445 research outputs found
Coefficients for the Farrell-Jones Conjecture
We introduce the Farrell-Jones Conjecture with coefficients in an additive
category with G-action. This is a variant of the Farrell-Jones Conjecture about
the algebraic K- or L-Theory of a group ring RG. It allows to treat twisted
group rings and crossed product rings. The conjecture with coefficients is
stronger than the original conjecture but it has better inheritance properties.
Since known proofs using controlled algebra carry over to the set-up with
coefficients we obtain new results about the original Farrell-Jones Conjecture.
The conjecture with coefficients implies the fibered version of the
Farrell-Jones Conjecture.Comment: 21 page
Novikov-Shubin invariants for arbitrary group actions and their positivity
We extend the notion of Novikov-Shubin invariant for free G-CW-complexes of
finite type to spaces with arbitrary G-actions and prove some statements about
their positivity. In particular we apply this to classifying spaces of discrete
groups.Comment: 18 pages, metadata change
Commuting homotopy limits and smash products
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 Boekstedt, Hsiang
and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map
for the family of virtually cyclic subgroups.Comment: 23 page
Algebraic K-theory of group rings and the cyclotomic trace map
We prove that the Farrell-Jones assembly map for connective algebraic
K-theory is rationally injective, under mild homological finiteness conditions
on the group and assuming that a weak version of the Leopoldt-Schneider
conjecture holds for cyclotomic fields. This generalizes a result of
B\"okstedt, Hsiang, and Madsen, and leads to a concrete description of a large
direct summand of in terms
of group homology. In many cases the number theoretic conjectures are true, so
we obtain rational injectivity results about assembly maps, in particular for
Whitehead groups, under homological finiteness assumptions on the group only.
The proof uses the cyclotomic trace map to topological cyclic homology,
B\"okstedt-Hsiang-Madsen's functor C, and new general isomorphism and
injectivity results about the assembly maps for topological Hochschild homology
and C.Comment: To appear in Advances in Mathematics. 77 page
- β¦