Mackey Functors Research Articles

COHOMOLOGICAL DIMENSION OF MACKEY FUNCTORS FOR INFINITE GROUPS

We consider the cohomology of Mackey functors for infinite groups and define the Mackey-cohomological dimension of a group G. We relate this dimension to other cohomological dimensions such as the Bredon cohomological dimension and the relative cohomological dimension. In particular we show that for virtually torsion free groups the Mackey cohomological dimension is equal to both the relative cohomological dimension and the virtual cohomological dimension.

Clifford theory for Mackey algebras

We develop a Clifford theory for Mackey algebras. For simple Mackey functors, using their classification we prove Mackey algebra versions of Clifford's theorem and the Clifford correspondence. Let μ R ( G ) be the Mackey algebra of a finite group G over a commutative unital ring R, and let 1 N be the unity of μ R ( N ) where N is a normal subgroup of G. Observing that 1 N μ R ( G ) 1 N is a crossed product of G / N over μ R ( N ) , a number of results concerning group graded algebras are extended to the context of Mackey algebras, including Fong's theorem, Green's indecomposibility theorem and some reduction and extension techniques for indecomposable Mackey functors.

Equivariant Universal Coefficient and Künneth Spectral Sequences

This paper constructs hyper‐homology spectral sequences of Z‐graded and ROG‐graded Mackey functors that compute Ext and Tor over G‐equivariant S‐algebras A∞ ring spectra) for finite groups G. These specialize to universal coefficient and Künneth spectral sequences. 2000 Mathematics Subject Classification 55N91 (primary), 55P43, 55U20, 55U25 (secondary).

Mackey functors and functorial extensions

Given a finite group X and a normal subgroup G of X, we show that any Mackey functor M for X induces another Mackey functor M ˜ for X associated to G. We then consider the question, whether there exists a map M ˜ → M extending elements from M ( G ) to M ( X ) and compatible with the restriction maps. In the case that the order of G and the index of G in X are relatively prime, we give a sufficient condition for the existence of such a map, using canonical induction formulae.

Fibred permutation sets and the idempotents and units of monomial Burnside rings

We study the units of monomial Burnside rings and the idempotents of monomial Burnside algebras. Introducing a tenduction map, we realise the unit group and the torsion unit group as a Mackey functor.

Cohomological Mackey functors in number theory

We prove relations between the evaluations of cohomological Mackey functors over complete discrete valuation rings or fields and apply this to Mackey functors that arise naturally in number theory. This provides relations between λ- and μ-invariants in Iwasawa theory, between Mordell–Weil groups, Shafarevich–Tate groups, Selmer groups and zeta functions of elliptic curves, and between ideal class groups and regulators of number fields.

A note on the equivariant Dold–Thom theorem

In this note we prove a version of the classical Dold–Thom theorem for the RO( G)-graded equivariant homology functors H ∗ G(−; M ̄ ) , where G is a finite group, M is a discrete Z[G] -module, and M̱ is the Mackey functor associated to M. In the case where M= Z with the trivial G-action, our result says that, for a G-CW-complex X, and for a finite dimensional G-representation V, there is a natural isomorphism [S V, Z 0(X)] G≅H V G(X; Z ), where Z 0(X) denotes the free abelian group on X.

Induction theorems of surgery obstruction groups

Let G G be a finite group. It is well known that a Mackey functor { H ↦ M ( H ) } \{ H \mapsto M(H) \} is a module over the Burnside ring functor { H ↦ Ω ( H ) } \{ H \mapsto \Omega (H) \} , where H H ranges over the set of all subgroups of G G . For a fixed homomorphism w : G → { − 1 , 1 } w : G \to \{ -1, 1 \} , the Wall group functor { H ↦ L n h ( Z [ H ] , w | H ) } \{ H \mapsto L_n^h ({\mathbb Z}[H], w|_H) \} is not a Mackey functor if w w is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside ring functor as well as over the Grothendieck-Witt ring functor { H ↦ G W 0 ( Z , H ) } \{ H \mapsto {\mathrm {GW}}_0 ({\mathbb Z}, H) \} . In fact, we prove a more general result, that the functor assigning the equivariant surgery obstruction group on manifolds with middle-dimensional singular sets to each subgroup of G G is a module over the Burnside ring functor as well as over the special Grothendieck-Witt ring functor. As an application, we obtain a computable property of the functor described with an element in the Burnside ring.

