## Algebraic & Geometric Topology

### The LS category of the product of lens spaces

Alexander N Dranishnikov

#### 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 $p$, $q > n∕2$. 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 $q$.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 5 (2015), 2983-3008.

Dates
Revised: 17 February 2015
Accepted: 20 February 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841039

Digital Object Identifier
doi:10.2140/agt.2015.15.2985

Mathematical Reviews number (MathSciNet)
MR3426700

Zentralblatt MATH identifier
1346.55004

#### Citation

Dranishnikov, Alexander N. The LS category of the product of lens spaces. Algebr. Geom. Topol. 15 (2015), no. 5, 2983--3008. doi:10.2140/agt.2015.15.2985. https://projecteuclid.org/euclid.agt/1510841039

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