\(S^{2}\)- and \(P^{2}\)-category of manifolds. (English) Zbl 1244.57037
Let \(K\) be a CW-complex. This work deals with a generalization of the Lusternik-Schnirelmann category, denoted by \(\text{cat}_K(-)\) and due to M. Clapp and D. Puppe [Trans. Am. Math. Soc. 298, 603–620 (1986; Zbl 0618.55003)]. In the paper under review, the authors give a complete classification of topological \(n\)-manifolds \(M\) such that \(\text{cat}_{S^2}(M)=2\) and a complete list of the fundamental groups of \(n\)-manifolds such that \(\text{cat}_{{\mathbb P}^2}(M)=2\). For this determination, they use a graph \({\mathcal G}\), determined by the components of a categorical open cover of \(M\), and whose fundamental group is isomorphic to \(\pi_1(M)\).
Reviewer: Daniel Tanré (Villeneuve d’ Ascq)
MSC:
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
| 57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |
| 57N15 | Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010) |
| 57M30 | Wild embeddings |
| 55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |
Keywords:
Lusternik-Schnirelmann category; coverings of \(n\)-manifolds with open \(S^2\)-contractible or \({\mathbb P}^2\)-contractible subsetsCitations:
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