500 research outputs found
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
Index, eta and rho-invariants on foliated bundles
We study primary and secondary invariants of leafwise Dirac operators on
foliated bundles. Given such an operator, we begin by considering the
associated regular self-adjoint operator on the maximal Connes-Skandalis
Hilbert module and explain how the functional calculus of encodes both
the leafwise calculus and the monodromy calculus in the corresponding von
Neumann algebras. When the foliation is endowed with a holonomy invariant
transverse measure, we explain the compatibility of various traces and
determinants. We extend Atiyah's index theorem on Galois coverings to these
foliations. We define a foliated rho-invariant and investigate its stability
properties for the signature operator. Finally, we establish the foliated
homotopy invariance of such a signature rho-invariant under a Baum-Connes
assumption, thus extending to the foliated context results proved by Neumann,
Mathai, Weinberger and Keswani on Galois coverings.Comment: 65 page
Cylindrical Wigner measures
In this paper we study the semiclassical behavior of quantum states acting on
the C*-algebra of canonical commutation relations, from a general perspective.
The aim is to provide a unified and flexible approach to the semiclassical
analysis of bosonic systems. We also give a detailed overview of possible
applications of this approach to mathematical problems of both axiomatic
relativistic quantum field theories and nonrelativistic many body systems. If
the theory has infinitely many degrees of freedom, the set of Wigner measures,
i.e. the classical counterpart of the set of quantum states, coincides with the
set of all cylindrical measures acting on the algebraic dual of the space of
test functions for the field, and this reveals a very rich semiclassical
structure compared to the finite-dimensional case. We characterize the
cylindrical Wigner measures and the \emph{a priori} properties they inherit
from the corresponding quantum states.Comment: 59 page
Dynamics measured in a non-Archimedean field
We study dynamical systems using measures taking values in a non-Archimedean
field. The underlying space for such measure is a zero-dimensional topological
space. In this paper we elaborate on the natural translation of several
notions, e.g., probability measures, isomorphic transformations, entropy, from
classical dynamical systems to a non-Archimedean setting.Comment: 12 page
The two definitions of the index difference
Given two metrics of positive scalar curvature metrics on a closed spin
manifold, there is a secondary index invariant in real -theory. There exist
two definitions of this invariant, one of homotopical flavour, the other one
defined by a index problem of Atiyah-Patodi-Singer type. We give a complete and
detailed proof of the folklore result that both constructions yield the same
answer. Moreover, we generalize this to the case of two families of positive
scalar curvature metrics, parametrized by a compact space. In essence, we prove
a generalization of the classical "spectral-flow-index theorem" to the case of
families of real operators.Comment: Revised versio
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