M.Sc Student | Levi Shay |
---|---|

Subject | On the Distance between Homotopy Classes of Maps from the Sphere to a Convex Surface |

Department | Department of Mathematics |

Supervisor | Professor Itai Shafrir |

Full Thesis text |

Certain Sobolev
spaces of *S*^{n}-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 *H*^{1}*(S*^{1}*,S*^{1}*)*
and proved that the distance is not attained. They also proved analogous formula
for the distance in the space *W*^{1,p}*(S*^{1}*,S*^{1}*)*
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 *S*^{1} with a curve *C*
which is the boundary of a convex body *K* in *R*^{2} of
class *C*^{2}_{}. 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 *R*^{3} of class *C*^{2}_{}.
In each of these cases we find an explicit formula for the distance and treat
the question of attainability of the distance.