On a theorem of Eilenberg in simplicial Bredon-Illman cohomology with local coefficients
aa r X i v : . [ m a t h . A T ] M a y On a theorem of Eilenberg in simplicial Bredon-Illmancohomology with local coefficients
Goutam Mukherjee and Debasis Sen
Abstract
We prove simplicial version of a classical theorem of Eilenberg in the equivariantcontext and give an alternative description of the simplicial version of Bredon-Illmancohomology with local coefficients, as introduced in[16], to derive a spectral sequence.
Keywords: Simplicial sets, Kan fibration, universal cover, local coefficients,group action, equivariant cohomology.
For spaces with group actions the analogue of cohomology with local coefficients is theBredon-Illman cohomology with local coefficients, as introduced in [13]. This is based on thenotion of fundamental groupoid of a space equipped with a group action. A classical theoremof Eilenberg states that cohomology with local coefficients of a space can be described by thecohomology of an invariant subcomplex of the cochain complex of its universal cover, wherethe universal cover is equipped with the action of the fundamental group of the base space.An equivariant analogue of this result was proved in [13]. Recently, in [16], we introducedequivariant simplicial cohomology with local coefficients, which is the simplicial version ofBredon-Illman cohomology with local coefficients and proved a classification theorem. Thecorresponding non-equivariant result was proved in [9], [7], [3]. In this paper we deriveEilenberg’s theorem for equivariant simplicial cohomology with local coefficients. This isbased on the notion of universal covering complexes of one vertex Kan complexes [8]. Inequivariant context, the role of the universal cover is played by a contravariant functor fromthe category of canonical orbits to the category of one vertex Kan complexes. Finally, wegive an alternative description of equivariant simplicial cohomology with local coefficientsvia the notion of cohomology of a small category following [15] and use it to derive a spectralsequence.The paper is organized as follows. In Section 2, we recall some standard results andfix notations. The notion of equivariant local coefficients of a simplicial set equipped witha simplicial group action is based on fundamental groupoid. In Section 3, we recall theseconcepts and the definition of simplicial version of Bredon-Illman cohomology with localcoeffic
The second author would like to thank CSIR for its support.MSC(2000) : 55U10, 55N91,55N25, 55T10, 57S30 Preliminaries on G -simplicial sets In this section we set up our notations and recall some standard facts [11].Let ∆ be the category whose objects are ordered sets[ n ] = { < < · · · < n } , n ≥ , and morphisms are non-decreasing maps f : [ n ] −→ [ m ] . There are some distinguishedmorphisms d i : [ n − −→ [ n ] , ≤ i ≤ n , called cofaces and s i : [ n + 1] −→ [ n ] , ≤ i ≤ n ,called codegeneracies, defined as follows: d i ( j ) = j, j < i and d i ( j ) = j + 1 , j ≥ i, ( n > , ≤ i ≤ n ); s i ( j ) = j, j ≤ i, and s i ( j ) = j − , j > i, ( n ≥ , ≤ i ≤ n ) . These maps satisfy the standard cosimplicial relations.A simplicial object X in a category C is a contravariant functor X : ∆ −→ C . Equivalently,a simplicial object is a sequence { X n } n ≥ of objects of C , together with C -morphisms ∂ i : X n −→ X n − and s i : X n −→ X n +1 , ≤ i ≤ n, verifying the following simplicial identities: ∂ i ∂ j = ∂ j − ∂ i , ∂ i s j = s j − ∂ i , if i < j,∂ j s j = id = ∂ j +1 s j ,∂ i s j = s j ∂ i − , i > j + 1; s i s j = s j +1 s i , i ≤ j. A simplicial map f : X −→ Y between two simplicial objects in a category C , is a collectionof C -morphisms f n : X n −→ Y n , n ≥ , commuting with ∂ i and s i .In particular, a simplicial set is a simplicial object in the category of sets. Throughout S will denote the category of simplicial sets and simplicial maps.For any n -simplex x ∈ X n , in a simplicial set X , we shall use the notation ∂ ( i ,i , ··· ,i r ) x to denote the simplex ∂ i ∂ i · · · ∂ i r x obtained by applying the successive face maps ∂ i r − k on x , where 0 ≤ i r − k ≤ n − k, ≤ k ≤ r − . Recall that the simplicial set ∆[ n ], n ≥
0, is defined as follows. The set of q -simplices is∆[ n ] q = { ( a , a , · · · , a q ); where a i ∈ Z , ≤ a ≤ a ≤ · · · ≤ a q ≤ n } . The face and degeneracy maps are defined by ∂ i ( a , · · · , a q ) = ( a , · · · , a i − , a i +1 , · · · , a q ) , s j ( a , · · · , a q ) = ( a , · · · , a i , a i , · · · , a q ) . Alternatively, the set of k -simplices can be viewed as the contravariant functor∆[ n ]([ k ]) = Hom ∆ ([ k ] , [ n ]) , the set of ∆-morphisms from [ k ] to [ n ]. The only non-degenerate n -simplex is id : [ n ] −→ [ n ]and is denoted by ∆ n . In the earlier notation, it is simply, ∆ n = (0 , , · · · , n ) . It is well known that if X is a simplicial set, then for any n -simplex x ∈ X n there is aunique simplicial map x : ∆[ n ] −→ X with x (∆ n ) = x . Often by an n -simplex in a simplicialset X we shall mean either an element x ∈ X n or the corresponding simplicial map x. We have simplicial maps δ i : ∆[ n − → ∆[ n ] and σ i : ∆[ n + 1] → ∆[ n ] for 0 ≤ i ≤ n defined by δ i (∆ n − ) = ∂ i (∆ n ) and σ i (∆ n +1 ) = s i (∆ n ). The boundary subcomplex ∂ ∆[ n ] of∆[ n ] is defined as the smallest subcomplex of ∆[ n ] containing the faces ∂ i ∆ n , i = 0 , , ..., n .2 efinition 2.1. Let G be a discrete group. A G -simplicial set is a simplicial object in thecategory of G -sets. More precisely, a G -simplicial set is a simplicial set { X n ; ∂ i , s i , ≤ i ≤ n } n ≥ such that each X n is a G -set and the face maps ∂ i : X n −→ X n − and the degeneracymaps s i : X n −→ X n +1 commute with the G -action. A map between G-simplicial sets is asimplicial map which commutes with the G-action. Definition 2.2. A G -simplicial set X is called G -connected if each fixed point simplicial set X H , H ⊆ G , is connected. Definition 2.3.
