On category weight and its applications
The author develops and applies the concept of category weight introduced by \textit{E. Fadell} and \textit{S. Husseini} [Bol. Soc. Mat. Mex., II. Ser. 37, No. 1-2, 151-161 (1992; Zbl 0848.55001)]. For recall, the category weight of a cohomology class \(u\), \(\text{cwgt}(u)\) is the maximum \(k\) such that \(u| A=0\) for every \(cw\)-subspace \(A\) with \(\text{cat}_XA<k\). The author shows that for any closed connected stably parallelizable PL manifold with \(4\leq\dim M\leq 2(\text{cat} M-2)\), there is a class \(u\) such that \(\text{cwgt} (u)= \text{cat} M\). This has a lot of corollaries: If \(f:N\to M\) is a map of degree 1 of closed manifolds with the above properties, then \(\text{cat} N\geq \text{cat} M\); on the other hand, the Ganea conjecture holds for those manifolds: \(\text{cat} (M\times S^r) =\text{cat}(M)+1\).
scientific article; zbMATH DE number 1307822