Heil, Wolfgang; Wang, Dongxu On 3-manifolds with torus or Klein bottle category two. (English) Zbl 1301.57016 Can. Math. Bull. 57, No. 3, 526-537 (2014). This paper concerns a generalization of the Lusternik-Schnirelmann category introduced by M. Clapp and D. Puppe [Trans. Am. Math. Soc. 298, 603–620 (1986; Zbl 0618.55003)]: Let \(M\) be an \(n\)-dimensional manifold and \(K\) be a closed \(k\)-manifold, \(0\leq k\leq n-1\). A subset \(W\) in \(M\) is \(K\)-contractible if the inclusion of \(W\) in \(M\) factors up to homotopy through \(K\). The \(K\)-category of \(M\), cat\(_K(M)\), is the smallest number of open \(K\)-contractible subsets of \(M\) that can cover \(M\). J.C. Gomez-Larranaga, F. Gonzalez-Acuna and W. Heil have determined the class of 3-manifolds \(M\) for which cat\(_{K}M= 2\) for \(K= S^1, S^2\) or \(P^2\). In this paper the authors give a classification of all 3-manifolds with cat\(_KM= 2\) when \(K\) is a torus or a Klein bottle.The result follows from a more general classification of 3-dimensional manifolds for which cat\(_{\mathcal K}M = 2\). Here a subset \(W\) in \(M\) is \(\mathcal K\)-contractible is the image of \(\pi_1(W, *)\) in \(\pi_1(M,*)\), for any base point, is a subgroup of a quotient of \(\pi_1(K)\) which is also the fundamental group of a \(3\)-manifold. Then cat\(_{\mathcal K}M\) is the minimal number of open \(\mathcal K\)-contractible subsets in a covering of \(M\). Reviewer: Yves Félix (Louvain-La-Neuve) Cited in 1 Document MSC: 57N10 Topology of general \(3\)-manifolds (MSC2010) 55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) 57M27 Invariants of knots and \(3\)-manifolds (MSC2010) 57N16 Geometric structures on manifolds of high or arbitrary dimension Keywords:Lusternik-Schnirelmann category; 3-manifolds; \(K\)-contractible sets Citations:Zbl 0618.55003 PDFBibTeX XMLCite \textit{W. Heil} and \textit{D. Wang}, Can. Math. Bull. 57, No. 3, 526--537 (2014; Zbl 1301.57016) Full Text: DOI