On Gromov's conjecture for totally non-spin manifolds

2014/02/18 by Bolotov, Dmitry, Dranishnikov, Alexander
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1402.4510

Abstract

Gromov's Conjecture states that for a closed n-manifold M with positive scalar curvature the macroscopic dimension of its universal covering M satisfies the inequality dim_mc M≤ n-2\citeG2. We prove this inequality for totally non-spin n-manifolds whose fundamental group is a virtual duality group with vcd≠ n. In the case of virtually abelian groups we reduce Gromov's Conjecture for totally non-spin manifolds to the vanishing problem whether H_n(Tⁿ)⁺= 0 for the n-torus Tⁿ where H_n(Tⁿ)⁺⊂ H_n(Tⁿ) is the subgroup of homology classes which can be realized by manifolds with positive scalar curvature.

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