Description of G-bundles over G-spaces with quasi-free proper action of discrete group II
aa r X i v : . [ m a t h . K T ] D ec Description of G -bundles over G -spaces withquasi-free proper action of discrete group II Morales Mel´endez ∗ , QuitzehNovember 18, 2018 This problem naturally arises from the Connor-Floyd’s description of the bor-disms with the action of a group G using the so-called fix-point constructionThis construction reduces the problem of describing the bordisms to two simplerproblems: a) description of the fixed-point set (or, more generally, the station-ary point set), which happens to be a submanifold attached with the structureof its normal bundle and the action of the same group G , however, this actioncould have stationary points of lower rank; b) description of the bordisms oflower rank with an action of the group G . We assume that the group G isdiscrete.Lets ξ be an G -equivariant vector bundle with base M . ξ y M Where the action of the group G is quasi-free over the base with normal sta-tionary subgroup H < G and there is no more fixed points of the action of thegroup H in the total space of the bundle ξ .According [1, p.1] the bundle ξ separates as the sum of its G -subbundles: ξ ≈ M k ξ k where the index runs over all (unitary) irreducible representations ρ k : H −→ U ( V k )of the group H and, as a H -bundle ξ k can be presented as the tensor product: ξ ≈ M k η k O V k , ∗ Partially supported by the grant 299388 of the mexican National Council for Science andTechnology (CONACyT) H over the bundles η k is trivial, V k denotes thetrivial bundle with fiber V k and with fiberwise action of the group H , definedusing the linear representation ρ k .The particular case ξ = η k N V k was described in the previous article [1].According [1, p.14] the bundle ξ k can be obtained as the inverse image of amapping f k : M/G −→ B Aut G ( X k )where G = G/H, X k = G × ( F k ⊗ V k ) is the canonical model and Aut G ( X k )is the group of equivariant automorphisms of the space X k as a vector G -bundleover the base G . So, the bundle ξ can be given by a mapping f : M/G −→ Y k B Aut G ( X k ) . Consider the vector bundle over the discrete base G X ρ = G × M k ( F k ⊗ V k ) ! . (1)Define a fiberwise action G × X ρ → X ρ by the formula φ ([ g ] , g ) : [ g ] × M k ( F k ⊗ V k ) ! → [ g g ] × M k ( F k ⊗ V k ) ! φ ([ g ] , g ) = L k (cid:0) Id ⊗ ρ k ( u ( g g ) u − ( g )) (cid:1) = ρ ( u ( g g ) u − ( g )) . (2) Definition 1
The bundle X ρ −→ G with the just defined action is called thecanonical model for the representation ρ . By Aut G ( X ρ ) we denote the group of equivariant automorphisms of thecanonical model X ρ as a vector G -bundle over the base G with fiber L k ( F k ⊗ V k )and canonical action of the group G . Lemma 1
There exists a monomorphism i : Aut G ( X ρ ) −→ Y k Aut G ( X k ) Proof.
As before, an element of the group Aut G ( X ρ ) is an equivariant mapping A a such that the pair ( A a , a ) defines a commutative diagram X ρ A a −→ X ρ y y G a −→ G , ∈ Aut G ( G ) ≈ G , a [ g ] = [ ga ] , [ g ] ∈ G . By the lemma 1 [1] applied to group of automorphisms Aut G ( X ρ ), for A a ∈ Aut G ( X ρ ), we have A a | X k : X k −→ X k . Note that this restriction is G -equivariant.Define i : Aut G ( X ρ ) −→ Y k Aut G ( X k )by the formula i ( A a ) = ( A a | X k ) k . This is clearly a homomorphism: it is a product of restrictions over invariantsubspaces. Lets prove that it is injective. If A a | X k = Id X k , then A a = Id X because X = ⊕ k X k .In order to prove that the image of i is closed, note that it coincides withthose automorphisms which commute with the inclusion∆ × Id : X ρ −→ Y k X k i.e. if an element ( A a k k ) k ∈ Q k Aut G ( X k ) leaves the image of X ρ invariant, thenits restriction defines an element in Aut G ( X ρ ) and the diagram X ρ A a −→ X ρ y yQ k X k ( A akk ) k −→ Q k X k commutes, i.e. a k = a ∀ k and A a | X k = A ak . In other words Y k pr k ◦ i ( A a ) = ∆( a )where pr k : Aut G ( X k ) −→ G is the epimorphism of lemma 2 [1, p. 9] and∆ : G ֒ → Y k G . Then i (Aut G ( X ρ )) = Y k pr − k ∆( G ) . orollary 1 It takes place an exact sequence of groups → Y k GL ( F k ) ϕ −→ Aut G ( X ρ ) pr −→ G → Proof.
