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1,700 |
A Comparison of Continuously Controlled and Controlled K-theory
|
math.KT
|
We define an unreduced version of the e-controlled lower $K$-theoretic groups
of Ranicki and Yamasaki, and Quinn. We show that the reduced versions of our
groups coincide (in the inverse limit and its first derived, $\lim^1$) with
those of Ranicki and Yamasaki. We also relate the controlled groups to the
continuously controlled groups of Anderson and Munkholm, and to the Quinn
homology groups of Quinn.
|
math
|
1,701 |
Cohomology of uniformly powerful p-groups
|
math.KT
|
Studies the cohomology of p-central, powerful, p-groups with a certain
extension property. These groups are naturally associated to Lie algebras. The
paper develops a machinery that calculates the first few terms of the Bockstein
spectral sequence in terms of the associated Lie algebras. This is then used to
obtain results on the integral cohomology of these groups.
|
math
|
1,702 |
Lifting Lie algebras over the residue field of a discrete valuation ring
|
math.KT
|
Studies among other things, the question of whether a Lie algebra over
Z/(p^k)Z lifts to one over Z/(p^(k+1))Z. An obstruction theory is developed and
examples of Fp-Lie algebras which don't lift to Lie algebras over Z/p^2Z are
discussed. An example of an application of the result: A Fp-Lie algebra L with
H^3(L, ad)=0 will lift to a p-adic Lie algebra.
|
math
|
1,703 |
Analytic cyclic cohomology
|
math.KT
|
We prove excision in entire and periodic cyclic cohomology and construct a
Chern-Connes character for Fredholm modules over a C*-algebra without
summability restrictions, taking values in a variant of Connes's entire cyclic
cohomology.
Before these results can be obtained, we have to sort out some fundamental
questions about the class of algebras on which to define entire cyclic
cohomology. The right domain of definition for entire cyclic cohomology is the
category of complete bornological algebras. For these algebras, we define a
bivariant cohomology theory, called analytic cyclic cohomology, that contains
Connes's entire cyclic cohomology as a special case.
The definition of analytic cyclic cohomology is based on the Cuntz-Quillen
approach to cyclic cohomology theories using tensor algebras and X-complexes.
The appropriate completion of the tensor algebra that yields analytic cyclic
cohomology can be understood using an appropriate notion of analytic
nilpotence.
In addition, we develop the elementary theory of analytic cyclic cohomology
(smooth homotopy invariance, stability, Chern character in K-theory).
|
math
|
1,704 |
On the K-theory of local fields
|
math.KT
|
The authors establish a connection between the Quillen K-theory of certain
local fields and the de Rham-Witt complex of their rings of integers with
logarithmic poles at the maximal ideal. They consider fields K that are
complete discrete valuation fields of characteristic zero with perfect residue
fields k of characteristic p > 2. They evaluate the K-theory with
Z/p^v-coefficients of K, and verify the Lichtenbaum-Quillen conjecture for K.
|
math
|
1,705 |
Infinitesimal K-theory
|
math.KT
|
In this paper we study the fiber F of the rational Jones-Goodwillie character
$$ F:=\hofiber(ch:K^\rat(A)@>>>HN^\rat(A)) $$ going from K-theory to negative
cyclic homology of associative rings. We describe this fiber F in terms of
sheaf cohomology. We prove that, for $n\ge 1$, there is an isomorphism: $$
\pi_n(F)\cong H^{-n}_{inf}(A,K^\rat) $$ between the homotopy of the fiber and
the hypercohomology groups of $K^\rat$ on a non-commutative version of
Grothendieck's infinitesimal site.
|
math
|
1,706 |
Cyclic homology of commutative algebras over general ground rings
|
math.KT
|
We consider commutative algebras and chain DG algebras over a fixed
commutative ground ring $k$ as in the title. We are concerned with the problem
of computing the cyclic (and Hochschild) homology of such algebras via free
DG-resolutions $\Lambda V @>>> A$. We find spectral sequences
$$E^2_{p,q}=H_p(\Lambda V\otimes\Gamma^q(dV))\Rightarrow HH_{p+q}(\Lambda V)$$
and $${E'}^2_{\pq}=H_p(\Lambda V\otimes\Gamma^{\le q}(dV)) \Rightarrow
HC_{p+q}(\Lambda V)$$ The algebra $\Lambda V\otimes\Gamma(dV)$ is a divided
power version of the de Rham algebra; in the particular case when $k$ is a
field of characteristic zero, the spectral sequences above agree with those
found by Burghelea and Vigu\'e (Cyclic homology of commutative algebras I,
Lecture Notes in Math. {\bf 1318} (1988) 51-72), where it is shown they
degenerate at the $E^2$ term. For arbitrary ground rings we prove here (Theorem
2.3) that if $V_n=0$ for $n\ge 2$ then $E^2=E^\infty$. From this we derive a
formula for the Hochschild homology of flat complete intersections in terms of
a filtration of the complex for crystalline cohomology, and find a description
of ${E'}^2$ also in terms of crystalline cohomology (theorem 3.0). The latter
spectral sequence degenerates for complete intersections of embedding dimension
$\le 2$ (Corollary 3.1). Without flatness assumptions, our results can be
viewed as the computation Shukla (cyclic) homology (T. Pirashvili, F.
Waldhausen; Mac Lane homology and topological Hochschild homology, J. Pure
Appl. Algebra{\bf 82} (1992) 81-98).
|
math
|
1,707 |
On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology
|
math.KT
|
Cuntz and Quillen have shown that for algebras over a field $k$ with
$char(k)=0$, periodic cyclic homology may be regarded, in some sense, as the
derived functor of (non-commutative) de Rham (co-)homology. The purpose of this
paper is to formalize this derived functor analogy. We show that the
localization ${Def}^{-1}\Cal{PA}$ of the category $\Cal{PA}$ of countable
pro-algebras at the class of (infinitesimal) deformations exists (in any
characteristic) (Theorem 3.2) and that, in characteristic zero, periodic cyclic
homology is the derived functor of de Rham cohomology with respect to this
localization (Corollary 5.4). We also compute the derived functor of rational
$K$-theory for algebras over $\Bbb Q$, which we show is essentially the fiber
of the Chern character to negative cyclic homology (Theorem 6.2).
|
math
|
1,708 |
On the Leibniz cohomology of vector fields
|
math.KT
|
I. M. Gelfand and D. B. Fuks have studied the cohomology of the Lie algebra
of vector fields on a manifold. In this article, we generalize their main tools
to compute the Leibniz cohomology, by extending the two spectral sequences
associated to the diagonal and the order filtration. In particular, we
determine some new generators for the diagonal Leibniz cohomology of the Lie
algebra of vector fields on the circle.
|
math
|
1,709 |
Equivariant K-groups of spheres with actions of involutions
|
math.KT
|
We calculate the R(G)-algebra structure on the reduced equivariant K-groups
of two-dimensional spheres on which a compact Lie group G acts as involutions.
