Small Cover and Halperin-Carlsson Conjecture -II

Details Small Cover and Halperin-Carlsson Conjecture -II Small Cover and Halperin-Carlsson Conjecture -II

2010-10-24

arxiv.org/abs/1010.4929v1

Mathematics

Algebraic Topology

11 pages

Scientific paper

For a small cover Q^n and any principal (Z_2)^m-bundle M^n over Q^n, it was shown in a previous work of the author that the total sum of Z_2-Betti numbers of M^n is at least 2^m. In this paper, we prove that when M^n is connected, the total sum of Z_2-Betti numbers of such an M^n exactly equals 2^m if and only if M^n is homeomorphic to a product of spheres, and Q^n in this case must be a generalized real Bott manifold (or equivalent, Q^n is a small cover over a product of simplices).

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