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On 3-manifolds with Torus or Klein Bottle Category Two

Published online by Cambridge University Press:  20 November 2018

Wolfgang Heil  and
Dongxu Wang
Wolfgang Heil
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA e-mail: [email protected]@gmail.com
Dongxu Wang
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA e-mail: [email protected]@gmail.com
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Abstract

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A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$ is a torus or Klein bottle if the inclusion $W\,\to \,M$ factors homotopically through a map to $K$. The image of ${{\pi }_{1}}\left( W \right)$ (for any base point) is a subgroup of ${{\pi }_{1}}\left( M \right)$ that is isomorphic to a subgroup of a quotient group of ${{\pi }_{1}}\left( K \right)$. Subsets of $M$ with this latter property are called ${{\mathcal{G}}_{K}}$-contractible. We obtain a list of the closed 3-manifolds that can be covered by two open ${{\mathcal{G}}_{K}}$-contractible subsets. This is applied to obtain a list of the possible closed prime 3-manifolds that can be covered by two open $K$-contractible subsets.

Keywords

57N1055M3057M2757N16Lusternik–Schnirelmann categorycoverings of 3-manifolds by open K-contractible sets
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 3 , 01 September 2014 , pp. 526 - 537
Copyright
Copyright © Canadian Mathematical Society 2014

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