Lower bounds for the L.-S. category of products

Abstract:

Halperin and Lemaire introduced L.-S. category type invariants ${\text {left-}}\operatorname {Mcat} (A)$ and ${\text {right-}}\operatorname {Mcat} (A)$(/l) for certain differential algebras $(A,d)$. In particular, they proved that if $(A,d) = {C^{\ast }}(S,k)$ is the $k$-valued singular cochains on $1$-connected space $S$, then these invariants are lower bounds for the classical category ${\text {cat}}(S)$. We use an explicit model for Ganea’s space due to Felix, Halperin, Lemaire, and Thomas to prove $\operatorname {lMcat} (A \otimes B) \leqslant \operatorname {lMcat} (A) + e(B)$, over any field, where $e$ denotes Toomer’s invariant. This proves Ganea’s conjecture for Mcat over fields of arbitrary characteristic.