Polyhedral neighborhoods in triangulated manifolds

This note is an outline of some of the author's recent work concerning triangulated manifolds. A combinatorial structure is never assumed; indeed with this condition added, our results are mostly corollaries to well known theorems. The purpose of these investigations is two-fold: it is possible that they may lead to a proof of the cellularity of vertex stars in manifolds (a result that would have critical implications for the theory) ; but also, should a noncombinatorial triangulation of a manifold be found, they might serve as a starting point for the local study of such examples. Our tools are: I. The generalized Schoenflies theorem of Brown and Mazur [2 ; 3] . II . Let M be a, compact Hausdorff space which is the union of two open sets each of which is a homeomorph of E; then M is homeomorphic to S (we write M«?5). This is an immediate consequence of I. III . If the cone over Y (=C(Y)) is w-euclidean at the vertex, then the suspension of F ( = 5 (7 ) ) is topologically S. This proposition of Mazur [3] follows from II . The join of spaces X and Y is written X o F. The kth barycentric subdivision of a polyhedron P is denoted by P. Let (K, L) be a polyhedral pair. The stellar neighborhood of Lin K ( = N(K> L)) is the union of all open simplexes of K with vertices in L. The closure of N(Ky L) is represented by St(K, L) (read star in K ol L). For a simplex w in K let Lk(i£, w) be the link of w in if, and Cl(K, w) ( = w o Lk(K, w)) be the cluster of w in K. For a simplex w = u ov let D be the set of midpoints of segments from u to v and let B(w, u) be the union of all straight segments x o p in w with xCiu and p(ED. If L is full in K define the barrel neighborhood B(Ky L) of L in K as the union of all sets B(w, u) with w and u simplexes of St (if, L) and L, respectively. If K is homogeneous (in the sense of [l]) then the double of K, or 2K, consists of K and a disjoint copy K' with their combinatorial boundaries canonically identified. A quotient space of X whose only possible nondegenerate element is F will be written X/ Y. A subset A of an w-manifold is cellular if it is the intersection of w-cells (C»)