Homology manifold

Definition
Suppose $$n$$ is a nonnegative integer. A topological space $$X$$ is said to be a homology manifold of dimension $$n$$ if it satisfies both the following conditions:


 * $$X$$ is a defining ingredient::locally compact space.
 * For every point $$x \in X$$, the homology groups for the pair $$(X; X \setminus \{ x \})$$ are as follows:

$$H_i(X; X \setminus \{ x \}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & i = n \\ 0, & i \ne n\end{array} \right.$$

We can similarly define the concept of a homology $$G$$-manifold of dimension $$n$$ for any abelian group: instead of looking at the homology groups, we look at the homology groups with coefficients in $$G$$. The usual definition of homology manifold therefore equals homology $$G$$-manifold.