Equivalence Between The Set Of Lipschitz Homotopy Classes And The Set Of Continuous Homotopy Classes

In this paper,we discuss the density of Lipschitz maps in the space of continuous maps,and then prove that a natural function from the set of Lipschitz homotopy classes to the set of continuous homotopy classes is a bijection under some reasonable assumptions on the target metric space.In this bijection,the injection part expands the scope of positive answer to the existence problem of Lipschitz homotopy.This thesis consists of five sections,and the arrangements are as follows:In section 1,we summarize the research background,research significance and latest research results of homotopy classes,and we also introduce the main results of this thesis.In section 2,we fix our notations and collect some basic facts from topology and metric geometry that will be used in what follow,and then define several concepts of homotopy.In section 3,we introduce several equivalence relations and the corresponding sets of equivalent classes that appear in Theorem 1.1 and 1.2.In section 4,we give a proof of Theorem 1.2.In section 5,we give a proof of Theorem 1.1.