For odd n, the CW complex (S^n∨S^n) with a specific 2n-cell attachment is homotopy equivalent to a topological manifold, and the paper claims it is smoothable only in exceptional dimensions linked to the Kervaire invariant.
On spherical fibrations and Poincare complexes
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abstract
In this paper, we prove that certain spherical fibrations over certain CW-complexes are stably fibre homotopy equivalent to $\mm{TOP}$-spherical fibrations (see Definition 1,1). Applying this result, we get a sufficient condition for whether a Poincar$\mm{\acute{e}}$ complex is of the homotopy type of a topological manifold. Moreover, we present the classification for some highly connected manifolds by the homotopy types of highly connected Poincar$\mm{\acute{e}}$ complexes.
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Manifold structures on highly connected Poincar\'e complexes
For odd n, the CW complex (S^n∨S^n) with a specific 2n-cell attachment is homotopy equivalent to a topological manifold, and the paper claims it is smoothable only in exceptional dimensions linked to the Kervaire invariant.