Group homomorphisms of small cohomological dimension

Abstract

We characterize group epimorphisms of cohomological dimension one under the additional assumption that their Lusternik–Schnirelmann category is one. We further establish an Eilenberg–Ganea type theorem for epimorphisms between one-relator groups. Finally, we construct examples of epimorphisms between finite groups that have cohomological dimension two but infinite Lusternik–Schnirelmann category.