The Lusternik-Schnirelmann category of a space \(X,\) \(\text{cat}(X),\) is the least nonnegative integer \(n\) such that \(X\) admits a cover constituted by \(n\) open subsets that are contractible in \(X.\) The motivation of this numerical homotopy invariant is that, for any compact differentiable manifold \(M\), \(\text{cat}(M)\) provides a lower bound of the number of critical points of any differentiable map \(f:M\rightarrow \mathbb{R}.\) A useful generalization of \(\text{cat}(-)\) is given by M. Clapp and D. Puppe [Trans. Am. Math. Soc. 298, 603–620 (1986; Zbl 0618.55003)]. Namely, given a nonempty class of spaces \(\mathcal{A},\) a subset \(W\) of a space \(X\) is said to be \(\mathcal{A}\)-contractible (in \(X\)) whenever the inclusion \(W\subseteq X\) factorizes, up to homotopy, through some space \(K\in \mathcal{A}.\) Then the \(\mathcal{A}\)-category of \(X,\) \(\text{cat}_{\mathcal{A}}(X),\) is the minimal number of open \(\mathcal{A}\)-categorical subsets that are needed to cover \(X\) (or infinity if such a cover does not exist). As Clapp and Puppe showed, when \(M\) is a smooth manifold, \(\text{cat}_{\mathcal{A}}(M)\) gives information about the topology of the critical sets of \(M.\) A particular case of Clapp-Puppe’s notion is when \(\mathcal{A}=\{K\}\) consists of a unique space \(K.\) In this case \(\text{cat}_{\mathcal{A}}(X)\) is simply denoted as \(\text{cat}_K(X)\).
The paper under review is basically a survey of the research on the relationship between \(\text{cat}_K(M)\) for certain cases of the space \(K\) (such as the circle, the 2-sphere, the projective plane, a wedge of circles, a wedge of 2-spheres or a wedge of projective planes) and the geometric topology of the manifold \(M\), particularly focused on the 3-dimensional case. The authors also consider a certain generalization of \(\text{cat}_K(-)\) as follows: Consider \(\mathcal{G}\) a nonempty class of groups and let \(M\) be a manifold. A subset \(W\) of \(M\) is said to be \(\mathcal{G}\)-contractible if for every base point \(*\in W\) the image of \(\pi _1(W,*)\) in \(\pi _1(M,*)\) belongs to the class \(\mathcal{G}.\) Then, \(\text{cat}_{\mathcal{G}}(M)\) is the smallest number of open \(\mathcal{G}\)-contractible subsets of \(M\) that cover \(M\) (or infinity when such a cover does not exist). Similarly as done for \(\text{cat}_K(-),\) the authors analyze \(\text{cat}_{\mathcal{G}}(M^3)\) for compact 3-manifolds for certain choices of \(\mathcal{G}\) (among them we can mention the class of amenable groups or the class of solvable groups, for instance). Other similar studies about geometry \(\mathcal{K}\)-category or fillings are also considered.