Local flatness of combinatorial manifolds in codimension one
William A. LaBach · Proceedings of the American Mathematical Society · 1967
PROOF. Let x be any point of K and let v be a vertex of K containing x in the interior of its star, St(v, K). Without loss of generality, we may assume that v is the origin in Rn+l. The radial projection r of the link, Lk(v, K), of v in K on Sn is a combinatorial (n-1)-sphere in Sn whose cells are geodesic simplexes on Sn. By the main theorem of [2], there is a homeomorphism h of Sn (onto itself) taking r(Lk(v, K)) onto Sn-1. Let h* denote the radial extension of h to a homeomorphism of Rn+'. Then h* maps St(v, K) into Rn. Thus K is locally flat in R+1.