Transverse field implies normal microbundle

Abstract

1. Definition. Let s denote a simplex (open) of M. By P(s) we will denote the subset (possibly empty) in Gn,p (the Grassmannian manifold of all p-planes through the origin in Rn+p) consisting of all p-planes P, such that, if H is the orthogonal n-plane to P in Rn+p, then the orthogonal projection q: Rn+p-*H, restricted to St(s, M)(the closed star of s in M) is a homeomorphism carrying St(s, M) onto an open set in H. If the p-plane P belongs to P(s), then P is said to be transversal to M at m, where m is any point of s. A continuous map g: M-*Gn,p is a transversefield, if the set g(s) is contained in the set P(s), for every simplex s of M.