Stable systolic inequalities and cohomology products

2002/04/14 by Victor Bangert, Bangert, Victor, Mikhail G. Katz +2 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #math.DG #math.GT #math.MG #msc:53C23 #msc:55Q15

paper · pdf · doi:10.48550/arxiv.math/0204181

26 pages. To appear in Communications on Pure and Applied Mathematics

arxiv created 2002/04/14 · arxiv updated 2009/11/30

Abstract

Multiplicative relations in the cohomology ring of a manifold impose constraints upon its stable systoles. Given a compact Riemannian manifold (X,g), its real homology H_*(X,R) is naturally endowed with the stable norm. Briefly, if h∈ H_k(X,R) then the stable norm of h is the infimum of the Riemannian k-volumes of real cycles representing h. The stable k-systole is the minimum of the stable norm over nonzero elements in the lattice of integral classes in H_k(X,R). Relying on results from the geometry of numbers due to W. Banaszczyk, and extending work by M. Gromov and J. Hebda, we prove metric-independent inequalities for products of stable systoles, where the product can be as long as the real cup length of X.

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