On exotic characteristic classes for spherical fibrations (Q1070574)

Similar to Thom's method of using Steenrod squares to produce Stiefel- Whitney classes for sphere bundles, the author uses secondary cohomology operations, based upon admissible relations in the mod 2 Steenrod algebra \({\mathcal A}\), to produce a lot of exotic characteristic classes for spherical fibrations. These exotic classes are primitive and vanish for sphere bundles. Some of them are identified with algebra generators of \(H^*(BSG; {\mathbb{Z}}/(2)),\) originally given by Milgram, and their behaviour under the action of \({\mathcal A}\) is computed (here BSG is the classifying space for spherical fibrations as defined by Stasheff). Finally an interesting Wu formula is proven for Poincaré duality spaces all whose Stiefel-Whitney classes vanish.