101,833 research outputs found
A new approach to topological T-duality for principal torus bundles
No full textWe introduce a new `Thom class\u27 formulation of topological T-duality for principal torus bundles. This definition is equivalent to the established one of Bunke, Rumpf, and Schick but has the virtue of removing the global assumptions on the H-flux required in the old definition. With the new definition, we provide easier and more transparent proofs of the classification of T-duals and generalise the local formulation of T-duality for circle bundles by Bunke, Schick, and Spitzweck to the torus case.29 pages. v2: final version with spelling corrections and added reference, accepted in Reviews in Mathematical Physic
Equivariant Topological T-Duality
No full textTopological T-duality is a relationship between pairs (E, P ) over a fixed
space X, where E over X is a principal torus bundle and P over E is a twist,
such as a gerbe of principal PU(H)-bundle. This is of interest to topologists
because of the T-duality transformation: a T-duality relation between pairs (E,
P ) and (F, Q ) comes with an isomorphism (with degree shift) between the
twisted K-theory of E and the twisted K-theory of F. We formulate topological
T-duality in the equivariant setting, following the definition of Bunke, Rumpf,
and Schick. We define the T-duality transformation in equivariant K-theory and
show that it is an isomorphism for actions of compact Lie groups, equal to its
own inverse and uniquely characterized by naturality and a normalization for
trivial situations
Equivariant Topological T-Duality
No full textTopological T-duality is a relationship between pairs (E, P ) over a fixed space X, where E over X is a principal torus bundle and P over E is a twist, such as a gerbe of principal PU(H)-bundle. This is of interest to topologists because of the T-duality transformation: a T-duality relation between pairs (E, P ) and (F, Q ) comes with an isomorphism (with degree shift) between the twisted K-theory of E and the twisted K-theory of F. We formulate topological T-duality in the equivariant setting, following the definition of Bunke, Rumpf, and Schick. We define the T-duality transformation in equivariant K-theory and show that it is an isomorphism for actions of compact Lie groups, equal to its own inverse and uniquely characterized by naturality and a normalization for trivial situations.36 pages, v2: added discussion of uniqueness of T-duality transform, correction of typos, update of references. v3 Added reference on physics background. Stressed that we deal with circle bundle case -final version, to appear in Communications in Mathematical Physic
The surgery exact sequence, K-Theory and the signature operator
No full textThe main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe.
Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large scale index theory for Dirac type operators that are perturbed by lower order operators
A K-theoretic proof of Boutet de Monvel's index theorem for boundary value problems
No full textWe study the C*-closure U of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with boundary partial derivative X not equal phi. We find short exact sequences in K-theory 0 -> K-i (C(X)) -> K-i(U/R) ->(P) K1-i(C-0(T*X degrees)) -> 0, i = 0,1, which split, so that K-i(U/R) congruent to K-i(C(X)) circle plus K1-i(Co(T*X degrees)). Using only simple K-theoretic arguments and the Atiyah-Singer index theorem, we show that the Fredhohn index of an elliptic element in A is given by ind A = ind(t)(p[A])), where [A] is the class of A in K-1(U/R) and ind(t) is the topological index, a relation first established by Boutet de Monvel by different methods
L2-torsion of Hyperbolic Manifolds of Finite Volume
No full textSuppose M¯ is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the L2-topological torsion of M¯ and the L2-analytic torsion of the Riemannian manifold M are equal. In particular, the L2-topological torsion of M¯ is proportional to the hyperbolic volume of M, with a constant of proportionality which depends only on the dimension and which is known to be nonzero in odd dimensions [HS]. In dimension 3 this proves the conjecture [Lü2, Conjecture 2.3] or [LLü, Conjecture 7.7] which gives a complete calculation of the L2-topological torsion of compact L2-acyclic 3-manifolds which admit a geometric JSJT-decomposition.¶In an appendix we give a counterexample to an extension of the Cheeger-Müller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes
Rho-classes, index theory and Stolz' positive scalar curvature sequence.
No full textIn 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.
Higson and Roe introduced a K-theory exact sequence in K-Theory. The K-theory groups in question are the home of interesting (secondary) invariants, in particular the rho-class of a metric of positive scalar curvature.
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
The Local Structure of T-Duality Triples
No full textIn der Literatur findet man verschiedene mathematische Zugänge, um T-Dualität zu beschreiben. Wir zeigen, daß der C*-algebraische Zugang von Mathai und Rosenberg äquivalent ist zur topologischen Beschreibung von Bunke und Schick. Im C*-algebraischen Zugang geht es darum, verschränkte Produkte abelscher C*-dynamischer Systeme zu verstehen, wohingegen der topologische Zugang auf sogenannten T-Dualitätstripeln basiert. Wir geben eine explizite Konstruktion, basierend auf der lokalen Struktur der zugrundeliegenden Objekte, an, die diese ineinander überführt. Die dabei verwendeten Methoden sind allgemein genug, um eine Theorie für beliebige zweit abzählbare, lokal kompakte, abelsche Gruppen mit diskreter,kokompakter Untergruppe zu formulieren.There are different approaches for a mathematical understanding of T-duality. We show that the C*-algebraic approach of Mathai and Rosenberg is equivalent to the topological approach of Bunke and Schick. The C*-algebraic approach is based on the understanding of crossed product C*-algebras, whereas the topological point of view involves the notion of so-called T-duality triples. We decode the local structure of the respective underlying objects and give an explicit construction how to transform them into each other. The methods used are general enough to formulate a theory for all second countable, locally compact, abelian groups with discrete cocompact subgroup
Positive scalar curvature due to the cokernel of the classifying map
No full textThis paper contributes to the classification of positive scalar curvature metrics up to bordism and up to concordance. Let M be a closed spin manifold of dimension ≥ 5 which admits a metric with positive scalar curvature. We give lower bounds on the rank of the group of psc metrics over M up to bordism in terms of the corank of the canonical map KO*(M) → KO*(Bπ1(M)), provided the rational analytic Novikov conjecture is true for π1(M)
Bordism, rho-invariants and the Baum–Connes conjecture
No full textLet � 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 and fundamental group�; (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group�. The invariants we consider are more precisely the Atiyah–Patodi–Singer ( APS) rho-invariant associated to a pair of finite dimensional unitary representations 1; 2W � ! U.d/, theL 2-rho-invariant of Cheeger–Gromov, the delocalized eta-invariant of Lott for a non-trivial conjugacy class of � which is finite. We prove that all these rho-invariants vanish if the group � is torsion-free and the Baum–Connes map for the maximal group C*-algebra is bijective. This condition is satisfied, for example, by torsion-free amenable groups or by torsion-free discrete subgroups of SO.n;1 / and SU.n;1/. For the delocalized invariant we only assume the validity of the Baum–Connes conjecture for the reduced C*-algebra. In addition to the examples above, this condition is satisfied e.g. by Gromov hyperbolic groups or by cocompact discrete subgroups of SL.3; C/. 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 reestablishes this result and also extends it to the delocalized eta-invariant of Lott. The proof exploits in a fundamental way results from bordism theory as well as various generalizations of the APS-index theorem; it also embeds these results in general vanishing phenomena for degree zero higher rho-invariants (taking values inA=ŒA;A � for suitable C*-algebrasA). We also obtain precise information about the eta-invariants in question themselves, which are usually much more subtle objects than the rho-invariants
- …