In particular, the reduced equivariant K-groups are trivial if G is abelian,
which shows that the previous Y. Yang's calculation in [Yan95] is not true.
|
math
|
1,710 |
Equivariant Cyclic Cohomology of H-Algebras
|
math.KT
|
We define an equivariant $K_0$-theory for \textit{Yetter-Drinfeld} algebras
over a Hopf algebra with an invertible antipode. We then show that this
definition can be generalized to all Hopf-module algebras. We show that there
exists a pairing, generalizing Connes' pairing, between this theory and a
suitably defined Hopf algebra equivariant cyclic cohomology theory.
|
math
|
1,711 |
A New Cyclic Module for Hopf Algebras
|
math.KT
|
We define a new cyclic module, dual to the Connes-Moscovici cyclic module,
for Hopf algebras, and give a characteristric map for the coaction of Hopf
algebras. We also compute the resulting cyclic homology for cocommutative Hopf
algebras, and some quantum groups.
|
math
|
1,712 |
From Mennicke symbols to Euler class groups
|
math.KT
|
Bhatwadekar and Raja Sridharan have constructed a homomorphism of abelian
groups from an orbit set Um(n,A)/E(n,A) of unimodular rows to an Euler class
group. We suggest that this is the last map in a longer exact sequence of
abelian groups. The hypothetical group G that precedes Um(n,A)/E(n,A) in the
sequence is an orbit set of unimodular two by n matrices over the ring A. If n
is at least four we describe a partially defined operation on two by n
matrices. We conjecture that this operation describes a group structure on G if
A has Krull dimension at most 2n-6. We prove that G is mapped onto a subgroup
of Um(n,A)/E(n,A) if A has Krull dimension at most 2n-5.
|
math
|
1,713 |
Hopf Algebra Equivariant Cyclic Homology and Cyclic Homology of Crossed Product Algebras
|
math.KT
|
We introduce the cylindrical module $A \natural \mathcal{H}$, where
$\mathcal{H}$ is a Hopf algebra and $A$ is a Hopf module algebra over
$\mathcal{H}$. We show that there exists an isomorphism between
$\mathsf{C}_{\bullet}(A^{op} \rtimes \mathcal{H}^{cop})$ the cyclic module of
the crossed product algebra $A^{op} \rtimes \mathcal{H}^{cop} $, and $\Delta(A
\natural \mathcal{H}) $, the cyclic module related to the diagonal of $A
\natural \mathcal{H}$. If $S$, the antipode of $\mathcal{H}$, is invertible it
follows that $\mathsf{C}_{\bullet}(A \rtimes \mathcal{H}) \simeq \Delta(A^{op}
\natural \mathcal{H}^{cop})$. When $S$ is invertible, we approximate
$HC_{\bullet}(A \rtimes \mathcal{H})$ by a spectral sequence and give an
interpretation of $ \mathsf{E}^0, \mathsf{E}^1$ and $\mathsf{E}^2 $ terms of
this spectral sequence.
|
math
|
1,714 |
A proof of the Baum-Connes conjecture for reductive adelic groups
|
math.KT
|
Let F be a global field, A its ring of adeles, G a reductive group over F. We
prove the Baum-Connes conjecture for the adelic group G(A).
|
math
|
1,715 |
K-Theory Past and Present
|
math.KT
|
A brief account of K-theory written in honour of Friedrich Hirzebruch
|
math
|
1,716 |
Para-Hopf algebroids and their cyclic cohomology
|
math.KT
|
We introduce the concept of {\it para-Hopf algebroid} and define their cyclic
cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf
algebras. Para-Hopf algebroids are closely related to, but different from, Hopf
algebroids. Their definition is motivated by attempting to define a cyclic
cohomology theory for Hopf algebroids in general. We show that many of Hopf
algebraic structures, including the Connes-Moscovici algebra
$\mathcal{H}_{FM}$, are para-Hopf algebroids.
|
math
|
1,717 |
Cyclic Cohomology of Crossed Coproduct Coalgebras
|
math.KT
|
We extend our work in~\cite{rm01} to the case of Hopf comodule coalgebras. We
introduce the cocylindrical module $C \natural^{} \mathcal{H}$, where
$\mathcal{H}$ is a Hopf algebra with bijective antipode and $C$ is a Hopf
comodule coalgebra over $\mathcal{H}$. We show that there exists an isomorphism
between the cocyclic module of the crossed coproduct coalgebra $C >
\blacktriangleleft \mathcal{H} $ and $\Delta(C \natural^{}\mathcal{H}) $, the
cocyclic module related to the diagonal of $C \natural^{} \mathcal{H}$. We
approximate $HC^{\bullet}(C > \blacktriangleleft \mathcal{H}) $ by a spectral
sequence and we give an interpretation for $ \mathsf{E}^0, \mathsf{E}^1$ and
$\mathsf{E}^2 $ terms of this spectral sequence.
|
math
|
1,718 |
Cohomology of trivial extensions of Frobenius algebras
|
math.KT
|
We obtain a decomposition for the Hochschild cochain complex of a split
algebra and we study some properties of the cohomology of each term of this
decomposition. Then, we consider the case of trivial extensions, specially of
Frobenius algebras. In particular, we determine completely the cohomology of
the trivial extension of a finite dimensional Hopf algebra. Finally, as an
application, we obtain a result about the Hochschild cohomology of Frobenius
algebras.
|
math
|
1,719 |
Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras
|
math.KT
|
In this paper we construct a cylindrical module $A \natural \mathcal{H}$ for
an $\mathcal{H}$-comodule algebra $A$, where the antipode of the Hopf algebra
$\mathcal{H}$ is bijective. We show that the cyclic module associated to the
diagonal of $A \natural \mathcal{H}$ is isomorphic with the cyclic module of
the crossed product algebra $A \rtimes \mathcal{H}$. This enables us to derive
a spectral sequence for the cyclic homology of the crossed product algebra. We
also construct a cocylindrical module for Hopf module coalgebras and establish
a similar spectral sequence to compute the cyclic cohomology of crossed product
coalgebras.
|
math
|
1,720 |
Hochschild homology and cohomology of generalized Weyl algebras
|
math.KT
|
We compute Hochschild homology and cohomology of a class of generalized Weyl
algebras (for short GWA, defined by Bavula in St.Petersbourg Math. Journal 1999
4(1) pp. 71-90). Examples of such algebras are the n-th Weyl algebras, U(sl_2),
primitive quotients of U(sl_2), and subalgebras of invariants of these algebras
under finite cyclic groups of automorphisms. We answer a question of Bavula -
Jordan (Trans. A.M.S. 353 (2) 2001 pp. 769 -794) concerning the generator of
the group of automorphism of a GWA. We also explain previous results on the
invariants of Weyl algebras and of primitive quotients.
