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




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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.



Any PL-homology-manifold is homotopy equivalent to a manifold
On the revised question:


I am not sure what you mean by doing the same in smooth category but
there are PL -manifolds that are not homotopy equivalent to smooth ones, see e.g. [M. Davis and J-C. Hausmann, Aspherical manifolds without smooth or PL structure, Springer Lecture Notes in Math. 1370, (1989), 135--142] available [here](https://people.math.osu.edu/davis.12/old_papers/Springer/Hausmann.pdf). See also example 2.1 in [here](https://people.math.osu.edu/lafont.1/KnotObstruct2.pdf). 


As for your question 4 you seem to be asking whether the identity map of a homology sphere $S$ extends to a homotopy equivalence between the cone $CS$ on $S$ and a contractible manifold $X$ with $\partial X=S$. This is an obstruction theory problem and (I think) there is no obstruction because all obstruction cocycles lie in relative cohomology groups with coefficients in $\pi\_\*(X)$ or $\pi\_\*(CS)$, which all vanish.


Perhaps by "doing it in the smooth category" you meant finding a smoothing on your manifold minus a star of the singular locus, and then replacing the remaining cones by smooth contractible manifolds. This might not be possible. The issue is that the smoothing must be such that all boundary homology spheres bound smooth contractible manifolds. For example, if a homotopy sphere smoothly bounds a contractible manifold, then removing a small ball from the interior of the contractible manifold gives an $h$-cobordism between the given homotopy sphere and the standard sphere. In higher dimensions the $h$-cobordism is trivial, so the homotopy sphere cannot be exotic.


