Stable homotopy classification of BGp̌

Martino, John; Priddy, Stewart

doi:10.1016/0040-9383(94)00040-r

Cited by 19 publications

(17 citation statements)

References 10 publications

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“…For the second part, given an automorphism f of P, we can form a commutative square Theorem 8 (Martino-Priddy [11]) Let G and G be finite groups. The following are equivalent:…”

Section: Proposition 10mentioning

confidence: 99%

Lectures on the stable homotopy of BG

Priddy¹

2007

Geometry and Topology Monographs

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“…For the second part, given an automorphism f of P, we can form a commutative square Theorem 8 (Martino-Priddy [11]) Let G and G be finite groups. The following are equivalent:…”

Section: Proposition 10mentioning

confidence: 99%

Lectures on the stable homotopy of BG

Priddy¹

2007

Geometry and Topology Monographs

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“…Thus there is a one-to-one correspondence between the indecomposable summands and the simple modules of the ring of stable self-maps. This correspondence is explored in both [2] and [9] (see also [10]). It turns out that modular representation theory plays a crucial role: if P is a Sylow p-subgroup of G then each indecomposable summand of BG ∧ p originates in BQ for some subgroup Q P and corresponds to a simple F p Aut(Q) module.…”

Section: Introductionmentioning

confidence: 99%

Group extensions and automorphism group rings

Martino

Priddy

2003

Homology, Homotopy and Applications

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“…The paper [21] gives an explicit formula for the multiplicity of a wedge summand as the rank of a certain matrix defined in terms of subgroups and conjugations, while the paper [4] attempts a more abstract description of the simple modules. For further work in this area, see Martino and Priddy [22,23], Nishida [26], Priddy [27,28]. Some explicit calculations appear in Dietz [11], Dietz and Priddy [12], Martino and Priddy [20].…”

Section: This Multiplication Makes A(g G) Into a Noncommutative Noetmentioning

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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Stable homotopy classification of BGp̌

Cited by 19 publications

References 10 publications

Lectures on the stable homotopy of BG

Lectures on the stable homotopy of BG

Group extensions and automorphism group rings

Stably splitting BG

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