6,683 research outputs found

    Iterated doubles of the Joker and their realisability

    Get PDF
    Let A(1)\mathcal{A}(1)^* be the subHopf algebra of the mod~22 Steenrod algebra A\mathcal{A}^* generated by Sq1\mathrm{Sq}^1 and Sq2\mathrm{Sq}^2. The \emph{Joker} is the cyclic A(1)\mathcal{A}(1)^*-module A(1)/A(1){Sq3}\mathcal{A}(1)^*/\mathcal{A}(1)^*\{\mathrm{Sq}^3\} which plays a special r\^ole in the study of A(1)\mathcal{A}(1)^*-modules. We discuss realisations of the Joker both as an A\mathcal{A}^*-module and as the cohomology of a spectrum. We also consider analogous A(n)\mathcal{A}(n)^*-modules for n2n\geq2 and prove realisability results (both stable and unstable) for n=2,3n=2,3 and non-realisability results for n4n\geq4.Comment: Minor changes and corrections. A version will appear in Homology, Homotopy and Application

    On the cohomology of loop spaces for some Thom spaces

    Full text link
    In this paper we identify conditions under which the cohomology H^*(\Omega M\xi;\k) for the loop space ΩMξ\Omega M\xi of the Thom space MξM\xi of a spherical fibration ξB\xi\downarrow B can be a polynomial ring. We use the Eilenberg-Moore spectral sequence which has a particularly simple form when the Euler class e(\xi)\in H^n(B;\k) vanishes, or equivalently when an orientation class for the Thom space has trivial square. As a consequence of our homological calculations we are able to show that the suspension spectrum ΣΩMξ\Sigma^\infty\Omega M\xi has a local splitting replacing the James splitting of ΣΩMξ\Sigma\Omega M\xi when MξM\xi is a suspension.Comment: Final version, minor change

    Power operations in KK-theory completed at a prime

    Get PDF
    We describe the action of power operations on the pp-completed cooperation algebras K^\vee_0 K = K_0(K)\sphat_p for KK-theory at a prime~pp, and K^\vee_0 KO = K_0(KO)\sphat_2.Comment: Version 6: final update, to appear in special issue of the Tbilisi Mathematical Journal on Homotopy Theory, Spectra, and Structured Ring Spectr

    Characteristics for E∞ ring spectra

    Get PDF
    We introduce a notion of characteristic for connective p-local E∞ ring spectra and study some basic properties. Apart from examples already pointed out by Markus Szymik, we investigate some examples built from Hopf invariant 1 elements in the stable homotopy groups of spheres and make a series of conjectures about spectra for which they may be characteristics; these appear to involve hard questions in stable homotopy theory

    Frobenius Green functors

    Full text link
    These notes provide an informal introduction to a type of Mackey functor that arises naturally in algebraic topology in connection with Morava KK-theory of classifying spaces of finite groups. The main aim is to identify key algebraic aspects of the Green functor structure obtained by applying a Morava KK-theory to such classifying spaces.Comment: Corrections, minor improvements in Appendix, additional reference

    The production cycles of the Scottish construction industry, 1802-2002

    Get PDF
    The revival of Scotland's national Parliament has focussed attention on potential differences in institutions and industries north of the border, compared to the rest of the UK. The Scottish construction industry, as with its counterparts anywhere, has developed enormously over the past two centuries and has experienced fluctuations due to internal and external influences. Much has been written about business cycles and building cycles relating to the construction industry in England, but this does not give a useful context for studies of the Scottish industry. This analysis of long-term time-series data is part of a larger project, looking historical aspects of the construction industry in Scotland, particularly the place of women in the industry, in order to establish aspects of the context and economic climate in which women found roles in the industry. This paper aims to use a wide range of data over the period in order to consider the Scottish experience. Has the Scottish construction industry's output demonstrated a cyclic nature in the last two centuries? What influences any such cycles? How do the cycles and any influences compare with the rest of the United Kingdom? Whereas most previous commentators, such as Cairncross, Rodgers and Glendinning , have mainly used housebuilding statistics as a tool for discussing the business cycles of the construction industry, this paper has gathered statistics for a wider range of construction activities. These are employed to show how the Scottish construction industry has its own pattern of long and short cycles. The patterns of boom-bust cycles associated with external events (such as wars or financial depression) or government intervention (such as housing policy or subsidies) can clearly be identified and compared with cycle patterns and shapes due to other effects such as industry structural features and credit availability

    Invertible modules for commutative S\mathbb{S}-algebras with residue fields

    Full text link
    The aim of this note is to understand under which conditions invertible modules over a commutative S-algebra in the sense of Elmendorf, Kriz, Mandell and May give rise to elements in the algebraic Picard group of invertible graded modules over the coefficient ring by taking homotopy groups. If a connective commutative S-algebra R has coherent localizations (R_*)_m for every maximal ideal m in R_*, then for every invertible R-module U, U_* is an invertible graded R_*-module. In some non-connective cases we can carry the result over under the additional assumption that the commutative S-algebra has `residue fields' for all maximal ideals m in R_* if the global dimension of R_* is small or if R is 2-periodic with underlying Noetherian complete local regular ring R_0.Comment: Revised version. One serious flaw correcte
    corecore