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 KK-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 E∞ ring spectra

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

BP: close encounters of the E∞ kind

Inspired by Stewart Priddy’s cellular model for the <i>p</i>-local Brown–Peterson spectrum <i>BP</i>, we give a construction of a <i>p</i>-local <i>E</i>∞ ring spectrum <i>R</i> which is a close approximation to <i>BP</i>. Indeed we can show that if <i>BP</i> admits an <i>E</i>∞ structure then these are weakly equivalent as <i>E</i>∞ ring spectra. Our inductive cellular construction makes use of power operations on homotopy groups to define homotopy classes which are then killed by attaching <i>E</i>∞ cells

E∞ ring spectra and elements of Hopf invariant 1

The 2-primary Hopf invariant 1 elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper, we explore some properties of the E∞ ring spectra obtained from certain iterated mapping cones by applying the free algebra functor. In fact, these are equivalent to Thom spectra over infinite loop spaces related to the classifying spaces BSO, BSpin, BString. We show that the homology of these Thom spectra are all extended comodule algebras of the form A∗A(r)∗ P∗ over the dual Steenrod algebra A∗ with A∗A(r)∗F2 as an algebra retract. This suggests that these spectra might be wedges of module spectra over the ring spectra HZ, kO or tmf; however, apart from the first case, we have no concrete results on this

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