Sphere fibrations over highly connected manifolds

2023/05/11 by Samik Basu, Basu, Samik, Aloke Kr. Ghosh +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2305.06738

openalex publication_date 2023/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct sphere fibrations over (n-1)-connected 2n-manifolds such that the total space is a connected sum of sphere products. More precisely, for n even, we construct fibrations S^n-1 → #^k-1(Sⁿ × S^2n-1) → M_k, where M_k is a (n-1)-connected 2n-dimensional Poincaré duality complex which satisfies H_n(M_k)≅ ℤ^k, in a localized category of spaces. The construction of the fibration is proved for k≥ 2, where the prime 2, and the primes which occur as torsion in π_2n-1(Sⁿ) are inverted. In specific cases, by either assuming n is small, or assuming k is large we can reduce the number of primes that need to be inverted. Integral results are obtained for n=2 or 4, and if k is bigger than the number of cyclic summands in the stable stem π_n-1^s, we obtain results after inverting 2. Finally, we prove some applications for fibrations over N# M_k, and for looped configuration spaces.

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