|
math
|
1,721 |
Algebra structure on the Hochschild cohomology of the ring of invariants of a Weyl algebra under a finite group
|
math.KT
|
Let $A_n$ be the $n$-th Weyl algebra, and let
$G\subset\Sp_{2n}(\C)\subset\Aut(A_n)$ be a finite group of linear
automorphisms of $A_n$. In this paper we compute the multiplicative structure
on the Hochschild cohomology $\HH^*(A_n^G)$ of the algebra of invariants of
$G$. We prove that, as a graded algebra, $\HH^*(A_n^G)$ is isomorphic to the
graded algebra associated to the center of the group algebra $\C G$ with
respect to a filtration defined in terms of the defining representation of $G$.
|
math
|
1,722 |
Homology stability for symplectic groups
|
math.KT
|
In this paper the homology stability for symplectic groups over a ring with
finite stable rank is established. First we develop a `nerve theorem' on the
homotopy type of a poset in terms of a cover by subposets, where the cover is
itself indexed by a poset. We use the nerve theorem to show that a poset of
sequences of isotropic vectors is highly connected, as conjectured by Charney
in the eighties.
|
math
|
1,723 |
The obstruction to excision in K-theory and in cyclic homology
|
math.KT
|
Let $f:A \to B$ be a ring homomorphism of not necessarily unital rings and
$I\triangleleft A$ an ideal which is mapped by f isomorphically to an ideal of
B. The obstruction to excision in K-theory is the failure of the map between
relative K-groups $K_*(A:I) \to K_*(B:f(I))$ to be an isomorphism; it is
measured by the birelative groups $K_*(A,B:I)$. We show that these are
rationally isomorphic to the corresponding birelative groups for cyclic
homology up to a dimension shift. In the particular case when A and B are
$\Q$-algebras we obtain an integral isomorphism.
|
math
|
1,724 |
Homology stability for unitary groups
|
math.KT
|
In this paper homology stability for unitary groups over a ring with finite
unitary stable rank is established. Homology stability of symplectic groups and
orthogonal groups appears as a special case of our results.
|
math
|
1,725 |
KK-theory of C*-categories and the analytic assembly map
|
math.KT
|
We define KK-theory spectra associated to C*-categories and look at certain
instances of the Kasparov product at this level. This machinery is used to give
a description of the analytic assembly map as a natural map of spectra.
|
math
|
1,726 |
Comparisons between periodic, analytic and local cyclic cohomology
|
math.KT
|
We compute periodic, analytic and local cyclic cohomology for convolution
algebras of compact Lie groups in order to exhibit differences between these
theories. A surprising result is that the periodic and analytic cyclic
cohomology of the smooth convolution algebras differ, although these algebras
have finite homological dimension.
|
math
|
1,727 |
Unitaire multiplicatif K-moyennable
|
math.KT
|
Nous generalisons la theorie de la K-moyennabilite au cas d'un unitaire
multiplicatif regulier V. Nous montrons que si (H,V,U) est un systeme de Kac
K-moyennable, alors pour toute S-algebre A, les algebres $ A\times_{m}\hat S$
(produit croise maximal) et $ A\times \hat S$ (produit croise reduit) sont
KK-equivalentes ou S est la $C^*$-algebre de Hopf reduite associee a V.
|
math
|
1,728 |
K-theory of Solvable Groups
|
math.KT
|
We first prove that the Whitehead group of a torsion-free virtually solvable
linear group vanishes. Next we make a reduction of the fibered isomorphism
conjecture from virtually solvable groups to a class of virtually solvable
Q-linear groups. Finally we prove an L-theory analogue for elementary amenable
groups.
|
math
|
1,729 |
Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product
|
math.KT
|
We prove that the category of Hopf bimodules over any Hopf algebra has enough
injectives, which enables us to extend some results on the unification of Hopf
bimodule cohomologies of [T1,T2] to the infinite dimensional case. We also
prove that the cup-product defined on these cohomologies is graded-commutative.
|
math
|
1,730 |
Functional equations of higher logarithms
|
math.KT
|
We give the first genuine 2-variable functional equation for the
7--logarithm. We investigate and relate identities for the 3-logarithm given by
Goncharov and Wojtkowiak and deduce a certain family of functional equations
for the 4-logarithm.
|
math
|
1,731 |
G-Structure on the cohomology of Hopf algebras
|
math.KT
|
We prove that Ext^*_A(k,k) is a Gerstenhaber algebra, where A is a Hopf
algebra. In case A=D(H) is the Drinfeld double of a finite dimensional Hopf
algebra H, our results implies the existence of a Gerstenhaber bracket on
H^*_{GS}(H,H). This fact was conjectured by R. Taillefer in math.KT0207154. The
method consists in identifying Ext^*_A(k,k) as a Gerstenhaber subalgebra of
H^*(A,A) (the Hochschild cohomology of A).
|
math
|
1,732 |
K-theory of stratified vector bundles
|
math.KT
|
We show that the Atiyah-Hirzebruch K-theory of spaces admits a canonical
generalization for stratified spaces. For this we study algebraic constructions
on stratified vector bundles. In particular the tangent bundle of a stratified
manifold is such a stratified vector bundle.
|
math
|
1,733 |
Finite group extensions and the Baum-Connes conjecture
|
math.KT
|
In this note, we exhibit a method to prove the Baum-Connes conjecture (with
coefficients) for extensions with finite quotients of certain groups which
already satisfy the Baum-Connes conjecture. Interesting examples to which this
method applies are torsion-free finite extensions of the pure braid groups,
e.g. the full braid groups, or certain fundamental groups of complements of
links in S^3.
|
math
|
1,734 |
Hochschild cohomology of Frobenius algebras
|
math.KT
|
Let k be a field and let A be a Frobenius algebra over k. Assume that the
Nakayama automorphism of A associated to a Frobenius homomorphism of A has
finite order m, and k has a m-th primitive root of unity. Then, A has a natural
Z/mZ-gradation. Consider the decomposition of the Hochschild cohomology HH*(A),
of A with coefficients in A, induced by this gradation. We prove that just the
0-degree component of HH*(A) is non trivial. Moreover, we prove that if A is a
strongly Z/mZ-graded algebra, then Z/mZ acts on the Hochschild cohomology
HH*(A_0), of the 0-degree component of A, and HH*(A) is the set of invariants
of this action.
|
math
|
1,735 |
Higher complex torsion and the framing principle
|
math.KT
|
This paper contains a long summary of the basic properties of higher FR
torsion. An attempt is made to simplify the constructions from my book Higher
Franz-Reidemeister Torsion (IP/AMS Studies in Advanced Math 31). Some new basic
theorems are also proved such as the Framing Principle in full generality. This
is used to compute the higher torsion for bundles with closed even dimensional
fibers. We construct a higher complex torsion for bundles with almost complex
fibers. This is shown to generalize the real even dimensional higher FR
torsion. We also show that the higher complex torsion is a multiple of
generalized Miller-Morita-Mumford classes.
