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
>>>