Two G -maps f, g : K → L between two G -simplicial sets are G -homotopicif there exists a G -map F : K × ∆[1] → L such that F ◦ ( id × δ ) = f, F ◦ ( id × δ ) = g. The map F is called a G -homotopy from f to g and we write F : f ≃ G g. If i : K ′ ⊆ K isan inclusion of subcomplex and f, g agree on K ′ then we say that f is G -homotopic to g relative to K ′ if there exists a G homotopy F : f ≃ G g such that F ◦ ( i × id ) = α ◦ pr , where α = f | ′ K = g | ′ K and pr : K ′ × ∆[1] −→ K ′ is the projection onto the first factor. In thiscase we write F : f ≃ G g ( rel K ′ ) . Definition 2.4. A G -simplicial set is a G -Kan complex if for every subgroup H ⊆ G thefixed point simplicial set X H is a Kan complex. Remark 2.5.
Recall ([1], [5]) that the category G S of G -simplicial sets and G -simplicialmaps between G -simplicial sets has a closed model structure [17], where the fibrant objectsare the G -Kan complexes and cofibrant objects are the G -simplicial sets. From this it followsthat G -homotopy on the set of G -simplicial maps K → L is an equivalence relation, forevery G -simplicial set K and G -Kan complex L . More generally, relative G -homotopy is anequivalence relation if the target is a G -Kan complex. We consider
G/H × ∆[ n ] as a simplicial set where ( G/H × ∆[ n ]) q = G/H × (∆[ n ]) q withface and degeneracy maps as id × ∂ i and id × s i . Note that the group G acts on G/H by lefttranslation. With this G -action on the first factor and trivial action on the second factor G/H × ∆[ n ] is a G -simplicial set.A G -simplicial map σ : G/H × ∆[ n ] → X is called an equivariant n -simplex of type H in X . Remark 2.6.
We remark that for a G -simplicial set X, the set of equivariant n -simplicesin X is in bijective correspondence with n-simplices of X H . For an equivariant n -simplex σ ,the corresponding n -simplex is σ ′ = σ ( eH, ∆ n ) . The simplicial map ∆[ n ] −→ X H , ∆ n σ ′ will be denoted by σ. We shall call σ degenerate or non-degenerate according as the n -simplex σ ′ ∈ X Hn isdegenerate or non-degenerate.Recall that the category of canonical orbits, denoted by O G , is a category whose objectsare cosets G/H , as H runs over the all subgroups of G . A morphism from G/H to G/K is a G -map. Recall that such a morphism determines and is determined by a subconjugacyrelation g − Hg ⊆ K and is given by ˆ g ( eH ) = gK . We denote this morphism by ˆ g [2].3 efinition 2.7. A contravariant functor from O G to S is called an O G -simplicial set. Amap between O G -simplicial sets is a natural transformation of functors. We shall denote the category of O G -simplicial sets by O G S . The notion of O G -groups or O G -abelian groups has the obvious meaning replacing S by G rp or A b. For a G -simplicial set X , with a G -fixed 0-simplex v , we have an O G -group πX definedas follows. For any subgroup H of G , πX ( G/H ) := π ( X H , v )and for a morphism ˆ g : G/H −→ G/K, g − Hg ⊆ K , πX (ˆ g ) is the homomorphism infundamental groups induced by the simplicial map g : X K −→ X H . Definition 2.8. An O G -group π is said to act on an O G -simplicial set (group or abeliangroup) X if for every subgroup H ⊆ G , π ( G/H ) acts on X ( G/H ) and this action is naturalwith respect to maps of O G . Thus if φ ( G/H ) : π ( G/H ) × X ( G/H ) −→ X ( G/H ) denotes the action of π ( G/H ) on X ( G/H ) then for each subconjugacy relation g − Hg ⊆ K, φ ( G/H ) ◦ ( π (ˆ g ) × X ( b g )) = X (ˆ g ) ◦ φ ( G/K ) . In this section we recall [16] the notion of fundamental groupoid of a G -simplicial set X , thenotion of equivariant local coefficients on X and the definition of simplicial Bredon-Illmancohomology with local coefficients.We begin with the notion of fundamental groupoid. Recall [7] that the fundamentalgroupoid πX of a Kan complex X is a category having as objects all 0-simplexes of X anda morphism x −→ y in πX is a homotopy class of 1-simplices ω : ∆[1] −→ X rel ∂ ∆[1] suchthat ω ◦ δ = y , ω ◦ δ = x . If ω represents an arrow from x to y and ω represents an arrowfrom y to z , then their composite [ ω ] ◦ [ ω ] is represented by Ω ◦ δ , where the simplicialmap Ω : ∆[2] −→ X corresponds to a 2-simplex, which is determined by the compatible pair( ω ′ , , ω ′ ). For a simplicial set X the notion of fundamental groupoid is defined via thegeometric realization and the total singular functor.Suppose x H and y K are equivariant 0-simplices of type H and K , respectively, andˆ g : G/H → G/K is a morphism in O G , given by a subconjugacy relation g − Hg ⊆ K , g ∈ G, so that ˆ g ( eH ) = gK . Moreover suppose that we have an equivariant 1-simplex φ : G/H × ∆[1] → X of type H such that φ ◦ ( id × δ ) = x H , φ ◦ ( id × δ ) = y K ◦ (ˆ g × id ) . Then, in particular, φ ′ is a 1-simplex in X H such that ∂ φ ′ = x ′ H and ∂ φ ′ = gy ′ K , notationsare as in Remark 2.6. Observe that the 0-simplex gy ′ K in X H corresponds to the composition G/H × ∆[0] ˆ g × id → G/K × ∆[0] y K −→ X φ is a G -homotopy x H ≃ G y K ◦ (ˆ g × id ). Definition 3.1.
