The rational Lusternik-Schnirelmann category of products and of Poincaré duality complexes
Let \(X\) be a simply connected CW complex having finite-dimensional rational homology in each degree. The rational LS category of \(X\) (denoted by \(\text{cat}_0X\)) is the LS category of its rationalization \(X_{\mathbb{Q}}\). In this paper, the authors show that \(\text{cat}_0X\) depends only on the structure of \(C_*(X; \mathbb{Q})\) as a module over the cochain algebra \(C^*(X; \mathbb{Q})\) and they give a homological recipe for computing \(\text{cat}_0X\). Two theorems are easily obtained from the above-mentioned result: Theorem 1: \(\text{cat}_0(X\times Y)= \text{cat}_0X+ \text{cat}_0Y\): Theorem 3: If \(X\) is a Poincaré duality complex, then \(\text{cat}_0X= e_0(X)\), where \(e_0\) is Toomer's invariant.
scientific article; zbMATH DE number 1133064