A definition of exotic characteristic classes of spherical fibrations

Abstract

The object of this paper is to define certain characteristic cohomology classes for spherical fibrations which are zero on vector bundles and to show that the classes defined are not always zero. For an introductory survey of characteristic classes and spherical fibrations the reader is referred to [12] and the references therein. Briefly there is a 'structure group' G and a classifying space BG for spherical fibrations and similar spaces (SG and BSG) for oriented spherical fibrations. SG has the homotopy type of lim,_~o~ (f2S)1, where (f2S)1 is the space of degree 1 basepoint preserving maps of to itself. The cohomology of all four spaces has recently been computed by Milgram ([I0]), May ([8]) and Tsuchiya ([18]). My object is to define certain classes e k e H r ~ I ( B S G ; Zp) for p an odd prime (where r = 2 p 2 ) and ekeH2~l l (BG; Z2) which I will refer to as exotic classes. In order to simplify notation I will only deal with the case o fp odd, but all of the theorems herein can be proved for p = 2 with the obvious changes in notation. All cohomology groups will have Zp coefficients unless otherwise indicated. The definition given here is similar to one given by Peterson in [13] and to a definition of e~ given by Gitler-Stasheff in [4]. In each case the exotic classes are defined in terms of twisted secondary cohomology operations (TSCO's) acting on the Thom class u e H * M S G , where MSG is the Thorn space of the universal bundle over BSG. TSCO's were introduced by Thomas ([16]) and axiomatized by McClendon ([9]). They are a generalization of ordinary secondary operations to the category of topological pairs (X, V) over a fixed space Y. The analogue of the Steenrod algebra in this category is A (Y) where A is the Steenrod algebra and A ( Y ) = H * Y| as a vector space with the multiplication appropriate to defining an A (Y) module structure on H* (X, V). TSCO's are derived from relations in A(Y) jus t as ordinary secondary operations are derived from relations in A. Indeed, ordinary secondary operations can be regarded as TSCO's for the special case Y=pt. Now the Thorn space of any oriented spherical fibration can be regarded as a pair over BSG, so relations in A (BSG) could be used to define characteristics classes on suitable spherical fibrations. In [13] Peterson defined an algebra injection O:A-~ A (BSG) with the property that O(a) annihilates the Thom class u of MSG