Define pr = ∆ − Q k pr k ◦ i . This is an epimorphism: let A ak ∈ pr k ( a ),then ( A ak ) k ∈ i (Aut G ( X ρ )).We define the monomorphism ϕ : Y k GL ( F k ) −→ Aut G ( X ρ )using the monomorphisms ϕ k : GL ( F k ) −→ Aut G ( X k )by the formula ϕ = i − Y k ϕ k . Then pr ◦ ϕ = 1 and if a = 1, then Q k pr k ◦ i ( A ) = ∆(1). This means that, forevery k , there is a B ∈ GL ( F k ) such that ϕ k ( B k ) = A | X k , i.e. ϕ ( B k ) k = A .Denote by Vect G ( M, ρ ) the category of G -equivariant vector bundles ξ overthe base M with quasi-free action of the group G over the base and normalstationary subgroup H < G .Then, by lemma 1 and the observations on p. 13 in [1] in terms of homotopywe have Vect G ( M, ρ ) ≈ Y k [ M, BU ( F k )] (3)Denote by Bundle( X, L ) the category of principal L -bundles over the base X . Theorem 1
There exists an inclusion
Vect G ( M, ρ ) −→ Bundle(
M/G , Aut G ( X ρ )) . (4) Proof.
We already have a monomorphismVect G ( M, ρ ) −→ Bundle(
M/G , Y k Aut G ( X k ))i.e. Y k Vect G ( M, ρ k ) −→ Y k Bundle(
M/G , Aut G ( X k ))see [1, p. 13]A bundle ξ ∈ Vect G ( M, ρ ) is given by transition functionsΨ αβ ( x ) ∈ Y k Aut G ( X k )4ith the property that there exist h α,k ( x ) ∈ G such that h − α,k ( x ) pr k ◦ Ψ αβ ( x ) h α,k ( x )does not depends on k . Lets show that can be found transition functions withthe property that Y k pr k Ψ αβ ( x ) = ∆( a αβ ( x ))for some cocycle a αβ ( x ).Because the group G is discrete, for an atlas of connected charts withconnected intersections, we can assume that pr k ◦ Ψ αβ ( x ) = a αβ,k and h α,k ( x ) = h α,k ∈ G do not depend on x and, therefore, h − α,k ( x ) pr k ◦ Ψ αβ ( x ) h α,k ( x ) = a αβ does not depend on k nor x . Let H α,k ∈ Aut G ( X k ) such that pr k ( H α,k ) = h α,k . Then, Y k pr k ( H − α,k Ψ αβ ( x ) H α,k ) = ∆( a αβ ) . Theorem 2
If the space X is compact , then Bundle( X, Aut G ( X ρ )) ≈ G M ∈ Bundle(
X,G ) Vect G ( M, ρ ) . (5) Proof.
We will follow the proof theorem 3 in [1, p. 14]. Given a bundle ξ ∈ Bundle( X, Aut G ( X ρ )) with transition functionsΨ αβ ( x ) ∈ Aut G ( X ρ )we obtain transition functions pr ◦ i ◦ Ψ αβ ( x ) ∈ Aut G ( G ) ≈ G , defining an element M ∈ Bundle(
X, G ) together with a projection ξ −→ M .Changing the fibers Aut G ( X ρ ) by X ρ , we obtain an action of the group G , thatreduces over the base to the factor group G .Lets rewrite this in terms of homotopy. Corollary 2
If the space X is compact, then [ X, Aut G ( X ρ )] ≈ G M ∈ [ X,BG ] Y k [ M, BU ( F k )] . (6)5 The case when the subgroup
H < G is notnormal
Consider an equivariant vector G -bundle ξ over the base Mξ y pM. Let
H < G be a finite subgroup. Assume that M is the set of fixed points ofthe conjugation class of this subgroup, more accurately M = [ [ g ] ∈ G/N ( H ) M gHg − , (7)and that there is no more fixed points of the conjugation class of H in the totalspace of the bundle ξ ; here we have denoted by M H the set of fixed points ofthe action of the subgroup H over the space M , N ( H ) the normalizer of thegroup H in G and we are using the equality gM H = M gHg − and the fact that lHl − = gHg − if and only if g − l ∈ N ( H ).Lets denote by F ξ the family of subgroups of G having non-trivial fixedpoints in the total space of the bundle ξ , i.e. F ξ = { K < G | ξ K = ∅} . This is a partial ordered set by inclusions and is closed under the action of thegroup G by conjugation . Also, the action G × F ξ −→ F ξ ( g, K ) gKg − preserves the order. Definition 2
We will say that
H < G is the unique, up to conjugation, maximalsubgroup for the G -bundle ξ if every conjugate gHg − is maximal in F ξ and thereis no more maximal elements in this family. In this section will assume in any case, that
H < G is the unique, up toconjugation, maximal subgroup.
