Maps with 0-dimensional critical set

Philip Throop Church and James Timourian

Vol. 57 (1975), No. 1, 59–66

DOI: 10.2140/pjm.1975.57.59

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, 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^a(d = 2,3,⋯), 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.

Under the additional hypothesis that dim(f(R_p−1(f))) ≦ p − 2 this result was proved in an earlier paper of the authors. They show here that dim(R_p−1(f)) ≦ 0 implies somethin g like dim(f(R_p−1(f))) ≦ p − 2.

Mathematical Subject Classification

Primary: 57D35

Secondary: 57D70

Milestones

Received: 7 February 1973

Published: 1 March 1975