Cohomological dimension, self-linking, and systolic geometry

2008/07/31 by Dranishnikov, Alexander N., Katz, Mikhail G., Rudyak, Yuli B.
#53C23 #55M30 #57N65 #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0807.5040

Abstract

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 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.

Related

• Stable Systolic Category of Manifolds and the Cup-length
• Systolic freedom of orientable manifolds
• A systolic inequality for 2-complexes of maximal cup-length and systolic area of groups
• Volume of Riemannian manifolds, geometric inequalities, and homotopy theory
• Small values of Lusternik-Schnirelmann and systolic categories for manifolds