The LS category of the product of lens spaces

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 $cat\left(L_p^n\times L_q^n\right)$ 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 $cat\left(L_p^n\times L_q^n\right)\le 2n-2$ for all odd $n$ and odd relatively prime $p$ and $q$.