The Segal conjecture for elementary abelian p-groups

Adams, J. F.; Gunawardena, Jeremy; Miller, Haynes

doi:10.1016/0040-9383(85)90014-x

Cited by 97 publications

(89 citation statements)

References 7 publications

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“…The fact that M satisfies the axioms listed above is now a consequence of the following easilyverified isomorphisms of bisets: for axiom (1), (2), if α : G → G is inner then G G G with G acting via α on one side and straight multiplication on the other side is isomorphic to G G G with G acting via straight multiplication on both sides; for axiom (3)…”

Section: Definitions Of Globally Defined Mackey Functors and Preliminmentioning

confidence: 99%

“…We define a generalization of the double Burnside ring, which has its origins in [37], [27], [25] and [1]. Let G and H be finite groups.…”

Section: Definitions Of Globally Defined Mackey Functors and Preliminmentioning

confidence: 99%

“…We next define a category C X ,Y R (D) which is a more general form of a category defined in [27,1] where it is called the Burnside category. In generality it was considered by Bouc [7].…”

Section: Definitions Of Globally Defined Mackey Functors and Preliminmentioning

confidence: 99%

See 2 more Smart Citations

Stratifications and Mackey Functors II: Globally Defined Mackey Functors

Webb¹

2010

J K-Theor

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Abstract. We describe structural properties of globally defined Mackey functors related to the stratification theory of algebras. We show that over a field of characteristic zero they form a highest weight category and we also determine precisely when this category is semisimple. This approach is used to show that the Cartan matrix is often symmetric and non-singular, and we are able to compute finite parts of it in some instances. We also develop a theory of vertices of globally defined Mackey functors in the spirit of group representation theory, as well as giving information about extensions between simple functors.

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Section: Definitions Of Globally Defined Mackey Functors and Preliminmentioning

confidence: 99%

“…We define a generalization of the double Burnside ring, which has its origins in [37], [27], [25] and [1]. Let G and H be finite groups.…”

Section: Definitions Of Globally Defined Mackey Functors and Preliminmentioning

confidence: 99%

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Stratifications and Mackey Functors II: Globally Defined Mackey Functors

Webb¹

2010

J K-Theor

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“…The Segal conjecture was proved first for the cyclic group of order two by Lin [19], then for the cyclic groups of odd prime order by Gunawardena [13], for general finite cyclic groups by Ravenel [29], for elementary abelian 2-groups by Gunnar Carlsson [6], for odd elementary abelian groups by Adams, Gunawardena and Miller [2], and finally for general finite groups by Gunnar Carlsson [7]. An unstable proof along entirely different lines can be found in Lannes [17].…”

Section: R(g)/imentioning

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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“…Z-module, A(G, G) is free, with a base of elements which may be written 9j*. Here i runs over the inclusions of subgroups i: H-*G, and i* corresponds to the transfer map Tr.BG + ->BH+; 9 runs over homomorphisms 9:H->G, and 9^ corresponds to the induced map B9+: BH + ->BG + [1,Section 9;7,p. 433].…”

Section: M->lim Mmentioning

confidence: 99%

Atomic spaces and spectra

Adams

Kuhn

1989

Proceedings of the Edinburgh Mathematical Society

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1. The subject-matter of this paper is in some sense known; but we will try to organise, explain and reprove it, and to give examples.In essence, a space or spectrum X is "atomic" if a map / : X-*X may be proved to be an equivalence by a simple, computable test applied in one dimension; this goes back to [4] (published as [5]) and first appeared in print in [12]. That it is useful to prove X atomic and then apply the fact has been amply shown, beginning with [3].This notion is related to two others. Unique factorisation results for spaces and spectra have been considered in [6,9,14]. Here one needs the notion of an "irreducible" or "indecomposable" object X, and a slightly stronger notion of "prime".We first show that the case of "spaces" and the case of "spectra" can be considered together, by concentrating on the fact that the hom-set [X,X] is (under suitable assumptions) a profinite monoid. In this case we show that the "weaker" condition implies the "stronger", as follows.(a) If X is indecomposable then its hom-set [_X, AT] is "good", and (b) if [X,AT] is "good" then X is both "atomic" and "prime".We give some illustrative examples, including some which arise "in nature" as stable summands of classifying spaces BG. We conclude with the proofs.Related results have been obtained by M. C. Crabb and J. R. Hubbuck; we are grateful to them for letters, and also to F. R. Cohen and F. P. Peterson.2. First we unify the two cases to be considered. We will comment in Section 3.

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The Segal conjecture for elementary abelian p-groups

Cited by 97 publications

References 7 publications

Stratifications and Mackey Functors II: Globally Defined Mackey Functors

Stratifications and Mackey Functors II: Globally Defined Mackey Functors

Stably splitting BG

Atomic spaces and spectra

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