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Master's Dissertation
DOI
https://doi.org/10.11606/D.55.2011.tde-22032011-090041
Document
Author
Full name
Northon Canevari Leme Penteado
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2011
Supervisor
Committee
Manzoli Neto, Oziride (President)
Hartmann Júnior, Luiz Roberto
Lucas, Laercio Aparecido
Title in Portuguese
O produto cartesiano de duas esferas mergulhado em uma esfera em codimensão um
Keywords in Portuguese
Cohomologia
Dualidades
Grupo fundamental
h-cobordismo
Homologia
Mergulho de variedades
Número interseção
Abstract in Portuguese
James W. Alexander, no artigo[1],mostra que se tivermos um mergulho PL f : 'S POT. 1' × 'S POT. 1' 'S POT. 3', então o fecho de uma das componentes conexas de 'S POT. 3' f('S POT. 1' × 'S POT. 1') é homeomorfo a um toro sólido, isto é, homeomorfo a 'S POT. 1' × 'D POT. 2'. Este teorema ficou conhecido por Teorema do toro de Alexander. Nesta dissertação, estamos detalhando a demonstração deste teorema feita em[25] que é diferente da demonstração apresentada em [1]. Mais geralmente, para um mergulho diferenciável f : 'S POT. p' × 'S POT. q' 'S POT. p + q+1' , demonstra-se que o fecho de uma das componentes conexasde 'S POT. p +q + 1' f('S POT. p' × 'S POT. q') é difeomorfo a 'S POT. p' × 'D POT. q + 1' se p q 1 e p + q 'DIFERENTE DE' 3 ou se p = 2 e q = 1 um dos fechos será homeomorfo a 'S POT. 2' × 'D POT. 2' , nesta dissertação estaremos também detalhando estas demonstrações feita em [20]
Title in English
Product of two spheres embedded in sphere in codimension one
Keywords in English
Cohomology
Duality
Embedding of manifolds
Fundamental group
h-cobordim
Homology
Intersection number
Abstract in English
James W. Alexander shows in[1] that the closure of one of the two connected components of 'S POT. 3'f( 'S POT. 1 × 'S POT. 1') is homeomorphic to a solid torus 'S POT. 1' × 'D POT. 2' , where f : 'S POT. 1' ×' SPOT. 1' 'S POT. 3' is a PL embedding. This result became known as Alexanders torus theorem. In this dissertation we are detailing the proof of this theorem made in[25] which is different from the demonstration presented in[1]. More generally, when considering a smooth embeding f : 'S POT. p' × 'S POT. q' ' SPOT. p+q+1' , it is demonstrated that the closure of one of the two connected components 'S POT. p+q+1' f ('S POT. p' × 'S POT. q' ) is diffeomorphic to 'S POT. p' × 'D POT. q+1' if p q 1 and p+q 'DIFFERENT OF' 3 or if p = 2 and q = 1 one of the closures will be homeomorphic to 'S POT. 2' × 'D POT. 2'. In this work we are also detailing the proves made in[20]
 
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Publishing Date
2011-03-22
 
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