Cohomological dimension, self-linking, and systolic geometry

Given a closed manifold M, we prove the upper bound of (n+d)/2 for the length of a product of systoles that can form a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov’s systolic inequalities. Here n is the dimension of M, while d is the is the cohomological dimension of its fundamental group. We apply this upper bound to show that, in the case of a 4-manifold, the Lusternik–Schnirelmann category is an upper bound for such length. 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 the typical fiber of its Abel–Jacobi map to the 2-torus.