Higher cohomology operations, exotic characteristic classes and spherical fibrations (Q1911824)

Let \(BSG\) be the classifying space for spherical fibrations. Let \(p\) be an odd prime and let all cohomology have coefficients in \(\mathbb{Z}/p\). We have \(j:BSO \to BSG\) and the \(J\)-homomorphism \(J_*: \pi_* (BSO) \to \pi_*(BSG)\). Tsuchiya computed \(H^*(BSG) \cong P(q_i) \otimes E(\beta q_i) \otimes C\), where the \(q_i\) are Wu classes. Sullivan gave a \(p\)-local splitting \(BSG_{(p)} \simeq BJ_p \times B \text{Coker} J_p\). The author proves a conjecture of Peterson that \(H^* (\text{Coker} J_p) \cong C\) as algebras over the mod \(p\) Steenrod algebra. Cohomology classes of \(H^* (BSG)\) which lie in the kernel of \(j^*\) and which are not generated by Wu classes are the exotic (primary) characteristic classes. The author gives a criterion for proving the nontriviality of higher order exotic characteristic classes.