Families of Structures on Spherical Fibrations

Let SF(n) be the usual monoid of orientation- and base point-preserving self-equivalences of the n-sphere \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{S}^n}$$ \end{document}n. If Y is a (right) SF(n)-space, one can construct a classifying space B(Y, SF(n), *)=Bn for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{S}^n}$$ \end{document}n-fibrations with Y-structure, by making use of the two-sided bar construction. Let k: Bn→BSF(n) be the forgetful map. A Y-structure on a spherical fibration corresponds to a lifting of the classifying map into Bn. Let Ki=K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left( {{\mathbb{Z}_2 }} \right)$$ \end{document}, i) be the Eilenberg–Mac Lane space of type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left( {{\mathbb{Z}_2 }} \right)$$ \end{document}, 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=Wn is a space homotopy equivalent to ∏i≥1Ki. Applying results due to Booth, Heath, Morgan and Piccinini, we prove that the universal family is a spherical fibration over the space map{Bn, Bn}×Bn. Furthermore, we point out the significance of this space for secondary characteristic classes. Finally, we calculate the cohomology of Bn.