Equivariant homology and cohomology of groups

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.     

Uniform K-homology theory

We define a uniform version of analytic K-homology theory for separable, proper metric spaces. Furthermore, we define an index map from this theory into the K-theory of uniform Roe C^∗-algebras, analogous to the coarse assembly map from analytic K-homology into the K-theory of Roe C^∗-algebras. We show that our theory has a Mayer-Vietoris sequence. We prove that for a torsion-free countable discrete group Γ, the direct limit of the uniform K-homology of the Rips complexes of Γ, , is isomorphic to , the left-hand side of the Baum-Connes conjecture with coefficients in ?^∞Γ. In particular, this provides a computation of the uniform K-homology groups for some torsion-free groups. As an application of uniform K-homology, we prove a criterion for amenability in terms of vanishing of a “fundamental class”, in spirit of similar criteria in uniformly finite homology and K-theory of uniform Roe algebras.     

Isomorphisms of Certain Locally Nilpotent Finitary Groups and Associated Rings

For any chain Γ the ring NT(Γ,K) of all finitary Γ-matrices ‖a _ij ‖ _i,jεΓ over an associative ring K with zeros on and above the main diagonal is locally nilpotent and hence radical. If R′=NT(Γ′,K′),R=NT(Γ,K) and either |Γ|<∞ or K is a ring with no zero-divisors, then isomorphisms between rings R and R′, their adjoint groups and associated Lie rings are described.     

An equivariant higher index theory and nonpositively curved manifolds

In this paper, we define an equivariant higher index map from to K_∗(C^∗Γ(X)) if a torsion-free discrete group Γ acts on a manifold X properly, where C^∗Γ(X) is the norm closure of all locally compact, Γ-invariant operators with finite propagation. When Γ acts on X properly and cocompactly, this equivariant higher index map coincides with the Baum-Connes map [P. Baum, A. Connes, K-theory for discrete groups, in: D. Evens, M. Takesaki (Eds.), Operator Algebras and Applications, Cambridge Univ. Press, Cambridge, 1989, pp. 1-20; P. Baum, A. Connes, N. Higson, Classifying space for proper actions and K-theory of group C^∗-algebras, in: C^∗-Algebras: 1943-1993, San Antonio, TX, 1993, in: Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240-291]. When Γ is trivial, this equivariant higher index map is the coarse Baum-Connes map [J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (497) (1993); J. Roe, Index Theory, Coarse Geometry, and the Topology of Manifolds, CBMS Reg. Conf. Ser. Math., vol. 90, Amer. Math. Soc., Providence, RI, 1996]. If X is a simply-connected complete Riemannian manifold with nonpositive sectional curvature and Γ is a torsion-free discrete group acting on X properly and isometrically, we prove that the equivariant higher index map is injective.     

A Fourier–Mukai approach to the K-theory of compact Lie groups

Let G be a compact, connected, simply-connected Lie group. We use the Fourier–Mukai transform in twisted K-theory to give a new proof of the ring structure of the K-theory of G.     

The localization sequence for the algebraic <Emphasis Type="Italic">K</Emphasis>-theory of topological <Emphasis Type="Italic">K</Emphasis>-theory

We verify a conjecture of Rognes by establishing a localization cofiber sequence of spectra \(K(\mathbb{Z})\to K(ku)\to K(KU) \to\Sigma K(\mathbb{Z})\) for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence of a dévissage theorem identifying the K-theory of the Waldhausen category of finitely generated finite stage Postnikov towers of modules over a connective \(A_\infty\) ring spectrum R with the Quillen K-theory of the abelian category of finitely generated \(\pi_{0}R\)-modules.     

Algebraic K-theory of special groups

Following the introduction of an algebraic K-theory of special groups in [Dickmann and Miraglia, Algebra Colloq. 10 (2003) 149-176], generalizing Milnor's mod 2 K-theory for fields, the aim of this paper is to compute the K-theory of Boolean algebras, inductive limits, finite products, extensions, SG-sums and (finitely) filtered Boolean powers of special groups. A parallel theme is the preservation by these constructions of property [SMC], an analog for the K-theory of special groups of the property “multiplication by l(-1) is injective” in Milnor's mod 2 K-theory (see [Milnor, Invent. Math. 9 (1970) 318-344]).     

Bass’ NK groups and cdh-fibrant Hochschild homology

The K-theory of a polynomial ring R[t] contains the K-theory of R as a summand. For R commutative and containing ?, we describe K _*(R[t])/K _*(R) in terms of Hochschild homology and the cohomology of Kähler differentials for the cdh topology. We use this to address Bass’ question, whether K _n (R)=K _n (R[t]) implies K _n (R)=K _n (R[t ₁,t ₂]). The answer to this question is affirmative when R is essentially of finite type over the complex numbers, but negative in general.     

Isomorphism Conjecture for homotopy K-theory and groups acting on trees

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.     

Boundary amenability of relatively hyperbolic groups

Let K be a fine hyperbolic graph and Γ be a group acting on K with finite quotient. We prove that Γ is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group Γ acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations.     

