Critical value sets of generic mappings

Goo Ishikawa, Satoshi Koike and Masahiro Shiota

Vol. 114 (1984), No. 1, 165–174

DOI: 10.2140/pjm.1984.114.165

Abstract

Let Y be a real analytic set. The subset of Y consisting of all points where the local dimension of Y is maximal is called the main part of Y . A subset Y ′ of a real analytic manifold N is called a main semi-analytic set if Y ′ is the main part of some analytic set in a neighborhood of each point of N. In this paper it is shown that any proper C^∞ mapping between analytic manifolds can be approximated by an analytic mapping in the Whitney topology so that the critical value set is a main semi-analytic set. An analogue holds true for the algebraic case too.

Mathematical Subject Classification 2000

Primary: 58C27

Secondary: 32B20

Milestones

Received: 19 April 1982

Revised: 9 June 1983

Published: 1 September 1984