Subject: Re: response re Lie gps From: Johannes Huebschmann Date: Fri, 24 Feb 2006 11:43:50 +0100 (CET) Dear Lisa and Mike Mike's calculation yields the cohomology of the homotopy quotient of G^n relative to the conjugation action. But what does it tell us about the cohomology of the ordinary quotient? For example, for n=1, the quotient is a contractible space whereas the Hochschild homology of R[G] coincides with the cohomology of G x BG or, equivalently, with the cohomology of the free loops on the classifying space of G - the classifying space of the loop group on G. Johannes HUEBSCHMANN Johannes Professeur de Mathematiques USTL, UFR de Mathematiques UMR 8524 Laboratoire Paul Painleve F-59 655 Villeneuve d'Ascq Cedex France http://math.univ-lille1.fr/~huebschm TEL. (33) 3 20 43 41 97 On Thu, 23 Feb 2006, Don Davis wrote: > Subject: Re: question abt Lie gps > From: mjh@math.mit.edu (Michael J. Hopkins) > Date: Wed, 22 Feb 2006 16:56:00 -0500 (EST) > > Lisa > > There's a paper from a few years back by Brylinski (I think it is in > K-theory) computing K_G(G). His argument builds on Hodgkin's proof > that K(G) is torsion free. Both arguments are sneaky versions of the > Eilenberg-Moore spectral sequence. The technique leads to the > cohomology computation you're asking about (at least with rational > coefficients). > > The trick is to write > > K_G(G) = K_(G\times )(G\times G) > > where G\time G is acting on itself by > > (a,b)(x,y) = (a x b^(-1), b y a^(-1)). This holds because each G\times G orbit meets G \times 1, and the > stabilizer of G\times 1 is the diagonal. This equivalence can also be > thought of as the equivalence between flat G-bundles on S1 and flat > G-bundles on an annulus. Anyway, now use the Kunneth spectral sequence > > Tor^R[G\times G] (K_{G\times G} (G), K_{G\times G} (G)) ==> > K_{G\times G}(G\times G) = K_G(G). > > By a similar "orbit" argument, > > K_{G\times G}(G) = K_G(pt) = R[G]. So the Tor calculation is just > the Hochshild homology of the representation ring. It follows that > > K_G(G) = \Omega^*_R(G) -- the deRham complex of the representation ring. > > Since R(G) is smooth, this is free as an R(G)-module. One now gets, by Kunneth again > > K_G(G^n) = \Omega^*_R(G) \tensor_{R(G)} ...\tensor_R(G) \Omega^*_R(G). > > For the Borel construction, replace R(G) by its completion at the augmentation > ideal. The same argument works in cohomology with rational coefficients. If you work at the chain level, rather than the cohomology level, you > get more or less the same picture with any coefficients. > > Mike > >>> Subject: question for the topology mailing list >>> From: Lisa Jeffrey >>> Date: Wed, 22 Feb 2006 15:02:56 -0500 (EST) >>> >>> Let G be a compact connected Lie group (I am particularly interested in >>> the case G=SU(2)). Let G act on itself by conjugation. Can anyone >>> tell me the homotopy type of the quotient space G^n/G, or at least its >>> cohomology? I suspect this is well known but I don't know where to look >>> for it. Can anyone supply a reference? >>> >>> Many thanks, Lisa Jeffrey >>>