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Topology/Geometry Seminar
Thursday, October 18, 2007
5:00pm,
150 Hawthorn
Marco Varisco
SUNY Binghampton
On the Isomorphism Conjecture in Algebraic K-Theory and Topological
Cyclic Homology
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
groupalgebras, 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.
We meet weekly on Thursday afternoons, with the third meeting of each month in Altoona. If you are interested in giving a talk in our seminar please contact one of the coordinators for the Spring 2006 semester: Aissa Wade and Wojciech Dorabiala.
