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Doctoral Thesis
DOI
Document
Author
Full name
Rodrigo Figueiredo
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2017
Supervisor
Committee
Bianconi, Ricardo (President)
Brunner, Andreas Bernhard Michael
Lopes, Vinicius Cifú
Mariano, Hugo Luiz
Miraglia Neto, Francisco
Title in English
O-minimal De Rham cohomology
Keywords in English
De Rham cohomology
O-minimal manifolds
O-minimal structures
Pfaffian closure
Abstract in English
The aim of this dissertation lies in establishing an o-minimal de Rham cohomology theory for smooth abstract-definable manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function, by following the classical de Rham cohomology. We can specify the o-minimal cohomology groups and attain some properties as the existence of Mayer-Vietoris sequence and the invariance under smooth abstract-definable diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must, working in a tame context that defines sufficiently many primitives, assume the validity of a statement related to Bröcker's problem.
Title in Portuguese
Cohomologia de De Rham o-minimal
Keywords in Portuguese
Cohomologia de De Rham
Estruturas o-minimais
Fecho pfaffiano
Variedades o-minimais
Abstract in Portuguese
O objetivo desta tese reside em estabelecer uma cohomologia de De Rham o-minimal para variedades definíveis abstratas lisas em uma expansão o-minimal do corpo ordenado dos reais, a qual admite decomposição celular lisa e define a função exponencial, seguindo a cohomologia de De Rham clássica. Além de especificarmos os grupos da cohomologia de Rham o-minimal, obtemos algumas propriedades, como a existência da sequência de Mayer-Vietoris e a invariância sob difeomorfismos definíveis abstratos lisos. Todavia, a fim de lograrmos a invariância de nossa cohomologia o-minimal sob homotopia definível abstrata devemos, além de trabalhar num contexto moderado no qual muitas primitivas são definidas, assumir a validade de uma asserção relacionada ao problema de Bröcker.
 
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Phd_Thesis__RF.pdf (1.07 Mbytes)
Publishing Date
2019-05-31
 
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