In the paper under review, the author presents an interesting new approach to the Lusternik-Schnirelmann category, LS-category henceforth, and its relative notions. These are well know numerical invariants of the homotopy type of spaces. This new approach is done by introducing the categorical length of a space (or of a pair), a homotopy version of the classical Fox categorical sequence: A categorical sequence of a space \(X\) is a sequence of closed subspaces \(F_0\subset\dots\subset F_m\) such that \(F_0\) is contractible in \(X\), \(F_m\) has the homotopy type of \(X\), and for each \(i\), the diagonal \(\Delta_i: F_i\to F_i\times F_I\subset F_m\subset F_m\) can be deformed to \(F_{i-1}\times F_m\cup F_m\times *\) relative to \(F_{i-1}\). Here \(F_{i-1}\) is identified with its diagonal image in \(F_{i-1}\times F_{i-1}\subset F_{i-1}\times F_m\cup F_m\times *\). Then, it is proved that the minimum of the lengths of categorical sequences of a space coincides with the LS-category of the space. A relative notion of this fact, which in particular contains the relative versions of Berstein-Ganea and Fadell-Husseini, is also proved. After that, the author defines, in this new environment, higher Hopf invariants and shows whether the vanishing of these invariants implies that the categorical length is not modified by certain cone attachments. Then, the author finds interesting lower bounds to this new version of relative LS-category, analogues to the classical cup length and category weight, in terms of any generalized cohomology theory.
The paper ends with some examples and computations.
For the entire collection see [Zbl 1162.00013].