Victor Groth: The Brown Representability Theorem and Representing Reduced Cohomology Theories

Abstract.

The goal of this paper is to state and prove the Brown representability theorem, named after Edgar Brown, which gives sufficient conditions for when a functor from the homotopy category of connected based CW complexes to the category of pointed sets is representable. The theorem will then be used to prove a bijective correspondence between reduced cohomology theories on CW complexes and \(\Omega\)-spectra.

The proof presented is based on the proof by Macerato and Slaoui in their article "The Brown Representability theorem, old and new" [6] with general lemmas that can be generalized to the \(\infty\)-categorical version of the theorem which we will not touch on. However, we correct an oversight in the paper of Macerato and Slaoui as they neglect to mention the crucial assumption of restricting to connected CW complexes.

Finding the correspondence between reduced cohomology theories and \(\Omega\)-spectra is what Edgar Brown himself did in his paper ”Cohomology theories“ [1] with the first proof of his theorem.