Showing papers in "K-theory in 2006"
TL;DR: The Atiyah-Hirzebruch spectral sequence as discussed by the authors relates ordinary singular cohomology to complex K-theory, in a way that is explained further in this paper.
Abstract: In recent years much attention has been given to a certain spectral sequence relating motivic cohomology to algebraic K-theory [Be, BL, FS, V3]. This spectral sequence takes on the form H(X,Z(− q 2 )) ⇒ K(X), where the H(X ;Z(t)) are the bi-graded motivic cohomology groups, and K(X) denotes the algebraic K-theory of X . It is useful in our context to use topologists’ notation and write K(X) for what K-theorists call K−n(X). The above spectral sequence is the analog of the classical Atiyah-Hirzebruch spectral sequence relating ordinary singular cohomology to complex K-theory, in a way that is explained further below. It is well known that there are close similarities between motivic homotopy theory and the equivariant homotopy theory of Z/2-spaces (cf. [HK1, HK2], for example). In fact there is even a forgetful map of the form (motivic homotopy theory over R) → (Z/2-equivariant homotopy theory),
101 citations
90 citations
TL;DR: In this paper, the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that was recently constructed on quantum SU(2) was discussed.
Abstract: We discuss the local index formula of Connes–Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.
54 citations
TL;DR: The flasque model structure as discussed by the authors is a combination of the projective and injective model structures on presheaves, and it can be used to avoid technical and annoying difficulties for different reasons.
Abstract: By now it is well known that there are two useful (objectwise or local) families of model structures on presheaves: the injective and projective. In fact, there is at least one more: the flasque. For some purposes, both the projective and the injective structure run into technical and annoying (but surmountable) difficulties for different reasons. The flasque model structure, which possesses a combination of the convenient properties of both structures, sometimes avoids these difficulties.
44 citations
TL;DR: In this paper, a family index theorem in K-theory is given for the setting of Atiyah, Patodi and Singer of a family of Dirac operators with spectral bound-ary condition.
Abstract: A families index theorem in K-theory is given for the setting of Atiyah, Patodi and Singer of a family of Dirac operators with spectral bound- ary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred cusp, pseudodierential opera- tors on the fibres (with boundary) of a fibration; a version of Poincare duality is also shown in this setting.
42 citations
TL;DR: In this paper, a family of non-commutative 3-sphere deformations was constructed by using a Heegaard splitting of the topological 3sphere as a guiding principle.
Abstract: We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C∗-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C∗-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts.
40 citations
TL;DR: In this paper, a new version of the bivariant k-theory is defined on the category of all locally convex algebras and its invariants are determined using a natural extension for the Weyl algebra.
Abstract: We introduce a new version $kk^{\rm alg}$ of bivariant $K$-theory that is defined on the category of all locally convex algebras. A motivating example is the Weyl algebra $W$, i.e. the algebra generated by two elements satisfying the Heisenberg commutation relation, with the fine locally convex topology. We determine its $kk^{\rm alg}$-invariants using a natural extension for $W$. Using similar methods the $kk^{\rm alg}$-invariants can be determined for many other algebras of similar type.
39 citations
TL;DR: Brown and Ellis as mentioned in this paper generalised this corrected result to derive formulae of Hopf type for the n-fold Cech derived functors of the lower central series functors Z_k.
Abstract: In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Cech derived functors of the lower central series functors Z_k. The paper ends with an application to algebraic K-theory.
32 citations
TL;DR: In this paper, the authors considered the homotopy coniveau tower construction for an arbitrary cohomology theory on smooth varieties over a field or a Dedekind domain.
Abstract: We consider the "homotopy coniveau tower" for an arbitrary cohomology theory on smooth varieties over a field or a Dedekind domain. This tower is a generalization of the construction used by Bloch-Lichtenbaum and Friedlander-Suslin in their studies of the spectral sequence from motivic cohomology to K-theory. Our main result is that an application of the classical Chow's moving lemma and some categorical constructions make the homotopy coniveau tower strictly functorial when the base is a field, and functorial in the homotopy category when the base is a Dedekind domain.
28 citations
TL;DR: In this article, the authors give a proof of the cobordism invariance of the index of elliptic pseudodifferential operators on sigma-compact manifolds, where, in the noncompact case, the operators are assumed to be multiplication outside a compact set.
Abstract: We give a proof of the cobordism invariance of the index of elliptic pseudodifferential operators on sigma-compact manifolds, where, in the non-compact case, the operators are assumed to be multiplication outside a compact set. We show that, if the principal symbol class of such an elliptic operator on the boundary of a manifold X has a suitable extension to K^1(TX), then its index is zero. This condition is incorporated into the definition of a cobordism group for non-compact manifolds, called here ``cobordism of symbols''.