Let X be a G -Kan complex. The fundamental groupoid Π X is a categorywith objects equivariant -simplices x H : G/H × ∆[0] → X of type H , as H varies over all subgroups of G . Given two objects x H and y K in Π X , amorphism from x H −→ y K is defined as follows. Consider the set of all pairs (ˆ g, φ ) where ˆ g : G/H → G/K is a morphism in O G , given by a subconjugacy relation g − Hg ⊆ K , g ∈ G so that ˆ g ( eH ) = gK and φ : G/H × ∆[1] → X is an equivariant -simplex such that φ ◦ ( id × δ ) = x H , φ ◦ ( id × δ ) = y K ◦ (ˆ g × id ) . The set of morphisms in Π X from x H to y K is a quotient of the set of pairs mentionedabove by an equivalence relation ‘ ∼ ‘ , where (ˆ g , φ ) ∼ (ˆ g , φ ) if and only if g = g = g (say) and there exists a G -homotopy Φ :
G/H × ∆[1] × ∆[1] −→ X of G -homotopies suchthat Φ : φ ≃ G φ (rel G/H × ∂ ∆[1] ). Since X is a G -Kan complex, by Remark 2.5, ∼ is an equivalence relation. We denote the equivalence class of (ˆ g, φ ) by [ˆ g, φ ] . The set ofequivalence classes is the set of morphisms in Π X from x H to y K .The composition of morphisms in Π X is defined as follows. Given two morphisms x H y K z L ✲ [ˆ g ,φ ] ✲ [ˆ g ,φ ] their composition [ˆ g , φ ] ◦ [ˆ g , φ ] is [ d g g , ψ ] : x H −→ z L , where the first factor is thecomposition G/H G/K G/L ✲ ˆ g ✲ ˆ g and ψ : G/H × ∆[1] −→ X is an equivariant -simplex of type H as described below. Let x be a -simplex in the Kan complex X H determined by the compatible pair of -simplices ( g φ ′ , , φ ′ ) so that ∂ x = g φ ′ and ∂ x = φ ′ . Then ψ is given by ψ ( eH, ∆ ) = ∂ x . It is proved in [16] that the composition is well defined.For a version of fundamental groupoid of a G -space, we refer [10] and [13].Observe that if X is a G -simplicial set then S | X | is a G -Kan complex, where for anyspace Y , SY denotes the total singular complex and for any simplicial set X , | X | denotesthe geometric realization of X . Definition 3.2.
For any G -simplicial set X , we define the fundamental groupoid Π X of X by Π X := Π S | X | . Remark 3.3.
If G is trivial then Π X reduces to fundamental groupoid πX of a simplicial setX. Again, for a fixed H, the objects x H together with the morphisms x H → y H with identityin the first factor, constitute a subcategory of Π X which is precisely the fundamental groupoid πX H of X H . Moreover, a morphism [ˆ g, φ ] from x H to y K , corresponds to the morphism [ φ ] in the fundamental groupoid πX H of X H from x ′ H to ay ′ K , where φ is as in 2.6. Suppose ξ is a morphism in πX H from x to y given by a homotopy class [ ω ] , where ω : ∆[1] −→ X H represents the -simplex in X H from x to y . Let x H and y H be the objects in πX H definedrespectively by x H ( eH, ∆ ) = x, y H ( eH, ∆ ) = y. hen we have a morphism [ id, ω ] : x H −→ y H in Π X , where ω ( eH, ∆ ) = ω (∆ ) . We shalldenote this morphism corresponding to ξ by bξ. Definition 3.4.
An equivariant local coefficients on a G -simplicial set X is a contravariantfunctor from Π X to the category A b of abelian groups. Next, we briefly describe the simplicial version of Bredon-Illman cohomology with localcoefficients as introduced in [16].Let X be a G -simplicial set and M an equivariant local coefficients on X . For eachequivariant n -simplex σ : G/H × ∆[ n ] → X, we associate an equivariant 0-simplex σ H : G/H × ∆[0] → X given by σ H = σ ◦ ( id × δ (1 , , ··· ,n ) ) , where δ (1 , , ··· ,n ) is the composition δ (1 , , ··· ,n ) : ∆[0] δ → ∆[1] δ → · · · δ n → ∆[ n ] . The j -th face of σ is an equivariant ( n − H , denoted by σ ( j ) , and is definedby σ ( j ) = σ ◦ ( id × δ j ) , ≤ j ≤ n. Remark 3.5.
