Secondary characteristic classes in $K$-theory

Abstract

The object of this paper is to develop a general theory of secondary characteristic classes and to study secondary characteristic classes that arise in K-theory. Secondary characteristic classes are particularly adapted to studying embedding problems. Massey, and Peterson and Stein developed and exploited secondary characteristic classes in ordinary cohomology theory [11], [17], [18]. On the other hand, Atiyah [3] showed that certain primary characteristic classes yi in K-theory give good results for nonembedding of projective spaces. The characteristic classes we develop here were motivated by the desire to define a secondary operation when the top yi-class vanishes, in analogy with the operations which arise from the relation Sqi(wn) = wi u wn where wi is the ith Whitney class of an n-plane bundle. The viewpoint we will take in this paper is that, in a general cohomology theory, secondary characteristic classes arise in two ways: from a relation between characteristic classes and cohomology operations, or from the degeneration of the Gysin sequence. The organization of the paper is as follows. The first section is preliminary and collects results on spectra, functional operations, representation theory, and K-theory. In ?2 we give the various definitions of secondary characteristic classes in the setting of general cohomology theories and principal G-bundles. In ?3 we develop the crucial Peterson-Stein formula relating a functional operation in the universal example and the universal secondary characteristic class. We apply this formula in ?4 and ?5 to study the indeterminacy for the universal secondary characteristic classes and to study the relationship between the various definitions given in ?2. In ?6 we discuss the secondary characteristic classes in K-theory which arise from the relation /k(A _) =k -1, and in ?7 and ?8 we carry out some computations involving these operations. We should note that throughout this paper H* will always be a generalized cohomology theory. We hope that a forthcoming paper will deal with the application of these operations to embedding problems. Most of the material in this paper appeared in the author's doctoral thesis at Harvard University written under the direction of Professor Raoul Bott. The author wishes to express his gratitude to Professor Bott and also to Professors Donald Anderson and Al Vasquez.