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Systolic category, cohomological dimension, and self-linking

Given a closed manifold M , we prove the upper bound catsys(M) ≤ dim M+cd(π1M) 2 for the systolic category of M , where “cd” is the cohomological dimension. We apply this upper bound to deduce the inequality catsys(M) ≤ catLS(M) between the systolic category and the Lusternik–Schnirelmann category of 4-manifolds. Furthermore we obtain a lower bound for the systolic category of M with b1(M) = 2 in the presence of a nontrivial self-linking class of the typical fiber of its Abel–Jacobi map to the 2-torus.