Note that σ ( j ) H = σ H for j > , and σ (0) H = σ ◦ ( id × δ (0 , , ··· ,n ) ) . Let C nG ( X ; M ) be the group of all functions f defined on equivariant n -simplexes σ : G/H × ∆[ n ] → X such that f ( σ ) ∈ M ( σ H ) with f ( σ ) = 0 , if σ is degenerate. We have amorphism σ ∗ = [ id, α ] in Π X from σ H to σ (0) H induced by σ , where α : G/H × ∆[1] −→ X is given by α = σ ◦ ( id × δ (2 , ··· ,n ) ) . Define a homomorphism δ : C nG ( X ; M ) → C n +1 G ( X ; M ) f δf where for any equivariant ( n + 1)-simplex σ of type H ,( δf )( σ ) = M ( σ ∗ )( f ( σ (0) )) + Σ n +1 j =1 ( − j f ( σ ( j ) ) . A routine verification shows that δ ◦ δ = 0 . Thus { C ∗ G ( X ; M ) , δ } is a cochain complex. Weare interested in a subcomplex of this cochain complex as defined below.Let η : G/H × ∆[ n ] → X and τ : G/K × ∆[ n ] → X be two equivariant n -simplexes.Suppose there exists a G -map ˆ g : G/H −→ G/K, g − Hg ⊆ K, such that τ ◦ (ˆ g × id ) = η. Then η and τ are said to be compatible under ˆ g . Observe that if η and τ are compatibleas described above then η is degenerate if and only if τ is degenerate. Moreover notice thatin this case, we have a morphism [ˆ g, k ] : η H → τ K in Π X , where k = η H ◦ ( id × σ ) , where σ : ∆[1] −→ ∆[0] is the map as described in Section 2. Let us denote this induced morphismby g ∗ . 6 efinition 3.6. We define S nG ( X ; M ) to be the subgroup of C nG ( X ; M ) consisting of allfunctions f such that if η and τ are equivariant n-simplexes in X which are compatible under ˆ g then f ( η ) = M ( g ∗ )( f ( τ )) . If f ∈ S nG ( X ; M ) then one can verify that δf ∈ S n +1 G ( X ; M ) . Thus we have a cochain complex S G ( X ; M ) = { S nG ( X ; M ) , δ } . Definition 3.7.
Let X be a G-simplicial set with equivariant local coefficients M on it. Thenthe n -th Bredon-Illman cohomology of X with local coefficients M is defined by H nG ( X ; M ) = H n ( S G ( X ; M )) . Let X be a one vertex Kan complex. For any x ∈ X , we denote by [x] the element of π = π ( X, v ) represented by the 1-simplex x where v is the unique vertex of X . Recall that([6], [8]) the universal covering complex e X of X is defined as follows: e X n = π × X n with the face maps ∂ i ( γ, x ) = ( γ, ∂ i x ) , < i ≤ n, x ∈ X n , γ ∈ π∂ ( γ, x ) = ([ ∂ (2 , , ··· ,n ) x ] γ, ∂ x ) , where ∂ (2 , , ··· ,n ) x = ∂ ∂ · · · ∂ n x . The degeneracy maps are s i ( γ, x ) = ( γ, s i x ) 0 ≤ i ≤ n. Then p : e X −→ X, p being the first projection, has the usual properties of universal covering.Any map f : X −→ Y of such complexes induces a map ˜ f : e X −→ e Y by ˜ f ( γ, x ) =( f ∗ ( γ ) , f ( x )) , where f ∗ : π ( X ) −→ π ( Y ) is the homomorphism of fundamental groupsinduced by f . Remark 4.1.
We note that given any two -simplexes x = ( γ , v ) and x = ( γ , v ) in e X, there is a unique homotopy class of -simplexes ω such that ∂ ω = x , ∂ ω = x , as e X issimply connected. We may represent this class by ω = ( γ , ω ω − ) where ω i represents γ i , i = 1 , . The fundamental group π ( X ) operates on e X freely by( σ, ( γ, x )) ( γσ − , x ) . This action is natural with respect to maps of complexes and an analogue of EilenbergTheorem holds. The purpose of this section is to prove an equivariant version of this result.We define a contravariant functor from the category of canonical orbits to the category ofone vertex Kan complexes as follows. 7et X be a one vertex G -Kan complex. We denote the G -fixed vertex by v . For anysubgroup H of G , let p H : g X H → X H be the universal cover of X H . The left translation a : X K → X H , corresponding to aG-map ˆ a : G/H → G/K , a − Ha ⊆ K, induces a simplicial map ˜ a : g X K → g X H suchthat p H ◦ ˜ a = a ◦ p K . This defines an O G -Kan complex e X by setting e X ( G/H ) = g X H and e X (ˆ a ) = ˜ a . This is called the universal O G -covering complex of X . This is simplicial analogueof O G -covering space as introduced in [13]. We refer [10] for a more general version, called’universal covering functor’. For any