|
math
|
1,736 |
Algebraic cobordism
|
math.KT
|
Together with F. Morel, we have constructed in \cite{CR, Cobord1, Cobord2} a
theory of {\em algebraic cobordism}, an algebro-geometric version of the
topological theory of complex cobordism. In this paper, we give a survey of the
construction and main results of this theory; in the final section, we propose
a candidate for a theory of higher algebraic cobordism, which hopefully agrees
with the cohomology theory represented by the $\P^1$-spectrum $MGL$ in the
Morel-Voevodsky stable homotopy category.
|
math
|
1,737 |
Norm varieties and algebraic cobordism
|
math.KT
|
We outline briefly results and examples related with the bijectivity of the
norm residue homomorphism. We define norm varieties and describe some
constructions. We discuss degree formulas which form a major tool to handle
norm varieties. Finally we formulate Hilbert's 90 for symbols which is the hard
part of the bijectivity of the norm residue homomorphism, modulo a theorem of
Voevodsky.
|
math
|
1,738 |
Algebraic K-theory of mapping class groups
|
math.KT
|
We prove that the Fibered Isomorphism Conjecture of T. Farrell and L. Jones
holds for various mapping class groups. In many cases, we explicitly calculate
the lower algebraic K-groups, showing that they do not always vanish.
|
math
|
1,739 |
Algebra cohomology over a commutative algebra revisited
|
math.KT
|
The aim of this paper is to give a relatively easy bicomplex which computes
the Shukla, or Quillen cohomology in the category of associative algebras over
a commutative algebra $A$, in the case when $A$ is an algebra over a field.
|
math
|
1,740 |
A characterization of the Dirac Dual Dirac Method
|
math.KT
|
Let G be a discrete, torsion free group with a finite dimensional classifying
space BG. We show that the existence of a gamma-element for such G is a metric,
that is, coarse, invariant of G. We also obtain results for groups with
torsion. The method of proof involves showing that a group G possesses a
gamma-element if and only if a certain coarse (co)-assembly map is an
isomorphism.
|
math
|
1,741 |
The Baum-Connes Conjecture via Localisation of Categories
|
math.KT
|
We redefine the Baum-Connes assembly map using simplicial approximation in
the equivariant Kasparov category. This new interpretation is ideal for
studying functorial properties and gives analogues of the assembly maps for all
equivariant homology theories, not just for the K-theory of the crossed
product. We extend many of the known techniques for proving the Baum-Connes
conjecture to this more general setting.
|
math
|
1,742 |
The cyclic homology and K-theory of certain adelic crossed products
|
math.KT
|
The multiplicative group of a global field acts on its adele ring by
multiplication. We consider the crossed product algebra of the resulting action
on the space of Schwartz functions on the adele ring and compute its
Hochschild, cyclic and periodic cyclic homology. We also compute the
topological K-theory of the C*-algebra crossed product.
|
math
|
1,743 |
Homology stability for Unitary groups II
|
math.KT
|
In this note the homology stability problem for hyperbolic unitary groups
over a local ring with an infinite residue field is studied.
|
math
|
1,744 |
Third homology of general linear groups
|
math.KT
|
The third homology group of GL_n(R) is studied, where R is a `ring with many
units' with center Z(R). The main theorem states that if K_1(Z(R))_Q \simeq
K_1(R)_Q, (e.g. R a commutative ring or a central simple algebra), then
H_3(GL_2(R), Q) --> H_3(GL_3(R), Q) is injective. If R is commutative, Q can be
replaced by a field k such that 1/2 is in k. For an infinite field R (resp. an
infinite field R such that R*=R*^2), we get a better result that H_3(GL_2(R),
Z[1/2] --> H_3(GL_3(R), Z[1/2]) (resp. H_3(GL_2(R), Z) --> H_3(GL_3(R), Z)) is
injective. As an application we study the third homology group of SL_2(R) and
the indecomposable part of K_3(R).
|
math
|
1,745 |
The Hochschild cohomology ring modulo nilpotence of a monomial algebra
|
math.KT
|
For a finite dimensional monomial algebra $\Lambda$ over a field $K$ we show
that the Hochschild cohomology ring of $\Lambda$ modulo the ideal generated by
homogeneous nilpotent elements is a commutative finitely generated $K$-algebra
of Krull dimension at most one. This was conjectured to be true for any finite
dimensional algebra over a field by Snashall-Solberg.
|
math
|
1,746 |
The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory
|
math.KT
|
We give a survey of the meaning, status and applications of the Baum-Connes
Conjecture about the topological K-theory of the reduced group C^*-algebra and
the Farrell-Jones Conjecture about the algebraic K- and L-theory of the group
ring of a (discrete) group G.
|
math
|
1,747 |
Induction Theorems and Isomorphism Conjectures for K- and L-Theory
|
math.KT
|
The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the
algebraic K- and L-theory of the group ring and the topological K-theory of the
reduced group C^*-algebra of a group G in terms of these functors for the
virtually cyclic subgroups or the finite subgroups of G. By induction theory we
want to reduce these families of subgroups to a smaller family, for instance to
the family of subgroups which are either finite hyperelementary or extensions
of finite hyperelementary groups with infinite cyclic kernel or to the family
of finite cyclic subgroups. Roughly speaking, we extend the induction theorems
of Dress for finite groups to infinite groups.
|
math
|
1,748 |
A Controlled Approach to the Isomorphism Conjecture
|
math.KT
|
We use a hocolim approach to the Isomorphism Conjecture in K-Theory to
analyze the case of groups of the form $G\rtimes Z$ and $G_1*_{G}G_2$. As an
important corollary we prove that the isomorphism conjecture in K-Theory holds
for a finitely generated free group.
|
math
|
1,749 |
The Baum-Connes conjecture, noncommutative Poincare duality and the boundary of the free Group
|
math.KT
|
Every hyperbolic group acts continuously on its Gromov boundary. One can form
the corresponding cross-product C*-algebra A. We show that there always exists
a canonical Poincare duality map from the K-theory of A to the K-homology of A.
We show that this map is an isomorphism when the group in question is the free
group on two generators. There is a direct connection between our constructions
and the Baum-Connes Conjecture, and we use the latter to deduce our result.
|
math
|
1,750 |
Twisted $K$-theory
|
math.KT
|
Twisted complex $K$-theory can be defined for a space $X$ equipped with a
bundle of complex projective spaces, or, equivalently, with a bundle of
C$^*$-algebras. Up to equivalence, the twisting corresponds to an element of
$H^3(X;\Z)$. We give a systematic account of the definition and basic
properties of the twisted theory, emphasizing some points where it behaves
differently from ordinary $K$-theory. (We omit, however, its relations to
classical cohomology, which we shall treat in a sequel.) We develop an
equivariant version of the theory for the action of a compact Lie group,
proving that then the twistings are classified by the equivariant cohomology
group $H^3_G(X;\Z)$. We also consider some basic examples of twisted $K$-theory
classes, related to those appearing in the recent work of
Freed-Hopkins-Teleman.
|
math
|
1,751 |
Une structure de categorie de modeles de Quillen sur la categorie des dg-categories
|
math.KT
|
We construct a cofibrantly generated Quillen model structure on the category
of small differential graded categories.
