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Introduction

In Chapter 14 we showed that ordinary cohomology with coefficients in $G$ is represented by Eilenberg–MacLane spaces: $H^n(Y;G)\cong[Y,K(G,n)]\text{,}$ or in the based reduced form $\widetilde H^n(Y;G)\cong[Y,K(G,n)]_*\text{.}$ Brown’s representability theorem explains why this phenomenon extends to generalized cohomology. Its hypotheses express two familiar properties: classes on a wedge are specified independently on its summands, and compatible classes can be glued across a pushout. We will state the theorem and use the suspension axiom to assemble representing spaces into a spectrum. Throughout, generalized cohomology means reduced cohomology on based CW complexes, with the wedge axiom.

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