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