H-Semidirect Products

Abstract

AbstractThe concept of H-semidirect product structure on a grouplike space is introduced. It is shown that the loop space ΩX of any based CW-complex X is the H-semidirect product of the identity path-component of ΩX with π1,X. The set of free homotopy classes of maps into a Hsemidirect product inherits the structure of a semidirect product. This leads to new results concerning the nilpotency of homotopy classes of maps into a group-like space.