Vol. 48, No. 1, 1973

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Differentiable open maps of (p + 1)-manifold to p-manifold

Philip Throop Church and James Timourian

Vol. 48 (1973), No. 1, 35–45
Abstract

Let f : Mp+1 Np be a C8 open map with p 1, let Rp1(f) be the critical set of f, and let

dim(Rp− 1(f)∩ f−1(y)) ≦ 0

for each y Np. Then (1.1) there is a closed set X Mp+1 such that dimf(X) p2 and, for every x Mp+1 X, there is a natural number d(x) with f at x locally topologically equivalent to the map

ϕd(x) : C × Rp−1 → R × Rp−1

defined by

ϕd(x)(z,t1,⋅⋅⋅ ,tp−1) = (ℛ(zd(x)),t1,⋅⋅⋅ ,tp−1)

((zd(x)) is the real part of the complex number zd(x)).

Mathematical Subject Classification
Primary: 57D35
Milestones
Received: 21 June 1972
Published: 1 September 1973
Authors
Philip Throop Church
James Timourian