Description of G-bundles over G-spaces with quasi-free proper action of discrete group II

Description of G -bu ndles o v er G -spaces with quasi-free prop er action of dis crete group I I Morales Mel ´ endez ∗ , Quitzeh No ve mber 18, 201 8 1 The setting of the problem This problem naturally a rises f r om the Connor-Floyd’s description of the bo r- disms with the action of a gr oup G using the s o-called fix-p oin t construction This construction reduces the pro blem of d e scribing th e bo rdisms to t wo simpler problems: a) description o f the fixed-p oin t se t (or, mor e generally , the s tation- ary point set), which happens to be a subma nif o ld attac hed with the s tructure of its normal bundle and the action of the same group G , how e ver, this action could have stationary p o in ts of low er rank; b) description of the b ordisms o f low er rank with an ac tio n of th e group G . W e assume tha t the group G is discrete. Lets ξ be an G -equiv ariant vector bundle with base M . ξ   y M Where the actio n of the group G is quasi-fre e ov er the ba se with normal sta- tionary subgro up H < G and ther e is no mor e fixed p oints of the action of the group H in the total space of the bundle ξ . According [1, p.1] the bundle ξ separa tes as the sum of its G -subbundles: ξ ≈ M k ξ k where the index runs o ver all (unitary) irreducible representations ρ k : H − → U ( V k ) of the group H a nd, a s a H -bundle ξ k can be presented as the tenso r pr oduct: ξ ≈ M k η k O V k , ∗ Pa r tially supp orted by the gr an t 299388 of the mexican National Council for Science and T echnolog y (CONA CyT) 1 where the a ction of the group H ov er the bundles η k is trivial, V k denotes the trivial bundle with fib er V k and with fib erwise actio n of the g roup H , defined using the linear represe n tation ρ k . The par ticular case ξ = η k N V k was describ ed in the previous article [1]. According [1, p.1 4 ] the bundle ξ k can b e obta ined as the in verse image of a mapping f k : M /G 0 − → B Aut G ( X k ) where G 0 = G/H, X k = G 0 × ( F k ⊗ V k ) is the canonical model and Aut G ( X k ) is the gro up of equiv ariant automo rphisms of the space X k as a vector G -bundle ov er the base G 0 . So , the bundle ξ can b e giv en by a mapping f : M /G 0 − → Y k B Aut G ( X k ) . Consider the vector bundle ov er the discrete base G 0 X ρ = G 0 × M k ( F k ⊗ V k ) ! . (1) Define a fib erwise action G × X ρ → X ρ by the formula φ ([ g ] , g 1 ) : [ g ] × M k ( F k ⊗ V k ) ! → [ g 1 g ] × M k ( F k ⊗ V k ) ! φ ([ g ] , g 1 ) = L k  Id ⊗ ρ k ( u ( g 1 g ) u − 1 ( g ))  = ρ ( u ( g 1 g ) u − 1 ( g )) . (2) Definition 1 The bund le X ρ − → G 0 with the j u st define d action is c al le d the c anonic al mo del for the r epr esentation ρ . By Aut G ( X ρ ) w e deno te t he g roup of equiv aria n t automorphisms of the canonical mo de l X ρ as a vector G -bundle ov er the base G 0 with fiber L k ( F k ⊗ V k ) and canonical action of the group G . Lemma 1 Ther e exists a monomorphism i : Aut G ( X ρ ) − → Y k Aut G ( X k ) Pro of. As b efore, a n element of the gro up Aut G ( X ρ ) is a n eq uiv ar ian t mapping A a such that the pair ( A a , a ) defines a comm utative diagram X ρ A a − → X ρ   y   y G 0 a − → G 0 , 2 a ∈ Aut G ( G 0 ) ≈ G 0 , a [ g ] = [ g a ] , [ g ] ∈ G 0 . By the lemma 1 [1] applied to group of a utomorphisms Aut G ( X ρ ), for A a ∈ Aut G ( X ρ ), we hav e A a | X k : X k − → X k . Note that this restriction is G -equiv ariant. 