M.Sc Thesis
M.Sc Student Levi Shay On the Distance between Homotopy Classes of Maps from the Sphere to a Convex Surface Department of Mathematics Professor Itai Shafrir

Abstract

Certain Sobolev spaces of Sn-valued functions can be written as a disjoint union of homotopy classes. The study of the distance between two distinct homotopy classes seems to be initiated by Rubinstein and Shafrir. They computed the distance for the space H1(S1,S1) and proved that the distance is not attained. They also proved analogous formula for the distance in the space W1,p(S1,S1) for different values of p. The objective of this thesis is to generalize the above results in several directions. The simplest generalization is obtained when we replace the target S1 with a curve C which is the boundary of a convex body K in R2 of class C2. A second, more substantial direction of generalization consists of studing the distance between homotopy classes for maps between spheres of higher dimensions. The last generalization we make can be viewed as a combination of the previous ones in which we study the distance between homotopy classes for maps from the sphere to a surface which is the boundary of a convex body K in R3 of class C2. In each of these cases we find an explicit formula for the distance and treat the question of attainability of the distance.