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G-Sets DeNition LECTURE ABOUT MACKEY FUNCTORS SHLOMI AGMON G is assumed to be a nite group throughout. G-Sets Denition. A G-set S is a set with an action of G on it. A morpishm of G- sets is a function (of sets) f : S ! T which commutes with the action, f(gx) = gf(x) 8x 2 S 8g 2 G. We will denote by G − set the category 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 2 S, denote by Gx it's stabilizer, Gx = fg 2 G s:t: gx = xg. It is a subgroup ∼ of G. So an orbit satises Gx = G=Gx , a G-sets isomorphism. We threfore obtain: Theorem. Any -set decomposes to a disjoint union ∼ ` G . G S S = orbits x =Gx Any functor 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 relations between them: G Denition. The orbit category OG of G is dened by Obj = f =H s:t: H ≤ Gg (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 2 G xed be a map, and denote f(H) = aK, a 2 G. For it to be an equivariant map (a G-set map), we need: • (Equivariant) f(gH) = gf(H) = gaK for any g 2 G. • (Well dened) f(ghH) = f(gH) 8h 2 H. Thus ghaK = gaK for any h 2 H, meaning Ha ⊆ K. Ha ⊆ K a −1 a 1 Notation: H := a Ha, H := aHa . Denote a^ : G=H ! G=K, a^ (gH) = gaK. −1 Then a^ : G=H ! G=K is equivariant, and a^ = ^b i aK = bK, meaning a b 2 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 a^ Ra Ra ' G=Ha / G=K quotient a a Ra ◦ π (gH) = Ra (g K) = gaK =a ^ (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 2 N (H). Since inclusions are irrelevant in this case, we obtain an iso- morphism N(H) G G =H ≈ HomOG ( =H; =H) given by correspondence with right translations, −1 (the inverse because a 7! Ra ^ ^ RaRb = Rba , or more generally a^b = 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 -morphisms are the trivial Zp P = Z =Zp Zp = Z =e Zp ones and a (right-)multiplication by for each P ! P; Zp ! P a^ : Zp ! Zp a . a 2 Zp Mackey Functors - motivation and definition2 For a cohomology theory h∗, the groups h∗ (point) are called it's coecients, by analogy with the case of singular cohomology with coecients. Classical cohomol- Proposition (3.19 from [Hatcher]). If a natural transformation 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 axiom, 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 homotopy 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-module isomorphisms: 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 commutative diagram 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 axioms, 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 := fx 2 K s:t: g (x) = xg is a subcomplex for each g 2 G. G-complex Since g−1 is the inverse to g's action, we get that any g 2 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 homotopies 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 2 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 ) = fg : L ! gL ⊆ L ; g 2 Gg 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 = fg 2 G s:t: g leaves L pointwise fixedg. If gL ⊆ L , denote by f θ : K ! OG the map 0 induced by . Then g , which precisely means that L ! L g GL0 ⊂ GL 0 G G , is a map .
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