[Binghamton Geometry/Topology Seminar] with abstract

[Ross Geoghegan]

                   BINGHAMTON GEOMETRY/TOPOLOGY SEMINAR


   Date: Thursday October 26, 2006

   Time:  2.50 pm

   Place: Library North 2205 followed by coffee/tea in the Anderson
          Reading Room.

   Speaker: Marco Varisco (Binghamton) 

   Title: On the Isomorphism Conjecture in Algebraic K-Theory and
          Topological Cyclic Homology

   (This is the first of two talks.)

Abstract: 

In this series of two talks I will report on joint work with
Wolfgang Lueck, Holger Reich, and John Rognes.

In the first talk 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 normally 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 Boeksted, 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.

In the second talk I will 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 Boekstedt-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.


An elementary and introductory talk "On the Whitehead group: from
topology to algebra and back" will be given in the Toeholds in
Topology Seminar on Tuesday, October 24, 5:20-6:30pm in LN2205



NOTE:  The seminar has a webpage where the semester's program is listed:

http://math.binghamton.edu/MATH/dept/topsem/index.html

It can also be linked from the Department's Home Page.
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