Subject: Re: question
Date: Tue, 1 Jul 2003 15:14:26 +0200 (CEST)
From: Marek Golasinski
You wrote:
=========
Subject: Simplicial sets of finite type (a forum question)
Date: Mon, 30 Jun 2003 19:26:09 +0400 (MSD)
From: ssp@pdmi.ras.ru
Simplicial sets of finite type
Given a simply connected CW-space
having finitely many n-cells for every n (for example, CP^infty),
is it homotopy equivalent to the geometric realization
of a simplicial set having finitely many n-simplices for every n?
The answer "yes" is claimed
in Bousfield's - Kan's "Homotopy limits..." (Ch. V, Lemma 7.5);
the proof refers to the Wall's paper
"Finiteness conditions for CW-complexes" published in Annals of Math.
This reference seems to be incorrect,
just because Wall does not consider simplicial sets.
Does anybody know a reliable reference for this assertion?
Is it true?
====================
Well, I think the following hold:
(1) if $X$ a $CW$-complex then there is a homotopy equivalence
$|SX|\simeq X$, where $S$ and $|-|$ are the singular and
the geometric realization functors, respectively;
(2) if $X$ is a $CW$-complex having finitely many n-cells
for every n then following Wall's procedure (i.e. replacing
$CW$-complexes by simplicial sets) from Ann. of Math. 81 (1965),
56-69 we can find a simplicial set $Y$ having finitely
many n-simplices for every n and a simplicial homotopy
equivalence $SX\simeq Y$.
(3) Consequently, a homotopy equivalence of $CW$-complexes
$|SX|\simeq |Y|\simeq X$ follows.
I hope that works.
Marek Golasinski
N. Copernicus University
Torun, Poland