Subject: Re: Two questions
Date: Fri, 11 Jan 2002 14:54:45 +0000
From: Ronnie Brown
To: Don Davis
In reply to the second part of the first question, this is a special case
of `cogluing homotopy equivalences'
(with P.R. HEATH), ``Coglueing homotopy equivalences'', {\em Math. Z.} 113
(1970) 313-362.
which gives conditions for a pullback of homotopy equivalences to be a
homotopy equivalence. The result can be proved in abstract homotopy
theories - see the book by H Kamps and T Porter.
Ronnie Brown
Don Davis wrote:
>
> Two postings: A question about homotopy equivalences, and a
> question about Riemann surfaces........DMD
> _____________________________________
>
> Subject: Question for discussion group
> Date: Thu, 10 Jan 2002 15:52:28 +0100 (MET)
> From: Rasmus Ejlers Moegelberg
>
> I have a question for the discussion group. A theorem of Milnor says
> that
> if A and B are CW-complexes and h:A->B is any map between them, then the
>
> homotopy fiber of h is homotopy equivalent to a CW-complex. Is it true,
> that if A and B are only homotopy equivalent to CW-complexes, then the
> homotopy fiber of h is homotopy equivalent to a CW-complex?
>
> I have a different question, that would answer the above question for
> me:
> Suppose we have a (strictly) commutative diagram of maps:
>
> A ----h----> B
> | |
> v v
> A' ---h'---> B'
>
> where the two vertical maps are homotopy equivalences. Is the induced
> map
> from the homotopy fiber of h to the homotopy fiber of h' a homotopy
> equivalence, and not just a weak homotopy equivalence?
>
> Rasmus Ejlers Mogelberg,
> Student, University of Copenhagen, Denmark
>
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