Subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebras

Carlisle, David; Kuhn, Nicholas J.

doi:10.1016/0021-8693(89)90073-2

Cited by 31 publications

(36 citation statements)

References 5 publications

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“…, x n ] to questions about stable splittings in homotopy theory. In this work, an important role is played by a commuting action on S of the mod p Steenrod algebra; some references are Smith [29, Chapters 10,11], Carlisle and Kuhn [7], two surveys by Wood [32] and [33, §7], and the papers of Doty and Walker [9,10,11]. We have not seen how to use these results in attacking Parabolic Conjectures 1.1 and 1.2.…”

Section: This Means One Can Regard Conjecture 12 As Being About Hilbmentioning

confidence: 99%

Invariants of GL_n(𝔽_q) in polynomials modulo Frobenius powers

Lewis¹,

Reiner²,

Stanton³

2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Section: This Means One Can Regard Conjecture 12 As Being About Hilbmentioning

confidence: 99%

Invariants of GL_n(𝔽_q) in polynomials modulo Frobenius powers

Lewis¹,

Reiner²,

Stanton³

2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Since the Steinberg representation is projective and self-dual (in characteristic p), the cohomology of this spectrum is the image of the cohomology of the Thom space B(Z/pZ) [MtP83], and then again in a more "algebraic" fashion (at the prime 2) by Carlise and Kuhn in [CK89]. In particular, it is well-known that St H * B(Z/pZ) k is free over A k−1 .…”

Section: Cohomology and Relation With Mitchell's Spectramentioning

confidence: 99%

Iterates of the suspension map and Mitchell’s finite spectra with $A_k$-free cohomology

Arone

1998

Mathematical Research Letters

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Abstract. We study certain cross-effects of the unstable homotopy of spheres. These cross-effects were constructed by Weiss, for different purposes, in the context of "Orthogonal calculus". We show that Mithchell's finite spectra with A kfree cohomology (constructed in [Mt85]) arise naturally as stabilizations of Weiss' cross-effects. In particular, we find that after a suitable Bousfield localization, our cross-effects, which capture meaningful information about the unstable homotopy of spheres, are homotopy equivalent to the infinite loop spaces associated with Mitchell's spectra. This last result is a partial generalization of the main result of Mahowald and the author in [AM97]. IntroductionLet X be a topological space. We are interested in studying the difference between the homotopy type of X and that of S 2 X, the double suspension of X (for technical reasons the statements turn out to be a little cleaner if one works with double suspensions rather than single suspensions). One naive way to compare X and S 2 X would be by means of the map X → S 2 X given by taking the smash product of the identity map on X with the inclusion S 0 → S 2 . Of course, this map is null-homotopic and thus is unlikely to provide useful information about the difference between X and S 2 X. A much better idea is to consider the Freudental (double) suspension map w 1 : X → Ω 2 S 2 X. Let F 1 (X) be the homotopy fiber of this map. We regard F 1 (X) as measuring the difference between the homotopy type of X and that of S 2 X. We would like to iterate the process and find higher analogues of the suspension map and higher iterated differences of the suspension map. Thus at the second stage we would like to find a suitable way to compare F 1 (X) with F 1 (S 2 X). Obviously, there exists a natural map F 1 (X) → Ω 2 F 1 (S 2 X). However, this map turns out to be null-homotopic, just as the map X → S 2 X is null-homotopic. It follows from the 1991 Mathematics Subject Classification. 55P40, 55P42, 55P65.

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“…There remains also the interesting and largely unsolved problem of understanding the cohomology of the wedge summands. Some surprising work of Carlisle and Kuhn [5] relates the representation theoretic indexing of the factors to their Morava K-theories, via freeness over finite subalgebras of the Steenrod algebra.…”

Section: Here G Runs Over the Equivalence Classes Of Maximal Subgroumentioning

confidence: 99%

Stably splitting BG

Benson

1996

Bull. Amer. Math. Soc.

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Abstract. In the early nineteen eighties, Gunnar Carlsson proved the Segal conjecture on the stable cohomotopy of the classifying space BG of a finite group G. This led to an algebraic description of the ring of stable self-maps of BG as a suitable completion of the "double Burnside ring". The problem of understanding the primitive idempotent decompositions of the identity in this ring is equivalent to understanding the stable splittings of BG into indecomposable spectra. This paper is a survey of the developments of the last ten to fifteen years in this subject. The Segal conjectureThe nineteen eighties saw a number of major breakthroughs in homotopy theory. Among the most spectacular are the proofs of the nilpotence conjecture by Devinatz, Hopkins and Smith [10], the Sullivan conjecture by Haynes Miller [24], and the Segal conjecture by Gunnar Carlsson [7]. Two of these three concern the role of finite groups in homotopy theory: the Sullivan conjecture is "unstable", while the Segal conjecture is "stable" with respect to suspension. This report is about the consequences of the Segal conjecture for the stable splittings of the classifying space of a finite group.We begin by setting the scene. Around 1960, Atiyah [3] calculated the Ktheory of the classifying space of a finite group. There is a natural map from the character ring R(G) to K 0 (BG) which sends a complex representation V to the corresponding vector bundle EG× G V → BG. Atiyah proved that this map induces an isomorphismwhere the completionn is with respect to the augmentation ideal I = Ker(dim : R(G) → Z). He also showed that K 1 (BG) = 0. The Segal conjecture is the corresponding statement for stable cohomotopy, in which the representation ring is replaced by the Burnside ring A(G). This is the 1991 Mathematics Subject Classification. Primary 55P.

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Subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebras

Cited by 31 publications

References 5 publications

Invariants of GL_n(𝔽_q) in polynomials modulo Frobenius powers

Invariants of GL_n(𝔽_q) in polynomials modulo Frobenius powers

Iterates of the suspension map and Mitchell’s finite spectra with $A_k$-free cohomology

Stably splitting BG

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Subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebras

Cited by 31 publications

References 5 publications

Invariants of GLn(𝔽q) in polynomials modulo Frobenius powers

Invariants of GLn(𝔽q) in polynomials modulo Frobenius powers

Iterates of the suspension map and Mitchell’s finite spectra with $A_k$-free cohomology

Stably splitting BG

Contact Info

Product

Resources

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Invariants of GL_n(𝔽_q) in polynomials modulo Frobenius powers

Invariants of GL_n(𝔽_q) in polynomials modulo Frobenius powers