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. _________________________________________________________________ Join the world’s largest e-mail service with MSN Hotmail. http://www.hotmail.com