Volume 15, issue 5 (2015)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
The LS category of the product of lens spaces

Alexander N Dranishnikov

Algebraic & Geometric Topology 15 (2015) 2983–3008
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 Lpn × Lqn with relatively prime p and q. We have computed cat(Lpn × Lqn) for values p, q > n2. It turns out that our computation supports the conjecture.

For spin manifolds M we establish a criterion for the equality catM = dimM 1, which is a K–theoretic refinement of the Katz–Rudyak criterion for catM = dimM. We apply it to obtain the inequality cat(Lpn × Lqn) 2n 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