The complete stable splitting for the classifying space of a finite group

Martino, John; Priddy, Stewart

doi:10.1016/0040-9383(92)90067-r

Cited by 34 publications

(55 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(This old version was once again distributed in 1990 as MSRI preprint series 00425-91 [32].) During this period, there has been some progress in the subject [27,28,41,20,26,21]. Such updated progress has also occurred in the current version of this paper.…”

Section: Introductionmentioning

confidence: 84%

See 1 more Smart Citation

Hecke algebras and cohomotopical Mackey functors

Minami

1999

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 84%

“…For X = EG, Theorem 6.6 was called the "Minami-Webb formula" in [27,28]. There is a recent interesting preprint of Martino and Priddy [29] on this subject.…”

Section: Proof Of (Ii)mentioning

confidence: 99%

Hecke algebras and cohomotopical Mackey functors

Minami

1999

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The multiplicity of a given stable homotopy type as a wedge summand is equal to the dimension of the corresponding simple module over its endomorphism ring, which is a finite field of characteristic p. Unfortunately, the representation theory of these algebras is not easy to study, though the papers of Martino and Priddy [21] and Benson and Feshbach [4] make some progress in this direction. 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.…”

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

confidence: 99%

“…The problem is not completely solved even in this case, but rather, it reduces to a well-known problem in modular representation theory-the determination of the simple modules in characteristic p for the finite general linear groups GL(n, F p ). In the remaining sections, we explain some of the ideas of Nishida [26], Martino and Priddy [21] and Benson and Feshbach [4] in the general case.…”

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

confidence: 99%

“…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].…”

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

confidence: 99%

See 1 more Smart Citation

Stably splitting BG

Benson

1996

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

Automorphisms of p-Groups Given as Cyclic-by-Elementary Abelian Central Extensions

Dietz

2001

Journal of Algebra

View full text Add to dashboard Cite

show abstract

The complete stable splitting for the classifying space of a finite group

Cited by 34 publications

References 5 publications

Hecke algebras and cohomotopical Mackey functors

Hecke algebras and cohomotopical Mackey functors

Stably splitting BG

Automorphisms of p-Groups Given as Cyclic-by-Elementary Abelian Central Extensions

Contact Info

Product

Resources

About