Local 𝐻-maps of 𝐵𝑈 and applications to smoothing theory

Abstract

ABSTRACT. When localized at an odd prime p, the classifying space PL/O for smoothing theory splits as an infinite loop space into the product CxN where C = Cokernel (J) and N is the fiber of a p-local H-map BU -> BU. This paper studies spaces which arise in this latter fashion, computing the cohomology of their Postnikov towers and relating their fc-invariants to properties of the defining self-maps of BU. If Y is a smooth manifold, the set of homotopy classes [Y, N] is a certain subgroup of resmoothings of Y, and the ^-invariants of N generate obstructions to computing that subgroup. These obstructions can be directly related to the geometry of Y and frequently vanish. Introduction.When localized at an odd prime p, the classifying spaces for smoothings and for surgery theory can be written as products involving the p-local factors BO, C (the cokernel of J), and the fibers of certain B-maps BU -► BU. This paper studies the latter spaces, computing the cohomology of their Postnikov towers and relating the fc-invariants to the underlying geometry they classify. In smoothing theory there is an infinite loop space decomposition PL/O ss G x N where N is the fiber of a self-map of BU whose homotopy is the p-torsion in the groups of homotopy spheres bounding parallelizable manifolds. We show that for many smooth manifolds Y the resmoothings classified by N can be effectively computed and given a "characteristic variety" sort of description.Any B-map /: BU -► BU is determined up to homotopy by the characteristic sequence A = (Ai,A2,...), where /» is multiplication by Xj on Ti2j(BU) = Ztpy Because of the infinite loop space decomposition BU ss W x Q2W x ■■ ■ x U2p~4W [4,46] where n^W) = Zip) in dimensions t = 2j(p -1), we may write / as a product of B-maps with corresponding factorization of the fiber