Our proof is topological, in that we use properties of the push-forward map in K-theory defined by Atiyah and Singer, to reduce it to R^n. In particular, we generalize the invariance of the index with respect to the push-forward map to the non-compact case, and obtain an extension of the K-theoretical index formula of Atiyah and Singer to operators that are multiplication outside a compact set. Our results hold also for G-equivariant operators, where G is a compact Lie group.
22 citations
TL;DR: In this article, it was shown that the deformation K-theory spectrum has an Atiyah-Hirzebruch spectral sequence arising from a spectrum level ltration.
Abstract: For nitely generated groups G and H, we prove that there is a weak equivalence KG^ku KH ' K(G H) of ku-algebra spectra, where K denotes the \unitary deformation K-theory" functor. Additionally, we give spectral sequences for computing the homotopy groups of KG and HZ^ku KG in terms of connective K-theory and homology of spaces of G-representations. The underlying goal of many programs in algebraic K-theory is to understand the algebraic K-groups of a eld F as being built from the K-groups of the algebraic closure of the eld, together with the action of the absolute Galois group. Specically , Carlsson's program (see (2)) is to construct a model for the algebraic K-theory spectrum using the Galois group and the K-theory spectrum of the algebraic closure F. In some specic instances, the absolute Galois group of the eld F is explicitly the pronite completion ^ G of a discrete group G. (For example, the absolute Galois group of the eld k(z) of rational func- tions, where k is an algebraically closed of characteristic zero, is the pronite completion of a free group.) In the case where F contains an algebraically closed subeld, the pronite completion of a \deformation K-theory" spectrum KG is conjecturally equivalent to the pronite completion of the algebraic K-theory spectrum KF . Additionally, it would be advantageous for this description to be compatible with the motivic spectral sequence. This deformation K- theory spectrum has an Atiyah-Hirzebruch spectral sequence arising from a spectrum level ltration. The ltration quotients are spectra built from isomorphism classes of representations of the group. It is hoped that this ltration is related to the motivic spectral sequence, and that this relation would give a greater understanding of the rela- tionships between Milnor K-theory, Galois cohomology, and the repre- sentation theory of the Galois group. We now outline the construction of deformation K-theory. To a nitely generated group G one associates the categoryC of nite dimen- sional unitary representations of G, with morphisms being equivariant isometric isomorphisms. Elementary methods of representation theory
TL;DR: In this paper, it was shown that for certain Frechet algebras the Chern character provides an isomorphism between these functors, and applied to prove that the Hecke algebra and the Schwartz algebra of a reductive p-adic group have isomorphic periodic cyclic homology.
Abstract: Using similarities between topological K-theory and periodic cyclic ho- mology we show that, after tensoring with C, for certain Frechet algebras the Chern character provides an isomorphism between these functors. This is applied to prove that the Hecke algebra and the Schwartz algebra of a reductive p-adic group have isomorphic periodic cyclic homology. Later an appendix was added, to deal with infinite direct products of algebras. Mathematics Subject Classification (2000) 19D55, 19L10, 22E50, 46L80
TL;DR: In this article, the homology stability problem for hyperbolic unitary groups over a local ring with an infinite residue field is studied, and the authors show that it is NP-hard.
Abstract: In this note the homology stability problem for hyperbolic unitary groups over a local ring with an infinite residue field is studied.
TL;DR: In this paper, a realization of K-theory with R/Z coefficients is presented, and the authors prove an R-Z index theorem for the case of R/K.
Abstract: This paper provides a realization of K-theory with R/Z coefficients and proves an R/Z index theorem.
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TL;DR: The I-diagrams of simplicial sets as discussed by the authors is a diagram of equivalences, in the sense that any morphism α : i → j of I is mapped to a weak equivalence α∗ : Xi → Xj of simplified sets.
Abstract: Suppose that I is a small category, and that X : I → S is a functor taking values in simplicial sets. Such functors are also commonly called I-diagrams of simplicial sets. Suppose further that X is a diagram of equivalences in the sense that any morphism α : i → j of I is mapped to a weak equivalence α∗ : Xi → Xj of simplicial sets. Then it is a fundamental and well known result of Quillen [11], [4, IV.5.7] that each pullback square
TL;DR: In this paper, a simplified version of Voevodsky's proof of the Milnor-Kato conjecture is presented, which does not need to consider elds transcendental over F and makes no use of motivic cohomology at all.
Abstract: m ) from the Milnor K-theory of the eld F modulo m to the Galois cohomology of F with cyclotomic coecien ts is an isomorphism (here, as usually, m denotes the group of m-roots of unity in F ). One can see that it suces to verify this conjecture in the case when m is a prime number. Several years ago V. Voevodsky outlined his approach to the Milnor{Kato conjecture in the preprint [10]. The rst step of his argument dealt with a prime number ‘, a eld F having no nite extensions of degree prime to ‘, and an integer n > 1 such that the group K M n+1 (F ) is ‘-divisible. Assuming that the conjecture holds in degree less or equal to n for m = ‘ and any eld containing F , Voevodsky was proving that H n+1 (GF; Z=‘) = 0. The main goal of this paper is to give a simplied elementary version of Voevodsky’s proof of this step. In particular, we do not need to consider elds transcendental over F and we make no use of motivic cohomology at all. Our main result is formulated as follows.
