From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4444 Path: news.gmane.org!not-for-mail From: Johannes Huebschmann Newsgroups: gmane.science.mathematics.categories Subject: Query Date: Thu, 17 Jul 2008 10:35:20 +0200 (CEST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019950 13331 80.91.229.2 (29 Apr 2009 15:45:50 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:45:50 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Jul 17 12:48:56 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 17 Jul 2008 12:48:56 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KJVgO-0000ph-Oz for categories-list@mta.ca; Thu, 17 Jul 2008 12:46:16 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 11 Original-Lines: 52 Xref: news.gmane.org gmane.science.mathematics.categories:4444 Archived-At: Dear All Given a Lie group G and a G-representation V, according to Hochschild-Mostow, the ordinary Eilenberg-Mac Lane construction, suitably interpreted in terms of smooth functions, yields a differentiably injective resolution of V over G. More precisely, the Eilenberg-Mac Lane construction (dual bar construction) arises here as the differentiable cosimplicial G-module having, in degree p, the space of smooth V-valued maps on a product of p+1 copies of G, with the ordinary coface and codegeneracy operators. Suppose now that G is connected and finite-dimensional and let K be a maximal compact subgroup. Hochschild-Mostow have also shown that the V-valued differential forms on G/K then yield an injective resolution of V over G as well. This kind of construction actually goes back to van Est. The standard procedure yields comparison maps between the two resolutions. In degree zero the comparison is, of course, achieved by the obvious map from C^{\infty}(G/K,V) to C^{\infty}(G,V) induced by the projection from G to G/K and by the obvious map from C^{\infty}(G,V) to C^{\infty}(G/K,V) induced by integration over K. Does anybody know whether, in the literature, the constituents of a comparison map in higher degrees have been spelled out explicitly somewhere? Many thanks in advance Best regards 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 (33) 3 20 43 42 33 (secretariat) (33) 3 20 43 48 50 (secretariat) Fax (33) 3 20 43 43 02 e-mail Johannes.Huebschmann@math.univ-lille1.fr