Differentiable maps with 0-dimensional critical set. I

Philip Throop Church and James Timourian

Vol. 41 (1972), No. 3, 615–630

DOI: 10.2140/pjm.1972.41.615

Abstract

Let f : Mⁿ → N^p be Cⁿ with n − p = 0 or 1, let p ≧ 2, and let R_p−1(f) be the critical set of f. If dim(R_p−1(f)) ≦ 0 and dim(f(R_p−1(f))) ≦ p − 2, then (1.1) at each x ∈ Mⁿ, f is locally topologically equivalent to one of the following maps: (a) the projection map ρ : Rⁿ → R^p, (b) σ : C → C defined by σ(z) = z^d(d = 2,S,…), where C is the complex plane, or (c) τ : C × C → C × R defined by τ(z,w) = (2z ⋅w,|w|² −|z|²), where w is the complex conjugate of w. In particular, either f is locally topologically equivalent to ρ at each x ∈ Mⁿ, or (n,p) = (2,2) or (4, 3).

Mathematical Subject Classification

Primary: 57D70

Milestones

Received: 13 March 1970

Revised: 17 January 1972

Published: 1 June 1972