Hochschild Constructions for Green Functors

Let G be a finite group, and R be a commutative ring. If A is a Green functor for G over R, and Γ is a crossed G-monoid, then the Mackey functor AΓ obtained by the Dress construction has a natural structure of Green functor, and its evaluation AΓ(G) is an R-algebra. This framework involves as special cases the construction of the Hochschild cohomology algebra of the group algebra from the ordinary cohomology functor, and the construction of the crossed Burnside ring from the ordinary Burnside functor. This article presents some properties of those Green functors A Γ, and the functorial relations between the corresponding categories of modules. As a consequence, a general product formula for the algebra A Γ(G) is stated.

Higher Limits via Subgroup Complexes

We study the higher derived functors of the inverse limit of a functor F: D --> Z_{(p)}-mod, where D is one of the standard categories which arise when studying the homotopy theory of the classifying space of a finite group G, e.g., the orbit category or the Quillen category of G. These higher limits are of importance e.g., for the study of maps between classifying spaces as well as for group cohomology. We show that these higher limits can be identified with the G-equivariant Bredon cohomology of the subgroup complex of p-subgroups in G (i.e., the nerve of the poset of p-subgroups in G) with values in a G-local coefficient system. We examine when smaller complexes can be used e.g., taking only p-radical subgroups, p-centric subgroups, elementary abelian p-subgroups or various subcollections thereof. Since the subgroup complexes are finite complexes, and often rather small, this provides concrete, computable formulas for these higher limits, generalizing earlier work of especially Jackowski-McClure-Oliver. As an application we look at the special case where all the higher limits vanish, as for example is the case for group cohomology. If F is a functor on the orbit category our formulas for the higher limits in this case yield five different expressions of F(G) in terms of values of F on proper subgroups. Two of these are `classical' namely Webb's exact sequence of Mackey functors and a formula for calculating stable elements, previously obtained using Alperin's fusion theorem. Examining this case also leads to improvements of sharpness results of homology decompositions due to Dwyer and others.

Stratifications and Mackey Functors I: Functors For a Single Group

Stratifications and Mackey Functors I: Functors For a Single Group

Non-additive exact functors and tensor induction for Mackey functors

Introduction Non additive exact functors Permutation Mackey functors Tensor induction for Mackey functors Relations with the functors ${\mathcal L}_U$ Direct product of Mackey functors Tensor induction for Green functors Cohomological tensor induction Tensor induction for $p$-permutation modules Tensor induction for modules Bibliography.

Operations in equivariant ℤ/p-cohomology

If G is a compact Lie group and M a Mackey functor, then Lewis, May and McClure [4] define an ordinary cohomology theory H*G(−; M) on G-spaces, graded by representations. In this article, we compute the ℤ/p-rank of the algebra of integer-degree stable operations [Ascr ]M, in the case where G=ℤ/p and M is constant at ℤ/p. We also examine the relationship between [Ascr ]M and the ordinary mod-p Steenrod algebra [Ascr ]p.The main result implies that while [Ascr ]M is quite large, its image in [Ascr ]p consists of only the identity and the Bockstein. This is in sharp contrast to the case with M constant at ℤ/p for q≠p; there [Ascr ]M≅[Ascr ]q.

Hecke algebras and cohomotopical Mackey functors

In this paper, we define the concept of the cohomotopical Mackey functor, which is more general than the usual cohomological Mackey functor, and show that Hecke algebra techniques are applicable to cohomotopical Mackey functors. Our theory is valid for any (possibly infinite) discrete group. Some applications to topology are also given.

An equivariant smash spectral sequence and an unstable box product

Let G G be a finite group. We construct a first quadrant spectral sequence which converges to the equivariant homotopy groups of the smash product X ∧ Y X \wedge Y for suitably connected, based G G -CW complexes X X and Y Y . The E 2 E^2 term is described in terms of a tensor product functor of equivariant Π \Pi -algebras. A homotopy version of the non-equivariant Künneth theorem and the equivariant suspension theorem of Lewis are both shown to be special cases of the corner of the spectral sequence. We also give a categorical description of this tensor product functor which is analogous to the description in equivariant stable homotopy theory of the box product of Mackey functors. For this reason, the tensor product functor deserves to be called an “unstable box product”.

