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] / 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]

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] / 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]