Homotopy Epimorphisms and Lusternik-Schnirelmann Category

Abstract

This paper examines the relationship of the Lusternik-Schnirelmann category and related numerical homotopy invariants to the epimorphisms in the homotopy category. The results are of the form: if $N$ is a numerical homotopy invariant and $f:X \to Y$ is an epimorphism, then under certain hypotheses $N(X) \geq N(Y)$. The Eckmann-Hilton dual of the main result is also included; as a corollary, a criterion is given for a categorical subobject of an $H$-space to be an $H$-space.