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.
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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.