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# On $3$-manifolds with Torus or Klein Bottle Category Two

Published:2013-10-12
Printed: Sep 2014
• Wolfgang Heil,
Department of Mathematics, Florida State University, Tallahassee, Florida 32306
• Dongxu Wang,
Department of Mathematics, Florida State University, Tallahassee, Florida 32306
 Format: LaTeX MathJax PDF

## Abstract

A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$ is a torus or Kleinbottle, if the inclusion $W\rightarrow M$ factors homotopically through a map to $K$. The image of $\pi_1 (W)$ (for any base point) is a subgroup of $\pi_1 (M)$ that is isomorphic to a subgroup of a quotient group of $\pi_1 (K)$. 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: Lusternik--Schnirelmann category, coverings of $3$-manifolds by open $K$-contractible sets
 MSC Classifications: 57N10 - Topology of general $3$-manifolds [See also 57Mxx] 55M30 - Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space 57M27 - Invariants of knots and 3-manifolds 57N16 - Geometric structures on manifolds [See also 57M50]

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