On 3-manifolds with torus or Klein bottle category two (Q2925370)

This paper concerns a generalization of the Lusternik-Schnirelmann category introduced by \textit{M. Clapp} and \textit{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.NEWLINENEWLINEThe 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 \(M\).