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 homomo rphism: it is a pro duct of restric t io ns over inv aria nt subspaces. Lets pr o ve that it is injective. If A a | X k = Id X k , then A a = Id X bec ause X = ⊕ k X k . In order to prove that the image of i is closed, note that it coincides with those automorphisms which co mm ute with the inclusion ∆ × Id : X ρ − → Y k X k i.e. if an elemen t ( A a k k ) k ∈ Q k Aut G ( X k ) leav es the image of X ρ inv ariant, then its restrictio n defines an element in Aut G ( X ρ ) and the diagram X ρ A a − → X ρ   y   y Q k X k ( A a k k ) k − → Q k X k commutes, i.e. a k = a ∀ k and A a | X k = A a k . In other words Y k pr k ◦ i ( A a ) = ∆( a ) where pr k : Aut G ( X k ) − → G 0 is the epimorphism of lemma 2 [1, p. 9] and ∆ : G 0 ֒ → Y k G 0 . Then i (Aut G ( X ρ )) = Y k pr − 1 k ∆( G 0 ) . 3 Corollary 1 It takes plac e an ex a ct se quenc e of gr oups 1 → Y k GL ( F k ) ϕ − → Aut G ( X ρ ) pr − → G 0 → 1 Pro of. Define pr = ∆ − 1 Q k pr k ◦ i . This is an epimorphism: let A a k ∈ p r k ( a ), then ( A a k ) k ∈ i (Aut G ( X ρ )). W e define the monomor phism 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 − 1 Y k ϕ k . Then pr ◦ ϕ = 1 and if a = 1, then Q k pr k ◦ i ( A 1 ) = ∆(1). This means that, for every k , there is a B ∈ GL ( F k ) such that ϕ k ( B k ) = A 1 | X k , i.e. ϕ ( B k ) k = A 1 . Denote by V ect G ( M , ρ ) the catego ry of G -equiv ariant vector bundles ξ ov er the base M w ith quasi-free action of the group G over the base and normal stationary subgroup H < G . Then, by lemma 1 and the o bserv a t ions on p. 13 in [1] in ter ms of homotopy we hav e V ect G ( M , ρ ) ≈ Y k [ M , B U ( F k )] (3) Denote by Bundle( X , L ) the categ ory o f principa l L -bundles over the base X . Theorem 1 Ther e exists an inclus i on V ect G ( M , ρ ) − → Bundle( M /G 0 , Aut G ( X ρ )) . (4) Pro of. W e already hav e a monomorphism V ect G ( M , ρ ) − → Bundle( M / G 0 , Y k Aut G ( X k )) i.e. Y k V ect G ( M , ρ k ) − → Y k Bundle( M /G 0 , Aut G ( X k )) see [1, p. 13 ] A bundle ξ ∈ V ect G ( M , ρ ) is given by transition functions Ψ αβ ( x ) ∈ Y k Aut G ( X k ) 4 with the prop ert y that there exist h α,k ( x ) ∈ G 0 such tha t h − 1 α,k ( x ) pr k ◦ Ψ αβ ( x ) h α,k ( x ) do es not dep ends on k . Le ts show that can b e found transition functions with the prop ert y that Y k pr k Ψ αβ ( x ) = ∆( a αβ ( x )) for some co cycle a αβ ( x ). Because the group G 0 is discrete, for an atlas of co nnected charts with connected intersections, we ca n assume that p r k ◦ Ψ αβ ( x ) = a αβ ,k and h α,k ( x ) = h α,k ∈ G 0 do not dep end o n x a nd, therefor e, h − 1 α,k ( x ) pr k ◦ Ψ αβ ( x ) h α,k ( x ) = a αβ do es 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 − 1 α,k Ψ αβ ( x ) H α,k ) = ∆( a αβ ) . Theorem 2 If the sp ac e X is c omp act , then Bundle( X , Aut G ( X ρ )) ≈ G M ∈ Bun dle( X,G 0 ) V ect G ( M , ρ ) . (5) Pro of. W e will follo w t he pro of theore m 3 in [1 , p. 14]. Giv en a bundle ξ ∈ Bundle( X , Aut G ( X ρ )) with transitio n functions Ψ αβ ( x ) ∈ Aut G ( X ρ ) we o btain tra nsition functions pr ◦ i ◦ Ψ αβ ( x ) ∈ Aut G ( G 0 ) ≈ G 0 , defining an element M ∈ Bundle( X, G 0 ) together with a pr o jection ξ − → M . Changing the fib ers Aut G ( X ρ ) b y X ρ , we o bt a in an actio n of the group G , that