PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

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Title:On the Isomorphism Conjecture in Algebraic K-Theory and Topological Cyclic Homology
Seminar:Topology/Geometry Seminar
Speaker:Marco Varisco (SUNY Binghamton)
Abstract:
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.

Room Reservation Information

Room Number:150 Hawthorn, Altoona
Date:10 / 18 / 2007
Time:05:00pm - 06:00pm