Three quick responses to my posting, two of them more-or-less the same...DMD
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Subject: Re: example sought
From: Bill Dwyer <dwyer.1@nd.edu>
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
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Date: Fri, 26 Jan 2001 13:42:35 -0600 (CST)
From: Brayton Gray <brayton@math.uic.edu>
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
>

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From: Dan Kahn <kahn@veblen.math.nwu.edu>
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