The fibre of an iterative adjunction of cells (Q1358938)

The relative cone length of a pair \((Y,X)\), \(\text{cl}(Y,X)\), is the minimum number of mapping cones formed on maps from wedges of spheres needed to construct \(Y\) from \(X\). When \(X=*\), we simply refer to the cone length of \(Y\) and this notion, introduced by \textit{O. Cornea} [Topology 33, No. 1, 95-111 (1994; Zbl 0811.55004)], has proven important to the study of Lyusternik-Shnirel'man category and critical point theory. In this paper, the author relates relative cone length to a certain algebraic invariant associated to the inclusion \(i\: X \hookrightarrow Y\). Namely, let \(F\) denote the homotopy fibre of \(i\) and note that the holonomy of the fibration \(i\), \(\Omega Y \times F \to F\), provides \(\widetilde H_*(F;k)\) with an \(H_*(\Omega Y;k)\)-module structure. Then it is shown that \[ \text{grade}_{H_*(\Omega Y;k)}\widetilde H_*(F;k) \leq \text{cl}(Y,X)-1. \] \noindent Here the grade of a graded \(A\)-module \(M\) is the least integer \(p\) so that the graded \(\text{Ext}_A^p(M,A)\) is nonzero. Furthermore, equality above implies that the homological dimension of \(\widetilde H_*(F;k)\) as an \(H_*(\Omega Y;k)\)-module is \(\text{cl}(Y,X)- 1\). In particular, if \(Y\) is obtained from \(X\) by a single mapping cone (i.e. \(\text{cl}(Y,X)=1\)), then \(\widetilde H_*(F;k)\) is a free \(H_*(\Omega Y;k)\)-module (a result of the author and J.-C. Thomas). The methods used in the paper involve Adams-Hilton models and semi-free module theory. The author has a more expansive prepublication [Theorie homotopique des modules semi-libres, Recherches de Math. 47, Institut de Math. Pure et Appl. Univ. Catholique de Louvain] which contains the background for these ideas as well as other relations between cone length, category and grade.