Cohomological dimension, self-linking, and systolic geometry

Abstract

Given a closed manifold M, we prove the upper bound of $${1 \over 2}(\dim M + {\rm{cd}}({{\rm{\pi }}_1}M))$$ for the number of systolic factors in a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov’s systolic inequalities. Here “cd” is the cohomological dimension. We apply this upper bound to show that, in the case of a 4-manifold, the Lusternik-Schnirelmann category is an upper bound for the systolic category. Furthermore, we prove a systolic inequality on a manifold M with b 1(M) = 2 in the presence of a nontrivial self-linking class of a typical fiber of its Abel-Jacobi map to the 2-torus.