Loop theorem-sphere theorem: The Tower Construction

1980 ◽  
pp. 1-17
Keyword(s):  
Sphere Theorem ◽  
Loop Theorem

ULC Properties in Neighbourhoods of Embedded Surfaces and Curves in E3

1973 ◽  
Vol 25 (1) ◽  
pp. 31-73 ◽  
Author(s):  
J. W. Cannon
Keyword(s):  
Normal Subgroup ◽  
Approximation Theorem ◽  
Sphere Theorem ◽  
Curves And Surfaces ◽  
Local Homology ◽  
Loop Theorem ◽  
Embedded Surfaces ◽  
Nontrivial Normal Subgroup

In this paper we derive those properties of topologically embedded curves and surfaces in E3 which can be obtained without use of Bing's Side Approximation Theorem [3] for surfaces. The local homology and homotopy properties studied classically play the largest role in the paper, but the final geometrization of some of the results requires theorems such as the PL Schoenflies Theorem, Dehn's Lemma, the Loop Theorem, the Sphere Theorem, and Waldhausen's generalization of the Loop Theorem (n.b., one application of Waldhausen's theorem (in (3.4)) requires use of the nontrivial normal subgroup in the statement of that theorem).

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