Subject: Re: comments & clarification
Date: Thu, 9 May 2002 09:29:58 -0500 (CDT)
From: Brayton Gray
With rergard to inverse limits in the category of spaces, it is clear that
the inverse limit of CW complexes does not have to be the homotopy type of
a CW complex. Consider a countable product of copies of S^2. The homology
is not finitely generated, but if this were homotopy equivalent to a CW
complex, the image of such an equivalence would lie in a finite
subcomplex, being compact. One of the advantages of the category of
simplicial sets is that you avoid this problem.
>
> Subject: clarification
> Date: Thu, 9 May 2002 09:25:38 +0100 (GMT Daylight Time)
> From: Andrey Lazarev
>
> Perhaps I should clarify my question. Let D:I-->Top
> be a diagram of topological spaces (say, pointed, compactly
> generated, weakly Hausdorff). Here the indexing category I is not
> necessarily a directed set. Then we could form its homotopy (inverse)
> limit holim(D). I insist that the diagram D consist of CW-complexes.
>
> Question: is it true that holim(D) has homotopy type of a CW-complex?
>
> Note that it is important that we take the inverse limit. In the case
> of the homotopy direct limit the answer is positive and is a more or
> less a model category theory result. (hocolim of the object-wise
> cofibrant diagram is cofibrant.)
>
____________________________________________
Subject: Lazarev's question
Date: Thu, 9 May 2002 13:27:04 -0400 (Eastern Daylight Time)
From: "Nicholas J. Kuhn"
If X is a space, the infinite product
X x X x X x ...,
with the usual product topology, is the limit of the tower of fibrations
X <-- X x X <-- X x X x X <--
and is thus homotopy equivalent to the holimit of this tower.
If X is also a finite CW complex then the infinite product is compact,
(and thus already has the compactly generated topology).
However, if X is not contractible, then X will NOT have the homotopy
type of a CW complex. The simplest case would be X = two points.
The notion of a CW complex dates from a 1949 article by JHC Whitehead,
and examples like this seem to have been known from the beginning.
Milnor's 1959 TAMS article "On spaces having the homotopy type of a CW
complex" is a fine starting place for searching (backwards) through the
literature.
Nick Kuhn