Cohomological dimension, self-linking, and systolic geometry

Alexander N. Dranishnikov; Mikhail G. Katz; Yuli B. Rudyak

doi:10.1007/s11856-011-0075-8

Cohomological dimension, self-linking, and systolic geometry

Alexander N. Dranishnikov, Mikhail G. Katz, Yuli B. Rudyak

Department of Mathematics

University of Florida

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Given a closed manifold M, we prove the upper bound of 1/2(dim M + cdπ₁M)) for the number of systolic factors in a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov's systolic inequalities. Here "cd" is the cohomological dimension. We apply this upper bound to show that, in the case of a 4-manifold, the Lusternik-Schnirelmann category is an upper bound for the systolic category. Furthermore, we prove a systolic inequality on a manifold M with b₁(M) = 2 in the presence of a nontrivial self-linking class of a typical fiber of its Abel-Jacobi map to the 2-torus.

Original language	English
Pages (from-to)	437-453
Number of pages	17
Journal	Israel Journal of Mathematics
Volume	184
Issue number	1
DOIs	https://doi.org/10.1007/s11856-011-0075-8
State	Published - Aug 2011

Bibliographical note

Funding Information:
∗ Supported by the NSF, grant DMS-0604494. ∗∗ Supported by the Israel Science Foundation (grants 84/03 and 1294/06) and the BSF (grant 2006393). † Supported by the NSF, grant 0406311. Received May 26, 2009 and in revised form October 11, 2009

Funding

∗ Supported by the NSF, grant DMS-0604494. ∗∗ Supported by the Israel Science Foundation (grants 84/03 and 1294/06) and the BSF (grant 2006393). † Supported by the NSF, grant 0406311. Received May 26, 2009 and in revised form October 11, 2009

Funders	Funder number
National Science Foundation	DMS-0604494
Bonfils-Stanton Foundation	2006393, 0406311
Israel Science Foundation	1294/06, 84/03

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title = "Cohomological dimension, self-linking, and systolic geometry",

abstract = "Given a closed manifold M, we prove the upper bound of 1/2(dim M + cdπ1M)) for the number of systolic factors in a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov's systolic inequalities. Here {"}cd{"} is the cohomological dimension. We apply this upper bound to show that, in the case of a 4-manifold, the Lusternik-Schnirelmann category is an upper bound for the systolic category. Furthermore, we prove a systolic inequality on a manifold M with b1(M) = 2 in the presence of a nontrivial self-linking class of a typical fiber of its Abel-Jacobi map to the 2-torus.",

author = "Dranishnikov, {Alexander N.} and Katz, {Mikhail G.} and Rudyak, {Yuli B.}",

note = "Funding Information: ∗ Supported by the NSF, grant DMS-0604494. ∗∗ Supported by the Israel Science Foundation (grants 84/03 and 1294/06) and the BSF (grant 2006393). † Supported by the NSF, grant 0406311. Received May 26, 2009 and in revised form October 11, 2009",

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AU - Katz, Mikhail G.

AU - Rudyak, Yuli B.

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AB - Given a closed manifold M, we prove the upper bound of 1/2(dim M + cdπ1M)) for the number of systolic factors in a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov's systolic inequalities. Here "cd" is the cohomological dimension. We apply this upper bound to show that, in the case of a 4-manifold, the Lusternik-Schnirelmann category is an upper bound for the systolic category. Furthermore, we prove a systolic inequality on a manifold M with b1(M) = 2 in the presence of a nontrivial self-linking class of a typical fiber of its Abel-Jacobi map to the 2-torus.

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Cohomological dimension, self-linking, and systolic geometry

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