Transverse Whitehead triangulations

Martin Scharlemann

Vol. 80 (1979), No. 1, 245–251

DOI: 10.2140/pjm.1979.80.245

Abstract

Suppose M and N are PL manifolds and f : M → N is a proper PL map. Triangulate M and N so that f is simplical and let X be the dual complex in N. Then for each open simplex σ in X, f⁻¹(σ) is a PL submanifold of M, so the stratification of N by the open simplices of X pulls back to a stratification of M. In other words, any such PL map can be regarded as a map of combinatorially stratified sets in which each n-stratum of therange is a disjoint union of copies of Rⁿ. Here we prove the analogous theorem for a smooth map f : M → N between smooth manifolds.

Mathematical Subject Classification 2000

Primary: 57R05

Milestones

Received: 29 December 1977

Published: 1 January 1979