El estudio del infinito a través del espacio exterior

Abstract

In this thesis we introduce the notion of exterior space to study the properties of the spaces at infinite. One of the main problems of the proper homotopy is that there are few limits and colimits to satisfy CM1 axiom of Quillen closed model category. The category we define, the exterior spaces category, provides a nice framework. The proper category can be considered as a full subcategory of the category of the exterior spaces. Moreover, the category of the exterior spaces satisfies all the axioms of closed model category. This way, we can study proper category by taking advantage of the techniques and results available in this axiomatic model. In particular, CM1, will allow to develop different homotopy constructions throughout finite limits and colimits. One of the main problems of the proper category is that, until the moment, some homotopy constructions as homotopy fibres or loop spaces can't be developed. The utilization of the exterior spaces allows the construction of homotopy fibres and loop spaces. In this work we have obtained a consecutive homotopy fibres sequence and loop spaces to study groups type Brown-Grossman and other sequence to study groups type Steenrod The category of the exterior spaces consists of exterior spaces as objects and exterior maps as morphisms. More in detail an exterior space is a topological space, (X, Tx), enriched with an additional structure we name externology. The definition of externology is suggested by the properties of the neighbourhoods at infinite (complements of closed compact sets) that will form the externology. An externology is a non empty collection of open subsets of X, we name them exterior open subsets, satisfying that the finite intersection of exterior open subsets is an exterior open subset and that if an exterior open subset is contained in an open subset, then the open subset is also an exterior open subset. An externology is a topology but the reciprocal it is not true. A map is said to be exterior if it is continuous respect to the topology and the externology. One of the main results is Theorem 2.1.6. In this result we prove that the category of the exterior spaces has a structure of closed model category taking as weak equivalences the morphisms which induce isomorphisms with respect to the Brown -Grossman homotopy groups, that we denominate exterior homotopy groups. We name this structure exterior as well as its associated notions. We also obtain a long exact sequence of a morphism in the category of the exterior spaces with exterior base succession and, as a particular case, the long exact sequence of a pair in the category of pairs the exterior spaces with exterior base sequence. In this thesis we also define the notion of N-complex. An N-complex is constructed inductively. The n-skeleton is constructed from the n-1 skeleton by gluing N-cells of dimension n, that are sequences of n-spheres (n greater or equals to zero) with exterior maps such as if has the weak topology and externology respect to n-skeletons filtration. We prove that N-complexes are exterior cofibrant objects. We prove a Whitehead theorem for N-complex and, as a particular case, a proper Whitehead theorem. We can also prove that it is possible to provide the category of the exterior spaces with other structure of closed model category, we will denominate cylindric, as well as its associated notions to differentiate from the ones defined before for the exterior structure of closed model category and that were named exterior. In other important result, Theorem 5.2.5., we prove that the category of the exterior spaces has a structure of closed model category taking as weak equivalences the morphisms which induce isomorphisms with respect to the Steenrod homotopy groups, that we denominate cylindric homotopy groups. Likewise we obtain a long exact sequence of a morphism in the category of the exterior spaces with exterior base ray and, as a particular case, the long exact sequence of a pair in the category of pairs of the exterior spaces with exterior base ray. In an analogous way to N-complexes, we define R-complexes, which play a similar role to CW-complexes in standard homotopy. We prove a Whitehead theorem in terms of cylindric groups and other results. Finally, we compare the closed model categories induced in the category of the exterior spaces with exterior base ray by the exterior and cylindric structures. The main result establishes an equivalence of categories, restricted to cofibrant cylindric objects, between the localised category of the category of the exterior spaces with exterior base ray obtained by inverting weak equivalences with the cylindric model category structure, and the localised category of the category of the exterior spaces with exterior base ray obtained by inverting weak equivalences with the exterior model category structure.