From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5920 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: Re: covering spaces and groupoids Date: Tue, 08 Jun 2010 22:18:45 +0100 Message-ID: References: <4C02A580.2000606@math.upenn.edu> <4C02A698.9090706@math.uchicago.edu> <4C04CB41.9080705@btinternet.com> <4C0502DB.5030603@math.uchicago.edu> , Reply-To: Ronnie Brown NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1276037410 26755 80.91.229.12 (8 Jun 2010 22:50:10 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 8 Jun 2010 22:50:10 +0000 (UTC) To: "F. William Lawvere" Original-X-From: categories@mta.ca Wed Jun 09 00:50:08 2010 connect(): No such file or directory 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 4.69) (envelope-from ) id 1OM7cW-0007wI-1V for gsmc-categories@m.gmane.org; Wed, 09 Jun 2010 00:50:08 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OM7AF-0004LL-Qf for categories-list@mta.ca; Tue, 08 Jun 2010 19:20:55 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5920 Archived-At: Dear Bill, You ask: In the applications of algebraic topology to topology, where do the=20 =91basepoints=92 originate? I'd like to give a different type of answer to that suggested in your=20 list. (yet another is : whoopee, we have a group!) After I first thought of the van Kampen theorem for the whole=20 fundamental groupoid I realised one wanted *computation*, and the whole=20 fundamental groupoid of a space was too big for that. The fundamental=20 group at a base point was too small in may cases, such as the circle.=20 The solution that seemed right to me, and which took a while to find,=20 was the fundamental groupoid on a set of base points chosen conveniently=20 according to the geometry of a given situation. Eldon Dyer said I ought=20 to take a hard line on this: if the connected space X is the union of=20 127 open sets whose intersections have 3,272 components, you do not want=20 to take a single base point! Such situations (even with infinitely many=20 components) occur in group theory applications, and analogously in=20 topology in connected covering spaces over a union of 2 open sets. I advocated the fundamental groupoid on a set of base points in my 1968=20 book `Elements of Modern Topology', but this concept has I think not=20 been mentioned in any later date algebraic topology text in English by=20 other authors. Also group theorists are happy with graphs and free=20 groups, but are very wary of the free groupoid on a graph! The only new=20 result from that book that has been taken up is the gluing theorem for=20 homotopy equivalences, but its origin is usually unacknowledged. Looking at the way this groupoid van Kampen theorem could be used, it=20 seemed amazing to me that one could obtain *complete* information on a=20 fundamental group by deducing that from knowledge of a larger structure,=20 for which one had colimit information, whereas my tries with nonabelian=20 cohomology gave only exact sequences. It seems that groupoids have the=20 advantages of structure in dimensions 0 and 1, and that this is needed=20 for what Grothendieck later called `integration of homotopy types'. Could one find analogous objects with structure in dimensions 0, 1,=20 ...,n? This question led (after many years, and with fortunate=20 collaborations) to higher dimensional van Kampen theorems, which gave=20 quite new information on homotopy invariants, some of them nonabelian,=20 e.g. second relative homotopy groups, triad homotopy groups, n-adic=20 Hurewicz Theorems. Such results were published in 1978, 1981 (with=20 Higgins), 1987 (with Loday). These theorems are, I think, not even=20 mentioned in any texts on algebraic topology (except mine). Some current=20 writers (Faria Martins, Kauffman, Ellis, Mikhailov,...) are using these=20 techniques. So for me the question is: why are people unwilling to throw off the=20 shackles of a single base point? I would welcome enlightenment. It may=20 be that it is found just too hard to fit this idea into what is=20 currently considered the `real world', and so to obtain new results. For=20 example, what happens to iterated loop space theory, with more than one=20 base point? Those interested in the sociology of science might like an excerpt from=20 a lecture by Alan MacKay on icosahedral symmetry at the LMS in 1985. He=20 said the reaction went through three phases: Phase 1) It is false. Phase 2) It is true, but unimportant. Phase 3) It is true, it is very important; and we have known it for years= ! All the best Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]