Strong LS category equals cone-length (Q1804905)

Let \(X\) be a pointed space with the homotopy type of a CW-complex, the LS-category of \(X\), \(\text{cat } X\), is the least number \(n\) such that there are \(n+1\) open subsets of \(X\), whose union covers \(X\), each of them being contractible to the base point in \(X\). The strong LS-category, \(\text{Cat } X\), was introduced by \textit{T. Ganea} [Ill. J. Math. 11, 417-427 (1967; Zbl 0149.407)]\ as an approximation of \(\text{cat } X\); that is the least integer \(n\) such that a CW-complex of the same homotopy type as \(X\) can be covered by \(n+1\) self-contractible subcomplexes. Ganea relied the strong LS-category to the construction of a CW-complex by proving that \(\text{Cat } X\) is the least integer \(n\) such that there are cofibrations sequences \(Z_i\to X_i\to X_{i+1}\), \(0\leq i< n\), \(X_0\) contractible, \(X_n\simeq X\). These two notions, \(\text{cat } X\) and \(\text{Cat } X\), are really close; they satisfy the inequalities \(\text{cat } X\leq \text{Cat } X\leq \text{cat } X+1\) [\textit{F. Takens}, Compos. Math. 22, 175- 180 (1970; Zbl 0198.283)]. The author answers a question left open in this story: ``What kind of bricks, \(Z_i\), can be used in the definition of \(\text{Cat } X\)?'' He defines the cone-length, \(\text{Cl } X\), as the invariant obtained by requiring in the definition above, \(Z_i= \Sigma^i A_i\), for some space \(A_i\), and proves: ``\(\text{Cat } X= \text{Cl } X\), for \(X\) path connected''. In a previous paper [Trans. Am. Math. Soc. 344, No. 2, 835-848 (1994; Zbl 0813.55003)], the author established this result for simply connected rational spaces of finite type, by using algebraic models. In the paper under review, he uses homotopy pull-backs and push-outs in the spirit of \textit{M. Mather} [Can. J. Math. 28, 225-263 (1976; Zbl 0351.55005)]\ and \textit{H. J. Marcum} [Ill. J. Math. 24, 344-358 (1980; Zbl 0459.55009)].