Homology manifold

Remarks

Several variants are in use: some authors also ask that $X$ be finite-dimensional or an absolute neighbourhood retract, or use sheaf-theoretic homology or coefficients in a ring other than $\Z$.

Example: a topological $n$-manifold $M$ is a homology $n$-manifold. By excision, $H_i(M, M \setminus \set{x}) \cong H_i(\R^n, \R^n \setminus \set{0})$, and the long exact sequence of the pair gives $H_i(\R^n, \R^n \setminus \set{0}) \cong \tilde{H}_{i-1}(S^{n-1})$, which is $\Z$ for $i = n$ and $0$ otherwise.

Non-example: let $X$ be the wedge of two circles and $x$ the wedge point. Then $X \setminus \set{x}$ is a disjoint union of two open arcs, and the long exact sequence gives $H_1(X, X \setminus \set{x}) \cong \Z^3$, so $X$ is not a homology $1$-manifold.

The suspension of the Poincaré homology sphere is a homology $4$-manifold that is not a topological manifold. The reason is that a small neighbourhood of a suspension point, with that point removed, is not simply connected.