Is it true that any compact piecewise linear homology manifold is homotopically equivalent to a (smooth?) manifold of the same dimension?

Is it true that any compact piecewise linear homology manifold is homotopically equivalent to a (smooth?) manifold of the same dimension?

Let me say bit more since my question was wrongly understood.

1. Any link of homology manifold has to be a homoplogy sphere.

2. By double suspension every point on a simplex of dimension at least 1 is a manifold point (it has a neighborhood homeomorphic to an open set in $\mathbb R^n$.

3. Therefore we have a finite discrete set of topological singularities. We can remove an $\epsilon$-neighbborhood around each, its boundary is a homological sphere so we can patch the hole by contactable manifold with the same boundary.

4. It seems to be an answer in the topological category. Am I right?

5. I hope that starting with dimension 5 one can do the same in smooth category.