Date: Wed, 31 Jan 2001 12:14:10 +0000 From: Ronnie Brown Subject: More on X to X/A equiv A variant on these examples is to replace the Hawaiian earrings by the subset of [0,1] consisting of the points 1/n , n \in N, and 0. This example is in my 1968 book, second edition 1988 (out of print). The fact that X -> X/A is a homotopy equivalence if A is `good' and contractible is of course a special case of the gluing lemma for homotopy equivalences (same source), though it has an easier proof. Part of the reason for putting this lemma in the book was that it seems reasonable to give techniques for constructing homotopy equivalences before giving sophisticated invariants proving certain spaces are *not* equivalent. The proof by the way was derived by successive generalisations of the fact that a homotopy eqivalence of spaces (not necessarily pointed) induces isomorphisms of homotopy groups. Nowadays we would express the lemma by saying that a homotopy pushout of homotopy equivalences is a homotopy equivalence, and there are proofs in various abstract homotopy theories. The utility of homotopy pushouts brings me, at the risk of groans from readers, to an old theme. A homotopy pushout of groups is of course a groupoid, and not a group! See math.AT/0101220 Title: Free crossed resolutions for graph products and amalgamated sums of groups Authors: Ronald Brown, Manuel Bullejos, Timothy Porter for applications of this idea. Eldon Dyer once a long time ago emphasised to me the importance of homotopy pushouts in such contexts and I think I now understand what he was getting at! Ronnie Brown DON DAVIS wrote: > Three quick responses to my posting, two of them more-or-less the same...DMD > _____________________________ > > Subject: Re: example sought > From: Bill Dwyer > Date: 26 Jan 2001 14:35:07 -0500 > > Here's a candidate. Let U be the Hawaiian earring (the union of the > circles of radius 1/n with centers (1/n,0), n\ge 1). Let p be the > point (0,0) in this space. Let V be the cone on U, which contains U, > and therefore p. Let X be the space obtained from two copies of V by > joining one copy of p to the other one by a closed interval (two coned > earrings attached by a thread); the subspace A is then the closed > interval. Collapsing A to a point seems to give the one-point union > of two copies of V, and this space is not simply-connected (Spanier, > Chapter I, Problem G.7). > > Bill > ______________________________ > Date: Fri, 26 Jan 2001 13:42:35 -0600 (CST) > From: Brayton Gray > Subject: Re: example sought > > The simplest example I know is the faux circle made out of the closure of > the curve y=sin(1/x),0<=x<=1/pi and a curve connecting the point (0.1/pi) > to the point (0,0) in such a way that it does not intersect the sin curve > except at the endpoints. This is X. A is the line segment from (0,-1) to > (0,1). In this case X/A is homeomorphic to a circle whilethe fundamental > group of X is trivial. > > Brayton Gray > > On Fri, 26 Jan 2001, DON DAVIS wrote: > > > This posting is by your moderator, Don Davis. > > For my beginning algebraic topology course I > > would like a nice example of a closed subspace > > A of a topological space X such that A is > > contractible but X and X/A do not have the same > > homotopy type. The purpose of such an example > > would be to highlight the significance of the > > result that says that such a thing cannot happen > > if A is a subcomplex of a CW complex X. > > > > Don Davis > > dmd1@lehigh.edu > > > > ______________________________________ > From: Dan Kahn > Date: Fri, 26 Jan 2001 13:45:45 -0600 (CST) > > I think this is an example - which has the advantage of using an example > that is famous for other reasons: > > Let X and Y both be the unreduced cone on the Hawaiian earrings space. > Let Z be the one point union of X and Y - joined at the points of the > base where the circles are tangent. > > This the a classic example of a wedge of contractible spaces which is > not itself contractible. > > Let the contractible subspace A be the union of the line segments > running from wedge point to the two cone points. The result should be > contractible. > > Write me if this isn't clearly expressed. > > Regards, > > Daniel S. Kahn > kahn@math.northwestern.edu -- Prof R. Brown, School of Informatics, Mathematics Division, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382681 fax: +44 1248 361429 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ (Links to survey articles: Higher dimensional group theory Groupoids and crossed objects in algebraic topology) Symbolic Sculpture and Mathematics: http://www.cpm.informatics.bangor.ac.uk/sculmath/ Centre for the Popularisation of Mathematics http://www.cpm.informatics.bangor.ac.uk/ Raising Public Awareness of Mathematics http://www.cpm.informatics.bangor.ac.uk/rpamath/