On the algebraic approximation of Lusternik-Schnirelmann category

Abstract

Algebraic approximations have proved to be very useful in the investigation of Lusternik-Schnirelmann category. In this paper the L.-S. category and its approximations are studied from the point of view of abstract homotopy theory. We introduce three notions of L.-S. category for monoidal cofibration categories, i.e., cofibration categories with a suitably incorporated tensor product. We study the fundamental properties of the abstract invariants and discuss, in particular, their behaviour with respect to cone attachments and products. Besides the topological L.-S. category the abstract concepts cover classical algebraic approximations of the L.-S. category such as the Toomer invariant, rational category, and the A- and M-categories of Halperin and Lemaire. We also use the abstract theory to introduce a new algebraic approximation of L.-S. category. This invariant which we denote by $\ell$ is the first algebraic approximation of the L.-S. category which is not necessarily $\leq 1$ for spaces having the same Adams-Hilton model as a wedge of spheres. For a space $X$ the number $\ell (X)$ can be determined from an Anick model of $X$. Thanks to the general theory one knows \textit{a priori} that $\ell$ is a lower bound of the L.-S. category which satisfies the usual product inequality and increases by at most 1 when a cone is attached to a space.