A Delocalization Property for Assembly Maps and an Application in Algebraic K-Theory of Group Rings

Let be a group, F the free -module on the set of finite order elements in , with acting by conjugation, and the ring extension of by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada% WcaaqaaiaaigdaaeaatCvAUfKttLearyGqLXgBG0evaGqbciab-5ga% UbaaieaacaGFLbGaaGOmaiaabc8acqWFPbqAcaqGVaGae8NBa42aaq% qaaeaacqGHdicjcqaHZoWzcqGHiiIZcqqHtoWrcaqGGaGaae4Baiaa% bAgacaqGGaGaae4BaiaabkhacaqGKbGaaeyzaiaabkhacaqGGaGae8% NBa4gacaGLhWoaaiaawUhacaGL9baaaaa!563E!\[\left\{ {\frac{1}{n}e2{\text{\pi }}i{\text{/}}n\left| {\exists \gamma \in \Gamma {\text{ of order }}n} \right.} \right\}\]. For a ring R with , we build an injective assembly map , detected by the Dennis trace map. This is proved by establishing a delocalization property for the assembly map in Hochschild homology, namely providing a gluing of simpler assembly maps (i.e. localized at the identity of ) to build , and by delocalizing a known assembly map in K-theory to define . We also prove the delocalization property in cyclic homology and in related theories.     

Novikov Conjectures for Arithmetic Groups with Large Actions at Infinity

We construct a new compactification of a noncompact rank one globally symmetric space. The result is a nonmetrizable space which also compactifies the Borel–Serre enlargement X of X, contractible only in the appropriate ech sense, and with the action of any arithmetic subgroup of the isometry group of X on X not being small at infinity. Nevertheless, we show that such a compactification can be used in the approach to Novikov conjectures developed recently by G. Carlsson and E. K. Pedersen. In particular, we study the nontrivial instance of the phenomenon of bounded saturation in the boundary of X and deduce that integral assembly maps split in the case of a torsion-free arithmetic subgroup of a semi-simple algebraic Q-group of real rank one or, in fact, the fundamental group of any pinched hyperbolic manifold. Using a similar construction we also split assembly maps for neat subgroups of Hilbert modular groups.     

Tate''s Conjecture, Algebraic Cycles and Rational K-Theory in Characteristic p

The purpose of this article is to discuss conjectures on motives, algebraic cycles and K-theory of smooth projective varieties over finite fields. We give a characterization of Tate's conjecture in terms of motives and their Frobenius endomorphism. This is used to prove that if Tate's conjecture holds and rational and numerical equivalence over finite fields agree, then higher rational K-groups of smooth projective varieties over finite fields vanish (Parshin's conjecture). Parshin's conjecture in turn implies a conjecture of Beilinson and Kahn giving bounds on rational K-groups of fields in finite characteristic. We derive further consequences from this result.     

The Novikov Conjecture for Low Degree Cohomology Classes

We outline a twisted analogue of the Mishchenko–Kasparov approach to prove the Novikov conjecture on the homotopy invariance of the higher signatures. Using our approach, we give a new and simple proof of the homotopy invariance of the higher signatures associated to all cohomology classes of the classifying space that belong to the subring of the cohomology ring of the classifying space that is generated by cohomology classes of degree less than or equal to 2, a result that was first established by Connes and Gromov and Moscovici using other methods. A key new ingredient is the construction of a tautological C* _r (, )-bundle and connection, which can be used to construct a C* _r (, )-index that lies in the Grothendieck group of C* _r (, ), where is a multiplier on the discrete group corresponding to a degree 2 cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our theorem.     

Filtrations of Simplicial Functors and the Novikov Conjecture

We show that the Strong Novikov Conjecture for the maximal C^*-algebra C^*(π) of a discrete group π is equivalent to a statement in topological K-theory for which the corresponding statement in algebraic K-theory is always true. We also show that for any group π, rational injectivity of the full assembly map for K_*^t(C^*(π)) follows from rational injectivity of the restricted assembly map. (Received: February 2006)     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

The integral Novikov conjectures for S-arithmetic groups I

We prove the integral Novikov conjecture for torsion free S-arithmetic subgroups Γ of linear reductive algebraic groups G of rank 0 over a global field k. They form a natural class of groups and are in general not discrete subgroups of Lie groups with finitely many connected components. Since many natural S-arithmetic subgroups contain torsion elements, we also prove a generalized integral Novikov conjecture for S-arithmetic subgroups of such algebraic groups, which contain torsion elements. These S-arithmetic subgroups also provide a natural class of groups with cofinite universal spaces for proper actions. Partially Supported by NSF grants DMS 0405884 and 0604878.     