subgroup H , let ˜ v H ∈ g X H denote the zero simplex( e H , v ) , where e H is the identity element of πX ( G/H ) = π ( X H , v ) . Note that the map ˜ a induced by a : X K −→ X H maps ˜ v K to ˜ v H . The natural actions of πX ( G/H ) = π ( X H , v ) on e X ( G/H ) = g X H as H varies oversubgroups of G , define an action of the O G -group πX on e X .Suppose M is an equivariant local coefficients on X . We have an abelian O G -group M associated to M as described below.For any subgroup H of G , let v H be the object of type H in Π X defined by v H : G/H × ∆[0] → X, ( eH, ∆ ) v. Then for any morphism b g : G/H → G/K in O G , given by a subconjugacy relation g − Hg ⊆ K , we have a morphism [ b g, k ] : v H → v K Π X , where k : G/H × ∆[1] −→ X is given by k ( eH, ∆ ) = s v . Define M : O G → Ab by M ( G/H ) = M ( v H ) and M ( b g ) = M [ b g, k ].The abelian O G -group M comes equipped with a natural action of the O G group πX asdescribed below.Let α = [ φ ′ ] ∈ πX ( G/H ) = π ( X H , v ). Then the morphism [ id, φ ] : v H → v H where φ ( eH, ∆ ) = φ ′ (∆ ) , is an equivalence in the category Π X . This yields a group homomor-phism b : π ( X H , v ) → Aut Π X ( v H ) , α = [ φ ′ ] b ( α ) = [ id, φ ] . We remark that the composition of the fundamental group π ( X H , v ) coincides with themorphism composition of Π X , contrary to the usual topological composition in the funda-mental group. The composition of the map b with the group homomorphism Aut Π X ( v H ) → Aut A b ( M ( v H )) which sends α ∈ Aut Π X ( v H ) to [ M ( α )] − defines the action of π ( X H , v ) on M ( G/H ). It is routine to check that this action is natural with respect to morphism of O G .We define cohomology groups of e X with coefficients in M using invariance of actions of πX on e X and M as follows.We have a chain complex n C n ( e X ) , ∂ n o in the abelian category C G defined by C n ( e X )( G/H ) := C n ( e X H ; Z )for every object G/H and for every morphism ˆ a : G/H −→ G/K , a − Ha ⊆ K, C n (ˆ a ) := ˜ a : C n ( e X K ; Z ) → C n ( e X H ; Z ), where C n ( e X H ; Z ) denotes the free abelian group generated bythe non-degenerate n -simplexes of e X H . The boundary map ∂ n : C n ( e X ) → C n − ( e X ) is the8atural transformation, defined by ∂ n ( G/H ) := ∂ n where ∂ n : C n ( e X H ; Z ) → C n − ( e X H ; Z )is the ordinary boundary map. Note that πX acts on the chain complex { C n ( e X ) , ∂ n } viaits action on e X . We now form the cochain complex n Hom πX ( C n ( e X ) , M ) , δ n o , where Hom πX ( C n ( e X ) , M ) consists of all natural transformations C n ( e X ) −→ M respecting theactions of πX and δ n f = f ◦ ∂ n +1 . Then the n th equivariant cohomology of e X with coefficientsin M is defined by H nπX,G ( e X ; M ) := H n ( Hom πX ( C ( e X ) , M )) . Theorem 4.2.
Let X be a one vertex G-Kan complex and M be an equivariant local coeffi-cients on X. Then H nG ( X ; M ) ∼ = H nπX,G ( e X ; M ) . Proof.
Recall that for any two 0-simplexes x, y ∈ g X H of the universal cover of the H -fixed point complex X H , there is a unique homotopy class of 1-simplexes ω with ∂ ω = x and ∂ ω = y . Let us denote this class by e ξ H ( x, y ) . In particular, if x = ˜ v H , then weshall write e ξ (˜ v H , y ) simply by e ξ H ( y ) . Upon projecting e ξ H ( x, y ) via p H we get an element ξ H ( x, y ) ∈ π ( X H , v ) . By Remark 3.3, ξ H ( x, y ) corresponds to an automorphism bξ H ( x, y )of v H in Π X. As before ξ H ( x, y ) will be denoted by ξ H ( y ) when x = ˜ v H . Define a map φ : S nG ( X ; M ) −→ Hom πX ( C n ( e X ) , M )as follows. Let f ∈ S nG ( X ; M ) and y be a non-degenerate n -simplex in g X H . Let σ be theequivariant n -simplex of type H in X such that σ ′ = p H ◦ y, where y : ∆[ n ] −→ g X H is thesimplicial map with y (∆ n ) = y. Then φ ( f ) ∈ Hom πX ( C n ( e X ) , M ) is given by φ ( f )( G/H )( y ) = M ( bξ H ( ∂ (1 , , ··· ,n ) y )) f ( σ ) . Recall that f ( σ ) ∈ M ( σ H ) and σ H in this case coincides with v H . We check that φ ( f )( G/H ) is equivariant with respect to the respective actions of πX ( G/H )on C n ( e X )( G/H ) and on M ( G/H ) . Let u ∈ πX ( G/H ) , y ∈ g X Hn and σ be the equivariant n -simplex determined by y as above. Then φ ( f )( G/H )( uy ) = M ( bξ H ( ∂ (1 , , ··· ,n ) uy )) f ( τ ) , where τ ′ = p H ◦ uy. By the definition of the action of πX ( G/H ) on C n ( g X H ; Z ), we have p H ◦ uy = p H ◦ y, hence τ ′ = σ ′ . It follows that φ ( f )( G/H )( uy ) = M ( bξ H ( ∂ (1 , , ··· ,n ) uy )) f ( σ ) . Now write e ξ H ( ∂ (1 , , ··· ,n ) uy ) as a composition e ξ H ( u ˜ v H , ∂ (1 , , ··· ,n ) uy ) ◦ e ξ H ( u ˜ v H )of morphisms in the fundamental groupoid of g X H . Observe that by Remark 4.1, ξ H ( u ˜ v H ) = u − and ξ H ( u ˜ v H , ∂ (1 , , ··· ,n ) uy ) = ξ H ( ∂ (1 , , ··· ,n ) y ) . Therefore M ( bξ H ( ∂ (1 , , ··· ,n ) uy )) = M ( bu ) − ◦ M ( bξ H ( ∂ (1 , , ··· ,n ) y )) . φ ( f )( G/H )( uy ) = M ( bu ) − φ ( f )( G/H )( y ) . It follows from the definition of the actionof πX ( G/H ) on M ( G/H ) that φ ( f )( G/H ) is equivariant.To check that φ ( f )( G/H ) : C n ( e X ) −→ M is natural, we have to show that M (ˆ g ) ◦ φ ( f )( G/K ) = φ ( f )( G/H ) ◦ ˜ g whenever g − Hg ⊆ K . Recall that by definition of M , M (ˆ g ) = M ( v H [ˆ g,k ] −−−→ v K ), where k : G/H × ∆[1] −→ X is given by k ( eH, ∆ ) = s v. Let y ∈ g X Kn and g − Hg ⊆ K. Let τ bean equivariant n -simplex of type K in X such that τ ′ = p K ◦ y . Then M (ˆ g ) ◦ φ ( f )( G/K )( y )= M ( v H [ˆ g,k ] −−−→ v K ) ◦ M ( bξ K ( ∂ (1 , , ··· ,n ) y )) f ( τ )= M ( v H [ˆ g,k ] −−−→ v K ) ◦ M ([ id G/K , ω ]) f ( τ )= M ([ id G/H , ω ] ◦ [ˆ g, k ]) f ( τ ) , where as in Remark 3.3, ω is the equivariant 1-simplex of type K in X such that ω ′ represents ξ K ( ∂ (1 , , ··· ,n ) y ) . On the other hand, φ ( f )( G/H ) ◦ ˜ g ( y )= φ ( f )( G/H )(˜ gy )= M ( bξ H ( ∂ (1 , , ··· ,n ) ˜ gy )) f ( σ )where σ ′ = p H ◦ ˜ gy = p H ◦ ˜ g ◦ y = g ◦ p K ◦ y = g ◦ τ ′ . In particular, σ and τ are compatible n -simplexes. Thus φ ( f )( G/H ) ◦ ˜ g ( y ) = M ( bξ H ( ∂ (1 , , ··· ,n ) ˜ gy )) ◦ M ( g ∗ ) f ( τ ) . Note that v is the only vertex in X which is G -fixed and hence g ∗ is a morphism from v H to v K and is given by [ˆ g, k ] where k = v H ◦ ( id G/H × σ ). Now observe that ξ H ( ∂ (1 , , ··· ,n ) ˜ gy ) = ξ H (˜ g∂ (1 , , ··· ,n ) y ) can be represented by gω ′ . As a consequence we may write bξ H ( ∂ (1 , , ··· ,n ) ˜ gy ) = [ id G/H , ω ◦ (ˆ g × id ∆[1] )] . Therefore φ ( f )( G/H ) ◦ ˜ g ( y )= M ([ id G/H , ω ◦ (ˆ g × id ∆[1] )]) ◦ M ([ˆ g, k ]) f ( τ )= M ([ˆ g, k ] ◦ [ id G/H , ω ◦ (ˆ g × id ∆[1] )]) f ( τ ) . From the definition of composition of morphism in Π X , we have[ id G/K , ω ] ◦ [ˆ g, k ] = [ˆ g, k ] ◦ [ id G/H , ω ◦ (ˆ g × id ∆[1] )] . Thus φ ( f ) is natural. 10ext we check that φ is a chain map. Let f ∈ S nG ( X ; M ) , y ∈ ^ X Hn +1 . Let σ denotes theequivariant ( n + 1)-simplex of type H corresponding to y as described before. Then φ ( δf )( G/H )( y )= M ( bξ H ( ∂ (1 , , ··· ,n +1) y ))( δf )( σ )= M ( bξ H ( ∂ (1 , , ··· ,n +1) y )) n M ( σ ∗ ) f ( σ (0) ) + Σ n +1 j =1 ( − j f ( σ ( j ) ) o . On the other hand, δφ ( f )( G/H )( y )=Σ ( n +1) i =0 ( − i φ ( f )( G/H )( ∂ i y )=Σ ( n +1) i =0 ( − i M ( bξ H ( ∂ (1 , , ··· ,n ) ∂ i y )) f ( σ ( i ) )= M ( bξ H ( ∂ (0 , , ··· ,n +1) y )) f ( σ (0) ) + Σ n +1 i =1 ( − i M ( bξ H ( ∂ (1 , , ··· ,n +1) y )) f ( σ ( i ) ) . Now note that since g X H is simply connected the morphism ξ H ( ∂ (0 , , ··· ,n +1) y ) in πX H canbe factored as ξ H ( ∂ (1 , , ··· ,n +1) y, ∂ (0 , , ··· ,n +1) y ) ◦ ξ H ( ∂ (1 , , ··· ,n +1) y )and bξ H ( ∂ (1 , , ··· ,n +1) y, ∂ (0 , , ··· ,n +1) y ) is precisely the morphism σ ∗ . Therefore bξ H ( ∂ (0 , , ··· ,n +1) y ) = σ ∗ ◦ bξ H ( ∂ (1 , , ··· ,n +1) y ) . Hence φ ( δf ) = δφ ( f ) . To show that φ is a chain isomorphism define a map ψ : Hom πX ( C n ( ˜ X ) , M ) → C nG ( X ; M )as follows. Let f ∈ Hom πX ( C n ( ˜ X ) , M ) and σ be a non-degenerate equivariant n -simplexin