Subject: simplicial sets From: Brayton Gray Date: Fri, 31 Dec 2004 14:01:40 -0600 To: John Rognes , dmd1@Lehigh.EDU, hatcher@math.cornell.edu CC: Brayton Gray I should have refreshed my memory a bit before I wrote on this topic. Simplicial sets used to be called complete semisimplicial complexes(css complexes); the word complete referred to the incorporation of degeneracy operations with the face operations, so a semisimplicial complex only had face operators (i.e, the hom sets in the domain category consisted of strict monomorphisms [n]--->[m] ). In my book, I discussed realizations of semisimplicial complexes, which I called semisimplicial CW complexes. They seemed natural and useful. They occur frequently, for example as dissections of surfaces. and their homology can be easily read off. This concept has been resurrected by Alan Hatcher and others and called Delta complexes, although the definition given by Hatcher is somewhat different in emphasis. (I emphasized fixed characteristic maps from the n simplex). I proved a result, which I assumed was the result that Barratt had claimed; I never spoke with him on the topic or saw his preprint. It was the following: 1) The barycentric subdivision of a semisimplicial CW complex is a regular semisimplicial CW complex. 2) The barycentric subdivision of a regular semisimplicial CW complex is a (geometrical) simplicial complex. Thus these spaces were not in any way exotic, and were convenient for doing homology calculations. I apologize if my posting caused any confusion. On Wednesday, Dec 29, 2004, at 03:48 US/Central, John Rognes wrote: >> Subject: simplicial complexes >> From: Brayton Gray >> Date: Wed, 22 Dec 2004 09:10:31 -0600 >> >> I cannot answer your questions, but do have related information >> that is not well known. It was asserted to me by Barratt that the >> second barycentric subdivision of the realization of a simplicial >> set is a simplicial complex. I believe that this fact was discovered >> by him but never published. I wrote down a proof of it in my homotopy >> theory book. > > > Dear Prof. Gray, > > (cc to the toplist) > > What do you mean when you say that "the second barycentric subdivision of > the realization of a simplicial set _is_ a simplicial complex" (my > emphasis)? If the simplicial set $X$ is the 2-simplex $\Delta2$ with its > boundary $\partial \Delta2$ collapsed to a point, then the second > Kan/normal/barycentric subdivision $Z = Sd2(X)$ still has non-degenerate > 2-simplices that are not embedded, i.e., $Z$ is not a simplicial complex > in the obvious way. > > Here I am assuming that by "barycentric subdivision" you mean the > endofunctor of simplicial sets that is left adjoint to Kan's $Ex$. The > situation is different if you are talking about Delta-sets/presimplicial > sets/ignoring degeneracies, but that does not seem to be the mainstream > interpretation of "simplicial set". > > Perhaps you mean that "the realization of a simplicial set can be > triangulated by a simplicial complex"? This was asserted in a 1956 > Princeton preprint of Barratt's, titled "Simplical and semisimplicial > complexes", by an argument involving two subdivisions, and some more. > But that mimeographed paper contains a serious mistake in Lemma 3.1, and > remained unpublished. > > The first correct proof of the above claim seems to be in: > > Fritsch, Rudolf; Puppe, Dieter > Die Homöomorphie der geometrischen Realisierungen einer semisimplizialen > Menge und ihrer Normalunterteilung. (German) > Arch. Math. (Basel) 18 (1967) 508-512. > > as "Korollar" on page 508. For a textbook reference in English, see Cor. > 4.6.12 in > > Fritsch, Rudolf; Piccinini, Renzo A. > Cellular structures in topology. > Cambridge Studies in Advanced Mathematics, 19. > Cambridge University Press, Cambridge, 1990. > > This topic has a history of incorrect proofs. One early case: > > Fritsch, Rudolf > Some remarks on S. Weingram: "On the triangulation of a > semisimplicial complex". > Illinois J. Math. 14 (1970) 529--535. > > A related noteworthy fact is that there is no _natural_ homeomorphism > $h_X : |Sd(X)| \cong |X|$ between the geometric realization of a > simplicial set and its barycentric subdivision, even when restricted to > the full subcategory of simplices. The paper of Fritsch and Puppe cited > above gives a counterexample, repeated on pages 124-125 in the book of > Fritsch and Piccinini. > > It seems to me that after 1970 or so, the algebraic topological interest > in simplicial sets shifted entirely to their homotopy theory, rather than > their topology, i.e., people were mostly concerned with (weak) homotopy > equivalences rather than homeomorphisms of the geometric realizations of > simplicial sets. > > Sincerely yours, > John Rognes >