Two responses to John Klein's question, both by guys named John.........DMD ________________________________________________________ Subject: Re: two questions From: John Rognes Date: Wed, 9 Mar 2005 16:09:08 +0100 (CET) On Wed, 9 Mar 2005, Don Davis wrote: >> Subject: Question for the list >> From: "John R. Klein" >> Date: Mon, 07 Mar 2005 09:47:03 +0200 >> >> In the year 2000, Max Karoubi posted a question to the list (see below) >> about the the homotopy fixed set of Galois action of Z_2 on the K-theory >> of Banach Algebras. There he mentions a "known" result that the >> inclusion >> >> BO --> (BU)^{hZ_2} >> >> is a homotopy equivalence. I'm trying to find a homotopy theoretic proof >> of this statement (i.e., no operator algebras permitted). Does anybody >> know of one? Dear John, I assume that by Z_2 you mean the group of order 2, acting on BU by complex conjugation. In: Mitchell, Stephen A. On the Lichtenbaum-Quillen conjectures from a stable homotopy-theoretic viewpoint. Algebraic topology and its applications, 163--240, Math. Sci. Res. Inst. Publ., 27, Springer, New York, 1994. the first example in Section 5 covers this result. Mitchell writes: "But in fact the results of Atiyah, M. F. $K$-theory and reality. Quart. J. Math. Oxford Ser. (2) 17 1966 367--386. imply that BO satisfies descent globally. Since this often cited implication is not particularly obvious, we sketch the argument ..." So the proper reference must be to Atiyah. Mitchell refers to Bill Dwyer for a more direct argument, left as an amusing exercise. One solution to this exercise goes as follows: The homotopy fixed point spectral sequence E2_{s,t} = H^s(Z/2; pi_t(KU)) ==> pi_{s+t}(KU^{hZ/2}) is conditionally convergent for general reasons (use the Milnor lim-lim1 sequence), and E2 = Z[a, v, v^{-1}]/(2a) with a in bidegree (-1,2), v in bidegree (0,4). The complex Adams e-invariant of eta in pi_1(S) is 1/2, so eta is detected by a. From eta4 = 0 in pi_*(S) (or eta3 = 0 in pi_*(BO)) you deduce that a4 (or a3) is a boundary, and the only possibility is d3(v) = a3. Thus E4 = E^infty and the spectral sequence is strongly convergent. The E^infty term is the associated graded of pi_*(KO), and from this isomorphism pi_8(KO) --> pi_8(KU) it is not hard to show that KO --> KU^{hZ/2} is a weak equivalence. Passing to the connected component of the underlying infinite loop spaces, you find that BO --> BU^{hZ/2} is a weak equivalence. Sections 2 and 3 of my former student's Ph.D. thesis: Østvær, Paul Arne Étale descent for real number fields. Topology 42 (2003), no. 1, 197--225. give more details, and variations (with Z/2^v coefficients). For some reason I went over it again for Proposition 5.3.1 in John Rognes math.AT/0502183 Galois extensions of structured ring spectra on http://arxiv.org/ . - John PS: The Abel prize for 2005 will be announced on March 17th. _______________________________________________________________ Subject: RE: two questions From: "John Greenlees" Date: Wed, 9 Mar 2005 15:16:40 -0000 Here is an answer to John Klein's question. Let G be a group of order 2. The map BO---> BU^{hG} is obtained by passing to G-fixed points >From map(S0, BU)---->map(EG_+,BU) (based maps). The fibre is G-map(S^{\infty \xi},BU) where \xi is the real line with G acting as -1. It thus suffices to see that S^{k\xi}---> S^{(k+3)\xi} induces a map zero in homotopy. Now BU (with conjugation action) represents part of Atiyah's K-theory with reality (perhaps you won't like me quoting this), so the required result is 3.7 of his `K-theory and reality' (1966) (he describes the proof as `linear algebra'). John