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Surfaces Embedded in M2 × S1

Published online by Cambridge University Press:  20 November 2018

William Jaco
William Jaco*
Affiliation:
University of Michigan, Ann Arbor, Michigan
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In this paper we study incompressible and injective (see § 2 for definitions) surfaces embedded in M2 × S1, where M2 is a surface and S1 is the 1-sphere. We are able to characterize embeddings which are incompressible in M2 × S1 when M2 is closed and orientable. Namely, a necessary and sufficient condition for the closed surface F to be incompressible in M2 × S1, where M2is closed and orientable, is that there exists an ambient isotopy ht, 0 ≦ t ≦ 1, of M2 × S1onto itself so that either

(i) there is a non-trivial simple closed curve J ⊂ M2 and h1(F) = J × S1, or

(ii) p\h1(F) is a covering projection of h1(F) onto M2, where p is the natural projection of M2 × S1onto M2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 553 - 568
Copyright
Copyright © Canadian Mathematical Society 1970

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