On the number of critical submanifolds of a function on a manifold (Q1345001)

For given closed manifolds \(P\) and \(M\) the author investigates differentiable functions \(M \to \mathbb{R}\) whose set of critical points is a disjoint union of submanifolds each homeomorphic to \(P\). Such a function is called a \(P\)-function. A lower bound for the number of these critical submanifolds is the \(P\)-category of \(M\) which is defined in analogy to the Ljusternik-Schnirelmann category. The author announces without proofs a number of results on \(S^ 1\text{-cat}(M)\) for \(\dim(M) \leq 3\). For instance, if \(\dim(M) = 2\) then \(S^ 1\text{-cat}(M)\) can be computed and, in addition, there are \(S^ 1\)-functions \(M \to \mathbb{R}\) having precisely \(S^ 1\text{-cat}(M)\) critical submanifolds homeomorphic to \(S^ 1\).