X of type H. Choose an n -simplex y in g X H such that p H ( y ) = σ ( eH, ∆ n ) . Then ψ ( f ) isgiven by ψ ( f )( σ ) = M ( bξ H ( ∂ (1 , , ··· ,n ) y )) − f ( G/H )( y ) . Suppose z is another n -simplex in g X H such that p H ( z ) = σ ( eH, ∆ n ) . Since π ( X H , v )acts transitively on each fibre of p H : g X H −→ X H , there exists an element u ∈ π ( X H , v )such that uy = z and hence u∂ (1 , , ··· ,n ) y = ∂ (1 , , ··· ,n ) z. Thus M ( bξ H ( ∂ (1 , , ··· ,n ) z )) − f ( G/H )( z )= M ( bξ H ( ∂ (1 , , ··· ,n ) uy )) − f ( G/H )( uy )= M ( bξ H ( ∂ (1 , , ··· ,n ) y )) − M ( bu ) − f ( G/H )( y )= M ( bξ H ( ∂ (1 , , ··· ,n ) y )) − f ( G/H )( y ) . The last equality follows from the observation M ( bξ H ( ∂ (1 , , ··· ,n ) uy )) = M ( bu ) − ◦ M ( bξ H ( ∂ (1 , , ··· ,n ) y )) , φ takes any cocycle in S nG ( X ; M ) into Hom πX ( C n ( ˜ X ) , M ) , in the first partof the proof. Thus the map ψ is well defined.We claim that ψ ( f ) ∈ S nG ( X ; M ) for any f ∈ Hom πX ( C n ( ˜ X ) , M ) . Let a − Ha ⊆ K and σ : G/H × ∆[ n ] −→ X and η : G/K × ∆[ n ] −→ X be equivariant n -simplexes such that η ◦ (ˆ a × id ) = σ so that they are compatible. We need to show that ψ ( f )( σ ) = M ( a ∗ ) ψ ( f )( η ) . Let y ∈ g X K be such that p K ( y ) = η ( eK, ∆ n ) . Then the n -simplex ˜ ay ∈ g X Hn is such that p H (˜ ay ) = ap K ( y ) = aη ( eK, ∆ n ) = η ( aK, ∆ n ) = σ ( eH, ∆ n ) . By our choice, we have ψ ( f )( η ) = M ( bξ K ( ∂ (1 , , ··· ,n ) y )) − f ( G/K )( y )and ψ ( f )( σ ) = M ( bξ H ( ∂ (1 , , ··· ,n ) ˜ ay )) − f ( G/H )(˜ ay ) . Since f : C n ( ˜ X ) −→ M is natural, we have f ( G/H )(˜ ay ) = M (ˆ a )( f ( G/K )( y )) . In the first part of the proof we have observed that a ∗ ◦ bξ H ( ∂ (1 , , ··· ,n ) ˜ ay ) = bξ K ( ∂ (1 , , ··· ,n ) y ) ◦ a ∗ . Moreover, recall that M (ˆ a ) = M ( a ∗ ) . Therefore M ( a ∗ ) ψ ( f )( η )= M ( a ∗ ) M ( bξ K ( ∂ (1 , , ··· ,n ) y )) − f ( G/K )( y )= M ( bξ K ( ∂ (1 , , ··· ,n ) y ) − ◦ a ∗ ) f ( G/K )( y )= M ( a ∗ ◦ bξ H ( ∂ (1 , , ··· ,n ) ˜ ay )) f ( G/K )( y )= M ( bξ H ( ∂ (1 , , ··· ,n ) ˜ ay )) − M ( a ∗ ) f ( G/K )( y )= M ( bξ H ( ∂ (1 , , ··· ,n ) ˜ ay )) − M (ˆ a ) f ( G/K )( y )= ψ ( f )( σ ) . It is routine to check that ψ is the inverse of φ . This completes the proof of the the theorem. The aim of this last section is to derive a version of Serre spectral sequence. To do this wegive an alternative description of equivariant simplicial cohomology with local coefiicients interms of cohomology of small categories.Let G be a discrete group and X a G -Kan complex. Then we have a category ∆ G ( X )described as follows. Its objects are G -simplicial maps σ : G/H × ∆[ n ] → X and a morphismfrom σ : G/H × ∆[ n ] → X to τ : G/K × ∆[ m ] → X is a pair (ˆ g, α ) where ˆ g : G/H → G/K is a G -map and α : ∆[ n ] → ∆[ m ] is a simplicial map such that τ ◦ (ˆ g × α ) = σ. There is a canonical12unctor v X : ∆ G ( X ) → Π( X ) which sends σ : G/H × ∆[ n ] → X to σ H = σ ◦ ( id × δ (1 , , ··· ,n ) ) . For a morphism (ˆ g, α ) in ∆ G ( X ), v X (ˆ g, α ) : σ H → τ K is the morphism [ˆ g, φ ] in Π( X ) where φ : G/H × ∆[1] → X is an equivariant 1-simplex of type H obtained as follows. Suppose τ ◦ ( id × δ (1 , ··· , d α (0) ··· ,m ) )( eK, ∆ ) = ω ∈ X K . Let x be a 2-simplex in X K determined by thecompatible pair of 1-simplices ( , s ∂ ω, ω ) . Then φ is given by φ ( eH, ∆ ) = g ( ∂ x ) . If X is any G -simplicial set then we define ∆ G ( X ) = ∆ G ( S | X | ) . For a small category C , let A b ( C ) be the category of all contravariant functors from C to A b with morphisms natural transformations of functors. Definition 5.1.
A functor M ∈ A b (∆ G ( X )) is said to be G -local if M = v ∗ X M ′ = M ′ ◦ v X for some M ′ ∈ A b (Π( X )) . For a G -local coefficients M , the equivariant cohomology of X with coefficients M is defined to be h ∗ G ( X ; M ) := H ∗ (∆ G ( X ); M ) , where the right hand side denotes the cohomology of the category ∆ G ( X ) , in the sense of[17]. Theorem 5.2.
