Surgery approach to Rudyak's conjecture (Q2125127)

As usual \(\mathrm{cat}(X)\) stands for the (reduced) Lusternik-Schnirelmann category. The paper is written in the context of smooth (or PL) manifolds and is about Rudyak's Conjecture [\textit{Y. B. Rudyak}, Topology 38, No. 1, 37--55 (1999; Zbl 0927.55005)]: If \(f \colon M \to N\) is a degree one map between closed manifolds, then \(\mathrm{cat} (M) \leq \mathrm{cat} (N)\). This conjecture is already proved for manifolds of dimension \(\leq 4\). Rudyak has also shown the following Theorem: Let \(f \colon M \to N\) be a map of degree one between closed stably parallelizable PL manifolds with \(N\) being \((r-1)\)-connected for some \(r \geq 1\); if \(N\) satisfies the inequality \(\dim N \leq 2r \mathrm{cat} (N) - 4\), then \(\mathrm{cat} (M) \geq \mathrm{cat} (N)\). Rudyak's approach is based on stable cohomotopy theory, while the present authors use surgery theory. With this different approach, the authors obtain a theorem that replaces the stably parallelizable assumption on the manifolds in Rudyak's theorem by the weaker condition that the map \(f\) is normal, and loosens the bound on dimension by one. More precisely, they obtain the following Theorem: Let \(f \colon M \to N\) be a normal map of degree one between closed smooth manifolds with \(N\) being \((r-1)\)-connected for some \(r \geq 1\); if \(N\) satisfies the inequality \(\dim N \leq 2r \mathrm{cat} (N) - 3\), then \(\mathrm{cat} (M) \geq \mathrm{cat} (N)\).