Polynomial equations of immersed surfaces

Selman Akbulut and Henry Churchill King

Vol. 131 (1988), No. 2, 209–217

DOI: 10.2140/pjm.1988.131.209

Abstract

If V is a nonsingular real algebraic set we say H_i(V ;Z₂) is algebraic if it is generated by nonsingular algebraic subsets of V .

Let V ³ be a 3-dimensional nonsingular real algebraic set. Then, we prove that any immersed surface in V ³ can be isotoped to an algebraic subset if and only if H_i(V : Z₂) i = 1,2 are algebraic. This isotopy above carries the natural stratification of the immersed surface to the algebraic stratification of the algebraic set. Along the way we prove that if V is any nonsingular algebraic set then any simple closed curve in V is 𝜖-isotopic to a nonsingular algebraic curve if and only if H₁(V : Z₂) is algebraic.

Mathematical Subject Classification 2000

Primary: 57R19

Secondary: 14G30

Milestones

Received: 23 September 1986

Revised: 3 March 1987

Published: 1 February 1988