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Master's Dissertation
DOI
10.11606/D.45.2014.tde-11022014-095205
Document
Author
Full name
Renato Vasconcellos Vieira
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2014
Supervisor
Committee
Goncalves, Daciberg Lima (President)
Barros, Tomas Edson
Cardona, Fernanda Soares Pinto
Title in Portuguese
Topologia algébrica não-abeliana
Keywords in Portuguese
$n$-Cubos cruzados de grupos
cat$^n$-grupos
teorema generalizado de Seifert-van Kampen
teoria de homotopia.
topologia algébrica
Abstract in Portuguese
O presente trabalho é uma apresentação de aplicações de estruturas da álgebra de dimensões altas para a teoria de homotopia. Mais precisamente mostramos que existe uma equivalência entre as categorias dos cat$^n$-grupos e a dos $n$-cubos cruzados de grupos, ambas equivalentes a categoria das $n$-categorias estritas internas à categoria de grupos, e uma certa subcategoria da categoria dos $n$-cubos fibrantes, os chamados $n$-cubos de Eilenberg-MacLane. Além disso existe uma equivalência entre uma localização dessa subcategoria e a categoria homotópica dos $(n+1)$-tipos homotópicos, o que sugere a utilidade de usar as estruturas algébricas apresentadas como invariantes topológicas. O teorema central dessa teoria, o teorema generalizado de Seifert-van Kampen, diz que o funtor dos $n$-cubos de fibração aos cat$^n$-grupos usado para mostrar a equivalência mencionada preserva o colimite de certos diagramas e que nesses casos conectividade é preservada, o que permite certas computações. Apresentaremos definições das estruturas algébricas mencionadas além de como calcular certos colimites na categoria de $n$-cubos cruzados de grupos, demonstraremos os teoremas principais da teoria e mostramos como usar esses resultados para generalizar resultados clássicos da topologia algébrica como o teorema de Blakers-Massey, o teorema de Hurewicz e a fórmula de Hopf para homologia de grupos.
Title in English
Non-abelian algebraic topology
Keywords in English
algebraic topology
cat$^n$-groups
Crossed $n$-cubes of groups
Generalized Seifert-van Kampen theorem
homotopy theory.
Abstract in English
The present work is a presentation of applications to homotopy theory of structures in higher dimensional algebra. More precisely we show how the categories of crossed $n$-cubes of groups and of cat$^n$-groups, both equivalent to the category of strict $n$-categories internal to the category of groups, are equivalent to a subcategory of the category of fibrant $n$-cubes, namely the Eilenberg-MacLane $n$-cubes. There is also an equivalence between a localization of the category of Eilenberg-MacLane $n$-cubes and the homotopy category of homotopy $(n+1)$-types, which suggests the usefulness of the presented algebraic structures as topological invariants. The central theorem of this theory, the generalized Seifert-van Kampen theorem, states that the functor from $n$-cube of fibrations to the cat$^n$-groups used to show the aforementioned equivalence preserves the colimit of certain diagrams, and in these cases connectivity is preserved, which permits some computations. We present definitions of the relevant algebraic structures and also how to calculate certain colimits in the category of crossed $n$-cubes of groups, we demonstrate the main theorems of the theory and then we show how to generalize classical results in algebraic topology like the Blakers-Massey theorem, Hurewicz theorem and Hopf's formula for the homology of groups.
 
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Publishing Date
2014-04-03
 
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