We describe a Cat-valued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2-nerve. We define a 2-category NHom whose objects are bicategories and whose 1-cells are normal homomorphisms of bicategories, in such a way that the 2-nerve construction becomes a full embedding of NHom in the 2-category of simplicial objects in Cat. This embedding has a left biadjoint, and we characterize its image. The 2-nerve of a bicategory is always a weak 2-category in the sense of Tamsamani, and we show that NHom is biequivalent to a certain 2-category whose objects are Tamsamani weak 2-categories.
We give an explicit description of the set of all factorization structures, or twisting maps, existing between the algebras k^2 and k^2, and classify the resulting algebras up to isomorphism. In the process we relate several different approaches formerly taken to deal with this problem, filling a gap that appeared in a recent paper by Cibils. We also provide a counterexample to a result concerning the Hochschild (co)homology appeared in a paper by J.A. Guccione and J.J. Guccione.
We apply the machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized to arbitrary X by Edidin-Graham) and an Atiyah-Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bloch-Lichtenbaum and Friedlander-Suslin.
We give examples of smooth projective complex varieties of dimension d >= 4 and primes rho such that the morphic cohomology group (LH5)-H-3(X)/l is infinite, and L-3 H-5(X). circle times Q is not finitely generated as a rational vector space. In particular, for these examples the semi-topological K-group K-1(sst)(X). circle times Q has infinite dimension.
We give a construction for twisted equivariant K-theory in the case of a proper action of a discrete group using twisted bundles. Our construction uses results of Lueck and Oliver to extend a construction of Adem and Ruan. We also show the existence of a Chern character to twisted Bredon cohomology. This gives a partial answer to the question of when you can construct twisted equivariant K-theory out of finite rank twisted bundles.
We establish an Excision type theorem for niceness of group structure on the orbit space of unimodular rows of length n modulo elementary action. This permits us to establish niceness for relative versions of results for the cases when n=d+1; d being the dimension of the base algebra. We then study and establish niceness for the case when n=d, and also establish a relative version, when the base ring is a smooth affine algebra over an algebraically closed field.
We study the double coset Gal((Q) over bar/ k)\Ext(1) (E((Q) over bar, Lambda)/Aut (Lambda) , and interpret our results as partially showing that the notion of a path on a complex elliptic curve E can be characterised algebraically. The proofs show that our results are just concise reformulations of Kummer theory for E as well as the description of the image of the Galois action on the Tate module. Namely, we prove (a),(b) below by showing they are equivalent to (c) which is well-known: (a) uniquely divisible abelian EndE-module extensions of the group E((Q) over bar) of algebraic points of an elliptic curve, by Lambda congruent to Z(2), lie in finitely many double cosets in Gal((Q) over bar/ k)\Ext(1) (E((Q) over bar, Lambda)/Aut (Lambda) (b) natural algebraic properties characterise the Poincare's fundamental groupoid of a complex elliptic curve, restricted to the algebraic points, (c) up to finite index, the image of the Galois action on the sequences (Pi)(i>0), j P-ij = P-i, i, j > 0 of points P-i is an element of E-k ((Q) over bar) is as large as possible with respect to linear relations between the coordinates of the points P-i's. Our original motivations come from model theory.
We first discuss how open/closed chord diagrams, both with and without marked points, act on appropriate Hochschild complexes possibly coupled with the two-sided cobar complex. Then, in the main part of the paper, we introduce the notion of a V_k-algebra and obtain suitable homotopy versions.
The aim of this note is to give a simplified proof of the surjectivity of the natural Milnor-Chow homomorphism $\rho: K^M_n(A) \to CH^n(A,n)$ between Milnor $K$-theory and higher Chow groups for essentially smooth (semi-)local $k$-algebras $A$ with $k$ infinite. It implies the exactness of the Gersten resolution for Milnor $K$-theory at the generic point. Our method uses the Bloch-Levine moving technique and some properties of the Milnor $K$-theory norm for fields.
Dwyer, Weiss, and Williams have recently defined the notions of parametrized topological Euler characteristic and parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and Reidemeister torsion of finite CW-complexes.
In a previous paper, we showed nonvaninishing of the universal index elements in the K-theory of the maximal C*-algebras of the fundamental groups of enlargeable spin manifolds. The underlying notion of enlargeability was the one from the first relevant paper of Gromov and Lawson, involving contracting maps defined on finite covers of the given manifolds. In the paper at hand, we weaken this assumption to the one in the second paper of Gromov and Lawson, where infinite covers are allowed. The new idea is the construction of a geometrically given C*-algebra with trace which encodes the information given by these infinite covers; along the way we obtain an easy proof of a relative index theorem relevant in this context.
We explicitly compute the lower algebraic K-theory of Gamma(3) a discrete subgroup of the group of isometrics of hyperbolic 3-space.
This paper provides a realization of K-theory with R/Z coefficients and proves an R/Z index theorem.
We prove the integral Novikov conjecture for torsion free S-arithmetic subgroups Gamma 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.
We show that a duality of the Hopf-cyclic homology and cohomology can be explained in terms of functors defined on a PROP for Hopf algebras.
Let F be a nonarchimedean local field and let GL(N) = GL(N,F). We prove the existence of parahoric types for GL(N). We construct representative cycles in all the homology classes of the chamber homology of GL(3).