-----
Nous construisons une structure de categorie de modeles de Quillen a
engendrement cofibrant sur la categorie des petites categories differentielles
graduees.
|
math
|
1,752 |
Isomorphism Conjecture for homotopy K-theory and groups acting on trees
|
math.KT
|
We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic
K-theory. In particular, we prove that if a group G acts on a tree and all
isotropy groups satisfy this conjecture, then G satisfies this conjecture. This
result can be used to get rational injectivity results for the assembly map in
the Farrell-Jones Conjecture in algebraic K-theory.
|
math
|
1,753 |
Relative cyclic homology of square zero extensions
|
math.KT
|
Let k be a characteristic zero field, C a k-algebra and M a square zero two
sided ideal of C. We obtain a new mixed complex, simpler that the canonical
one, giving the Hochschild and cyclic homologies of C relative to M. This
complex resembles the canonical reduced mixed complex of an augmented algebra.
We begin the study of our complex showing that it has a harmonic decomposition
like to the one considered by Cuntz and Quillen for the normalized mixed
complex of an algebra. We also give new proofs of two theorems of Goodwillie,
obtaining an improvement of one of them.
|
math
|
1,754 |
Hochschild duality, localization and smash products
|
math.KT
|
In this work we study the class of algebras satisfying a duality property
with respect to Hochschild homology and cohomology, as in [VdB]. More
precisely, we consider the class of algebras $A$ such that there exists an
invertible bimodule $U$ and an integer number $d$ with the property
$H^{\bullet}(A,M)\cong H_{d-\bullet}(A,U\ot_AM)$, for all $A$-bimodules $M$. We
will show that this class is closed under localization and under smash products
with respect to Hopf algebras satisfying also the duality property. We also
illustrate the subtlety on dualities with smash products developing in detail
the example $S(V)#G$, the crossed product of the symmetric algebra on a vector
space and a finite group acting linearly on $V$.
|
math
|
1,755 |
Entire cyclic homology of Schatten ideals
|
math.KT
|
Certain cocycles constructed by Connes are characters of $p$-summable
Fredholm modules. In this article, we establish some consequences of the
universal properties which these characters enjoy. Our main technical result is
that the entire cyclic cohomology of the p-th Schatten ideal L^p (respectively,
homology) is independent of p and isomorphic to the entire cyclic cohomology
(respectively, homology) of the ideal of trace class operators L^1.
|
math
|
1,756 |
The equivariant index theorem in entire cyclic cohomology
|
math.KT
|
Let G be a locally compact group acting smoothly and properly by isometries
on a complete Riemannian manifold M, with compact quotient. There is an
assembly map which associates to any G-equivariant K-homology class on M, an
element of the topological K-theory of a suitable Banach completion B of the
convolution algebra of continuous compactly supported functions on G. The aim
of this paper is to calculate the composition of the assembly map with the
Chern character in the entire cyclic homology of B. We prove an index theorem
reducing this computation to a cup-product in bivariant entire cyclic
cohomology. As a consequence we obtain an explicit localization formula which
includes, as particular cases, the equivariant Atiyah-Segal-Singer index
theorem when G is compact, and the Connes-Moscovici index theorem for
G-coverings when G is discrete. The proof is based on the bivariant Chern
character introduced in previous papers.
|
math
|
1,757 |
Loday--Quillen--Tsygan Theorem for Coalgebras
|
math.KT
|
In this paper we prove that Loday--Quillen--Tsygan Theorem generalizes to the
case of coalgebras. Specifically, we show that the Chevalley--Eilenberg--Lie
homology of the Lie coalgebra of infinite matrices over a coassociative
coalgebra $C$ is generated by the cyclic homology of the underlying coalgebra
$C$ as an exterior algebra.
|
math
|
1,758 |
Equivariant periodic cyclic homology
|
math.KT
|
We define and study equivariant periodic cyclic homology for locally compact
groups. This can be viewed as a noncommutative generalization of equivariant de
Rham cohomology. Although the construction resembles the Cuntz-Quillen approach
to ordinary cyclic homology, a completely new feature in the equivariant
setting is the fact that the basic ingredient in the theory is not a complex in
the usual sense. As a consequence, in the equivariant context only the periodic
cyclic theory can be defined in complete generality. Our definition recovers
particular cases studied previously by various authors. We prove that bivariant
equivariant periodic cyclic homology is homotopy invariant, stable and
satisfies excision in both variables. Moreover we construct the exterior
product which generalizes the obvious composition product. Finally we prove a
Green-Julg theorem in cyclic homology for compact groups and the dual result
for discrete groups.
|
math
|
1,759 |
A new description of equivariant cohomology for totally disconnected groups
|
math.KT
|
We consider smooth actions of totally disconnected groups on simplicial
complexes and compare different equivariant cohomology groups associated to
such actions. Our main result is that the bivariant equivariant cohomology
theory introduced by Baum and Schneider can be described using equivariant
periodic cyclic homology. This provides a new approach to the construction of
Baum and Schneider as well as a computation of equivariant periodic cyclic
homology for a natural class of examples. In addition we discuss the relation
between cosheaf homology and equivariant Bredon homology. Since the theory of
Baum and Schneider generalizes cosheaf homology we finally see that all these
approaches to equivariant cohomology for totally disconnected groups are
closely related.
|
math
|
1,760 |
K- and L-theory of the semi-direct product of the discrete 3-dimensional Heisenberg group by Z/4
|
math.KT
|
We compute the group homology, the topological K-theory of the reduced
C^*-algebra, the algebraic K-theory and the algebraic L-theory of the group
ring of the semi-direct product of the three-dimensional discrete Heisenberg
group by Z/4. These computations will follow from the more general treatment of
a certain class of groups G which occur as extensions 1-->K-->G-->Q-->1 of a
torsionfree group K by a group Q which satisfies certain assumptions. The key
ingredients are the Baum-Connes and Farrell-Jones Conjectures and methods from
equivariant algebraic topology.
|
math
|
1,761 |
Outer authomorphisms and the Jacobian
|
math.KT
|
A graphs of rank n (homotopy equivalent to a wedge of n circles) without
``separating edges'' has a canonical n-dimensional compact C^1 manifold
thickening. This implies that the canonical homomorphism f:Out(F_n)-> GL(n,Z)
is trivial in rational cohomology in the stable range answering a question
raised by Hatcher and Vogtmann [6]. Another consequence of the construction is
the existence of higher Reidemeister torsion invariants for IOut(F_n)=ker f.
