Victor Groth: Basic Homotopy Theory and the Brown Representability Theorem

Abstract

 My goal is to introduce the necessary preliminaries in homotopy theory to understand the Brown Representability theorem. The theorem, named after Edgar Brown, gives sufficient conditions for when a contravariant functor from the homotopy category of connected CW complexes to pointed sets is represented by a CW complex. Since the theorem is beautiful and has interesting applications in cohomology theory, I wish to introduce it to anyone that has just started reading topology and talk about its connection to general cohomology.