A homotopy sphere is a topological $n$-manifold $M$ such that $\pi_i(M) \cong \pi_i(S^n)$ for all $i$.

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.

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:

Are there homotopy/homology spheres which admit no smooth structure?

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.

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:

Are there homotopy/homology spheres which admit no smooth structure?