Tittel: Bott periodicity via simplicial spaves and its applicatio to K-theory
Veileder: Gereon Quick
Sammendrag: In this article we will explain the proof by Bruno Harris of the famous Bott Periodicity theorem and show how it applies to K-theory. Section one will review some concepts that will be important for proving the Bott Periodicity theorem. Afterwards, in section two we will follow Bruno Harris proof and prove that $\Omega^2 (BU \times \mathbb{Z})$ is homotopy equivalent to $BU \times \mathbb{Z}$. The statement will be proven in three parts, namely
$$\Omega^2 (BU \times \mathbb{Z}) \overset{I}\simeq \Omega (U) \overset{II}\simeq \Omega (B \underset{n}\coprod BU_n) \overset{III}\simeq BU\times \mathbb{Z}.$$