Homology manifolds and homogeneous compacta

A non-trivial separable metric space $X$ is called an almost homology $n$-manifold if the homology groups $H_k(X,X\mathrm{\backslash }\{x\},\mathbb{Z})$ are trivial for all $x\in X$ and all $k=0,1,\mathrm{.}\mathrm{.},n-1$. We provide a necessary and sufficient condition locally compact homogeneous $ANR$-spaces or strongly locally homogeneous $ANR$-spaces to be almost homology $n$-manifolds.