In this talk I will report on joint work with Wolfgang Luck, Holger Reich, and John Rognes. First I will introduce and motivate a conjecture of Tom Farrell and Lowell Jones, known as the isomorphism conjecture in algebraic K-theory. Whitehead groups, and more generally algebraic K-theory groups of group-algebras, are fundamental tools for studying manifolds (of sufficiently high dimension), but they are usually very hard to calculate. The Farrell-Jones conjecture predicts that they are isomorphic to the (equivariant, generalized) homology groups of certain universal spaces, which are much more amenable to computations.
I will then state and explain our result about the rational injectivity part of the Farrell-Jones conjecture which generalizes a famous theorem of Marcel Boksted, Wu Chung Hsiang, and Ib Madsen, and in particular its corollary for Whitehead groups. A connection with Schneider's generalization of) the Leopoldt conjecture in algebraic number theory will also be highlighted.
At last I will briefly outline the proof of our theorem. The main ingredients are so-called trace maps from algebraic K-theory to other "easier" theories, like Hochschild homology and topological cyclic homology,
invented by Bokstedt-Hsiang-Madsen. Along the way we will also prove quite general isomorphism and injectivity results for assembly maps in topological Hochschild homology and topological cyclic homology
with arbitrary coefficients. |