Noncommutative localization in algebraic L-theory

Given a noncommutative (Cohn) localization A→σ⁻¹A which is injective and stably flat we obtain a lifting theorem for induced f.g. projective σ⁻¹A-module chain complexes and localization exact sequences in algebraic L-theory, matching the algebraic K-theory localization exact sequence of Neeman-Ranicki [Amnon Neeman, Andrew Ranicki, Noncommutative localisation in algebraic K-theory I, Geom. Topol. 8 (2004) 1385-1425] and Neeman [Amnon Neeman, Noncommutative localisation in algebraic K-theory II, Adv. Math. 213 (2007) 785-819].     

On Algebraic K-theory Categorical Groups

Homotopy categorical groups of any pointed space are defined via the fundamental groupoid of iterated loop spaces. This notion allows, paralleling the group case, to introduce the notion of K-categorical groups \mathbbK_iR\mathbb{K}_iR of any ring R. We also show the existence of a fundamental categorical crossed module associated to any fibre homotopy sequence and then, \mathbbK₁R\mathbb{K}_1R and \mathbbK₂R\mathbb{K}_2R are characterized, respectively, as the homotopy cokernel and kernel of the fundamental categorical crossed module associated to the fibre homotopy sequence FR\xrightarrowd_R BGLR\xrightarrowq_R BGLR ⁺ FR\xrightarrow{{d_{R} }}BGLR\xrightarrow{{q_{R} }}BGLR^{ + } As consequence, the 3th level of the Postnikov tower of the K-theory spectrum of R is classified by this categorical crossed module.     

A Note on K-Theory of Azumaya Algebras

For an Azumaya algebra A which is free over its centre R, we prove that K-theory of A is isomorphic to K-theory of R up to its rank torsions. We conclude that K _i (A, ?/m) = K _i (R, ?/m) for any m relatively prime to the rank and i ≥ 0. This covers, for example, K-theory of division algebras, K-theory of Azumaya algebras over semilocal rings, and K-theory of graded central simple algebras indexed by a totally ordered abelian group.     

K-theory of equivariant quantization

Using an equivariant version of Connes? Thom isomorphism, we prove that equivariant K-theory is invariant under strict deformation quantization for a compact Lie group action.     

Algebraic K-theory of groups wreath product with finite groups

The Farrell-Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for several classes of groups. For example, for discrete subgroups of Lie groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249-297], virtually poly-infinite cyclic groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249-297], Artin braid groups [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515-526], a class of virtually poly-surface groups [S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press] and virtually solvable linear group [F.T. Farrell, P.A. Linnell, K-Theory of solvable groups, Proc. London Math. Soc. (3) 87 (2003) 309-336]. We extend these results in the sense that if G is a group from the above classes then we prove the conjecture for the wreath product G?H for H a finite group. The need for this kind of extension is already evident in [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515-526; S.K. Roushon, The Farrell-Jones isomorphism conjecture for 3-manifold groups, math.KT/0405211, K-Theory, in press; S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press]. We also prove the conjecture for some other classes of groups.     

De la K-théorie algébrique vers la K-théorie hermitienne

In this Note, we introduce a new morphism between algebraic and hermitian K-theory. The topological analog is the Adams operation ${\psi }^2$ in real K-theory. From this morphism, we deduce a lower bound for the higher algebraic K-theory of a ring A in terms of the classical Witt group of the ring $A?A^{\mathrm{op}}$. To cite this article: M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Profinite groups, profinite completions and a conjecture of Moore

Let R be any ring (with 1), Γ a group and RΓ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an RΓ-module, whose restriction to RH is projective.Moore's conjecture (J. Pure Appl. Algebra 7(1976)287): Assume for every nontrivial element x in Γ, at least one of the following two conditions holds:

(M1)

〈x〉∩H≠{e} (in particular this holds if Γ is torsion free)

(M2)

ord(x) is finite and invertible in R.

Then M is projective as an RΓ-module.More generally, the conjecture has been formulated for crossed products R*Γ and even for strongly graded rings R(Γ). We prove the conjecture for new families of groups, in particular for groups whose profinite completion is torsion free.The conjecture can be formulated for profinite modules M over complete groups rings [[RΓ]] where R is a profinite ring and Γ a profinite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.     

K-Theory and the Quantization Commutes with Reduction Problem

The authors examine the quantization commutes with reduction phenomenon for Hamiltonian actions of compact Lie groups on closed symplectic manifolds from the point of view of topological K-theory and K-homology. They develop the machinery of K-theory wrong-way maps in the context of orbifolds and use it to relate the quantization commutes with reduction phenomenon to Bott periodicity and the K-theory formulation of the Weyl character formula.     

Tetravalent Edge-transitive Cayley Graphs of PGL （2, p）

t Let F = Cay（G, S）, R（G） be the right regular representation of G. The graph Г is called normal with respect to G, if R（G） is normal in the full automorphism group Aut（F） of F. Г is called a bi-normal with respect to G if R（G） is not normal in Aut（Г）, but R（G） contains a subgroup of index 2 which is normal in Aut（F）. In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL（2,p） are either normal or bi-normal when p ≠ 11 is a prime.