A General Theory of Canonical Induction Formulae

So far there exist three independent constructions of two different canonical versions of Brauer's induction theorem for complex characters due to V. Snaith, P. Symonds, and the author. “Canonical” in this context means functorial with respect to restrictions along group homomorphisms. In this article we axiomatize the situation in which the above canonical induction formulae are constructed. Mackey functors and related structures arise in this way naturally as a convenient language. This approach allows us to construct canonical induction formulae for arbitrary Mackey functors. In particular we obtain canonical induction formulae for the Brauer character ring, the group of projective characters, the ring of trivial source modules, and the ring of linear source modules. In most cases, it is not difficult to construct such formulae over the rational numbers. A much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers. We give a sufficient condition in the general framework of Mackey functors for a canonical induction formula to be integral. As an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings, as, for example, the span of linear characters in the case of the canonical Brauer induction formula. This will be used in a subsequent article in the case of Adams operations and Chern classes.

Mackey Functors with Chain Conditions

LetAbe a Green functor for a finite groupGand letMbe a leftA-module.Mis callednoetherian (artinian)if the lattice of all submodules ofMsatisfies ACC (respectively DCC).Mis calledtotally noetherian (totally artinian)ifMXis noetherian (respectively artinian) for every finiteG-setX. In this paper it is shown that the totally noetherian (totally artinian) leftA-modules behave in the same way as the classical noetherian (artinian) modules. The left noetherian (left artinian) Green functors are less well behaved but some properties from classical algebra still hold. It is shown that Cohen's theorem holds in this setting. Our main result shows that if a commutative Green functorAis artinian, then the ringsA(H)/Jac(A(H)) are semisimple artinian for all subgroupsHofG.

Rational Mackey functors for compact lie groups I

A Mackey functor M is a structure analogous to the representation ring functor H ↦ R(H) encoding good formal behaviour under induction and restriction. More explicitly, M associates an abelian group M(H) to each closed subgroup H of a fixed compact Lie group G, and to each inclusion K ⊆ H it associates a restriction map r e s K H : M ( H ) → M ( K ) and an induction map i n d K H : M ( K ) → M ( H ). This paper gives an analysis of the category of Mackey functors M whose values are rational vector spaces: such a Mackey functor may be specified by giving a suitably continuous family consisting of a Q π0(WG(H))-module V(H) for each closed subgroup H with restriction maps V(K) → V(K) whenever K is normal in K and K/K is a torus (a ‘continuous Weyl-toral module’). We show that the category of rational Mackey functors is equivalent to the category of rational continuous Weyl-toral modules. In Part II this will be used to give an algebraic analysis of the category of rational Mackey functors, showing in particular that it has homological dimension equal to the rank of the group. 1991 Mathematics Subject Classification: 19A22, 20C99, 22E15, 55N91, 55P42, 55P91.

Class Group Relations from Burnside Ring Idempotents

We show that for each finite cohomological Mackey functor on a finite groupGthere exist explicit relations in the category of finite abelian groups between the evaluations of the Mackey functor at all the subgroups, one for each conjugacy class of nonhypo-elementary subgroups ofG. Furthermore, we show that the class groups of the intermediate fields of a Galois extension of number fields form such a Mackey functor on the Galois group, thereby obtaining class group relations by using the presence of the structure of a cohomological Mackey functor.

The Structure of Mackey Functors

Keywords: blocks ; finite groups ; category of finite $G$-sets ; diagrams of $G$-sets ; category of Mackey functors ; category of modules ; path algebras ; quiver with relations ; composition factors ; Cartan matrices ; decomposition matrices ; Burnside rings ; projective Mackey functors ; orthogonal idempotents ; $p$-perfect subgroups ; vertices ; sources ; Green correspondents ; $p$-groups ; indecomposable projective Mackey functors ; source modules ; Hecke algebras ; restriction ; corestriction ; group cohomology ; finite representation type ; Brauer trees Reference CTG-ARTICLE-1995-002doi:10.2307/2154915 Record created on 2008-12-16, modified on 2017-05-12