Two responses to the question about direct limits........DMD
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Subject: Re: response and question
From: N P Strickland <N.P.Strickland@sheffield.ac.uk>
Date: Wed, 8 Sep 2004 10:02:31 +0100

>> Everybody believes that homotopy commutes with direct limits. Does it in
>> general?

No.  Let Y_k be the arc in the unit circle from angle 0 to 2 pi - 1/k,
and put X_k = S1/Y_k.  Then the colimit of the X_k's (at least in the
usual category of compactly generated weak hausdorff spaces) is a point,
so colim \pi_1(X_k) = Z is not the same as \pi_1(colim X_k).  If we
don't impose the hausdorff condition, then the colimit has two points,
and I'm not sure about its fundamental group.

Neil
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Subject: Re: response and question
From: Tom Goodwillie <tomg@math.brown.edu>
Date: Wed, 8 Sep 2004 11:43:36 -0400
>
> Subject: colimits and homotopy
> From: "Claude Schochet" <claude@math.wayne.edu>
> Date: Mon, 6 Sep 2004 21:25:38 -0400
>
> For toplist:
>
> Everybody believes that homotopy commutes with direct limits. Does it in
> general?
>
> To be precise, suppose that X is the direct limit of a directed (not
> necessarily countable) family X_\alpha of topological spaces with structural
> maps not necessarily 1-1. Then one would like that the natural map
>
>          dirlim \pi _*(X_\alpha ) \to \pi _*(X)
>
> would be an isomorphism.  If I assume nothing about the spaces and nothing
> about the structural maps other than continuity, is this true? Obviously the
> key point is that any map from a compact K to X should factor through some
> X_\alpha. This is true, for instance, if we have a countable increasing
> union with X having the weak topology and each X_n closed in the next, by
> Whitehead 1.6.3.


I don't believe that you even need "closed", as long as the spaces are Hausdorff, 
or even T_1:

Let K be a compact subset of X that is not contained in any X_\n. One can 
recursively make an increasing sequence of numbers {n_j} and points x_j in K 
such that x_j is not in X_{n_j}. Then the infinite set S={x_j} is closed in X 
because it intersects each X_n in a finite set. Every subset of S is closed for 
the same reason. Therefore S is infinite, discrete, and compact, a contradiction.

>  Is that the best result? Does it help to assume that each
> space is the homotopy type of a CW complex?


Here is an instructive example:

Let C be a circle. Let {A_n} be a sequence of closed arcs in C such that A_n is 
in the interior of A_{n+1} and such that the union U of all the A_n is the 
complement of a point in C.

Let X_n be C/A_n.

The colimit of the diagram of circles

X_1 -> X_2 -> ...

is C/U, a two-point space in which only one of the points is closed. Or if you 
prefer to form the colimit in the category of Hausdorff or T_1 spaces then the 
colimit is a point. Either way, pi_1 comes out wrong.

Tom Goodwillie