Vol. 41, No. 3, 1972

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ISSN: 0030-8730
Differentiable maps with 0-dimensional critical set. I

Philip Throop Church and James Timourian

Vol. 41 (1972), No. 3, 615–630
Abstract

Let f : Mn Np be Cn with n p = 0 or 1, let p 2, and let Rp1(f) be the critical set of f. If dim(Rp1(f)) 0 and dim(f(Rp1(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 Rp, (b) σ : C C defined by σ(z) = zd(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
Authors
Philip Throop Church
James Timourian