A measure of the deviation from there being fibrations between a pair of compact manifolds
Given two smooth manifolds \(M\) and \(N\), the \(\varphi\)-category of the pair \((M,N)\), denoted \(\varphi(M,N)\), is defined to be the minimum size of the critical set \(C(f)\) for any smooth map \(f\: M \to N\). The author notes that \(\varphi(M,N)=0\) implies that there is a submersion \(f\) which induces a fibering of \(M\) over \(N\). The homotopy sequence of the fibration then gives a relation between the homotopy groups of \(M\) and \(N\). This motivates the author's approach and he focuses on using various homotopy group constraints to obtain \(\varphi(M,N)=\infty\). The last section of the paper considers the equivariant situation.
scientific article; zbMATH DE number 5125298