Families of Structures on Spherical Fibrations

Abstract

Let SF(n) be the usual monoid of orientation- and base point-preserving self-equivalences of the n-sphere \({\mathbb{S}^n}\)n. If Y is a (right) SF(n)-space, one can construct a classifying space B(Y, SF(n), *)=B n for \({\mathbb{S}^n}\)n-fibrations with Y-structure, by making use of the two-sided bar construction. Let k: B n →BSF(n) be the forgetful map. A Y-structure on a spherical fibration corresponds to a lifting of the classifying map into B n . Let K i =K\(\left( {{\mathbb{Z}_2 }} \right)\), i) be the Eilenberg–Mac Lane space of type \(\left( {{\mathbb{Z}_2 }} \right)\), i). In this paper we study families of structures on a given spherical fibration. In particular, we construct a universal family of Y-structures, where Y=W n is a space homotopy equivalent to ∏i≥1K i . Applying results due to Booth, Heath, Morgan and Piccinini, we prove that the universal family is a spherical fibration over the space map{B n , B n }×B n . Furthermore, we point out the significance of this space for secondary characteristic classes. Finally, we calculate the cohomology of B n .