Georgia Institute of Technology College of Sciences

In a 1958 paper, Milnor observed that then new work by Bott allowed him to show that the n sphere admits a vector bundle with non-trivial top Stiefel-Whitney class precisely when n=1,2,4, 8. This can be interpreted as a calculation of the mod 2 Hurewicz map for the classifying space BO, which has the structure of an infinite loopspace. I have been studying Hurewicz maps for infinite loopspaces by showing how a filtration of the homotopy groups coming from stable homotopy theory (the Adams filtration) interacts with a filtration of the homology groups coming from infinite loopspace theory. There are some clean and tidy consequences that lead to a new proof of Milnor's theorem, and other applications.