A *homotopy sphere* is a topological $n$-manifold $M$ which is homotopy equivalent to $S^n$.


A *homology sphere* is a topological $n$-manifold $M$ such that $H\_i(M) \cong H\_i(S^n)$ for all $i$.


Note, by (one version of) Whitehead's Theorem, every simply connected homology sphere is a homotopy sphere. However, there are homology spheres which are not homotopy spheres, for example, the [Poincaré homology sphere](https://en.wikipedia.org/wiki/Homology_sphere#Poincar.C3.A9_homology_sphere).


I've seen in some references that homotopy/homology spheres are assumed to be smooth, but of course the smooth structure is not needed to define them. My question is whether or not this assumption is restrictive. That is:



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> Are there homotopy/homology spheres which admit no smooth structure?
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Are there non-smoothable homotopy/homology spheres?
Since the Poincaré conjecture is known in all dimensions any homotopy $n$-sphere is homeomorphic to $S^n$ and hence admits a smooth structure.


Any manifold of dimension $\le 3$ admits a smooth structure. 


Kirby-Siebenmann famously showed that a closed manifold $M$ of dimension $>4$ admits a PL structure if and only if a certain element of $H^4(M;\mathbb Z\_2)$ vanishes. The element is called the Kirby-Siebenmann class. Thus any homology sphere of dimension $\neq 4$ admits a PL structure.


Kervaire noted on p.71 of [Smooth homology spheres and their fundamental groups](http://www.maths.ed.ac.uk/~aar/papers/kervsmoo.pdf) that any PL homology sphere of dimension $\neq 3$ admits a smooth structure (because it bounds a contractible PL manifold and all such manifolds are smoothable). 


Thus the only remaining case is homology $4$-spheres. Some of them have large fundamental groups for which $4$-dimensional surgery is not yet known to work. It seems smoothability of such spheres is open. Note that any closed PL $4$-manifold admits a smooth structure. 


