6,683 research outputs found

### Iterated doubles of the Joker and their realisability

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

### On the cohomology of loop spaces for some Thom spaces

In this paper we identify conditions under which the cohomology H^*(\Omega
M\xi;\k) for the loop space $\Omega M\xi$ of the Thom space $M\xi$ of a
spherical fibration $\xi\downarrow B$ 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
$\Sigma^\infty\Omega M\xi$ has a local splitting replacing the James splitting
of $\Sigma\Omega M\xi$ when $M\xi$ is a suspension.Comment: Final version, minor change

### Power operations in $K$-theory completed at a prime

We describe the action of power operations on the $p$-completed cooperation
algebras K^\vee_0 K = K_0(K)\sphat_p for $K$-theory at a prime~$p$, 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 Eâ ring spectra

We introduce a notion of characteristic for connective p-local Eâ 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

These notes provide an informal introduction to a type of Mackey functor that
arises naturally in algebraic topology in connection with Morava $K$-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 $K$-theory
to such classifying spaces.Comment: Corrections, minor improvements in Appendix, additional reference

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

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 $\mathbb{S}$-algebras with residue fields

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

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