Examples of generalized homology spheres which are not triangulations of spheres

Question

A simplicial complex $\mathcal{K}$ is a generalized homology $n$-sphere if the following hold:

1. $\mathcal{K}$ has the same homology as $S^n$
2. For each non-empty simplex $\sigma \in \mathcal{K}$, $\mathcal{K} \backslash \sigma$ is acyclic (that is, has trivial reduced homology in all dimensions).

What examples are there of finite simplicial complexes which are generalized homology $n$-spheres but are not simply triangulations of $n$-spheres?

Answer 1

Take a triangulated 3-dimensional homology sphere which is not simply-connected, say, Poincare homology sphere. For a non-manifold example, take the suspension of a non-simply connected 3-dimensional homology sphere. For something even more exotic, take the double suspension of a non-simply connected 3-dimensional homology sphere.