W. Stephen Wilson’s Contributions to Homotopy Theory

This paper is a survey of Steve’s work on BP and periodic cohomology theories. It was presented as a talk given in March 2007 at a conference celebrating Steve’s 60th birthday. 1. Cohomology of Ω-spectra and Hopf rings Steve Wilson’s thesis, directed in 1972 by Frank Peterson, started unstable BP theory as we know it. Part 1 of his thesis appeared as [13], where he shows that the spaces in the Ω-spectrum for BP are as nice as one may expect. Theorem 1.1. Z(p)-cohomology of BP k, the k-th space in the Ω spectrum for BP , is torsion free. As a Hopf algebra it is a bi-free algebra; i.e., the homology and cohomology are free algebras. The original proof in his thesis goes back to Brown and Peterson’s definition of BP as the inverse limit of spaces inductively defined by a series of fibrations. Steve uses the Eilenberg-Moore spectral sequence of these Brown-Peterson fibrations to prove the theorem. This is not the way we now understand the homology of the spaces in the BP Ωspectrum, or for that matter the generalized homology of the spaces in the Ω-spectrum of any multiplicative homology theory. The relevant technology is the language of Hopf rings developed by Ravenel and Wilson in [10]. The idea here is to exploit the structure one has after packaging all of the spaces in the BP Ω-spectrum together. Milgram used this structure when he computed the homology of BG. The homology of the individual spaces are obviously H-spaces. But now we can mix the different factors together using the ring structure. This gives us two products, the ∗(loop space)-product and a ◦ product ◦ : BP∗(BPn)⊗BP∗(BP k) → BP∗(BPn+k). A Hopf ring is a graded Hopf algebra with an additional product satisfying a list of relations which makes it a ring object in the category of coalgebras. The collection {BP∗(BPn)} is an example. Received August 7, 2007, revised August 15, 2007; published on December 5, 2008. 2000 Mathematics Subject Classification: 55N22, 55Q51, 55S25.