Subject: Re: two more responses From: "John R. Klein" Date: Tue, 15 Mar 2005 10:54:33 +0200 To: Don Davis Dear Don, Here are some comments on the responses to my question. Thanks to those who contributed a response. It seems to me that there are two or maybe three flavors of proof of the result BO = (BU)^{hZ_2}: 1) A "completion theorem" type proof which ultimately boils down to the facts (1) BU is a Z_2-equivariant infinite loop space, and (2) the Euler class for that theory (+/- the reduced Hopf line bundle over P1) is nilpotent. This is the sort of proof that John Greenlees and Lisbeth Fajstrup outlined, and which I surmise to be more-or-less the one that John Rognes sketched. Fact (2) is proved in Atiyah's paper "K-theory and Reality" and the first fact is also implicit there. What is lacking in Atiyah's paper is the window dressing tech of equivariant spectra for the group Z_2 . 2) Another sort of proof, by Karoubi, is based on the theory of Banach Algebras (which I admittedly know very little about). As Max points out, his methods generalize to other situations. 3) Nitu Kitchloo and Steve Wilson give a proof using Lannes Theory. I don't know how that relates to approach (1). 4) Are there any other kinds of proofs? j.