A Counterexample for Lightning Flash Modules over E(e1,e2)

We give a counterexample to Theorem 5 in Section 18.2 of Margolis' book, "Spectra and the Steenrod Algebra", and make remarks about the proofs of some later theorems in the book that depend on it. The counterexample is a module which does not split as a sum of lightning flash modules and free modules.Comment: 2 pages. Revision corrects a typo in the definition of M(n

Differentials in the homological homotopy fixed point spectral sequence

We analyze in homological terms the homotopy fixed point spectrum of a T-equivariant commutative S-algebra R. There is a homological homotopy fixed point spectral sequence with E^2_{s,t} = H^{-s}_{gp}(T; H_t(R; F_p)), converging conditionally to the continuous homology H^c_{s+t}(R^{hT}; F_p) of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations beta^epsilon Q^i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the $^{2r}-term of the spectral sequence there are 2r other classes in the E^{2r}-term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to the E^infty-term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups C of T, and for the Tate- and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-27.abs.htm

Some Remarks on the Root Invariant

We show how the root invariant of a product depends upon the product of the root invariants, give some examples of the equivariant definition of the root invariant, and verify a weakened form of the algebraic Bredon-Löffler conjecture

Extended powers of manifolds and the Adams spectral sequence

The extended power construction can be used to create new framed manifolds out of old. We show here how to compute the effect of such operations in the Adams spectral sequence, extending partial results of Milgram and the author. This gives the simplest method of proving that Jones’ 30-manifold has Kervaire invariant one, and allows the construction of manifolds representing Mahowald’s classes η4 and η5, among others

Ext in the nineties

We describe a package of programs to calculate minimal resolutions, chain maps, and null homotopies in the category of modules over a connected algebra overe Z_2 and in the category of unstable modules over the mod 2 Steenrod algebra. They are available for free distribution and intended for use as an Adams spectral sequence \u27pocket calculator\u27. We provide a sample of the results obtained from them

Some root invariants and Steenrod operations in Ext_A(F2,F2)

We give the results of computations of root invariants in Ext over the Steenrod algebra through the 25-stem, with partial information through the 45-stem. This allows the computation of some new Steenrod operations as well

Ossa\u27s Theorem and Adams covers

We show that Ossa’s theorem splitting ku ∧ BV for elementary abelian groups V follows from general facts about ku ∧ BZ/2 and Adams covers. For completeness, we also provide the analogous results for ko ∧ BV