-annihilated elements in H_*(CP^∞ × CP^∞)

Abstract

We compute the subring of H*(CP∞ × CP∞; ) annihilated by the Steenrod algebra, , p being an odd prime. By calculating the subring's structure as a GL(2, )-space we may obtain information about the modular representations of that group.

$\mathcal{A}(p)$ generators for $H^*V$ and Singer's homological transfer

“…However, in Theorem 1.2 of [2], we computed the space M (2) whose dual has, as its basis, a minimal set of A(p)-generators for S(V * ), considered as a submodule of H * V , when k = 2. The methods of [2] were to define a right action of A(p) on homology, using the formula < ωθ, ζ >=< ω, θζ > where θ ∈ A(p), ω is a class in homology and ζ is a class in cohomology, and to calculate which homology classes were annihilated by all elements of A(p). We can extend these results to obtain the space of annihilated elements in H * (V ) by a brief consideration of the (dualised) action of β.…”

A new action of the Kudo–Araki–May algebra on the dual of the symmetric algebras, with applications to the hit problem

“…The operators k are left inverses of his operators k given by k .f / D k f 2 , which have been utilized by many researchers ( k is the k th elementary symmetric function, that is, c k (or w k ) in our notation), and have been adapted for mod 2 symmetric algebras by Janfada and Wood [9; 10] (see also Crossley's map [6] analogous to 2 for H .CP 1 CP 1 I F p / at odd primes).…”

“…We shall also discuss how our action incorporates on the duals of the finite symmetric algebras the operators introduced by Kameko [11], Crossley [6], and Janfada and Wood [9; 10] (see also Hà [7]). …”

Transfert algébrique et action du groupe linéaire sur les puissances divisées modulo 2

“…Probablement à cause des difficultés techniques, mis à part les raffinements du théorème de Wood et les travaux de Crossley [10,12] en caractéristique impair, aucune tentative d'aller plus loin dans cette direction n'a été enregistrée pendant les dix ans qui suivent. Ce n'est que très récemment (2002) qu'un calcul effectif de P A en degré d = 2 p+3 + 2 p+2 − 4 pour k = 4 a été réalisé par…”

“…La nécessité d'une telle analyse contribuera, nous l'espérons, à justifier la raison d'être de nos présents travaux. Nous signalons que le problème analogue en caractéristique impaire a été étudié par Crossley [10,12,11] (pour k 2).…”