These facts were first proved by the first author in [8] using different
methods.
|
math
|
1,762 |
Axioms for higher torsion invariants of smooth bundles
|
math.KT
|
We explain the relationship between various characteristic classes for smooth
manifold bundles known as ``higher torsion'' classes. We isolate two
fundamental properties that these cohomology classes may or may not have:
additivity and transfer. We show that higher Franz-Reidemeister torsion and
higher Miller-Morita-Mumford classes satisfy these axioms. Conversely, any
characteristic class of smooth bundles satisfying the two axioms must be a
linear combination of these two examples.
We also show how higher torsion invariants can be computed using only the
axioms. Finally, we explain the conjectured formula of S. Goette relating
higher analytic torsion classes and higher Franz-Reidemeister torsion.
|
math
|
1,763 |
On Compact and Fredholm Operators over C*-algebras and a New Topology in the Space of Compact Operators
|
math.KT
|
It is shown that the class of Fredholm operators over an arbitrary unital
$C^{*}$--algebra, which may not admit adjoint ones, can be extended in such a
way that this class of compact operators, used in the definition of the class
of Fredholm operators, contains compact operators both with and without
existence of adjoint ones. The main property of this new class is that a
Fredholm operator which may not admit an adjoint one has a decomposition into a
direct sum of an isomorphism and a finitely generated operator. In the space of
compact operators in the Hilbert space a new IM-topology is defined. In the
case when the $C^{*}$--algebra is a commutative algebra of continuous functions
on a compact space the IM-topology fully describe the set of compact operators
over the $C^{*}$--algebra without assumption of existence bounded adjoint
operators over the algebra. In the revised version of the paper the proof of
the theorem 8 has been added.
|
math
|
1,764 |
Euler characteristics and Gysin sequences for group actions on boundaries
|
math.KT
|
Let G be a locally compact group, let X be a universal proper G-space, and
let Z be a G-equivariant compactification of X that is H-equivariantly
contractible for each compact subgroup H of G. Let W be the resulting boundary.
Assuming the Baum-Connes conjecture for G with coefficients C and C(W), we
construct an exact sequence that computes the map on K-theory induced by the
embedding of the reduced group C*-algebra of G into the crossed product of G by
C(W). This exact sequence involves the equivariant Euler characteristic of X,
which we study using an abstract notion of Poincare duality in bivariant
K-theory. As a consequence, if G is torsion-free and the Euler characteristic
of the orbit space X/G is non-zero, then the unit element of the boundary
crossed product is a torsion element whose order is equal to the absolute value
of the Euler characteristic of X/G. Furthermore, we get a new proof of a
theorem of Lueck and Rosenberg concerning the class of the de Rham operator in
equivariant K-homology.
|
math
|
1,765 |
Some Fréchet algebras for which the Chern character is an isomorphism
|
math.KT
|
Using similarities between topological $K$-theory and periodic cyclic
homology we show that, after tensoring with $\mathbb C$, for certain Fr\'echet
algebras the Chern character provides an isomorphism between these functors.
This is applied to prove that the Hecke algebra and the Schwartz algebra of a
reductive $p$-adic group have isomorphic periodic cyclic homology.
The main theorem in the first version was incorrect for algebras related to
noncompact manifolds. This has no effect on the results concerning p-adic
groups.
In the appendix we show that an analogous cohomological result does hold in
the noncompact case.
|
math
|
1,766 |
The characteristic cohomology class of a triangulated category
|
math.KT
|
This is the final version of a series of papers uploaded in May 25, 2005. We
have splitted the long last paper of the previous version in two parts to make
it easier to understand. The results are essentially the same, although the
presentation has changed substantially. The first three papers have not
changed.
This is a collection of five papers on the foundation of triangulated
categories in the context of groupoid-enriched categories, termed track
categories, and characteristic cohomology classes. As a main result it is shown
that given an additive category A with a translation functor t: A --> A and a
class V in translation cohomology H^3(A,t) then two simple properties of V
imply that (A,t) is a triangulated category. The cohomology class V yields an
equivalence class (B,[s]) where B is a track category with homotopy category A
and [s] is the homotopy class of a pseudofunctor s: B --> B inducing t. The two
properties of V correspond to natural axioms on B and s which again imply that
(A,t) is a triangulated category.
The five papers of this volume depend on each other by cross references, but
each paper can be read independently of the others so that the reader is free
to choose one of the papers to start. Each paper has its own abstract,
introduction and literature.
|
math
|
1,767 |
Self-stabilization in certain infinite-dimensional matrix algebras
|
math.KT
|
Analytical tools to $K$-theory; namely, self-stabilization of rapidly
decreasing matrices, linearization of cyclic loops, and the contractibility of
the pointed stable Toeplitz algebra are discussed in terms of concrete
formulas. Adaptation to the *-algebra and finite perturbation categories is
also considered. Moreover, the finite linearizability of algebraically finite
cyclic loops is demonstrated.
|
math
|
1,768 |
Correspondences and index
|
math.KT
|
We define certain class of correspondences of polarized representations of
$C^*$-algebras. Our correspondences are modeled on the spaces of boundary
values of elliptic operators on bordisms joining two manifolds. In this setup
we define the index. The main subject of the paper is the additivity of the
index.
|
math
|
1,769 |
Detecting K-theory by cyclic homology
|
math.KT
|
We discuss which part of the rationalized algebraic K-theory of a group ring
is detected via trace maps to Hochschild homology, cyclic homology, periodic
cyclic or negative cyclic homology.
|
math
|
1,770 |
Cohomologie des algèbres de Krönecker générales
|
math.KT
|
The computation of the Hochschild cohomology $HH^*(T)=H^*(T,T)$ of a
triangular algebra $T=\pmatrix{A&M\cr 0&B\cr}$ was performed in {\bf[BG2]}, by
the means of a certain triangular complex. We use this result here to show how
$HH^*(T)$ splits in little pieces whenever the bimodule $M$ is decomposable. As
an example, we express the Hilbert-Poincar\'{e} serie $\sum\_{i=0}^\infty
dim\_K HH^i(T\_m)t^i$ of the "general" Kr\"{o}necker algebra
$T\_m=\pmatrix{A&M^m\cr 0&B\cr}$ as a function of $m\geq 1$ and those of $T$
(here the ground ring $K$ is a field and $dim\_K T<+\infty$). The Lie algebra
structure of $HH^1(T)$ is also considered.
|
math
|
1,771 |
Coefficients for the Farrell-Jones Conjecture
|
math.KT
|
We introduce the Farrell-Jones Conjecture with coefficients in an additive
category with G-action. This is a variant of the Farrell-Jones Conjecture about
the algebraic K- or L-Theory of a group ring RG. It allows to treat twisted
group rings and crossed product rings. The conjecture with coefficients is
stronger than the original conjecture but it has better inheritance properties.
