850 research outputs found
The trace on the K-theory of group C*-algebras
The canonical trace on the reduced C*-algebra of a discrete group gives rise
to a homomorphism from the K-theory of this C^*-algebra to the real numbers.
This paper addresses the range of this homomorphism. For torsion free groups,
the Baum-Connes conjecture and Atiyah's L2-index theorem implies that the range
consists of the integers. If the group is not torsion free, Baum and Connes
conjecture that the trace takes values in the rational numbers.
We give a direct and elementary proof that if G acts on a tree and admits a
homomorphism \alpha to another group H whose restriction to every stabilizer
group of a vertex is injective, then the range of the trace for G,
tr_G(K(C_r^*G)) is contained in the range of the trace for H, tr_H(K(C_r^*H)).
This follows from a general relative Fredholm module technique.
Examples are in particular HNN-extensions of H where the stable letter acts
by conjugation with an element of H, or amalgamated free products G=H*_U H of
two copies of the same groups along a subgroup U.Comment: Reference added. AMSLateX2e, 12 pages. Preprint-Series SFB Muenster,
No 66, to appear in Duke Math. Journa
Bordism, rho-invariants and the Baum-Connes conjecture
Let G be a finitely generated discrete group. In this paper we establish
vanishing results for rho-invariants associated to
(i) the spin-Dirac operator of a spin manifold with positive scalar curvature
(ii) the signature operator of the disjoint union of a pair of homotopy
equivalent oriented manifolds with fundamental group G.
The invariants we consider are more precisely
- the Atiyah-Patodi-Singer rho-invariant associated to a pair of finite
dimensional unitary representations.
- the L2-rho invariant of Cheeger-Gromov
- the delocalized eta invariant of Lott for a finite conjugacy class of G.
We prove that all these rho-invariants vanish if the group G is torsion-free
and the Baum-Connes map for the maximal group C^*-algebra is bijective. For the
delocalized invariant we only assume the validity of the Baum-Connes conjecture
for the reduced C^*-algebra.
In particular, the three rho-invariants associated to the signature operator
are, for such groups, homotopy invariant. For the APS and the Cheeger-Gromov
rho-invariants the latter result had been established by Navin Keswani. Our
proof re-establishes this result and also extends it to the delocalized
eta-invariant of Lott. Our method also gives some information about the
eta-invariant itself (a much more saddle object than the rho-invariant).Comment: LaTeX2e, 60 pages; the gap pointed out by Nigel Higson and John Roe
is now closed and all statements of the first version of the paper are proved
(with some small refinements
L2-determinant class and approximation of L2-Betti numbers
A standing conjecture in L2-cohomology is that every finite CW-complex X is
of L2-determinant class. In this paper, we prove this whenever the fundamental
group belongs to a large class of groups containing e.g. all extensions of
residually finite groups with amenable quotients, all residually amenable
groups and free products of these. If, in addition, X is L2-acyclic, we also
prove that the L2-determinant is a homotopy invariant. Even in the known cases,
our proof of homotopy invariance is much shorter and easier than the previous
ones. Under suitable conditions we give new approximation formulas for L2-Betti
numbers. Errata are added, rectifying some unproved statements about "amenable
extension": throughout, amenable extensions should be extensions with
\emph{normal} subgroups.Comment: amsLaTeX2e, 26 pages; v2: Errata are added, rectifying some unproved
statements about "amenable extension
Rho-classes, index theory and Stolz' positive scalar curvature sequence
In this paper, we study the space of metrics of positive scalar curvature
using methods from coarse geometry.
Given a closed spin manifold M with fundamental group G, Stephan Stolz
introduced the positive scalar curvature exact sequence, in analogy to the
surgery exact sequence in topology. It calculates a structure group of metrics
of positive scalar curvature on M (the object we want to understand) in terms
of spin-bordism of BG and a somewhat mysterious group R(G).
Higson and Roe introduced a K-theory exact sequence in coarse geometry which
contains the Baum-Connes assembly map, with one crucial term K(D*G) canonically
associated to G. The K-theory groups in question are the home of interesting
index invariants and secondary invariants, in particular the rho-class in
K_*(D*G) of a metric of positive scalar curvature on a spin manifold.
One of our main results is the construction of a map from the Stolz exact
sequence to the Higson-Roe exact sequence (commuting with all arrows), using
coarse index theory throughout.
Our main tool are two index theorems, which we believe to be of independent
interest. The first is an index theorem of Atiyah-Patodi-Singer type. Here,
assume that Y is a compact spin manifold with boundary, with a Riemannian
metric g which is of positive scalar curvature when restricted to the boundary
(and with fundamental group G). Because the Dirac operator on the boundary is
invertible, one constructs a delocalized APS-index in K_* (D*G). We then show
that this class equals the rho-class of the boundary.
The second theorem equates a partitioned manifold rho-class of a positive
scalar curvature metric to the rho-class of the partitioning hypersurface.Comment: 39 pages. v2: final version, to appear in Journal of Topology. Added
more details and restructured the proofs, correction of a couple of errors.
v3: correction after final publication of a (minor) technical glitch in the
definition of the rho-invariant on p6. The JTop version is not correcte
Various L2-signatures and a topological L2-signature theorem
For a normal covering over a closed oriented topological manifold we give a
proof of the L2-signature theorem with twisted coefficients, using Lipschitz
structures and the Lipschitz signature operator introduced by Teleman. We also
prove that the L-theory isomorphism conjecture as well as the C^*_max-version
of the Baum-Connes conjecture imply the L2-signature theorem for a normal
covering over a Poincar space, provided that the group of deck transformations
is torsion-free. We discuss the various possible definitions of L2-signatures
(using the signature operator, using the cap product of differential forms,
using a cap product in cellular L2-cohomology,...) in this situation, and prove
that they all coincide.Comment: comma in metadata (author field) added
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