The LS category of the product of lens spaces

Dranishnikov, Alexander N

doi:10.2140/agt.2015.15.2985

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The LS category of the product of lens spaces

Alexander N Dranishnikov

Algebraic & Geometric Topology 15 (2015) 2983–3008

DOI: 10.2140/agt.2015.15.2985

Abstract

We reduce Rudyak’s conjecture that a degree-one map between closed manifolds cannot raise the Lusternik–Schnirelmann category to the computation of the category of the product of two lens spaces $L_{p}^{n} \times L_{q}^{n}$ with relatively prime $p$ and $q$ . We have computed $cat (L_{p}^{n} \times L_{q}^{n})$ for values $p$ , $q > n ∕ 2$ . It turns out that our computation supports the conjecture.

For spin manifolds $M$ we establish a criterion for the equality $cat M = dim M - 1$ , which is a K–theoretic refinement of the Katz–Rudyak criterion for $cat M = dim M$ . We apply it to obtain the inequality $cat (L_{p}^{n} \times L_{q}^{n}) \leq 2 n - 2$ for all odd $n$ and odd relatively prime $p$ and $q$ .

Keywords

Lusternik–Schnirelmann category, lens spaces, inessential manifolds, ko-theory

Mathematical Subject Classification 2010

Primary: 55M30

Secondary: 55N15

References

Publication

Received: 15 October 2014

Revised: 17 February 2015

Accepted: 20 February 2015

Published: 10 December 2015

Authors

Alexander N Dranishnikov
Department of Mathematics University of Florida 358 Little Hall Gainesville, FL 32611-8105 USA

Volume 15, issue 5 (2015)