We use topological cyclic homology and the cyclotomic trace to detect elements in the rationalized higher algebraic K-theory groups of integral group rings. Modulo a deep conjecture in number theory, the so-called Schneider conjecture, and under mild homological finiteness conditions on the group, we prove that the connective algebraic K-theory assembly map for the family of virtually cyclic subgroups is rationally injective. This generalizes a result of Bökstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand inside the algebraic K-theory groups of an integral group ring in terms of group homology. Along the way we also prove integral splitting and isomorphism results for THH- and TC-assembly maps with coefficients in arbitrary connective ring spectra.
In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization which involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Bökstedt, Hsiang and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups.
The classical Witt groups of quadratic forms have been
generalized in algebraic topology, with Wall’s and
Ranicki’s L-groups for surgery theory, and in algebraic
geometry, with Knebusch’s Witt groups of schemes and
Balmer’s Witt groups of triangulated categories with
duality.
We introduce algebraic L-theory groups for
so-called pre-triangulated differential graded categories with
duality, generalizing and unifying the two aforementioned
approaches.
“Mathematics, taught and learned appropriately,
improves the mind and implants good habits of thought.”
George Pólya
Riley Visiting Assistant Professor
Department of Mathematical Sciences
Binghamton University,
SUNY
Binghamton, NY 13902-6000
Office: LN-2232
Office hours: MW 11:00–12:30
+1 (607) 777-3550 — Phone
+1 (607) 777-2450 — Fax
marco@math.binghamton.edu
math.binghamton.edu/marco/
Research interests: Algebraic and geometric topology, algebraic K- and L-theory.
Since 2006 I am Visiting Assistant Professor at Binghamton. Until then I was in the topology group at Münster (Germany), where I finished my doctorate under the supervision of Wolfgang Lück.