The category of a map and the grade of a module
The main purpose of the paper is to establish a relation between the Lusternik-Schnirelmann category \(\mathbf{ {cat} }f\) of a map \(f: Y\to X\) and certain homological invariants of \(H_ *(\Omega X)\)-module \(H_ *(F)\), where \(F\) is the homotopy fibre of \(f\). By definition \(\mathbf{ {cat} }f\) is the least \(m\leq\infty\) such that \(Y\) can be covered by \(m+1\) open sets \(U_ i\) with each \(f\mid U_ i\) homotopically constant. The homological invariant of \(H_ *(F)\), mentioned above, is the positive projective grade of \(f\), which is defined as the least \(m\) such that \(\text{Ext}^ m_{H_ *(\Omega X)} (H_ *(F),V)\neq 0\) for some \(H_ *(\Omega X)\)-projective \(V\) generated by \(V_{\geq 0}\). The main result of the paper is the theorem: if \(f: Y\to X\) is a map between path connected CW-complexes and \(H_ *(F)\) and \(H_ *(\Omega X)\) are both \(R\)-free (\(R\) is a principal ideal domain), then pos.proj.grade\(_{H_ *(\Omega X)} (H_ *(F))\leq\mathbf{ {cat} }f\). This result extends the earlier one [the authors with \textit{J.-M. Lemaire}, Invent. Math. 95, No. 2, 247-262 (1989; Zbl 0667.55007)] by passing from spaces to maps; extending to the non-simply connected case; removing the hypothesis that \(H_ *(X)\) be of finite type and that \(R\) is a field. As applications some structural properties of \(H_ *(\Omega X)\), as well as some properties of \(\pi_ 1(X)\) when \(X\) and its universal cover have homotopy type of finite complex, are given.
scientific article; zbMATH DE number 219373
scientific article