Applied Categorical Structures

, Volume 12, Issue 3, pp 225–243 | Cite as

Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

  • J. M. Garcia-Calcines
  • M. Garcia-Pinillos
  • L. J. Hernandez-Paricio


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 neighbourhoods 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.

proper homotopy theory exterior space closed model category proper homotopy groups Whitehead theorem 


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Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • J. M. Garcia-Calcines
    • 1
  • M. Garcia-Pinillos
  • L. J. Hernandez-Paricio
    • 2
  1. 1.Departamento de Matemática FundamentalUniversidad de La LagunaLa LagunaEspaña
  2. 2.Departamento de Matemáticas y ComputaciónUniversidad de La RiojaLogroñoEspaña

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