Mathematics

We prove that the graph C*-algebra C ∗ (E) of a trimmable graph E is U(1) -equivariantly isomorphic to a pullback C*-algebra of a subgraph C*-algebra C ∗ ( E ′′ ) and the C*-algebra of functions on a circle tensored with another subgraph C*-algebra C ∗ ( E ′ ) . This allows us to unravel the structure and K-theory of the fixed-point subalgebra C ∗ (E ) U(1) through the (typically simpler) C*-algebras C ∗ ( E ′ ) , C ∗ ( E ′′ ) and C ∗ ( E ′′ ) U(1) . As examples of trimmable graphs, we consider one-loop extensions of the standard graphs encoding respectively the Cuntz algebra O 2 and the Toeplitz algebra T . Then we analyze equivariant pullback structures of trimmable graphs yielding the C*-algebras of the Vaksman-Soibelman quantum sphere S 2n+1 q and the quantum lens space L 3 q (l;1,l) , respectively.

Let G:= S L 2 ? denote the affine Kac-Moody group associated to S L 2 and X ¯ the associated affine Grassmannian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of X ¯ . In the case of ordinary (non-equivariant) K -theory we find an explicit closed form for the structure constants. We also determine an inductive formula for the structure constants in the torus-equivariant cohomology ring, and use this formula to find closed forms for some of the structure constants.

The Baum-Connes conjecture predicts that a certain assembly map is an isomorphism. We identify the homotopy theoretical construction of the assembly map by Davis and Lück with the category theoretical construction by Meyer and Nest. This extends the result of Hambleton and Pedersen to arbitrary coefficients. Our approach uses abstract properties rather than explicit constructions and is formally similar to Meyer's and Nest's identification of their assembly map with the original construction of the assembly map by Baum, Connes and Higson.

We have been studying the index theory for some special infinite-dimensional manifolds with a "proper cocompact" actions of the loop group LT of the circle T, from the viewpoint of the noncommutative geometry. In this paper, we will introduce the LT-equivariant KK-theory and we will construct three KK-elements: the index element, the Clifford symbol element and the Dirac element. These elements satisfy a certain relation, which should be called the (KK-theoretical) index theorem, or the KK-theoretical Poincaré duality for infinite-dimensional manifolds. We will also discuss the assembly maps.

This survey paper introduces to a technique called Torsion Subcomplex Reduction (TSR) for computing torsion in the cohomology of discrete groups acting on suitable cell complexes. TSR enables one to skip machine computations on cell complexes, and to access directly the reduced torsion subcomplexes, which yields results on the cohomology of matrix groups in terms of formulas. TSR has already yielded general formulas for the cohomology of the tetrahedral Coxeter groups as well as, at odd torsion, of SL2 groups over arbitrary number rings. The latter formulas have allowed to refine the Quillen conjecture. Furthermore, progress has been made to adapt TSR to Bredon homology computations. In particular for the Bianchi groups, yielding their equivariant K-homology, and, by the Baum-Connes assembly map, the K-theory of their reduced C *-algebras. As a side application, TSR has allowed to provide dimension formulas for the Chen-Ruan orbifold cohomology of the complexified Bianchi orbifolds, and to prove Ruan's crepant resolution conjecture for all complexified Bianchi orbifolds.

We apply categorical machinery to the problem of defining anti-Yetter-Drinfeld modules for quasi-Hopf algebras. While a definition of Yetter-Drinfeld modules in this setting, extracted from their categorical interpretation as the center of the monoidal category of modules has been given, none was available for the anti-Yetter-Drinfeld modules that serve as coefficients for a Hopf cyclic type cohomology theory for quasi-Hopf algebras. This is a followup paper to the authors' previous effort that addressed the somewhat different case of anti-Yetter-Drinfeld contramodule coefficients in this and in the Hopf algebroid setting.

For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose M is a closed smooth spin manifold and M ˜ is a Γ -regular covering space of M . Let ⟨α⟩ be the conjugacy class of a non-identity element α∈Γ . Suppose { Γ i } is a sequence of finite-index normal subgroups of Γ that distinguishes ⟨α⟩ . Let π Γ i be the quotient map from Γ to Γ/ Γ i and ⟨ π Γ i (α)⟩ the conjugacy class of π Γ i (α) in Γ/ Γ i . If the scalar curvature on M is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of M ˜ at the conjugacy class ⟨α⟩ is equal to the limit of the delocalized eta invariants for the Dirac operators of M Γ i at the conjugacy class ⟨ π Γ i (α)⟩ , where M Γ i = M ˜ / Γ i is the finite-sheeted covering space of M determined by Γ i . In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of M Γ i at the conjugacy class ⟨ π Γ i (α)⟩ converges, under the assumption that the rational maximal Baum-Connes conjecture holds for Γ .

In this article we show how to build main aspects of our paper on globular weak (??n) -categories, but now for the cubical geometry. Thus we define a monad on the category CSets of cubical sets which algebras are models of cubical weak ??-categories. Also for each n?�N we define a monad on CSets which algebras are models of cubical weak (??n) -categories. And finally we define a monad on the category CSet s 2 which algebras are models of cubical weak ??-functors, and a monad on the category CSet s 4 which algebras are models of cubical weak natural ??-transformations.

We give a new proof of the universal property of K K G -theory with respect to stability, homotopy invariance and split-exactness for G a locally compact group, or a locally compact (not necessarily Hausdorff) groupoid, or a countable inverse semigroup which is relatively short and conceptual. Morphisms in the generators and relations picture of K K G -theory are brought to a particular simple form.

We introduce and analyze the concept of an assembly map from the original homotopy theoretic point of view. We give also interpretations in terms of surgery theory, controlled topology and index theory. The motivation is that prominent conjectures of Farrell-Jones and Baum-Connes about K- and L-theory of group rings and group C^*-algebras predict that certain assembly maps are weak homotopy equivalences.