Let X be a G -simplicial set and M be an equivariant local coefficients on X .Then there is an isomorphism H ∗ G ( X ; M ) ∼ = h ∗ G ( X ; M ) . (On the right we identify M with v ∗ X ( M ) ).Proof. Let N C denote the nerve of a small category C . Then as in ([12]) we let ˜ X be thebisimplicial set whose ( p, q ) simplices are triples ( u, α, σ ) where u = ([ n ] u −→ [ n ] → · · · u p −→ [ n p ]) ∈ N p (∆) α = ( G/H α −→ G/H → · · · α q −→ G/H q ) ∈ N q ( O G ) σ : G/H q × ∆[ n p ] → X is a G -simplicial map.The face and degeneracy maps on ˜ X are induced from those on N (∆) and N ( O G ). Thendiagonal( ˜ X ) ∼ = N (∆ G ( X )) . To every ( u, α, σ ) ∈ ˜ X p,q associate a G -simplicial map, σ = σ ◦ ( α q ◦ · · · ◦ α × u p ◦ · · · ◦ u ) : G/H × ∆[ n ] → X. Let C p,q ( X ; M ) denote the set of all functions on ˜ X p,q which sends an element ( u, α, σ ) of˜ X p,q to an element of M ( v X ( σ )). It follows quite easily that C p,q ( X ; M ) is a bicomplex withobvious differentials d h and d v induced from the face maps of ˜ X . Denote the total complexof C •• ( X, M ) by Tot C •• ( X ; M ). Let diag C •• ( X ; M ) be the cochain complex whose p th group is C p,p ( X ; M ) and differential is d h d v . Then by a result of Dold and Puppe ([4]) wehave H n (Tot C •• ( X ; M )) ∼ = H n (diag C •• ( X ; M )) . C p,p ( X ; M ) can be interpreted as the set of all functions on N (∆ G ( X )) which sendsa p -simplex ( τ → τ → · · · → τ p ) to an element of M ( v X ( τ )) and the differential on C p,p ( X ; M ) is just the differential induced from the face maps of N p (∆ G ( X )). Hence, H n (diag C •• ( X ; M )) ∼ = H n (∆ G ( X ); v ∗ X M ) = h nG ( X ; M ) . Recall that the spectral sequence associated to the p -filtration of the bicomplex C •• ( X ; M )converges to the cohomology of the total complex. Now proceeding as in [15], we may com-pute the E term of the spectral sequence. It turns out that E p,q = H q ( C p, • ( X ; M )) ∼ = Π u ∈ N p (∆) S n ( u ) G ( X ; M ) if q = 0= 0 if q > , where S n ( u ) G ( X ; M ) is a copy of S n p G ( X ; M ) for every u = ([ n ] → · · · → [ n p ]) . (See [15] fordetails.) Thus, H p (Tot C •• ( X ; M )) ∼ = H p (Π u ∈ N (∆) S n ( u ) G ( X ; M )) ∼ = H p (∆ op , S • G ( X ; M ))where S • G ( X ; M ) is the cosimplicial group which takes [ n ] to S nG ( X ; M ) with obvious faceand degeneracy maps induced from those on ∆. Then we know that ([12]), H p (∆ op ; S • G ( X ; M )) ∼ = H p ( S • G ( X ; M )) . Hence, H p (Tot C •• ( X ; M )) ∼ = H pG ( X ; M ) . We are now in a position to derive the required spectral sequence.Let
X, Y be G -simplicial sets and f : Y → X be a G -Kan fibration and M a G -localcoefficients on Y . For q ≥ h qG ( f, M ) : ∆ G ( X ) → A b as follows. For an object σ : G/H × ∆[ n ] → X of ∆ G ( X ), let σ ∗ ( Y ) be the G -simplicial setobtained by pulling back f along σ and define h qG ( f, M )( σ ) := h qG ( σ ∗ ( Y ); ˜ σ ∗ M ) , where ˜ σ : σ ∗ ( Y ) → Y is the canonical map. We claim that h qG ( f, M ) factors through v X yielding a G -local coefficients on X . To see this, first note that the following result holds([12], the proof of the Theorem 2.3). Theorem 5.3.
Let f : Y −→ X be a weak equivalence in G S . Then for any G -localcoefficients M on X , f induces an isomorphism h ∗ G ( X ; M ) ∼ = h ∗ G ( Y ; f ∗ M ) . f : Y −→ X is a weak equivalence in G S if and only if f H : Y H −→ X H isa weak equivalence in S . Next note that if u : ∆[ m ] −→ ∆[ n ] is a simplicial map then themap σ ( id × u )) ∗ ( Y ) −→ σ ∗ ( Y ) covering id × u : G/K × ∆[ m ] −→ G/K × ∆[ n ] is a weakequivalence in G S , as f is a G -Kan fibration. The claim now follows from the above result. Theorem 5.4.
For any G -Kan fibration f : Y → X and a G -local coefficients M on Y ,there is a natural spectral sequence with E -term E p,q = H pG ( X ; h pG ( f, M )) converging to H p + qG ( Y ; M ) . Proof.
The proof is parallel to the proof of Theorem 3.2, [12]. We only mention the essentialsteps. The G -Kan fibration f : Y → X induces a functor ∆ G ( f ) : ∆ G ( Y ) → ∆ G ( X ) and wehave a Grothendieck spectral sequence [18] H p (∆ G ( X ); h q (∆ G ( f ) / − ; M )) ⇒ H p + q (∆ G ( Y ); M ) . It is enough to show that the two contravariant functors h q (∆ G ( Y ) / − ; M ) and h qG ( f, M ) from∆ G ( X ) to A b are equivalent. For an object σ of ∆ G ( X ) , ∆ G ( f ) /σ is the comma category.Objects of ∆ G ( f ) /σ are pairs ( τ, u ) where τ ∈ Ob (∆ G ( Y )) and u : ∆ G ( f )( τ ) → σ is a mapin ∆ G ( X ) . Morphisms from ( τ, u ) to ( τ ′ , u ′ ) are maps α : τ → τ ′ such that u ′ ∆ G ( f )( α ) = u. A direct computation shows that there is a canonical equivalence of the categories∆ G ( f ) /σ ∼ = ∆ G ( σ ∗ ( Y )) , which is natural in σ. Hence we have natural isomorphism of functors h q (∆ G ( σ ∗ ( Y ); ˜ σ ∗ M )) ∼ = h q (∆ G ( f ) /σ ; M ) . The result now follows from Theorem 5.2.
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Goutam Mukherjee
Indian Statistical Institute, Kolkata-700108, India.e-mail: [email protected]