Group actions on to-grgupoids and crossed complexes, and the homotopy groups of orbit spaces

Abstract

This thesis is concerned with some algebraic and topological aspects of group actions on groupoids, w-groupoids and crossed complexes. One of our main aims is to obtain information on the homotopy groups of orbit spaces. Let A be a groupoid. w-groupold or crossed complex with an action of a group G. The algebraic part of the thesis concentrates on the orbit objects which are universal for G-morphisms into objects with trivial action. Algebraic descriptions are given for orbit groupolds and crossed complexes. Topological considerations arise as follows. We consider the fundamental groupoid of a space in dimension one, and the homotopy crossed complex of a filtered space in higher dimensions. When the space is equipped with a suitable G-actlon there is an action induced on the algebraic invariant. We prove that, under suitable conditions, the fundamental groupoid or homotopy crossed complex of the orbit space is the orbit object of the corresponding invariant of the space. In these cases the algebraic descriptions of orbit objects give information on certain relative homotopy groups of the orbit space. Finally we consider spaces equipped with a cover by subspaces, and various related groupoids. An application of G-groupoids is given to presentations of groups of homeomorphisms.