7,705 research outputs found
Iterated doubles of the Joker and their realisability
Let be the subHopf algebra of the mod~ Steenrod algebra
generated by and . The
\emph{Joker} is the cyclic -module
which plays a special
r\^ole in the study of -modules. We discuss realisations of
the Joker both as an -module and as the cohomology of a
spectrum. We also consider analogous -modules for
and prove realisability results (both stable and unstable) for and
non-realisability results for .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 of the Thom space of a
spherical fibration 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
has a local splitting replacing the James splitting
of when is a suspension.Comment: Final version, minor change
Power operations in -theory completed at a prime
We describe the action of power operations on the -completed cooperation
algebras K^\vee_0 K = K_0(K)\sphat_p for -theory at a prime~, 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 -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 -theory
to such classifying spaces.Comment: Corrections, minor improvements in Appendix, additional reference
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