Since known proofs using controlled algebra carry over to the set-up with
coefficients we obtain new results about the original Farrell-Jones Conjecture.
The conjecture with coefficients implies the fibered version of the
Farrell-Jones Conjecture.
|
math
|
1,772 |
Twisted K-theory and cohomology
|
math.KT
|
We explore the relations of twisted K-theory to twisted and untwisted
classical cohomology. We construct an Atiyah-Hirzebruch spectral sequence, and
describe its differentials rationally as Massey products. We define the twisted
Chern character. We also discuss power operations in the twisted theory, and
the role of the Koschorke classes.
|
math
|
1,773 |
Excision in Hopf cyclic homology
|
math.KT
|
In this paper we show that both variants of the Hopf cyclic homology has
excision under some natural homological conditions on the objects and the
coefficient module.
|
math
|
1,774 |
Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras
|
math.KT
|
A main contribution of this paper is the explicit construction of comparison
morphisms between the standard bar resolution and Bardzell's minimal resolution
for truncated quiver algebras (TQA's).
As a direct application we describe explicitely the Yoneda product and derive
several results on the structure of the cohomology ring of TQA's. For instance,
we show that the product of odd degree cohomology classes is always zero. We
prove that TQA's associated with quivers with no cycles or with neither sinks
nor sources have trivial cohomology rings. On the other side we exhibit a
fundamental example of a TQA with non trivial cohomology ring. Finaly, for
truncated polyniomial algebras in one variable, we construct explicit
cohomology classes in the bar resolution and give a full description of their
cohomology ring.
|
math
|
1,775 |
Smooth K-theory of locally convex algebras
|
math.KT
|
Smooth K-functors are introduced and the smooth K-theory of locally convex
algebras is developed. It is proved that the algebraic and smooth K-functors
are isomorphic on the category of quasi stable real (or complex) Frechet
algebras.
|
math
|
1,776 |
Sheaf theory for stacks in manifolds and twisted cohomology for S^1-gerbes
|
math.KT
|
This is the first of a series of papers on sheaf theory on smooth and
topological stacks and its applications. The main result of the present paper
is the characterization of the twisted (by a closed integral three-form) de
Rham complex on a manifold. As an object in the derived category it will be
related with the push-forward of the constant sheaf from a S^1-gerbe with
Dixmier-Douady class represented by the three-form. In order to formulate and
prove this result we develop in detail the foundations of sheaf theory for
smooth stacks.
|
math
|
1,777 |
Periodic cyclic homology of Hecke algebras and their Schwartz completions
|
math.KT
|
We show that the inclusion of an affine Hecke algebra in its Schwartz
completion induces an isomorphism on periodic cyclic homology.
|
math
|
1,778 |
On the K-theory of groups with finite asymptotic dimension
|
math.KT
|
It is proved that the assembly maps in algebraic K- and L-theory with respect
to the family of finite subgroups is injective for groups with finite
asymptotic dimension that admit a finite model for the classifying space for
proper actions. The result also applies to certain groups that admit only a
finite dimensional model for this space. In particular, it applies to discrete
subgroups of virtually connected Lie groups.
|
math
|
1,779 |
The Behavior of Nil-Groups under Localization and the Relative Assembly Map
|
math.KT
|
We study the behavior of the Nil-subgroups of K-groups under localization. As
a consequence we obtain that the relative assembly map from the family of
finite subgroups to the family of virtually cyclic subgroups is rationally an
isomorphism. Combined with the equivariant Chern character we obtain a complete
computation of the rationalized source of the K-theoretic assembly map in terms
of group homology and the K-groups of finite cyclic subgroups.
|
math
|
1,780 |
Homology of SL_n and GL_n over an infinite field
|
math.KT
|
The homology of GL_n(F) and SL_n(F) is studied, where F is an infinite field.
Our main theorem states that the natural map H_4(GL_3(F), k) --> H_4(GL_4(F),
k) is injective where k is a field with char(k) \neq 2, 3. For algebraically
closed field F, we prove a better result, namely, H_4(GL_3(F), Z) -->
H_4(GL_4(F), Z) is injective. We will prove a similar result replacing GL by
SL. This is used to investigate the indecomposable part of the K-group K_4(F).
|
math
|
1,781 |
Resolutions of free partially commutative monoids
|
math.KT
|
A free resolution of free partially commutative monoids is constructed and
with its help the homological dimension of these monoids is calculated.
|
math
|
1,782 |
Simplicial homotopy in semi-abelian categories
|
math.KT
|
We study Quillen's model category structure for homotopy of simplicial
objects in the context of Janelidze, Marki and Tholen's semi-abelian
categories. This model structure exists as soon as the base category A is
regular Mal'tsev and has enough regular projectives; then the fibrations are
the Kan fibrations of simplicial objects in A. When, moreover, A is
semi-abelian, weak equivalences and homology isomorphisms coincide.
|
math
|
1,783 |
Third Mac Lane cohomology via categorical rings
|
math.KT
|
It is proved that the third Mac Lane cohomology group of a ring R with
coefficients in a bimodule B classifies categorical rings having R as the ring
of isomorphism classes of objects and B as the bimodule of automorphisms of the
neutral object.
|
math
|
1,784 |
Equivariant local cyclic homology and the equivariant Chern-Connes character
|
math.KT
|
We define and study equivariant analytic and local cyclic homology for smooth
actions of totally disconnected groups on bornological algebras. Our approach
contains equivariant entire cyclic cohomology in the sense of Klimek, Kondracki
and Lesniewski as a special case and provides an equivariant extension of the
local cyclic theory developped by Puschnigg. As a main result we construct a
multiplicative Chern-Connes character for equivariant KK-theory with values in
equivariant local cyclic homology.
|
math
|
1,785 |
Chern character for totally disconnected groups
|
math.KT
|
In this paper we construct a bivariant Chern character for the equivariant
KK-theory of a totally disconnected group with values in bivariant equivariant
cohomology in the sense of Baum and Schneider. We prove in particular that the
complexified left hand side of the Baum-Connes conjecture for a totally
disconnected group is isomorphic to cosheaf homology. Moreover, it is shown
that our transformation extends the Chern character defined by Baum and
Schneider for profinite groups.
|
math
|
1,786 |
The RO(G)-graded coefficients of (Z/2)^n-equivariant K-theory
|
math.KT
|
In this note, we calculate all untwisted and twisted (Z/2)^n-equivariant
K-groups with compact supports of real finite-dimensional linear
representations of (Z/2)^n. The question was motivated by the question of
D-brane charges for orbifold type II string vacua.
