On the triangulation of the realization of a semisimplicial complex
S. Weingram · Illinois Journal of Mathematics · 1968
In some mimeographed notes of Barrett [1], in which he shows that the geometric realization of a semisimplicial complex has a simplicial subdivision, there appear to be several errors, one in the statement of the subdivision theorem and two more in the proof.This note will give (we hope) a correct statement and proof of this theorem (Theorem 1.1), and will draw as a conse- quence the theorem of h/[ilnor [4] that the homotopy groups of the realization S(X) of the singular complex of a space are naturally isomorphic to those of the space itself.We will write ssc for semisimplicial complex.Notation and terminology as in [4] or [5], except that we will denote the abstract n-simplex by Z(n), the gedmetric n-simplex by An.The main result is the following theorem.THEOREM 1.1.Let X be a ssc and XI its realization [4].Then there is a functor D from the category of ssc's to that of ordered simplicial complexes, a transformation of functors D 1, and, for each X, a map t DXI IX such that (i) t is a homeomorphism (and therefore a triangulation of IX I; (ii) t defines a subdivision of the CW complex IX I; and(iii) k(X) is homotopic to t by a homotopy F such that for each cell e of DX I, F maps e X I into the smallest cell x of X which contains t(I e I).We will give the proof later, in Sections 2, 3, and 4. COROLLARY 1.2.(Simplicialapproximationtheorem).Iff :1XI-* Ylis any map, then f is homotopic to the realization of a ss map of subdivisions of X and y I.If IX I, Y are finite and therefore metrizable, then for any pre- scribed e > O, the homotopy between f and its ss approximation can be chosen so that it does not displace a point outside of an e-disc.Apply the simplicial approximation theorem to the map f' DXI -* DY 1, where f' tlft | Some remarks about the singular complex of a space.Let X be a space, and S(X) its singular complex.Let px IS(X) --+ X be the map sending the point (P, x) of S(X)I into xn(P) [4].