MilnorK-theory is the simplest part of algebraicK-theory

We identify the MilnorK-theory of a field with a certain higher Chow group.     

On the algebraicK-theory of infinite product categories

Although theK-theory functor on the category of symmetric monoidal categories preserves finite products for essentially trivial reasons, this is not so in the case of infinite products. In this paper, we show that in factK-theory does preserve infinite products, but for non-trivial reasons.Supported in part by NSF DMS 9209714.     

AlgebraicK-theory of von Neumann algebras

To every von Neumann algebra, one can associate a (multiplicative) determinant defined on the invertible elements of the algebra with range a subgroup of the Abelian group of the invertible elements of the center of the von Neumann algebra. This determinant is a normalization of the usual determinant for finite von Neumann algebras of type I, for the type II₁-case it is the Fuglede-Kadison determinant, and for properly infinite von Neumann algebras the determinant is constant equal to 1. It is proved that every invertible element of determinant 1 is a product of a finite number of commutators. This extends a result of T. Fack and P. de la Harpe for II₁-factors. As a corollary, it follows that the determinant induces an injection from the algebraicK ₁-group of the von Neumann algebra into the Abelian group of the invertible elements of the center. Its image is described. Another group,K ₁ ^w (A), which is generated by elements in matrix algebras overA that induce injective right multiplication maps, is also computed. We use the Fuglede-Kadison determinant to detect elements in the Whitehead group Wh(G).Partially supported by NSF Grant DMS-9103327.     

The Product Formula in Unitary Deformation K-theory

For finitely generated groups G and H, we prove that there is a weak equivalence G H (G × H) of ku-algebra spectra, where denotes the “unitary deformation K-theory” functor. Additionally, we give spectral sequences for computing the homotopy groups of G and HG in terms of connective K-theory and homology of spaces of G-representations.     

Geometric modules and algebraicK-homology theory

Letp: XZ be a continuous map into a (proper) metric space. Using a variation on the geometric modules of Quinn, we associate top (and any reasonable ringR) an additive category (p, R). Mapsp, as above, are the objects of a category on which (-,R) becomes functorial. By composing with an open cone construction, we get a functor which associates to any topological space over a compact Lipschitz space an additive category. Finally, by using the algebraicK-theory spectrum for an additive category, we arrive at a functor which is our main object of study. We show that it is a homology theory in a suitable sense and we derive an Atiyah-Hirzebruch type spectral sequence for its calculation in many cases, including all triangulated objects. On our way, we show that the boundedK-theory of Pedersen and Weibel is essentially a special case of the boundedly controlledK-theory defined earlier by the authors and we establish a close connection, at least philosophically, between the latter theory and the K-theory with -control developed by Chapman, Ferry and Quinn.Partially supported by the NSF under grants numbered DMS-8504320 and DMS-8803149.Partially supported by the SNF (Denmark) under grants numbered 11-7062 and 11-7792.     

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.     

Continuously controlledA-theory

To anycontrolled space over the metric spaceB we can associate itsboundedly controlled algebraic K-theory, a functor designed to give information about the space of stable bounded concordances of manifolds homotopy equivalent toX. Generalizing a construction of D. R. Anderson, F. X. Connolly, S. Ferry, and E. K. Pedersen, we define another functor, calledcontinuously controlled A-theory, which depends only on thetopology of the control space, not itsmetric properties. In the special case whereB=R ₊, this functor is (more or less by definition) the same asproper A-theory. We prove that under certain conditions on the controlled space the natural transformation from boundedly controlledA-theory to continuously controlledA-theory is a weak homotopy equivalence, and hence defines a generalized homology theory. Continuously controlledK-theory is used in the approaches of G. Carlsson, E. K. Pedersen, and S. Ferry, S. Weinbergervto theK-theory Novikov conjecture.     

Relative MilnorK-theory

LetR be a commutative, semi-local ring,I ₁, ...,I _s ideals. In this paper, we define therelative Milnor K-groups of (R;I ₁, ...,I _s ),K _p ^M (R;I ₁, ...,I _s ), and show that these groups have many of the properties of the usual MilnorK-groups of a field. In particular, assuming a weak condition on the ideals, we show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic to the weightp portion of the relative QuillenK-groupK _p (R;I ₁, ...,I _s ), after inverting (p–1)!. We also define the relative group homology of GL _n (R;I ₁, ...,I _s ), and show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic toH _p (GL_p(R;I ₁, ...,I _s ))/Im(H _p (GL _p–1 (R;I ₁, ...,I _s ))). Finally, we consider a generalization to the relative setting of Kato's conjecture asserting that the Galois symbol gives an isomorphism fromK _p ^M (F)/l ^v to , and show that this relative version of Kato's conjecture implies the Quillen-Lichtenbaum conjectures asserting the Chern class:

Polynomial operations on complex K-theory

We adapt here the results of the author concerning polynomial operations on the 0th stable cohomotopy to the case of the 0th complex K-theory and consider polynomial operations : Kh, where h is a ring-valued contravariant functor, defined on finite CW-complexes, satisfying some properties. We construct a family of generating operations for the ring Pol(K,h) of all polynomial operations : Kh and doing so, we describe the additive structure of this ring in terms of the h(BU(n)'s. As an illustration of how polynomiality could be used to study operations in the setting of algebraic K-theory, we consider, from our point of view, the well known situation operations : KK on complex K-theory.     

