Subject: Re: finiteness
Date: Mon, 13 May 2002 15:02:25 +0000
From: "Avraham Goldstein"
To: dmd1@Lehigh.EDU
>From: Don Davis
>To: "dmd1@lehigh.edu (Don Davis)" <"dmd1"@lehigh.edu (Don Davis)>
>Subject: 2 responses
>Date: Mon, 13 May 2002 08:20:56 -0400
>
>2 postings: comments on last week's discussions........DMD
>________________________________________________
>
>Subject: finiteness
>Date: Fri, 10 May 2002 12:05:13 -0400
>From: Tom Goodwillie
>
> >In light of what I have learned I am now proposing a refinement of
> >the original question: what are the conditions on the indexing
> >category I ensuring that holim_I(D) is homotopy equivalent to a
> >CW-complex? The diagram D is supposed to be object-wise cellular. It
> >seems that the natural condition is that the nerve of D be a finite
> >simplicial set.
> >
> >Andrey
>
>That kind of finiteness is sufficient but not necessary.
>
>If I is (the one-object category corresponding to) a group,
>then I believe a sufficient condition is that BI should have
>finite CW homotopy type. So an infinite cyclic group is OK.
>
>We can say something more general. Think of holim(D) as the space of
>maps
>from EI to D where EI is any example of what might be called a
>free resolution of the point in the category of I-diagrams. I suppose
>that a sufficient condition is that there should be a finite
>free resolution of the point, if you know what I mean.
>That implies, but is stronger than, the condition that
>BI has finite CW homotopy type.
>
>As Vogt points out, a necessary condition is that the CW complex
>BI should be such that Map(BI,X) has CW homotopy type whenever X has.
>
>[Question: Which CW complexes, other than homotopy retracts
>of finite complexes, have that property?]
>___________________________________________________
It seems that the necessary and sufficient condition for Map(Y,X) to have CW
homotopy type, whenever X does, is that Y is a homotopy retcract of locally
finite CW complex...
Avi.
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