Spaces with Lusternik-Schnirelmann category \(n\) and cone length \(n + 1\)
The Lyusternik-Shnirel'man category of a space \(X\), denoted \(\text{cat} X\), was introduced as the minimum size of a covering by open sets which are contractible in \(X\). Fox gave an example showing that the notion of minimum size of a covering by open contractible sets is not a homotopy invariant. He introduced the strong category, denoted \(\text{Cat} X\), by taking the minimum of these numbers over all spaces of the same homotopy type. Ganea proved the equality between \(\text{Cat} X\) and the minimum number of mapping cones required to build \(X\) starting from a point. These invariants are relatively hard to determine and are linked by the inequality (due to Ganea and Takens): \(\text{cat} X \leq\text{Cat} X\leq (\text{cat} X)+1\). For a long time, the only examples where these two invariants were known to differ were co-H-spaces that are not suspensions, cf. [\textit{I. Bernstein} and \textit{P. J. Hilton}, Ill. J. Math. 4, 437-451 (1960; Zbl 0113.38301)]. In [Topology 38, No. 1, 189-196 (1999; Zbl 0927.55006)], \textit{N. Dupont} used Sullivan's minimal models to exhibit a rational space \(X\) with \(\text{cat} X=3\) and \(\text{Cat} X=4\) (cat and Cat are normalized by \(\text{cat} (*)=\text{Cat}(*)=0)\). In this interesting paper, the author constructs for each \(r>0\) a space \(X_r\) such that \(\text{cat} (X_r)=r\) and \(\text{Cat} (X_r)=r+1\). More precisely, he proves: Theorem: For every pair of positive integers \(t,r\) such that \(r>1\) and \(2t+ 1<2p-3\) there exist \(p\)-local spaces \(Q\) such that: (1) \(\text{cat} (Q\times S^n)=\text{cat}(Q)= \text{Cat} (Q\times S^n)= \text{Cat} (Q)=r\) for \(n>2t+1\), (2) \(\text{cat} (Q\times S^{2t+1}) =\text{cat} (Q)=r\) and \(\text{Cat} (Q \times S^{2t+1}) =r+1\). The proof uses two kinds of techniques: The spaces \(Q\) are obtained from a perturbation of the attaching map of the last cell of a fat wedge of spheres. For the choice of the perturbation, the author uses the method introduced by \textit{N. Iwase} [Bull. Lond. Math. Soc. 30, No. 6, 623-634 (1998; Zbl 0947.55006)] depending on homotopy classes with a non-trivial Hopf invariant which becomes trivial after some suspension. The control of Cat needs the elaboration of an obstruction theory to the fact that a space can be built in less than \(k\) cofibrations. The author uses the Adams-Hilton model [\textit{J. F. Adams} and \textit{P. J. Hilton}, Comment. Math. Helv. 30, 305-330 (1956; Zbl 0071.16403)] for the detection of a particular attaching map in the loop space. Note also that this paper contains the first written proof that \(\text{cat} X^{(i)} \leq\text{cat} X^{(i+1)}\) if \(X^{(j)}\) denotes the \(j\)-skeleton of a CW-complex \(X\).
scientific article; zbMATH DE number 1472312