From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4994 Path: news.gmane.org!not-for-mail From: Andrew Stacey Newsgroups: gmane.science.mathematics.categories Subject: Re: Topology on cohomology groups Date: Fri, 19 Jun 2009 22:50:42 +0200 Message-ID: Reply-To: Andrew Stacey NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1245608235 7927 80.91.229.12 (21 Jun 2009 18:17:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 21 Jun 2009 18:17:15 +0000 (UTC) To: Steve Vickers , Original-X-From: categories@mta.ca Sun Jun 21 20:17:13 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1MIRbJ-0004kC-Sf for gsmc-categories@m.gmane.org; Sun, 21 Jun 2009 20:17:10 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MIQzs-0006N6-KQ for categories-list@mta.ca; Sun, 21 Jun 2009 14:38:28 -0300 Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4994 Archived-At: On Fri, Jun 19, 2009 at 10:26:13AM +0100, Steve Vickers wrote: > A cohomology group can easily be an infinite power of the coefficient > group. But such a group has a natural non-discrete topology, namely the > compact-open (which in this case is also the product topology). > > Are there approaches to cohomology that, as part of the process, also > supply topologies on the cohomology groups? > I'm not sure if this is quite what you are looking for, but the topology on cohomology theories is given as an inverse limit (if I have my limits the correct way round) over the finite skeleta. This has an impact, for example, in the correct statement of the Kunneth theorem on cohomology of products (one has to complete the tensor product with respect to the topology). A fairly comprehensive and detail account is in Boardman and Boardman+Johnson+Wilson in the Handbook of Algebraic Topology: MR1361889 and MR1361900 (though it was known well before that). These papers are available online from Steve Wilson's homepage: http://www.math.jhu.edu/~wsw/ (scan right down to the bottom). Sarah Whitehouse and I also look at this in our paper 'The Hunting of the Hopf Ring' (arxiv:0711.3722, to appear in HHA). Andrew Stacey [For admin and other information see: http://www.mta.ca/~cat-dist/ ]