TL;DR: In this paper, the authors introduced several variants of the representation ring, built as subrings and quotients of the ring of virtual super-modules up to even isomorphisms.
Abstract: Given a Lie superalgebra \g, we introduce several variants of the representation ring, built as subrings and quotients of the ring R_{\Z_2}(\g) of virtual \g-supermodules (up to even isomorphisms). In particular, we consider the ideal R_{+}(\g) of virtual \g-supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring SR(\g) on which the parity reversal operator takes the class of a virtual \g-supermodule to its negative. We also construct representation groups built from ungraded \g-modules, as well as degree-shifted representation groups using Clifford modules. The full super representation ring SR^{*}(\g), including all degree shifts, is then a \Z_{2}-graded ring in the complex case and a \Z_{8}-graded ring in the real case.
Our primary result is a six-term periodic exact sequence relating the rings R^{*}_{\Z_2}(\g), R^{*}_{+}(\g), and SR^{*}(\g). We first establish a version of it working over an arbitrary (not necessarily algebraically closed) field of characteristic 0. In the complex case, this six-term periodic long exact sequence splits into two three-term sequences, which gives us additional insight into the structure of the complex super representation ring SR^{*}(\g). In the real case, we obtain the expected 24-term version, as well as a surprising six-term version of this periodic exact sequence.
TL;DR: In this article, the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group, was introduced.
Abstract: We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K_0 of the category of "phi-endomorphisms of finitely generated free RPi(G,X)-modules". We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds.
TL;DR: In this article, the authors give an elementary characterization of those abelian monoids M that are direct limits of countable sequences of finite direct sums of monoids of the form either (Z/nZ) ⊔ {0} or Z ⊆ {0}.
Abstract: We give an elementary characterization of those abelian monoids M that are direct limits of countable sequences of finite direct sums of monoids of the form either (Z/nZ) ⊔ {0} or Z ⊔ {0} This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of all subgroups of M (viewed as a semigroup), and it makes it pos- sible to express M as a certain submonoid of a direct product � × G, where � is a distributive semilattice with zero and G is an abelian group When applied to the monoids V (A) appearing in the nonstable K-theory of C*-algebras, our results yield a full description of V (A) for C*-inductive limits A of finite sums of full matrix algebras over either Cuntz algebras On, where 2 ≤ n < ∞, or corners of O1 by projections, thus extending to the case including O1 earlier work by the authors together with KR Goodearl
TL;DR: In this paper, the noncommutative Atiyah-Patodi-Singer index theorem was proved for the equivariant case and extended to the Atiyah/Patodi case.
Abstract: In [Wu], the noncommutative Atiyah-Patodi-Singer index theorem was proved. In this paper, we extend this theorem to the equivariant case.
TL;DR: In this article, the twisted Hochschild homology was used to calculate the dimension of quantum hyperplanes, and the dimension was then calculated using the twisted hochschild decomposition.
Abstract: We calculate the Hochschild dimension of quantum hyperplanes using the twisted Hochschild homology.
TL;DR: In this paper, the Grothendieck group of non-commutative analogues of projective space bundles is studied and generalized to intersection theory for quantum ruled surfaces, and a special case of this group is shown.
Abstract: We compute the Grothendieck group of non-commutative analogues of projective space bundles. Our results specialize to give the Grothendieck groups of non-commutative analogues of projective spaces, and specialize to recover the Grothendieck group of a usual projective space bundle over a regular noetherian separated scheme. As an application, we develop an intersection theory for quantum ruled surfaces.
TL;DR: In this paper, it was shown that K0(A) and K1(A), respectively, are isomorphic to Z and Z\oplus Z by computing the connecting mappings in the standard K-theory six-term exact sequences associated to \sigma and to \gamma.
Abstract: Let A denote the C*algebra of bounded operators on L2(R) generated by: (i) all multiplications a(M) by functions a\in C[-\infty,+\infty], (ii) all multiplications by 2\pi-periodic continuous functions and (iii) all Fourier multipliers F^{-1}b(M)F, where F denotes the Fourier transform and b is in C[-\infty,+\infty]. The Fredholm property for operators in A is governed by two symbols, the principal symbol \sigma and an operator-valued symbol \gamma. We give two proofs of the fact that K0(A) and K1(A) are isomorphic, respectively, to Z and Z\oplus Z: by computing the connecting mappings in the standard K-theory six-term exact sequences associated to \sigma and to \gamma. For the second computation, we prove that the image of \gamma is isomorphic to the direct sum of two copies of the crossed product of C[-\infty,+\infty] by an automorphism, and then use the Pimsner-Voiculescu exact sequence to compute its K-theory.