Topología algebraica moderadamente discontinua para gérmenes métricos subanaliticos

Resumen

We have developed a homology theory and a homotopy theory in the context of metric subanalytic germs. The former is called MD homology and is covered in Chapter 2, which contains a paper that is joined work with my PhD advisors Javier Fernández de Bobadilla and María Pe Pereira and with Edson Sampaio. The latter is called MD homotopy and is covered in Chapter 3. Both theories are functors from a category of germs of metric subanalytic spaces (resp. germs of metric subanalytic spaces that are punctured in a way that will be defined) to a category of commutative diagrams of groups. Similarly to classical homology and homotopy theories, the groups appearing in the target category are abelian in the homology theory for any degree and in the homotopy theory for higher degrees. The main objective was to construct an analytic invariant of real or complex analytic germs that would also contain information about the bi-Lipschitz geometry of the germ. We also had the objective to provide computational tools for that invariant. An optional objective was to concretely compute the invariant for some real or complex analytic germs. The realization of those objectives is given by the MD homology and the MD homotopy as described above. The MD homology shares several of the properties with the singular homology: it is invariant by suitable metric homotopies; it allows a relative and absolute Mayer-Vietoris long exact sequences for a suitable cover of the metric subanalytic germ; and as a consequence we have a certain theorem of excision and a ¿ech spectral sequence. The MD homotopy shares several of the properties of the ordinary homotopy theory of punctured topological spaces: it admits a Hurewicz homomorphism from the MD homotopy to the MD homology; in degree one, the Hurewicz homomorphism is an isomorphism when abelianizing the domain; and when the metric subanalytic germ fulfils a certain condition that softens the one of path-connectedness, it is independent from the choice of base point. The fact that the MD homology provides those computational tools mentioned above similarly to the tools in singular homology make it relatively well computable. We have given examples of computations of both the MD homology and the MD homotopy. In particular, we have given a concrete formula for the MD homology of complex plane algebraic curve germs equipped with the outer metric. That formula reveals how the MD homology recovers both, all Puiseux pairs of the branches of the curve, and the set of all contact numbers between two branches. In 1982, Teissier showed that the geometric type of a complex plane algebraic curve germ equipped with the outer metric coincides with its embedded topological type. Therefore, the MD homology is a complete invariant of irreducible complex plane algebraic curve germs equipped with the outer metric. Both the MD homology and the MD homotopy fulfil the main objective of constructing an analytic invariant of real or complex analytic germs. Indeed, both theories serve as a bi-Lipschitz subanalytic invariant. Therefore, in the context of real or complex analytic germs equipped with the inner or the outer metric, they are analytic invariants. Both theories also provide several powerful computational tools as mentioned above and therefore meet the second objective. In the context of the MD homology we have also attained the optional objective of computing the invariant for an important group of complex analytic germs: for all complex plane algebraic curve germs. Both theories seem very promising since they are rich invariants, as can be seen in the context of complex plane algebraic curve germs, and also well computable thanks to their various computational tools. Furthermore they provide a new and innovative approach to studying algebraic germs. Therefore we have the hope that the work done in this thesis might lay the ground for a new series of research in that direction.