AlgebraicK-theory of poset representations

LetS be a finite partially ordered set andk a field. This paper relates the algebraicK-theory ring of the category ofk-representations ofS to the Möbius ring ofS.     

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.     

Explicit weight two motivic cohomology complexes and algebraicK-theory

Beilinsonet al.'s motivic cohomology complex is related to a motivic complex (considered earlier by Lichtenbaum) formed using groups related to the scissors congruence groups. This involves an algebraic study of configurations of lines in projective spaces. One thereby obtains a more precise relationship between the cohomology of Beilinson's complex and lower dimensionalK-groups. Most of the proofs omitted in Beilinsonet al. are supplied here. There is also a list of certain torsional elements of the thirdK-group for fields between the rationals and the real cyclotomics.     

Personal Reminiscences of the Birth of Algebraic K-theory

These informal reminiscences, presented at the ICTP 2002 Conference on algebraic K-theory, recount the trajectory in the author's early research, from work on the Serre Conjecture (on projective modules over polynomial algebras), via ideas from algebraic geometry and topology, to the ideas and constructions that eventually contributed to the founding of algebraic K-theory. The solution of the Congruence Subgroup Problem is presented as a pivotal event.     

K-theory of twisted Grassmannians

We give a computation of the K-groups of Grassmannians and flag varieties over an arbitrary Noetherian base scheme. We also compute the K-groups of forms of Grassmannians and flag varieties associated to a sheaf of Azumaya algebras. One ingredient in the computation is the extension of the Bott theorem on the cohomology of line bundles on the flag variety (over Q) to a K ₀-Bott theorem valid over arbitrary Noetherian base schemes.Partially supported by the NSF.     

On the Milnor K-theory of the function fields of quasihomogeneous cones

Let V be a quasihomogeneous normal variety. The aim of this paper is to describe the Milnor K-theory of the function field of V in terms of the second residue homomorphisms associated with subvarieties and resolution data of V.Supported by KBN, 2 P301 010 06.     

Hodge decompositions of Loday symbols inK-theory and cyclic homology

In this paper, we study the Hodge decompositions ofK-theory and cyclic homology induced by the operations ^k and ^k , and in particular the decomposition of the Loday symbols x,y, ...z. Except in special cases, these Loday symbols do not have pure Hodge index. InK _n (A) they can project into every componentK _n ⁽ⁱ⁾ for 2in, and the projection of the Loday symbol x,y, ...,z intoK _n ⁽ⁿ⁾ is a multiple of the generalized Dennis-Stein symbol x,y, ...,z. Our calculations disprove conjectures of Beilinson and Soulé inK-theory, and of Gerstenhaber and Schack in Hochschild homology.Partially supported by National Security Agency grant MDA904-90-H-4019.Partially supported by National Science Foundation grant DMS-8803497.     

The Borel regulator map on pictures, II: An example from Morse theory

By doing parametrized Morse theory on circle bundles overS ², we produce pictures representing elements ofK ₃ of cyclotomic number fields. By computing the regulator map on these pictures, we show that they are rationally nontrivial.Research supported by NSF Grant No. MCS-90-02512.     

A non-Abelian K-theory and pseudo-isotopies of 3-manifolds

A non-Abelian version of algebraic K-theory, based on automorphism of free products rather than automorphisms of free modules, is considered and is related to pseudo-isotopies of 3-manifolds.Sometime after writing this paper I learned that some of the algebraic results in it were first proved by Gersten in [13]. Specifically each of Theorems 3.1, 3.2 and 5.1 in the special case when is the trivial group, and Theorem 3.3 and its corollaries are all results of Gersten. I have left the paper as it is for the sake of completeness and since the approach here often differs considerably from Gersten's.     

Additive L-theory

The cobordism groups of quadratic Poincaré complexes in an additive category with involution are identified with the Wall L-groups of quadratic forms and formations in , generalizing earlier work for modules over a ring with involution by avoiding kernels and cokernels.     

The Borel regulator map on pictures, I: A dilogarithm formula

We give a dilogarithm formula for the value of the Borel regulator map on an element ofK ₃ of the complex numbers given by a picture.This research is supported by NSF Grant No. MCS-90-02512.