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LECTURE ABOUT MACKEY

SHLOMI AGMON

G is assumed to be a nite throughout.

G-Sets Denition. A G- S is a set with an action of G on it. A morpishm of G- sets is a (of sets) f : S → T which commutes with the action, f(gx) = gf(x) ∀x ∈ S ∀g ∈ G. We will denote by G − set the of nite G-sets with left G-action (as dened). The basic building blocks of G-sets are the cosets G/H for H ≤ G. Given x ∈ S, denote by Gx it's stabilizer, Gx = {g ∈ G s.t. gx = x}. It is a subgroup ∼ of G. So an orbit satises Gx = G/Gx , a G-sets . We threfore obtain:

Theorem. Any -set decomposes to a disjoint union ∼ ` G . G S S = orbits x /Gx Any on G-sets which is determined by it's action on orbits can thus be studied in a simpler context. This motivated the following denition, which incorporates all the essential information about G-orbits and the between them:

G Denition. The orbit category OG of G is dened by Obj = { /H s.t. H ≤ G} (cosets, not quotients!), Mor=G-equivariant maps. OG denition

Classication of G-equivariant maps.1

Let H,K ≤ G be subgroups. Let Equivariant maps are f (gH) = gaK, f : G/H → G/K for a ∈ G xed be a map, and denote f(H) = aK, a ∈ G. For it to be an equivariant map (a G-set map), we need: • (Equivariant) f(gH) = gf(H) = gaK for any g ∈ G. • (Well dened) f(ghH) = f(gH) ∀h ∈ H. Thus ghaK = gaK for any h ∈ H, meaning Ha ⊆ K. Ha ⊆ K a −1 a 1 Notation: H := a Ha, H := aHa . Denote aˆ : G/H → G/K, aˆ (gH) = gaK. −1 Then aˆ : G/H → G/K is equivariant, and aˆ = ˆb i aK = bK, meaning a b ∈ K. So any equivariant map is uniquely determined by selecting a coset aK in G/K. a a The quotients G/Ha → G/K and G/H → G/aK for H ⊆ K and H ⊆ K are G G a a equivariant (easy). Also, right translation Ra : /H → /H , gH 7→ gHa = gaH ,

Date: 3/05/2012. I'd like to thank Emmanuel Farjoun, Tomer Schlank and Matan Prezma for assisting me with preparing this lecture. 1Following I.3 in [Bredon] 1 LECTURE ABOUT MACKEY FUNCTORS 2 is equivariant. We obtain that equivariant maps G G are precisely HomOG ( /H, /K) those induced by quotients and translations:

quotient G/H / G/aK aˆ Ra Ra  '  G/Ha / G/K quotient a a Ra ◦ π (gH) = Ra (g K) = gaK =a ˆ (gH) = π (ga H ) = π ◦ Ra (gH) G-maps are induced by inclusion + right In the particular case K = H, the well-denedness condition Ha ⊆ K = H translation implies a ∈ N (H). Since inclusions are irrelevant in this case, we obtain an iso- N(H) G G /H ≈ HomOG ( /H, /H) given by correspondence with right translations, −1 (the inverse because a 7→ Ra ˆ ˆ RaRb = Rba , or more generally aˆb = ba). Example. Take for a prime . Then the objects of are all the possible G = Zp p OZp orbits of a -action, p and p . The - are the trivial Zp P = Z /Zp Zp = Z /e Zp ones and a (right-)multiplication by for each P → P, Zp → P aˆ : Zp → Zp a . a ∈ Zp Mackey Functors - motivation and definition2 For a theory h∗, the groups h∗ (point) are called it's coecients, by with the case of singular cohomology with coecients. Classical cohomol- (3.19 from [Hatcher]). If a between unreduced ogy is determined cohomology theories on the category of CW pairs is an isomorphism when the CW by it's value at a pair is (point, ∅), then it is an isomorphism for all CW pairs. point If one also imposes the dimension , this narrows down even further the possible cohomology theories we may have:

