On the Structures of the Milnor K-Groups of Some Complete Discrete Valuation Fields

For a complete discrete valuation field K, the unit group of K has a natural decreasing filtration with respect to the valuation, and the graded quotients of this filtration are written in terms of the residue field. The Milnor K-group of a field is a generalization of the unit group. The Milnor K-group of K has a natural decreasing filtration of the same kind. However, if K is of mixed characteristics and has an absolute ramification index greater than one, the structure is not yet known. The aim of this paper is to determine the structure of the Milnor K-group of some special K, which are of mixed characteristics (0,p), whose residue fields are allowed to be imperfect, and which are of absolute ramification index p(p–1).     

ApproximatingK*(?) through degree five

We approximate theK-theory spectrum of the integers using a spectrum level rank filtration. By means of a certain poset spectral sequence, we explicitly compute the first three subquotients of this filtration. Assuming a conjecture about the filtration's rate of convergence, we conclude thatK ₄()=0 andK ₅() is a copy of (the Borel summand) plus two-torsion of order at most eight.     

Extremal K -Theory and Index for C*-Algebras

We develop the general theory for a new functor K _e on the category of C ^*-algebras. The extremal K-set, K _e (A), of a C ^*-algebra A is defined by means of homotopy classes of extreme partial isometries. It contains K ₁ (A) and admits a partially defined addition extending the addition in K ₁ (A), so that we have an action of K ₁ (A) on K _e (A). We show how this functor relates to K ₀ and K ₁, and how it can be used as a carrier of information relating the various K-groups of ideals and quotients of A. The extremal K-set is then used to extend the classical theory of index for Fredholm and semi-Fredholm operators.     

On a Class of Hungarian Semigroups and the Factorization Theorem of Khinchin

Let G be a connected reductive Lie group and K be a maximal compact subgroup of G. We prove that the semigroup of all K-biinvariant probability measures on G is a strongly stable Hungarian semigroup. Combining with the result [see Rusza and Szekely⁽⁹⁾], we get that the factorization theorem of Khinchin holds for the aforementioned semigroup. We also prove that certain subsemigroups of K-biinvariant measures on G are Hungarian semigroups when G is a connected Lie group such that Ad G is almost algebraic and K is a maximal compact subgroup of G. We also prove a p-adic analogue of these results.     

Algebraic K-Theory and the Conjectural Leibniz K-Theory

The analogy between algebraic K-theory and cyclic homology is used to build a program aiming at understanding the algebraic K-theory of fields and the periodicity phenomena in algebraic K-theory. In particular, we conjecture the existence of a Leibniz K-theory which would play the role of Hochschild homology. We propose a motivated presentation for the Leibniz K ₂-group ofa field.     

Semi-exactitude du bifoncteur de Kasparov équivariant

We define a notion of strong K-theoretic amenability for a locally compact group G. This notion coincides with the K-theoretic amenability of many groups. We prove that all results obtained concerning the behavior of KK(.,.) with respect to exact sequences are generalized to the case of KK _G (.,.) for G strongly K-amenable.     

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.     

Dihedral homology and Hermitian K-theory

We establish a rational isomorphism between certain relative versions of Hermitian K-theory and the dihedral homology of simplicial Hermitian rings. This is the dihedral analogue of Goodwillie's result for cyclic homology and algebraic K-theory. In particular, we describe involutions on (negative) cyclic homology and the K-theory of simplicial rings. We show that Goodwillie's map from K-theory to negative cyclic homology can be chosen to preserve involutions. By work of Burghelea and Fiedorowicz the invariants of the involution on K-theory can be identified with symmetric Hermitian K-theory. Finally, we show how the author's chain complex defining dihedral homology can be extended to the left to capture the invariants of the involution on negative cyclic homology.Supported by New Mexico State University, grant No. RC90-051.     

Nikulin involutions on <Emphasis Type="Italic">K</Emphasis>3 surfaces

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the 11-dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on Morrison–Nikulin involutions, these are Nikulin involutions which interchange two copies of E ₈(−1) in the Néron Severi group. The second author is supported by DFG Research Grant SA 1380/1-1.     

The structure of some classes of <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S ²ⁿ⁺¹ (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S ¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.     