Negative K-Theory of Generalized Quaternion Groups and Binary Polyhedral Groups

Classical group representation theory is used to determine which finite groups have finite negative K-theory. There follows a computation of the K _?1 of integral group rings ZG for all finite non-abelian subgroups of the group SU(2) of unit quaternions. In principle, the method applies to any finite group.     

Holomorphic K-Theory, Algebraic Co-cycles, and Loop Groups

In this paper we study the holomorphic K-theory of a projective variety. This K-theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory is built out of studying algebraic bundles over a variety up to algebraic equivalence. In this paper we will give calculations of this theory for flag like varieties which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors we will show that there is a rational isomorphism of graded rings between holomorphic K-theory and the appropriate morphic cohomology groups, in terms of algebraic co-cycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the symmetrized loop group U(n)/ _n where the symmetric group _n acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K-theory by inverting the Bott class, then rationally this is isomorphic to topological K-theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K-theory, and show that these obstructions vanish for generalized flag mani-folds.     

The Algebraic K-Theory of Operator Algebras

We the study the algebraic K-theory of C *-algebras, forgetting the topology. The main results include a proof that commutative C*-algebras are K-regular in all degrees (that is, all theirN ^T K _iand extensions of the Fischer-Prasolov Theorem comparing algebraic and topological K-theory with finite coefficients.     

K-theory of fine topological algebras, Chern character, and assembly

TheK-theory of the group algebra [] for a countable, discrete group is defined in terms of the simplicial ring of smooth simplices on [], where [] is given the fine topology with respect to its finite-dimensional, linear subspaces. The assembly map for this theory :K _* B K _* [] is studied and shown to be a rational injection. The proof uses the Connes-Karoubi Chern character fromK-theory of Banach algebras to cyclic homology, here generalized to any fine topological algebra, and proved to be multiplicative.     

Function Spaces and Continuous Algebraic Pairings for Varieties

Given a quasi-projective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of 'smooth curves') for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including (1) the existence of cap and join product pairings in topological cycle theory; (2) the agreement of cup product and intersection product for topological cycle theory; (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product; and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles.     

The Derived Picard Group is a Locally Algebraic Group

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic _K (A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic _K (A) is a locally algebraic group, and its identity component is Out⁰ _K (A). If B is a derived Morita equivalent algebra then DPic _K (A)DPic _K (B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.     

Algebraic Equivalence and Homology Classes of Real Algebraic Cycles

In this note we study the spectral properties of a multiplication operator in the space L_p(X)^m which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called “floating singularities”.     

The large scale topology and algebraic K-theory of arithmetic groups

New compactifications of symmetric spaces of noncompact type X are constructed using the asymptotic geometry of the Borel–Serre enlargement. The controlled K-theory associated to these compactifications is used to prove the integral Novikov conjecture for arithmetic groups.     

Calculation of Two-Primary Algebraic K -Theory of Some Group Rings

In this paper we use an excision result in algebraic K-theory and recent two-primary calculations of algebraic K-theory of number rings to obtain results on the two-primary algebraic K-theory of some group rings.     

Algebraic sets and coordinate groups for a free nilpotent group of nilpotency class 2

We give a complete classification of the algebraic sets and coordinate groups for the systems of equations in one variable over a free nilpotent group.     

The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms

The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373–382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric irreducible, non-singular plane algebraic curve then either or . In the former case is projectively equivalent to the curve with equation X^q+1Y+Y^q+1+X=0 studied by Pellikaan. Furthermore, the curve has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer subgroup, the set of its -rational points can be partitioned into finite projective planes . In this paper, the full automorphism group of such curves is determined. It turns out that is the normalizer of a Singer group in .     

On K3 of Witt Vectors of Length Two Over Finite Fields

We prove that for W₂ the Witt vectors of length two over the finite field , we have in characteristic at least 5 and for (3,f) = 1. The result is proved by using the identity and calculating the right term with a group homology spectral sequence. Some information on the spectral sequence is achieved by using the action of the outer automorphism of SL on the homology groups and recent results on K-groups of local rings and the ring of dual numbers over finite fields.     

Assembly maps,K-theory, and hyperbolic groups

Following Connes and Moscovici, we show that the Baum-Connes assembly map forK _*(C*v) is rationally injective when is word-hyperbolic, implying the Equivariant Novikov conjecture for such groups. Using this result in topologicalK-theory and Borel-Karoubi regulators, we also show that the corresponding generalized assembly map in algebraicK-theory is rationally injective.