Generalized Bernoulli numbers on the \(KO\)-theory (Q1917225)

This paper is devoted to the study of Bernoulli numbers defined for generalized cohomology theories. The author considers a ring spectrum \(E\) with unit. For an orientable and \(E\)-orientable vector bundle \(\alpha\) over a finite complex \(B\) he constructs out of the cohomology Thom class and the \(E\)-Thom class a characteristic class \(sh(\alpha)\) and, using this class, gives a general definition of Bernoulli numbers for \(\alpha\). Next, he considers the special case \(E = KO\) for various bundles on quaternionic projective spaces, quaternionic quasi-projective spaces and stunted quaternionic quasiprojective spaces. Finally, he shows how to use the results to study the factorization of the double transfer map for stunted quaternionic quasi-projective spaces.