Sobre invariantes de homotopía propia y sus relaciones

Abstract

Numerous mathematicians have introduced algebraic invariants for the study of proper homotopy since L. C. Siebenmann suggested in 1970 that for the study of non compact spaces the homotopy theory should be developed using proper maps instead of continuous maps. Some of these invariants and their relations are analyzed in this thesis. This work contains a detailed study of the properties of proper homotopy groups of Steenrod type and proper homology groups introduced previously by L. J. Hernández-J. I. Extremiana-M. T. Rivas. Some interesting theorems of the Hurewicz type are proved including the relative case and an explicit description of the kernel of the corresponding epimorphisms. We remark that a notion of proper CW-complex is also given in this thesis. The new notion extends the classical notion of CW-complex and permits to construct non compact spaces using only a finite number of proper cells. Some algorithms for computing proper homologies and different proper cellular approximation theorems are also obtained for proper CW-complexes.