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Stable systolic inequalities and cohomology products
Author(s) -
Bangert Victor,
Katz Mikhail
Publication year - 2003
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10082
Subject(s) - mathematics , infimum and supremum , cohomology , pure mathematics , multiplicative function , tensor product , norm (philosophy) , cohomology ring , manifold (fluid mechanics) , combinatorics , mathematical analysis , equivariant cohomology , mechanical engineering , political science , law , engineering
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 . © 2003 Wiley Periodicals, Inc.