The main goals of our research program are the construction of a
geometric homological equivariant Chern character for proper actions of a
totally disconnected locally compact group $G$. The main building block for
this construction will be the new cycle model of $K^G_*(X)$ which is another
major goal of this research project. The cycles will be of the form $(M,E,f)$,
and the main features will be that there is a suitable version of
K-orientation encoded in $(M,E)$. Here $E$, which is of K-theoretic character,
will involve maps to a convenient classifying space for K-theory with a good
Chern character cocycle. All together, this should allow to construct a Chern
character cycle out of this data for a cosheaf on $G\backslash M$ encoding the
representation rings of the isotropy subgroup (with cosheaf structure given by
induction of representations), then to be pushed to the correct target depending on $X$.
The program will need quite a few innovations to accommodate non-compact
non-discrete totally disconnected groups. But it will follow in suitable parts
the constructions for the profinite situation of \cite{BaumSchneider}, as well
as the constructions for the discrete case described with related methods
in \cite{Raven}.
Indeed, the constructions should be sufficiently flexible to combine with the
K-theoretic Chern character we plan to develop and to provide a geometrically
defined equivariant \emph{bivariant} Chern character for actions of totally
disconnected groups.
We envision applications of such a homological Chern character toward
equivariant index theory of a new kind for actions of p-adic groups.
Publications
We prove the following Lipschitz rigidity result in scalar curvature geometry. Let $M$ be a closed smooth connected spin manifold of even dimension $n$, let $g$ be a Riemannian metric of regularity $W^{1,p}$, $p > n$, on $M$ whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by $n(n-1)$, and let $f \colon (M,g) \to \mathbb{S}^n$ be a $1$-Lipschitz continuous (not necessarily smooth) map of non-zero degree to the unit $n$-sphere. Then $f$ is a metric isometry. This generalizes a result of Llarull (1998) and answers in the affirmative a question of Gromov (2019) in his "four lectures". Our proof is based on spectral properties of Dirac operators for low regularity Riemannian metrics and twisted with Lipschitz bundles, and on the theory of quasiregular maps due to Reshetnyak.
Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II58Profinite perspectives on l2-cohomology73Geometric Chern characters in p-adic equivariant K-theory
Team Members
Dr. Thorben Kastenholz
Universität Göttingen
thorben.kastenholz(at)mathematik.uni-goettingen.de
Prof. Dr. Thomas Schick
Project leader
Georg-August-Universität Göttingen
thomas.schick(at)math.uni-goettingen.de