arXiv subjects

Dmitry Bolotov

Publications and source records attributed to Dmitry Bolotov.

3 recordsLinked to original sources

arXiv2015

On Gromov's conjecture for totally non-spin manifolds

Gromov's Conjecture states that for a closed $n$-manifold $M$ with positive scalar curvature the macroscopic dimension of its universal covering $\tilde M$ satisfies the inequality $\dim_{mc}\tilde M\le n-2$\cite{G2}. We prove this inequality for totally non-spin $n$-manifolds whose fundamental group is a virtual duality group with $vcd\ne n$. In the case of virtually abelian groups we reduce Gromov's Conjecture for totally non-spin manifolds to the vanishing problem whether $H_n(T^n)^+= 0$ for the $n$-torus $T^n$ where $H_n(T^n)^+\subset H_n(T^n)$ is the subgroup of homology classes which can be realized by manifolds with positive scalar curvature.

math.GT↗

arXiv2009

Gromov's macroscopic dimension conjecture

In this note we construct a closed 4-manifold having torsion-free fundamental group and whose universal covering is of macroscopic dimension 3. This yields a counterexample to Gromov's conjecture about the falling of macroscopic dimension.

math.GT↗

arXiv2009

On Gromov's scalar curvature conjecture

We prove the Gromov conjecture on the macroscopic dimension of the universal covering of a closed spin manifold with a positive scalar curvature under the following assumptions on the fundamental group: 1. The Strong Novikov Conjecture holds for $π$. 2. The natural map $per:ko_n(Bπ)\to KO_n(Bπ)$ is injective.

math.GT↗