Theorem (4.59 from [Hatcher]). If h∗ is an unreduced cohomology theory on the category of CW pairs and hn (point) = 0 for n 6= 0, then there are natural isomor- phisms hn (X,A) ≈ Hn X,A; h0 (point) for all CW pairs (X,A) and all n. Replacing the basic Due to the invariance axiom, we had the (ordinary) cohomology of any building blocks single simplex determined by the coecients - cohomology of a point, hn (∆m) ≈ hn (point) . If we are to generalize this to the more general theme of G-cohomology, we should replace our basic building blocks by orbits. Since by homotopy in- ∗ n m n variance, a G-cohomology theory h should satisfy h (G/H × ∆ ) ≈ h (G/H) . A coecient system should also contain how they t together, meaning the G- equivariant maps between them. Mackey functor Denition. Let R be a ring, G a nite group. A Mackey functor for G, with values ∗ ∗ in R-Mod, is a bifunctor (M ,M∗): G − Set → R − Mod , with M contravariant and M∗ covariant, such that:

21.1 in [Bouc] LECTURE ABOUT MACKEY FUNCTORS 3

∗ • They coincide on objects: meaning, M∗ (X) = M (X) = M (X) for any G-set X. • For every pair of nite G-sets X,Y , let iX , iY denote the inclusions X,→ ` ∗ ∗ X Y ←-Y . Then the maps M (iX ) ⊕ M (iY ) and M∗ (iX ) ⊕ M∗ (iY ) are mutual inverse R- :

M∗(iX )⊕M∗(iY ) ) M (X) ⊕ M (Y ) ≈ M (X ` Y ) i

∗ ∗ M (iX )⊕M (iY ) • For any pullback diagram of G-sets γ T / Y

δ α   Z / X β there is a

M ∗(γ) M (T ) o M (Y )

M∗(δ) M∗(α)   M (Z) o M (X) M ∗(β) A morphism θ from a Mackey functor M to the Mackey functor N is a natural transformation of bifunctors. That is, a morphism θX : M (X) → N (X) for any G-set X, such that for any G-morphism f : X → Y the two corresponding diagrams (for the co- and contra- variant parts) commute. MackR (G) With these denitions, the Mackey functors for G over R form a category, de- noted MackR (G) or Mack (G). There are also alternative equivalent denitions of a Mackey functor which we will not present here; one in terms of several , and the other as a module for an R-algebra µR (G) of nite rank called the Mackey algebra (a representation). For more details, see 1.1 in [Bouc] or 3 in [TW].

Bredon Cohomology as motivation3 Denition. A G-complex is a CW-complex K together with an action of G on K by cellular maps (=sending the n-skeleton Kn to Kn) such that Kg := {x ∈ K s.t. g (x) = x} is a subcomplex for each g ∈ G. G-complex Since g−1 is the inverse to g's action, we get that any g ∈ G acts by an auto- morphism of the given CW structure. Kg is a sub-CW-complex. Example. 1 with the standard CW-structure can be a -complex only with the S Z3 trivial -action. To make it a -complex with the obvious non-trivial action (of Z3 Z3 rotating by 2π/3), we take the CW-structure with 3 zero-cells and 3 one-cells.

3Chapter I in [Bredon] LECTURE ABOUT MACKEY FUNCTORS 4

Same as with (the regular) HEP for a pair, a G-complex pair (K,L) has the G-equivariant HEP (homotopy replaced with a G-equivariant version). For any G-complex pair (K,L), the G-complex K/L is of the same equivariant homotopy type as K ∪ CL. Equivariant coho- An equivariant cohomology theory H∗ is simply a (unreduced) cohomology the- mology theories ory4, but with all maps and replaced by their equivariant counterparts. An interesting point worth mentioning is that if G is abelian, then the action of G induces on Hn (K,L) a G-module structure. This follows from the understanding of the relation N(H) G G , which reads G when is abelian. /H ≈ HomOG ( /H, /H) ≈ /H G We call an equivariant cohomology theory classical if it satises the dimension axiom, Hn (G/H) = 0 for all n 6= 0 and H ≤ G. Generic coecient A (generic) coecient system is a (contra-variant!) functor op .A system OG → Ab morphism of coecient systems is a natural transformation of such, thus all the coecients form a category CG. Let be a coecient system. Since N(H) G G , then M ∈ CG /H ≈ HomOG ( /H, /H) M (G/H) possess a natural N (H) /H module structure. In particular we see that M (G) has a G-module structure, and M (G/G) has the trivial module structure. K K Let K be a G-complex. From K we form a category K whose objects are nite subcomplexes of and whose morphisms are: 0 0 , K homK (L, L ) = {g : L → gL ⊆ L , g ∈ G} all the dierent maps from L to L0 induced by elements of G. op Dene a canonical (contravariant) functor θ : K → OG by θ (L) = G/GL, 0 op where GL = {g ∈ G s.t. g leaves L pointwise fixed}. If gL ⊆ L , denote by f θ : K → OG the map 0 induced by . Then g , which precisely that L → L g GL0 ⊂ GL 0 G G , is a map . Dene . gˆ : θ (L ) = /GL0 → /GL = θ (L) kGL0 7→ kgGL ∈ OG θ (f) =g ˆ Again we can see that any map is induced by right translation θ (L → gL) and inclusion θ (gL ⊆ L0). In order to proceed, we recall the construction of (ordinary) cellular homology: (see 2.2 in [Hatcher], for example) Cellular homology

