Cary Malkiewich: The transfer in algebraic K theory and THH

The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields.

Cary Malkiewich: The transfer in algebraic K-theory and THH

Abstract:
Let R → A be a map of rings (or ring spectra) and suppose that A is finitely generated projective (or perfect) as an R-module. Then in addition to the usual map on algebraic K-theory K(R) → K(A), there is a wrong-way "transfer" map K(A) → K(R). In particular, when E → B is a map of spaces whose homotopy fiber F is finitely dominated, this gives a wrong-way map on Waldhausen's functor A(B) → A(E). We will ask a few fundamental questions about this transfer, and present the beginning of a program to answer these questions using trace methods. Our main results concern the corresponding transfer on THH, which in the A-theory case is a stable map of free loop spaces LB+ → LE+. If there is time, we will also describe how our techniques are related to the study of fixed points of dynamical systems.