Small Cover and Halperin-Carlsson Conjecture -II

2010/10/24 by Yu Li, Li Yu, Yu, Li
Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #math.AT #math.CO #math.GT #msc:55R91 #msc:57R22 #msc:57S10 #msc:57S17

paper · pdf · doi:10.48550/arxiv.1010.4929

11 pages

arxiv created 2010/10/24 · arxiv updated 2010/10/26

Abstract

For a small cover Qⁿ and any principal (Z₂)^m-bundle Mⁿ over Qⁿ, it was shown in a previous work of the author that the total sum of Z₂-Betti numbers of Mⁿ is at least 2^m. In this paper, we prove that when Mⁿ is connected, the total sum of Z₂-Betti numbers of such an Mⁿ exactly equals 2^m if and only if Mⁿ is homeomorphic to a product of spheres, and Qⁿ in this case must be a generalized real Bott manifold (or equivalent, Qⁿ is a small cover over a product of simplices).

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