The approximation theorem and topological hochschild homology

Abstract

This work is divided in two halves. In the first half we introduce the basic notions of the theory of operads, including free algebras over an operad, a study of the little $n$-cube operads, and their action on iterated loop spaces. This leads to the proof of the approximation theorem by Peter May. In the second half we introduce the basic definitions and properties of spectra and its tensor product. We then define topological Hochschild homology and give the proof of Bökstedt’s periodicity theorem on the description of $\mathrm{THH}(\mathbb{F}_p)$ by Nikolaus and Krause, which relates it to a theorem by Hopkins and Mahowald using the approximation theorem of May.