Abstract:

The Brown Representability theorem states that a generalized cohomology theory is represented by a special sequence of topological spaces, called a spectrum, which is a fundamental object in stable homotopy theory. To motivate spectra and their cohomology theories, we will build enough foundations in simplicial homotopy theory to outline a proof of the Dold-Kan Correspondence, an adjoint equivalence between the category of nonnegatively graded chain complexes of abelian groups and the category of simplicial abelian groups. One application of Dold-Kan is an elegant and functorial construction of the classical Eilenberg-MacLane spaces, whose spectra represent singular cohomology. Time permitting, we will see that some of the basic properties of spectra are easily exhibited by the Eilenberg-MacLane spectra using only simplicial methods.

Key terms:

Algebraic topology, homotopy theory, simplicial sets, cohomology, spectra.