Vol. 57, No. 1, 1975

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Maps with 0-dimensional critical set

Philip Throop Church and James Timourian

Vol. 57 (1975), No. 1, 59–66
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, 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) = za(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(Rp1(f))) p 2 this result was proved in an earlier paper of the authors. They show here that dim(Rp1(f)) 0 implies somethin g like dim(f(Rp1(f))) p 2.

Mathematical Subject Classification
Primary: 57D35
Secondary: 57D70
Milestones
Received: 7 February 1973
Published: 1 March 1975
Authors
Philip Throop Church
James Timourian