Basic Homotopy

Abstract

In topology we study topological spaces and continuous functions from one topological space to another. In algebraic topology these objects are studied by assigning algebraic invariants to them. We assign groups, rings, vector spaces, or other algebraic objects to topological spaces and we assign homomorphisms of these objects to continuous functions. A basic equivalence relation called homotopy on the set of continuous functions from one topological space into another naturally arises in the study of these invariants. By investigating this relation we obtain interesting, deep, and sometimes surprising information about topological spaces and continuous functions and their algebraic representations. In addition, the relation of homotopy leads to new algebraic invariants for topological spaces and continuous functions.