On 3-manifolds with Torus or Klein Bottle Category Two

Heil, Wolfgang ; Wang, Dongxu

Canadian mathematical bulletin, Tome 57 (2014) no. 3, pp. 526-537

Résumé

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.

Détail

DOI : 10.4153/CMB-2013-035-3

Mots-clés : 57N10, 55M30, 57M27, 57N16, Lusternik–Schnirelmann category, coverings of 3-manifolds by open K-contractible sets