|
math
|
1,787 |
Algebraic K-theory of Fredholm modules and KK-theory
|
math.KT
|
This paper has been withdrawn because it is a duplicate of [math/0609208].
|
math
|
1,788 |
Algebraic K-theory of Fredholm modules and KK-theory
|
math.KT
|
Kasparov $KK$-groups $KK(A,B)$ are represented as homotopy groups of the
Pedersen-Weibel nonconnective algebraic $K$-theory spectrum of the additive
category of Fredholm $(A,B)$-bimodules for $A$ and $B$, respectively, a
separable and $\sigma$-unital trivially graded real or complex $C^*$-algebra
acted upon by a fixed compact metrizable group.
|
math
|
1,789 |
Inertia and delocalized twisted cohomology
|
math.KT
|
We show that the inertia stack of a topological stack is again a topological
stack. We further observe that the inertia stack of an orbispace is again an
orbispace. We show how a U(1)-banded gerbe over an orbispace gives rise to a
flat line bundle over its inertia stack. Via sheaf theory over topological
stacks it gives rise to the twisted delocalized cohomology of the orbispace.
With these results and constructions we generalize concepts, which are
well-known in the smooth framework, to the topological case. In the smooth case
we show, that our sheaf-theoretic definition of twisted delocalized cohomology
of orbispaces coincides with former definitions using a twisted de Rham
complex.
|
math
|
1,790 |
Modular Lattice for $C_{o}$-Operators
|
math.KT
|
We study modularity of the lattice Lat $(T)$ of closed invariant subspaces
for a $C_0$-operator $T$ and find a condition such that Lat $(T)$ is a modular.
Furthermore, we provide a quasiaffinity preserving modularity.
|
math
|
1,791 |
On exactness of long sequences of homology semimodules
|
math.KT
|
We investigate exactness of long sequences of homology semimodules associated
to Schreier short exact sequences of chain complexes of semimodules.
|
math
|
1,792 |
Comparison of spectral sequences involving bifunctors
|
math.KT
|
Suppose given functors A x A' -F-> B -G-> C between abelian categories, an
object X in A and an object X' in A' such that certain conditions hold. We show
that, E_1-terms exempt, the Grothendieck spectral sequence of the composition
of F(X,-) and G evaluated at X' is isomorphic to the Grothendieck spectral
sequence of the composition of F(-,X') and G evaluated at X. So instead of
"resolving X' twice", we may just as well "resolve X twice".
|
math
|
1,793 |
On K_1 of a Waldhausen category
|
math.KT
|
We give a simple representation of all elements in K_1 of a Waldhausen
category and prove relations between these representatives which hold in K_1.
|
math
|
1,794 |
Coarse and equivariant co-assembly maps
|
math.KT
|
We study an equivariant co-assembly map that is dual to the usual Baum-Connes
assembly map and closely related to coarse geometry, equivariant Kasparov
theory, and the existence of dual Dirac morphisms. As applications, we prove
the existence of dual Dirac morphisms for groups with suitable
compactifications, that is, satisfying the Carlsson-Pedersen condition, and we
study a K-theoretic counterpart to the proper Lipschitz cohomology of Connes,
Gromov and Moscovici.
|
math
|
1,795 |
Cyclic Cohomology and Higher Rank Lattices
|
math.KT
|
We give a new proof of the absence of non-trivial idempotents in the group
ring of torsion-free cocompact lattices in SL(n,C). It is based on the
following procedure. We lift the class of the trace in the cyclic cohomology of
the group ring to the crossed product of the smooth functions on the
Furstenberg boundary of SL(n,C) with the lattice. We then perform a Dirac-dual
Dirac method on smooth algebras in analytic cyclic cohomology. This is based on
a form of equivariant Bott periodicity under compact Lie groups in analytic
cyclic cohomology. We make crucial use of the Baum-Connes conjecture for
solvable Lie groups.
There is also a chapter in which we prove that the class of the unit in the
K-theory of the crossed product of the continuous functions on the visibility
boundary of the symmetric space of a real semisimple Lie group with
torsion-free discrete subgroups of that Lie group is not torsion if the lattice
is not cocompact. In case the lattice is cocompact, we show that the class of
unit is torsion if and only if the rank of the Lie group is the same as that of
a maximal compact subgroup.
|
math
|
1,796 |
An analytic index for Lie groupoids
|
math.KT
|
For a Lie groupoid there is an analytic index morphism which takes values in
the $K-$theory of the $C^*$-algebra associated to the groupoid. This is a good
invariant but extracting numerical invariants from it, with the existent tools,
is very difficult. In this work, we define another analytic index morphism
associated to a Lie groupoid; this one takes values in a group that allows us
to do pairings with cyclic cocycles. This last group is related to the
compactly supported functions on the groupoid. We use the tangent groupoid to
define our index as a sort of ''deformation''.
|
math
|
1,797 |
Orbifold index and equivariant K-homology
|
math.KT
|
We consider a invariant Dirac operator D on a manifold with a proper and
cocompact action of a discrete group G. It gives rise to an equivariant
K-homology class [D]. We show how the index of the induced orbifold Dirac
operator can be calculated from [D] via the assembly map. We further derive a
formula for this index in terms of the contributions of finite cyclic subgroups
of G. According to results of W. Lueck, the equivariant K-homology can
rationally be decomposed as a direct sum of contributions of finite cyclic
subgroups of G. Our index formula thus leads to an explicit decomposition of
the class [D].
|
math
|
1,798 |
Categorical aspects of bivariant K-theory
|
math.KT
|
This survey article on bivariant Kasparov theory and E-theory is mainly
intended for readers with a background in homotopical algebra and category
theory. We approach both bivariant K-theories via their universal properties
and equip them with extra structure such as a tensor product and a triangulated
category structure. We discuss the construction of the Baum-Connes assembly map
via localisation of categories and explain how this is related to the purely
topological construction by Davis and Lueck.
|
math
|
1,799 |
Homological algebra in bivariant K-theory and other triangulated categories
|
math.KT
|
Bivariant (equivariant) K-theory is the standard setting for non-commutative
topology. We may carry over various techniques from homotopy theory and
homological algebra to this setting. Here we do this for some basic notions
from homological algebra: phantom maps, exact chain complexes, projective
resolutions, and derived functors. We introduce these notions and apply them to
examples from bivariant K-theory.
An important observation of Beligiannis is that we can approximate our
category by an Abelian category in a canonical way, such that our homological
concepts reduce to the corresponding ones in this Abelian category. We compute
this Abelian approximation in several interesting examples, where it turns out
to be very concrete and tractable.
The derived functors comprise the second tableau of a spectral sequence that,
in favourable cases, converges towards Kasparov groups and other interesting
objects. This mechanism is the common basis for many different spectral
sequences. Here we only discuss the very simple 1-dimensional case, where the
spectral sequences reduce to short exact sequences.
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math
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