Lemma 2 If H = gHg − , then M H ∩ M gHg − = ∅ Proof.
If there is an x ∈ M H ∩ M gHg − then, the point x is fixed underthe action of the subgroup generated by H and gHg − , but this group is notcontained in any of the subgroups of the form lH − l, l ∈ G . If ξ K = ∅ , then ξ gKg − = gξ K = ∅ . emma 3 If the condition (7) holds, then the G -bundle ξ can be presented asa disjoint union of pair-wise isomorphic bundles with quasi-free action over thebase. More precisely ξ = G [ g ] ∈ G/N ( H ) ξ [ g ] , where ξ [ g ] = p − ( M gHg − ) is a vector bundle with quasi-free action of the group N ( gHg − ) and, for everyelement g ∈ G the mapping g : ξ [1] −→ gξ [1] = ξ [ g ] defines an equivariant isomorphism of this bundles, i.e. the diagram N ( H ) × ξ [1] −→ ξ [1] y s g × g y gN ( gHg − ) × ξ [ g ] −→ ξ [ g ] (8) commutes, where s g : N ( H ) −→ N ( gHg − ) = gN ( H ) g − , ( g, n ) gng − . Proof.
From lemma 2 it follows that M = G [ g ] ∈ G/N ( H ) M gHg − and, therefore, ξ = G [ g ] ∈ G/N ( H ) ξ [ g ] . Since the action of G is fiberwise, we have g · ξ [1] = ξ [ g ] for every g ∈ G .Restricting the projection ξ −→ M to the space ξ [ g ] , we obtain the bundle ξ [ g ] y pM gHg − . The bundle ξ [ g ] has an action of the normalizer N ( gHg − ): N ( gHg − ) × ξ [ g ] −→ ξ [ g ] , i.e. ξ [ g ] is a N ( gHg − )-bundle for every g ∈ G .Note that group conjugation s g : N ( H ) −→ N ( gHg − ) defines an isomor-phism between these groups that fits into the commutative diagram N ( H ) × ξ [1] −→ ξ [1] y y N ( gHg − ) × ξ [ g ] −→ ξ [ g ] . gng − · gx = g · nx . This means that the bundles ξ [1] and ξ [ g ] are naturallyand equivariantly isomorphic.Evidently, the mappings on the diagram (8) do not depend on the elements n ∈ N ( H ), but they depend on the element g ∈ G .The action of the group N ( H ) over the base M H reduces to the factor group N ( H ) /H : N ( H ) × ξ [1] −→ ξ [1] y y N ( H ) /H × M H −→ M H where, considering the maximality of the group H , the action N ( H ) /H × M −→ M is free and, by hypothesis, there is no more fixed of the action ofthe subgroup H in the total space of the bundle ξ , i.e. N ( H ) acts quasi-freelyover the base and has normal stationary subgroup H . Definition 3
If the condition (7) holds, we will say that the group G acts quasi-freely over the bundle ξ with (non-normal) stationary subgroup H . As we will see in theorem 3, for classifying purposes, it is enough to considerbundles with normal stationary subgroup.Let X ( ρ ) be the canonical model for the representation ρ : H −→ GL ( F ) withaction of the group N ( H ). Define a canonical model X ( ρ g ) for the representa-tion ρ g : gHg − s g − −→ H ρ −→ GL ( F ) ,s g ( n ) = gng − . The action of the group N ( gHg − ) over X ( ρ g ) is defined usingthe homomorphism of right gHg − -modules u g : gHg − s g − −→ H u −→ N ( H ) s g −→ N ( gHg − )by the formula (2).Let GX ( ρ ) := G [ g ] ∈ G/N ( H ) X ( ρ g )i.e. if lHl − = gHg − , then the spaces X ( ρ g ) and X ( ρ l ) coincide.This notation will be clear after the next lemma. Lemma 4