For a CW-complex X, the groups Hn (Xn,Xn−1) are free abelian with basis in one to one correspondence with the n-cells of X. They form the cellular chain dn complex ... → Hn (Xn,Xn−1) → Hn−1 (Xn−1,Xn) → ... . The cellular boundary map can be computed by the formula

n X n−1 dn (eα) = dαβ eβ β where is the degree of the map n−1 n−1 n−1 that is the composition dαβ Sα → X → Sβ of the attaching map of n with the quotient map collapsing n−1 n−1 to a eα X − eβ point. For a generic coecient system op , then is called a M : OG → Ab Mθ : K → Ab simple coecient system on K. CK ⊂ LC K A local coecient system on K is a covariant functor L : K → Ab. These form a category denoted LC K . Simple coecient systems form a subcategory CK ⊂ LC K . Denote by K (σ) the smallest subcomplex of K containing σ. For a coecient system L , denote L (σ) = L (K (σ)) and L (τ → σ) = L (K (τ) → K (σ)) when K (τ) ⊂ K (σ). Cq (K; L )

4Satisfying the axioms: homotopy invariance, LES for a pair, excision & disjoint union. LECTURE ABOUT MACKEY FUNCTORS 5

Let L : K → Ab be a local coecient system. Dene Cq (K; L ) to be all the functions f on the q-cells of K with f (σ) ∈ L (σ) ,   q  M  C (K; L ) := f : Hq (Kq,Kq−1) → L (σ) s.t. f (σ) ∈ L (σ)

 σ∈Hq (Kq ,Kq−1)  Dene the coboundary map δ : Cq (K; L ) → Cq+1 (K; L ) by X (δf)(σ) = dστ L (τ → σ) f (τ) τ

Note that this makes sense, since dστ is non-zero only when K (τ) ⊂ K (σ). So δf of a (q + 1)-dimensional cell σ is computed by pushing the coecients of the q-cells comprising it's boundary upwards to L (σ) along the diagram of abelian groups that L induces. δ2 = 0 (easy). G- action on Now we dene an of G on Cq (K; L ). For g ∈ G, f ∈ Cq (K; L ), Cq (K; L ) −1  −1  −1  −1  f g σ lies in L g σ . g induces a map K g σ → K (σ), so L (g) f g σ −1  q is in L (σ). Set g (f)(σ) = L (g) f g σ . It is immediate that g (f) ∈ C (K; L ), and that this indeed denes an action of G by chain maps. Writing g∗ for L (g), we obtain

g (f)(gσ) = g∗f (σ) G Thus the xed-point set Cq (K; L ) = {f ∈ Cq (K; L ) s.t g (f) = f ∀g ∈ G} := q is a sub chain complex. By the last equation, it consists exactly of the CG (K; L ) equivariant cochains: f (gσ) = g∗f (σ). Now dene the Bredon cohomology as the cohomology of this chain complex;

q q  ∗ G HG (K; L ) = H C (K; L )

For a generic coecient system M ∈ CG we write q q  ∗ G HG (K; M) = H C (K; Mθ) For a pair (K,L) with L xed under G's action we get the LES of a pair. References

[Bredon] Equivariant Cohomology Theories, G. E. Bredon, in LNM 34 (1967), Springer-Verlag. [Hatcher] Algebraic , A. Hatcher. [Alperin] Groups and Representations, J. L. Alperin and R. B. Bell, Springer. [Bouc] Green Functors and G-sets, S. Bouc, LNM 1671, Springer. [TW] The Structure of Mackey Functors, J. Thevenaz and P. Webb in AMS volume 347, 1995.