Polynomial equations of immersed surfaces

Polynomial Equations of Immersed Surfaces

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 surfa...

Harmonic Maps and Self - Dual Equations for Immersed Surfaces

Immersed Surfaces and Dehn Surgery

The problem of how many Dehn fillings on a torus boundary component T of a 3-manifold M will make a closed embedded essential surface F compressible has been settled. A slope β on T is a coannular slope if it is homotopic to some curve on F . As an embedded essential surface, F can have at most one coannular slope. If F has a coannular slope β on T , then by a result of Culler-Gordon-Luecke-Sha...

Projections of Immersed Surfaces and Regular Homotopy

This thesis is based on U. Pinkall’s study of the classification of immersions of compact surfaces into R3 up to regular homotopy. The main idea of the classification is to associate to any immersion f a quadratic form qf on the first homology group of the underlying surface Σ with Z2 coefficients, whose associated bilinear form is the nondegenerate intersection form in H1(Σ,Z2), having the pro...

On the Complexity of Immersed Normal Surfaces

Normal surface theory, a tool to represent surfaces in a triangulated 3-manifold combinatorially, is ubiquitous in computational 3-manifold theory. In this paper, we investigate a relaxed notion of normal surfaces where we remove the quadrilateral conditions. This yields normal surfaces that are no longer embedded. We prove that it is NP-hard to decide whether such a surface is immersed. Our pr...