The group G acts over the space GX ( ρ ) quasi-freely with (non-normal) stationary subgroup H and, under this action, the space GX ( ρ ) co-incides with the orbit of the subspace X ( ρ ) . In particular, we have the relations N ( H ) ( X ( ρ )) = X ( ρ ) and ( GX ( ρ )) gHg − = N ( gHg − ) /gHg − . roof. The action G × GX ( ρ ) → GX ( ρ ) is defined in the following way: for afixed g ∈ G define the mapping g : X ( ρ ) −→ X ( ρ g )as s g × Id : N ( H ) × F −→ N ( gHg − ) × F ( N ( H ) = N ( H ) /H ) and, if lHl − = gHg − , then the mapping l : X ( ρ ) −→ X ( ρ l )is chosen to make the diagramm X ( ρ g ) s − g × Id −→ X ( ρ ) k y l − gX ( ρ l ) l − −→ X ( ρ ) . (9)commutative, i.e. l = ( s g × Id ) ◦ ( g − l )where the mapping g − l : X ( ρ ) −→ X ( ρ ) = X ( ρ g − l ) (10)is the canonical left translation by the element g − l ∈ N ( H ). Corollary 3
There is an isomorphism g : Aut N ( H ) ( X ( ρ )) ≈ −→ Aut N ( gHg − ) ( X ( ρ g )) (11) that depends only on the class [ g ] ∈ G/N ( H ) . Proof.
We have a diagram (8) for ξ = GX ( ρ ). Such a diagram alwaysinduces an isomorphismAut N ( H ) ( X ( ρ )) ≈ −→ Aut N ( gHg − ) ( X ( ρ g ))by the rule A g A g − and, if l ∈ [ g ] ∈ G/N ( H ) then l − g ∈ N ( H ) commutes with A ∈ Aut N ( H ) ( X ( ρ )).Therefore g A g − = g ( g − l )( l − g ) A g − = g ( g − l ) A ( l − g ) g − = l A l − . Definition 4
The space GX ( ρ ) is called the canonical model for the case whenthe subgroup H < G is not normal.
Lemma 5
Aut G ( GX ( ρ )) ≈ Aut N ( H ) ( X ( ρ )) (12)9 roof. By definition, an element of the group Aut G ( X ) is an equivariantmapping A a such that the pair ( A a , a ) defines the commutative diagram X A a −→ X y y G/H a −→ G/H, that commutes with the canonical action, i.e. the mapping a ∈ Aut G ( G/H )satisfies the condition a ∈ Aut G ( G/H ) ≈ N ( H ) /H, a [ g ] = [ ga ] , [ g ] ∈ N ( H ) /H. Therefore, A a = ( A a [ g ]) [ g ] ∈ N ( H ) /H ∈ Aut N ( H ) ( X ( ρ )).The value of the operators ( A a [ g ]) [ g ] ∈ G/H can be calculated in terms of theoperator A a [1] as in lemma 2 from [1, p. 9].Denote by g Vect G ( M, ρ ) the category of vector bundles with quasi-free actionof the group G over the base M . Theorem 3 g Vect G ( M, ρ ) ≈ Vect N ( H ) ( M H , ρ ) . Proof.
From lemma 3 follows that the bundles ξ [1] and ξ [ g ] equivariantlyisomorphic and are given by mappings M gHg − /N ( gHg − ) −→ B Aut N ( gHg − ) ( X ( ρ g )) , and M H /N ( H ) −→ B Aut N ( H ) ( X ( ρ )) , that can be put in the commutative diagram M H /N ( H ) −→ B Aut N ( H ) ( X ( ρ )) y ¯ g y ¯ gM gHg − /N ( gHg − ) −→ B Aut N ( gHg − ) ( X ( ρ g )) . Here, g : ξ H −→ ξ gHg − is the action over the bundle ξ . The arrow on the rightside is induced by the isomorphism (11) and does not depend on the element g ∈ [ g ] ∈ G/N ( H ). References [1] Mishchenko A.S., Morales Mel´endez, Quitzeh.
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