The Volodin model for hermitian algebraic K-theory

We define the Volodin hermitian algebraic K-theory for a (discrete) ring with an involution and show that it is isomorphic to Karoubi's hermitian algebraic K-theory. We also construct the Volodin model X(R _*) of hermitian algebraic K-theory for a simplicial ring R _* and show that it is a homotopy fiber of the map B Ô(R _*)B Ô(R _*)⁺. We also prove the general linear version of this result, which has been claimed in the existing literature, but whose proof was overlooked.     

Constructions and Dévissage in Hermitian K-Theory

We first give the good definition of the Hermitian K-theory of an exact category with duality. Then we prove a Dévissage theorem which allows us to compare the Hermitian K-theory of a number field with the Hermitian K-theory of its integers.     

The MoravaK-theory of algebraicK-theory spectra

Ifn2 the MoravaK-theoryK(n) _* of an algebraicK-theory spectrumKX vanishes for any ring or schemeX. This is proved using thev _n -complexes of Hopkins and Smith, together with the following theorem. The natural mapf:Q ₀S⁰BGL⁺ factors through the space ImJ. In particularf _*: _* ^s K _* annihilates CokerJ. These results are closely related to the Lichtenbaum-Quillen conjectures.Partially supported by an NSF grant.     

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.     

Mixed Hodge structure in cyclic homology and algebraic K-theory, 1

Deligne defined the notion of a mixed Hodge structure (MHS) and proved that every quasiprojective variety over has a natural MHS on its cohomology. This paper establishes similar results for cyclic homology and the algebraic K-theory of simply connected quasi-projective varieties over . In the nonsimply connected case, an MHS is established on certain quotient groups of algebraic K-theory.Supported by a NSERC University Research Fellowship and operating grant.     

Computations of K- and L-Theory of Cocompact Planar Groups

The verification of the isomorphism conjectures of Baum and Connes and Farrell and Jones for certain classes of groups is used to compute the algebraic K- and L-theory and the topological K-theory of cocompact planar groups (=cocompact N.E.C-groups) and of groups G appearing in an extension where is a finite group and the conjugation -action on ⁿ is free outside . These computations apply, for instance, to two-dimensional crystallographic groups and cocompact Fuchsian groups.     

Algebras almost commuting with Clifford algebras in a II∞ factor

We are concerned here with certain Banach algebras of operators contained within a fixed II factor N. These algebras may be thought of as noncommutative classifying spaces for the functor Ext _* ^N The basic objects of study are the algebras A _kN (for n=1, 2,...). Here, we are given an essentially unique representation of the complex Clifford algebra C _k N and the elements of A _k are those operators in N which exactly commute with the first k–1 generators of C _k and also commute with the kth generator modulo a symmetric ideal N. Up to isomorphism, these algebras are periodic with period 2.We determine completely the homotopy types of A ₁ ^–1 and A ₂ ^–1 Here, A ₁ ^–1 is homotopy equivalent to the space of (Breuer) Fredholm operators in N, while A ₂ ^–1 is homotopy equivalent to the group K _N ^–1 ={x N^–1¦ x=1+k, k K_N}. We use these results to compute the K-theory of A ₁ and A ₂.For a fixed C ^*-algebra A, we define abelian groups G _k,N(A) of equivalence classes of homomorphisms AA _k. Letting N = M (H) for a II₁ factor M we define similar abelian groups G _k(A, M) where we replace N by L(E) for countably generated right Hilbert M-modules E with (left) actions C _k L(E). Using ideas of Skandalis, we show that G _k,NG_k(A, M) so that the G _k,N are stable half-exact homotopy functors because the G _k(·, M) are such.In general, we show that G _k(A, M)KK ^k(A, M) and so our theory fits neatly into Kasparov KK-theory. We investigate many interesting examples from our point of view.     

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

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.