Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a ‘system of open neighborhoods at infinity’ while an exterior map is a continuous map which is ‘continuous at infinity’. The category of spaces and proper maps is a subcategory of the category of exterior spaces. In this paper we show that the category of exterior spaces has a family of closed simplicial model structures, in the sense of Quillen, depending on a pair {T, T ′} of suitable exterior spaces. For this goal, for a given exterior space T , we construct the exterior T -homotopy groups of an exterior space under T. Using different spaces T we have as particular cases the main proper homotopy groups: the Brown-Grossman, Čerin-Steenrod, p-cylindrical, Baues-Quintero and Farrell-Taylor-Wagoner groups, as well as the standard (Hurewicz) homotopy groups. The existence of this model structure in the category of exterior spaces has interesting applications. For instance, using different pairs {T, T ′} , it is possible to study the standard homotopy type, the homotopy type at infinity and the global proper homotopy type.