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arxiv 2209.02236 v3 pith:GSYHKKUT submitted 2022-09-06 math.GT math.CO

Betti numbers and torsions in homology groups of double coverings

classification math.GT math.CO
keywords arrangementgroupsactuallycohomologydoublefiberhomologyinequality
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Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases associated with the Milnor fiber of the arrangement. Recently, an arrangement (the icosidodecahedral arrangement) with the following two peculiar properties was found: (i) the strict version of Papadima-Suciu's inequality holds, and (ii) the first integral homology of the Milnor fiber has a non-trivial $2$-torsion. In this paper, we investigate the relationship between these two properties for double covering spaces. We prove that (i) and (ii) are actually equivalent.

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