Parametrized Homotopy Theory and Fundamental Groupoids

Abstract

In general, smooth G-manifolds are not modeled on a single representation V, so, following the geometry, we need a way of encoding the varying local representations that do appear. The machinery for doing this was developed in detail in Costenoble et al. (Homology Homotopy Appl. 3:265–339, 2001) (electronic), Equivariant stable homotopy theory and related areas (Stanford, CA, 2000), where it was used to give a theory of equivariant orientations. It assembles the various local representations into what we call a representation of the fundamental groupoid. The fundamental groupoid $\Pi _{G}X$ of a G-space X, defined by tom Dieck in Transformation groups, vol 8. Walter de Gruyter & Co., Berlin, 1987, is a category who objects are the maps of orbits G∕H → X; the morphisms in this category are defined in Sect. 2.1. Representations of the fundamental groupoid can be thought of as the natural dimensions of G-vector bundles or G-manifolds, and provide the grading for the extension of ordinary homology and cohomology we will define in Chap. 3 The material from Costenoble et al. (Homology Homotopy Appl. 3:265–339, 2001) (electronic), Equivariant stable homotopy theory and related areas (Stanford, CA, 2000) that we need is recounted in Sect. 2.1.