HOMOTOPY

Author: GUTIÉRREZ MARÍN JAVIER JOSÉ.
Year: 2003.
University: BARCELONA [www.ub.es].
Place of defense: MATEMÁTICAS.
Place of preparation: UNIVERSIDAD DE BARCELONA.

Summary: The location is a technique well known in commutative algebra and geometry algebraica.Muchas of properties formal locations modules are shared by other similar nature transformations defined in other contextos.Este fact has led to a axiomatización the concept of funtor locational in arbitrary categories, with terminology similar to the algebra. The implementation of the localization topology algrebaica had its roots in the work of Serr and Adams, and began to formalize mainly thanks to the contributions of Sullivan and Quillen. The locations homológicas were the main transport route to homotopía stable, as well as the main tool for calculating groups homotopía stable areas for many years. In the last two decades has grown increasingly using techniques álgrabra commutative in homotopía estable.La theory homotopía stable focuses on the study of the spectra and captures an essential part of the properties momotópicas of spaces, regardless of the peculiar phenomena that occur in dimensions concretas.El treatment axiom stable category using the language of categories of models and categories trianguladas has led to new categories stable, as the category of spectrum semétricos or category of S-módulos The main objective of this report is to study the funtores location homoatopía stable, focusing primarily on the structures algebraic quse preserved under the action of these funtores.Uno of the central results of this work etablece that under appropriate assumptions, funtores Locating in the category homotópica stable transform spectra ring spectra ring i spectra modules on a ring spectra modules on the same spectrum ring (or even located on that spectrum). As a consequence of this is that gets locations conserving GEMs stable and that the location of a spectrum of Elinberg-Mac Lane tene at most two groups homotopía not trivales in size consecutivas.También characterized locations spectrum Elenberg-Mac Lane associated with the ring of integers, which have a single group homotopía non-zero structures rigid ring. This fact shows that the existence of a class of funtores locate itself is not equivalent. These and other recent results in localization theory addressing conservation structures under the action of localizaciones.Algunas of these structures can be included within the broader framework of algebra on opéradas.Las opéradas objects coding structure used algebraicas.Se early in the 70 tools in theory homnotopía to study spaces ties iterados.El survey of opéradas into categories monoidales symmetrical allowing important applications in algebra, topology and physics. In the latter part of the report dealing with the maintenance funtores locating structures defined as algebra on opéradas in a class of models simplicial and monoidal simétrica.Estas structures include spaces ties and spectra ring estrictos.El main result of this part states that in a class of models monoidal, funtores locate conserved structures algebra on opéradas simpliciales confibrantes.Este result shows that the location of an area of bonds is homotópicamente equivalent to a space of bonds and the location of a spectrum ring (strict) is homotópicamente equivalent to a spectrum ring (strict) how 8 do the fu 1c1 ntor locating switches with the suspension.

Author: MURO JIMENEZ FERNANDO.
Year: 2003.
University: SEVILLA [www.us.es].
Place of defense: FACULTAD DE MATEMATICAS.
Place of preparation: FACULTAD DE MATEMATICAS.

Summary: This thesis deals with the development program Whitehead in theory homotopía own. Specifically identifying models algebraic rate homotopía own stable CW-complejos locally compact simply related dimension of four with at least three final. The models finally built are given by the complex cellular chains in homotopía own along with a new invariant cohomológico, we call invariant of Steenrod. It defines a number funtores quadratic in the categories abelianas live modules homology own. Through some such funtores estimates the lower module Whitehead non-trivial in the light of homology. It defines invariants of James-Hopf in homotopía itself that allow us to build new invariant cohomológicos type cup-producto. It also defines, so homotópica, invariant cup-producto a complex chain bounded modules proyectivos finitamente generated on the ring space at the end of a tree. The invariant leads to the kind cup-producto in cohomología categories homotópicas complex chains. There will always be a detailed study of the theory of representations of the Rings space at the end of a tree. As a result, or for spaces with a maximum of three final, it builds a succession of long exact Bockstein in cohomología categories. This sequence makes it possible to calculate from cup-producto complex chains of the class cohomología categories determined by obstruction to the realization geometric morfismos between the complex chains of cellular homotopía own. It defines a funtor suspension of complex cross to quadratic that determines funtor suspension in the L-categoría complex quadratic totally free. Moreover, it is estimated explicitly co-H-estructura a complex quadratic suspended, tried out a structure that is strictly cogrupo. This will develop techniques that allow controlled algebra quadratic give a purely algebraic description of cup-producto complex chains. Using these results we perform various calculations explicit cohomología categories. Among other things tested that class cup-producto complex chains of command is not trivial for two spaces with three or more finals and that the kind of obstruction is trivial for spaces with a maximum of three final. The latter allows us to new invariant cohomológicos in homotopía own, we call on Steenrod, and allowing modeling algebra rates homotopía itself in the manner mentioned above.

Author: BUIJS MARTÍN URTZI.
Year: 2005.
University: MÁLAGA [www.uma.es].
Place of defense: UNIVERSIDAD DE MÁLAGA.
Place of preparation: UNIVERSIDAD DE MÁLAGA - UNIVERSIDAD DE GRANADA.

Summary: The objective of this thesis is to understand the rational behavior of certain construction-related areas and functions based free. The tool which is the starting point of this work is the model for the space functions developed by Beown and Szczarba using the funtor for conducting Bousfield-Gugenheim. From here develop models for evaluating applications, the point-based evaluation, implementation induced between two spaces of the other functions, and the inclusion of a component in the total space, the latter starting restrict all previous one individual component. Once developed these tools are studied algebraic groups homotopía rational components of the space functions in terms of referrals relating to a morfismo generalising results Lupton and Smsith and implementation, methods for working with groups Gottlieb, among others. Here is a complete description of Lie algebra in homotopía of the components of the space functions both as a free-based, using referrals concerning, as a corollary getting a isomorfismo between homotopía rational component of the continuing product tensor of cohomología of X and homotopía Y, generalizing a result known Vigué in which this same evidence in the case where the size of X is less than the connectivity and so that the space functions based escindan as a product of spaces Eilenberg Mac-Lane, rationally.