The LS-category of the product of lens spaces

TL;DR

This paper computes the Lusternik-Schnirelmann category of the product of certain lens spaces, providing evidence for Rudyak's conjecture and establishing new criteria for spin manifolds.

Contribution

It computes the LS-category of lens space products for specific parameters and supports Rudyak's conjecture, also introducing a K-theoretic criterion for the LS-category of spin manifolds.

Findings

Computed $cat(L^n_p\times L^n_q)$ for $p,q>n/2$

Supported Rudyak's conjecture that degree one maps do not raise LS-category

Established a K-theoretic criterion for $cat M=dim M-1$ in spin manifolds

Abstract

We reduced 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 $L^n_p\times L_q^n$ with relatively prime $p$ and $q$. We have computed $cat(L^n_p\times L^n_q)$ for values of $p,q>n/2$. 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 $cat(L^n_p\times L^n_q)\le 2n-2$ for all $n$ and odd relatively prime $p$ and $q$.