Polynomial equations of immersed surfaces

If V is a nonsingular real algebraic set we say H t (V;Z 2 ) * s 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 t (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 ε-isotopic to a nonsingular algebraic curve if and only if H 1 (V:Z 2 ) is algebraic.