How to use Haefliger's complexes of groups to prove Novikov Conjectures
by
Thomas Stiadle
Wells College

The algebraic K-theory assembly map, H_n(G; K(R)) --> K_n(RG), relates the homology of a group G with coefficients in the algebraic K-theory of the ring R to the algebraic K-theory of the group ring RG. The relationship is important in topology due to its connection with finiteness obstructions and simple homotopy theory. The algebraic K-theory Novikov Conjecture asserts that this map is a rational injection for various rings R and groups G.

This talk does not approach the conjecture directly; rather, I consider the K-theory of a complex of groups G(X) with respect to a ring R, K_n(RG(X)). For favorable rings and ``trivial'' complexes of groups, I can compute the corresponding assembly map H_n(G(X); K(R)) --> K_n(RG(X)).

In addition if a curious geometric condition on G(X) is met, this new assembly map is equivalent to the ordinary assembly map for the ``fundamental group'' G of G(X). As a consequence it follows that the Novikov Conjecture holds whenever R is the ring of integers in a number field and G admits a K(G, 1) whose universal cover contains arbitrarily large finite acyclic subcomplexes. For instance this method applies to prove the result for torsion-free hyperbolic groups.

Date received: April 15, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cabb-16.