Cone-length and Lusternik-Schnirelmann category
Let spaces \(X\) be of the homotopy type of path-connected pointed CW- complexes. Define \(Cl(X)\), cone-length of \(X\), as the minimal \(n\) -- if it exists -- such that there are cofibration sequences \[ \Sigma^ i Z^ i\to X^ i\to X^{i+1}, \qquad 0\leq i<n, \] \(X^ 0=*\), \(X_ n\sim X\) (\(\Sigma^ i\) the \(i\)-fold suspension, \(\Sigma^ 0 Z:= Z\)). If no such \(n\) exists let \(Cl(X): =\infty\). For 1-connected rational spaces this new cone-length coincides with the cone-length defined by Lemaire and Sigrist. The following is the main result of the geometric part of the paper: Let \(\text{cat} (X)=n\), then there exists a space \(Z\) such that \(Cl(X\vee \Sigma^ n Z)\leq n\). This implies \(Cl(X)\leq n+1\), and if \(k\geq \text{cat} (X)\), then \(\text{cat} (B_ k \Omega X)= \text{cat}(X)\) (where \(B_ k \Omega X\) is the \(k\)-th Ganea space of \(X\)). Moreover \(Cl(X \times Y)\leq Cl(X)+ Cl(Y)\). In the algebraic part of the paper, handling masterly the various algebraic models of 1-connected rational spaces \(X'\) of finite type over \(\mathbb{Q}\), the author proves \(Cl(X')= \text{nil} (X')\) (for such spaces \(X')\). Here \(\text{nil} (X')\) is the minimal \(n\) such that there exists an augmented commutative cochain algebra \(A\) over \(\mathbb{Q}\) with \(\overline {A}^{n+1} =0\) quasi isomorphic to the Sullivan minimal model of \(X'\). In subsequent papers the author shows that the 1-connected rational case all reasonable notions of cone-length coincide and that -- in general -- \(Cl(X)= \text{Cat} (X)\), the Ganea strong category of \(X\).
scientific article; zbMATH DE number 527440