Generalizations of J-homomorphism and the Rohlin theorem (Q1179314)

The author gives a generalized \(J\)-homomorphism, \(J: \pi_ n(SO)\to \pi_ n^ s\), where \(\pi_ n^ s\) is the \(n\)-th stable homotopy group of the sphere, and identifies its kernel as a cetain set of obstructions. This yields a generalization of some theorems of Milnor and Kervaire on divisibility properties of some Pontryagin numbers.