From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2454 Path: news.gmane.org!not-for-mail From: "John Baez" Newsgroups: gmane.science.mathematics.categories Subject: Euler characteristic versus homotopy cardinality Date: Wed, 1 Oct 2003 10:38:52 -0700 (PDT) Message-ID: <200310011738.h91Hcq022554@math-ws-n09.ucr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018672 4230 80.91.229.2 (29 Apr 2009 15:24:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:32 +0000 (UTC) To: categories@mta.ca (categories) Original-X-From: rrosebru@mta.ca Thu Oct 2 11:22:20 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 02 Oct 2003 11:22:20 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A54K3-0001Sp-00 for categories-list@mta.ca; Thu, 02 Oct 2003 11:20:52 -0300 X-Mailer: ELM [version 2.5 PL6] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 4 Original-Lines: 28 Xref: news.gmane.org gmane.science.mathematics.categories:2454 Archived-At: Dear Categorists - Some of you might be interested in this talk, since it's secretly about attempts to categorify the rational numbers. http://www.math.ucr.edu/home/baez/cardinality/ Euler Characteristic versus Homotopy Cardinality Abstract: Just as the Euler characteristic of a space is the alternating sum of the dimensions of its rational cohomology groups, the homotopy cardinality of a space is the alternating product of the cardinalities of its homotopy groups. The two quantities have many of the same properties, but it's hard to tell if they're the same, since like Jekyll and Hyde, they're almost never seen together: there are very few spaces for which the Euler characteristic and homotopy cardinality are both well-defined. However, in many cases where one is well-defined, the other may be computed by dubious manipulations involving divergent series - and the two then agree! We give examples of this phenomenon and beg the audience to find some unifying concept which has both Euler characteristic and homotopy cardinality as special cases.