Projections of Immersed Surfaces and Regular Homotopy

Abstract: 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 property that it depends only on the regular homotopy class of f. In the case of orientable surfaces qf turns out to be a Z2-quadratic form. In this thesis we construct the Z2-quadratic form using the notion of Spin - structure, and via D. Johnson’s correspondence between Spin - structures on a surface and Z2-quadratic forms on the first homology group of the surface. Then by studying the relation between surface immersions into 3-space and their projections to a 2-plane, we give a formula for computing the value of the quadratic form on any homology class c ∈ H1(Σ,Z2), which we will use to construct an example of two nonregularly homotopic immersions of the 2 - dimensional torus T2 into R3 with identical plane projections.