Polynomial equations of immersed surfaces

Abstract

If V is a nonsingular real algebraic set we say Ht(V;Z2) * s algebraic if it is generated by nonsingular algebraic subsets of V. Let V be a 3-dimensional nonsingular real algebraic set. Then, we prove that any immersed surface in V can be isotoped to an algebraic subset if and only if Ht (V: Z 2 ) / = 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 e-isotopic to a nonsingular algebraic curve if and only if H1(V:Z2) is algebraic.