Lusternik-Schnirelmann category of products with half-smashes
The Lusternik-Schnirelmann category (or just category) of a space \(X\), denoted as \(\mbox{cat}(X),\) is the least nonnegative integer \(n\) such that \(X\) admits a cover constituted by \(n + 1\) open subsets each of which is contractible to a point in \(X\). The original motivation of this numerical homotopy invariant is that, for any closed smooth manifold \(M\), \(\mbox{cat}(M)\) provides a lower bound of the number of critical points of any smooth map \(f : M\rightarrow \mathbb{R}.\) In the paper under review the authors prove that for any connected CW-complex \(X\) and any well-pointed path-connected space \(A\) satisfying a certain inequality which involves the dimension of \(X\), the connectivity of both spaces \(X\) and \(A\), and the category \(\mbox{cat}(X\times A)\), the following equality holds \[\mbox{cat}(X\times A)=\mbox{cat}(X\times (A\rtimes B))\] for every unpointed space \(B\). Here \(A\rtimes B\) denotes the half-smash product \(A\wedge B_+\). As a corollary, if \(n\) is less than or equal to three times plus one the connectivity of a well-pointed space \(A\), then \[\mbox{cat}(S^n\times (A\rtimes B))=\mbox{cat}(S^n\times A)\] for every unpointed space \(B\). This result enables the authors to provide a new list of examples that do not satisfy the Ganea condition.
scientific article; zbMATH DE number 7188029
scientific article