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Incompressible surfaces and the topology of 3-dimensional manifolds

Part of: Topological manifolds Low-dimensional topology

Published online by Cambridge University Press:  09 April 2009

Iain R. Aitchison  and
J. Hyam Rubinstein
Iain R. Aitchison
Affiliation:
Mathematics Department, University of Melbourne, Parkville, Victoria, Australia, 3052
J. Hyam Rubinstein
Affiliation:
Mathematics Department, University of Melbourne, Parkville, Victoria, Australia, 3052
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Abstract

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Existence and properties of incompressible surfaces in 3-dimensional manifolds are surveyed. Some conjectures of Waldhausen and Thurston concerning such surfaces are stated. An outline is given of the proof that such surfaces can be pulled back by non-zero degree maps between 3-manifolds. The effect of surgery on immersed, incompressible surfaces and on hierarchies is discussed. A characterisation is given of the immersed, incompressible surfaces previously studied by Hass and Scott, which arise naturally with cubings of non-positive curvature.

MSC classification

Secondary: 57N10: Topology of general $3$-manifolds 57M35: Dehn's lemma, sphere theorem, loop theorem, asphericity 57M50: Geometric structures on low-dimensional manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 1 , August 1993 , pp. 1 - 22
Copyright
Copyright © Australian Mathematical Society 1993

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