From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6707 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Wikipedia on Eilenberg-Mac Lane spaces Date: Sun, 12 Jun 2011 22:00:30 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1307938529 4139 80.91.229.12 (13 Jun 2011 04:15:29 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 13 Jun 2011 04:15:29 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Mon Jun 13 06:15:25 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QVyYe-0002Bk-QS for gsmc-categories@m.gmane.org; Mon, 13 Jun 2011 06:15:24 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:55573) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QVyWO-0000Vu-Ve; Mon, 13 Jun 2011 01:13:04 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QVyWO-0007tP-5s for categories-list@mlist.mta.ca; Mon, 13 Jun 2011 01:13:04 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6707 Archived-At: Something in Wikipedia on E.-M. spaces I think they've got not quite righ= t. The article in question: = http://en.wikipedia.org/wiki/Eilenberg%E2%80%93MacLane_space . The problem: after stating (more or less correctly) that "An important = property of K(G,n) is that, for any abelian group G, and any CW-complex X= , = the set [X, K(G,n)] of homotopy classes of maps from X to K(G,n) is in natural bijection with= = the n-th singular cohomology group H^n(X; G)" the article goes on to say (incorrectly) that "Since H^n(K(G,n); G) =3D = Hom(G,G), there is a distinguished element u {\in} H^n(K(G,n);G) = corresponding to the identity." Seems to me all that's justified here would be that 'the set [K(G,n), K(G,n)] of homotopy classes of maps from K(G,n) to itself is in natural bijection= = with H^n(K(G,n); G)', whence "there is a distinguished element u {\in} = H^n(K(G,n);G) corresponding to the identity." What exact role Hom(G,G) may have to play here might be of interest in it= s = own right, but there's no groundwork for that laid anywhere in this Wiki = article, and it's not germane to the Yoneda lemma instance being invoked.= Or am I missing something? In any event, I haven't the optimism or the enthusiasm to care to try = to revise this Wiki's text -- but I welcome any reader who has to do so. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]