gn.general topology - Homotopy equivalences and cores

Hi all,

Before asking my question, I need to fix some terms and notation.

Let $M$, $M'$ be locally compact, Hausdorff spaces, and $f:Mrightarrow M'$ a homotopy equivalence with homotopy inverse $g:M'rightarrow M$. Set for a natural number $n$, $C^n(M):=(gcirc f)^n(M)$. Similarly, we set $C^{n}(M'):=(fcirc g)^n(M')$.

I'll call the data $(f,g,M,M')$ finitely-cored if there exist $N,N'in mathbb{N}$ such that $(gcirc f)$ restricted to $C^N(M)$ is the identity and $(fcirc g)$ restricted to $C^{N'}(M')$ is also the identity.

My questions are the following:

1) What are the properties of $(f,g,M,M')$ such that it is finitely-cored?

2) Under what conditions (if at all) can $(f,g)$ be deformed to another homotopy equivalence $(f_1,g_1)$ (i.e. $fsim f_1$, $gsim g_1$ )such that $(f_1,g_1,M,M')$ is finitely-cored?

3) For $(f,g,M,M')$ not finitely-cored, define $C_{infty}(M):= cap_{n=1}^infty C^n(M)$ and similarly for $M'$.
Is it true that $C_{infty}(M)$ is homeomorphic to $C_{infty}(M')$? If not, I'd also like to know under what conditions this holds.

Thanks in advance for your replies. Any references are also much appreciated!

P.S: I've invented the term "finitely-cored" here to make the question easier to present, so if there is already a term for this object I apologise for my ignorance.