Splitting of Gysin extensions

Let X --> B be an orientable sphere bundle. Its Gysin sequence exhibits H^*(X) as an extension of H^*(B)-modules. We prove that the class of this extension is the image of a canonical class that we define in the Hochschild 3-cohomology of H^*(B), corresponding to a component of its A_infty-structure, and generalizing the Massey triple product. We identify two cases where this class vanishes, so that the Gysin extension is split. The first, with rational coefficients, is that where B is a formal space; the second, with integer coefficients, is where B is a torus.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-37.abs.htm

Hermitian K-theory of the integers

The 2-primary torsion of the higher algebraic K-theory of the integers has been computed by Rognes and Weibel. In this paper we prove analogous results for the Hermitian K-theory of the integers with 2 inverted (denoted by Z'). We also prove in this case the analog of the Lichtenbaum conjecture for the hermitian K-theory of Z' : the homotopy fixed point set of a suitable Z/2 action on the classifying space of the algebraic K-theory of Z' is the hermitian K-theory of Z' after 2-adic completion.Comment: 36 pages ; see also http://www.math.jussieu.fr/~karoubi/ and http://www.math.nus.edu.sg/~matberic

Homological realization of prescribed abelian groups via KK-theory

Using algebraic and topological K-theory together with complex C^*-algebras, we prove that every abelian group may be realized as the centre of a strongly torsion generated group whose integral homology is zero in dimension one and isomorphic to two arbitrarily prescribed abelian groups in dimensions two and three.Comment: 10 page

The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory

Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence in general, and an integral equivalence when X has no formally real residue field. We also show that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its \'etale version is an isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum conjecture in K-theory. Applications compute higher Grothendieck-Witt groups of complex algebraic varieties and rings of 2-integers in number fields, and hence values of Dedekind zeta-functions.Comment: 17 pages, to appear in Adv. Mat

Homological realization of prescribed abelian groups via K-theory

Using algebraic and topological K-theory together with complex C*-algebras, we prove that every abelian group may be realized as the centre of a strongly torsion generated group whose integral homology is zero in dimension one and isomorphic to two arbitrarily prescribed abelian groups in dimensions two and thre

Phasing-out tobacco: proposal to deny access to tobacco for those born from 2000

As a contribution to worldwide efforts towards a tobacco-free society, this paper considers the possibility of a long-term phasing-in of a total ban, by proposing that individuals born in or after the year 2000 have their supply of tobacco restricted. In conjunction, a survey that we have conducted in Singapore indicates strong public support (even among current smokers) for the proposal

Children’s and parents’ involvement in care order proceedings: a cross-national comparison of judicial decision-makers’ views and experiences

This paper presents the views of judicial decision-makers (n= 1794) in four child protection jurisdictions (England, Finland, Norway, and the USA (California)), about whether parents and children are provided with appropriate opportunities to participate in proceedings in their countries. Overall, the study found a high degree of agreement within and between the countries as regards the important conditions for parents and children´s involvement, although the four systems themselves are very different. There was less agreement about children’s involvement than parents’, and the court decision-makers from Norway and Finland were more likely to express doubts about this. Nevertheless, the main message from the judicial decision-makers is that they are relatively satisfied as to how parents and children´s involvement is handled in their countries. Whether or not this confidence is justified, the emphasis on achieving effective involvement of children and parents in court proceedings is likely to grow, with major implications for the workers, decision-makers and agencies involved