Local 𝐻-maps of 𝐵𝑈 and applications to smoothing theory

Abstract

When localized at an odd prime p p , the classifying space P L / O PL/O for smoothing theory splits as an infinite loop space into the product C × N C \times N where C = Cokernel ( J ) C = {\text {Cokernel}}\,(J) and N N is the fiber of a p p -local H H -map B U → B U BU \to BU . This paper studies spaces which arise in this latter fashion, computing the cohomology of their Postnikov towers and relating their k k -invariants to properties of the defining self-maps of B U BU . If Y Y is a smooth manifold, the set of homotopy classes [ Y , N ] [Y,\,N] is a certain subgroup of resmoothings of Y Y , and the k k -invariants of N N generate obstructions to computing that subgroup. These obstructions can be directly related to the geometry of Y Y and frequently vanish.