Homotopy Actions, Cyclic Maps and their Duals

An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The restriction F|_A: A to X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an H-space that are compatible with the H-structure. As a corollary, we prove that if any two actions F and F' of A on X have cyclic maps f and f' with Omega(f) = Omega(f'), then Omega(F) and Omega(F') give the same action of Omega(A) on Omega(X). We introduce a new notion of the category of a map g and prove that g is cocyclic if and only if the category is less than or equal to 1. From this we conclude that if g is cocyclic, then the Berstein-Ganea category of g is <= 1. We also briefly discuss the relationship between a map being cyclic and its cocategory being <= 1.Comment: 16 pages, LaTeX 2

Free Torus Actions and Two-Stage Spaces

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work.Comment: 17 pages, to appear in Math. Proc. Camb. Philos. So

The evaluation subgroup of a fibre inclusion

Given a fibration of simply connected CW complexes of finite type, we study the evaluation subgroup of the fibre inclusion as an invariant of fibre-homotopy type. For spherical fibrations, we show the evaluation subgroup may be expressed as an extension of the Gottlieb group of the fibre sphere provided the classifying map induces the trivial map on homotopy groups. We extend this result after rationalization: We show that the rationalized evaluation subgroup of the fibre inclusion decomposes as the direct sum of the rationalized Gottlieb group of the fibre and the rationalized homotopy group of the base if and only if the classifying map induces the trivial map on rational homotopy groups.Comment: 15 page