Nick Kuhn: Hurewicz maps for infinite loopspaces.

ABSTRACT: 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 such Hurewicz maps for generalized homology theories by relating the Adams filtration of the domain to an "augmentation ideal" filtration of the range. The key constructions are done in the world of commutative S--algebras and use topological Andre-Quillen homology to control obstructions. Also playing a role is composition structure developed recently by Luis Pereira and me, and, at odd primes, a bit of classic Goodwillie calculus. When specialized to ordinary mod p homology, my general results have some tidy consequences, with examples including including Milnor's theorem, a variant with ko replaced by tmf, and a new proof of irreducibility results of Steve Wilson.