#### Volume 15, issue 5 (2015)

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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 ${L}_{p}^{n}×{L}_{q}^{n}$ with relatively prime $p$ and $q$. We have computed $cat\left({L}_{p}^{n}×{L}_{q}^{n}\right)$ 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 $cat\left({L}_{p}^{n}×{L}_{q}^{n}\right)\le 2n-2$ for all odd $n$ and odd relatively prime $p$ and $q$.

##### Keywords
Lusternik–Schnirelmann category, lens spaces, inessential manifolds, ko-theory
Primary: 55M30
Secondary: 55N15