On spherical fibrations and Poincare complexes

Wen Shen

On spherical fibrations and Poincare complexes

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arxiv 2406.16019 v2 pith:S64OFRWV submitted 2024-06-23 math.GT math.AT

Wen Shen This is my paper

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keywords fibrationshomotopysphericalacutecertaincomplexesconnectedhighly

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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
math.AT 2025-07 reject novelty 6.0 of 10

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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