therefore $A \cup B \subset supp(fg)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \square$
Now for my particular question: 
As $f = aba^{-1}b^{-1}$ with $a, b \in Homeo(M)$ and $supp(f) = B$, 
then $\;supp(a) = ba^{-1}b^{-1}(B)$ 
$\therefore \;supp(a) = a(supp(a)) = aba^{-1}b^{-1}(B) = f(B) = B\;\;\;$ since a and f are injective.
Notice $\;supp(a) = supp(a^{-1})$ since a is self-homeomorphism. 
Let $\alpha \in M-B$, then $ga^{-1}g^{-1}(g(\alpha)) = ga^{-1}(\alpha) = g(\alpha)$ 
Therefore $supp(ga^{-1}g^{-1}) \subset g(B)\;$ since g is bijective.
Let $\beta \in B$, then $ga^{-1}g^{-1}(g(\beta)) = ga^{-1}(\beta) = g(\beta')$ with $\beta' \neq \beta$ and since g is injective, $g(\beta') \neq g(\beta)$ 
Therefore $g(B) \subset supp(ga^{-1}g^{-1})$ 
$\therefore supp(ga^{-1}g^{-1}) = g(B)$
As $B \cap g(B) = \emptyset$, by Lemma: 
$supp([a,g]) = supp(aga^{-1}g^{-1}) = B \cup g(B)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \square$
If you find any errors, feel free to let me know. Thank you.
