Trace Methods & Localization Sequences

We rapidly introduce the trace maps K \to THH and K \to TC as the master tool to understand algebraic K-theory. We indicate an application by sketching Antieau--Barthel--Gepner's argument showing that the localization sequence K(BP<n-1>) ---> K(BP<n>) ---> K(E(n)) is not a fiber sequence for n \geq 2. This answers a question of Rognes negatively.