reduces ov er the base to the factor gro up G 0 . Lets rewrite this in terms of homotopy . Corollary 2 If the sp ac e X is c omp act, then [ X , Aut G ( X ρ )] ≈ G M ∈ [ X,BG 0 ] Y k [ M , B U ( F k )] . (6) 5 2 The case when the subgroup H < G is not normal Consider an equiv ariant vector G -bundle ξ ov er the base M ξ   y p M . Let H < G b e a finite s ubgroup. Ass ume that M is the set of fixed p oin ts o f the conjugation class of this subgr oup, more accura tely M = [ [ g ] ∈ G/ N ( H ) M gH g − 1 , (7) and tha t there is no more fixed p o in ts of the conjugation class of H in the total space of the bundle ξ ; here we hav e denoted by M H the set of fixed p oin ts of the action of the subgr oup H ov e r the space M , N ( H ) the normalizer of the group H in G and we a re using the equalit y g M H = M gH g − 1 and the fact that l H l − 1 = g H g − 1 if and only if g − 1 l ∈ N ( H ). Lets denote by F ξ the family of subgro ups of G ha v ing non- trivial fixed po in ts in the total space of the bundle ξ , i.e. F ξ = { K < G | ξ K 6 = ∅} . This is a pa rtial order ed set b y inclusions and is closed under the action o f the group G b y conjuga tion 1 . Also , the action G × F ξ − → F ξ ( g , K ) 7→ g K g − 1 preserves the or der. Definition 2 W e wil l say t ha t H < G is the unique, up to c onjugation, maximal sub gr oup for the G -bund le ξ if every c onjugate g H g − 1 is maximal in F ξ and ther e is n o mor e maximal elements in this family. In this section will assume in any case, that H < G is the unique, up to conjugation, maximal subgro up. Lemma 2 If H 6 = g H g − 1 , t h en M H ∩ M gH g − 1 = ∅ Pro of. If t her e is an x ∈ M H ∩ M gH g − 1 then, the point x is fixed under the ac tion o f the subg r oup gener ated by H a nd g H g − 1 , but this gr oup is not contained in a n y of the subgroups of the form l H − 1 l , l ∈ G . 1 If ξ K 6 = ∅ , then ξ gK g − 1 = g ξ K 6 = ∅ . 6 Lemma 3 If the c ondition (7) holds, t h en the G -bund le ξ c an b e pr esente d as a disjoint union of p air-wise isomorphic bund les with quasi-fr e e action over the b ase. Mor e pr e cisely ξ = G [ g ] ∈ G/ N ( H ) ξ [ g ] , wher e ξ [ g ] = p − 1 ( M gH g − 1 ) is a ve ctor bu nd le with quasi-fr e e action of the gr ou p N ( gH g − 1 ) and, for every element g ∈ G the mapping g : ξ [1] − → g ξ [1] = ξ [ g ] defines an e qu i variant isomorphi sm of this bund les, i.e. the diagr am N ( H ) × ξ [1] − → ξ [1]   y s g × g   y g N ( g H g − 1 ) × ξ [ g ] − → ξ [ g ] (8) c ommutes, wher e s g : N ( H ) − → N ( g H g − 1 ) = g N ( H ) g − 1 , ( g , n ) 7→ g ng − 1 . Pro of. F rom lemma 2 it follows that M = G [ g ] ∈ G/ N ( H ) M gH g − 1 and, therefore, ξ = G [ g ] ∈ G/ N ( H ) ξ [ g ] . Since the action of G is fib erwise, we hav e g · ξ [1] = ξ [ g ] for every g ∈ G . Restricting the pro jection ξ − → M to the space ξ [ g ] , we obtain the bundle ξ [ g ]   y p M gH g − 1 . The bundle ξ [ g ] has an action of the normalizer N ( g H g − 1 ): N ( g H g − 1 ) × ξ [ g ] − → ξ [ g ] , i.e. ξ [ g ] is a N ( g H g − 1 )-bundle for every g ∈ G . Note that g roup conjugation s g : N ( H ) − → N ( g H g − 1 ) defines an isomor- phism b et ween these g roups that fits into the comm uta tiv e diagram N ( H ) × ξ [1] − → ξ [1]   y   y N ( g H g − 1 ) × ξ [ g ] − → ξ [ g ] . 7 i.e. gng − 1 · g x = g · nx . This means that the bundles ξ [1] and ξ [ g ] are naturally and equiv ariantly iso morphic. