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FShrike: assume $X$ compact for simplicity, and $X_{m-1}=X_{m-2}$. Collapsing $X_{m-2}$ results in the 1-point compactification of a manifold, namely $X-X_{m-2}$. By Poincar\'e duality, $H_m(X/X_{m-2})=H_m^{BM}(X-X_{m-2})=H^0(X-X_{m-2},O)$ where $O$ is the orientation sheaf. The latter group is $\mathbb{Z}$ or $0$, depending on whether or not $X-X_{m-2}$ is orientable. Assume it is. Then from the long exact sequence of $(X,X_{m-2})$ we deduce $H_m(X)=\mathbb{Z}$ using $H_{\geq m-1}(X_{m-2})=0$. Does this make sense?