Idempotency of Extensions via the Bicompletion

Let Top ₀ be the category of topological T ₀-spaces, QU ₀ the category of quasi-uniform T ₀-spaces, T : QU ₀ → Top ₀ the usual forgetful functor and K : QU ₀ → QU ₀ the bicompletion reflector with unit k : 1 → K. Any T-section F : Top ₀ → QU ₀ is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF. The literature considers important T-sections F that enjoy all three, or just one, or none of these properties. It is known that T(K,k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T(K,k)F is the reflection to Fix(TkF) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T(K,k)F is not idempotent, and a K-true F for which the coarsest associated T-preserving coreflector in QU ₀ is not stable under K. We dedicate this paper to the memory of Sérgio de Ornelas Salbany (1941–2005).     

The Higher K-Theory of Complex Varieties

Let X be a smooth complex variety, and let F be its function field. We prove that (after localizing at the prime 2) the K-groups of F are divisible above the dimension of X, and that the K-groups of X are divisible-by-finite. We also describe the torsion in the K-groups of F and X.     

A Basis of Bideterminants for the Coordinate Ring of the Orthogonal Group

We give a basis of bideterminants for the coordinate ring K[O(n)] of the orthogonal group O(n,K), where K is an infinite field of characteristic not 2. The bideterminants are indexed by pairs of Young tableaux which are O(n)-standard in the sense of King–Welsh. We also give an explicit filtration of K[O(n)] as an O(n,K)-bimodule, whose factors are isomorphic to the tensor product of orthogonal analogs of left and right Schur modules.     

The Algebraic K-Theory Spectrum of a 2-Adic Local Field

We explicitly determine the homotopy type of the 2-completed algebraic K-theory spectrum KF, where F is an arbitrary finite extension of the 2-adic rational numbers. The answer is formulated in terms of topological complex K-theory and the K-theory of suitable finite fields, suspended copies of which are glued together by connecting maps that depend on the Iwasawa theory of F.     

Multilinear Operators on C(K, X) Spaces

Given Banach spaces X, Yand a compact Hausdorff space K, we use polymeasures to give necessary conditions for a multilinear operator from C(K, X) into Yto be completely continuous (resp. unconditionally converging). We deduce necessary and sufficient conditions for Xto have the Schur property (resp. to contain no copy of c ₀), and for Kto be scattered. This extends results concerning linear operators.     

The weight twoK-theory of fields

We verify the weight-two portion of the Quillen-Lichtenbaum conjectures, relating the modn weight-twoK-groups of a field with the appropriate étale cohomology groups. This extends the work of Merkurjev and Suslin onK ₂ of a field, and the works of Suslin and the author on the indecomposableK ₃ of a field.Partially supported by the NSF.     

ContinuousK-theory

We study arithmetical properties of homotopy groups of thel-adic completion of Quillen'sK-theory space of number field, with a view on the Dwyer-Friedlander comparison map into étaleK-theory. The relation of these groups toK-theory is a complete analogy to the relation of continuous étale cohomology to étale cohomology. We identify the torsion subgroup of the resulting term with the subgroup of divisible elements inK _2n (F). We prove that this term is sent isomorphically into étaleK-theory, giving some further evidence for the Lichtenbaum-Quillen conjectures.     

Relative K-homology and K-homology of an ideal

We provide a straightforward proof of one of the main results of the Baum-Douglas K-homology theory: If A is a separable nuclear C ^*-algebra and I an ideal of A, then the natural restriction map K ⁰(A, I) K ⁰(I) is an isomorphism of Abelian groups.     

Equivariant K-Theory of Compact Connected Lie Groups Dedicated to Daniel Quillen on the Occasion of his Sixtieth Birthday

We compute the equivariant K-theory K _G ^* (G)for a compact connected Lie group Gsuch that ₁ (G)is torsion free (where Gacts on itself by conjugation). We prove that K _G ^* (G)is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a compact connected Lie group Gwith ₁ (G)torsion, namely PSU(3), and compute the corresponding equivariant K-theory.     

Index theory on locally homogeneous spaces

We describe how the equivariant K homology class of an invariant elliptic operator on a homogeneous space of a linear semisimple Lie group determines the L ²-index of the associated operator on a finite volume locally homogeneous space. The machinery of equivariant K homology and of KK theory can be used to prove theorems about L ²-indices. We give an application motivated by the problem of calculating multiplicities of subrepresentations of quasi-regular representations.Supported by the National Science Foundation under Grant No. DMS-8903472.Supported by the National Science Foundation under Grant No. DMS-8901436.