Vol. 80, No. 1, 1979

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Transverse Whitehead triangulations

Martin Scharlemann

Vol. 80 (1979), No. 1, 245–251

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, f1(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
Received: 29 December 1977
Published: 1 January 1979
Martin Scharlemann
Mathematics Department
UC Santa Barbara
Santa Barbara CA 93106-3080
United States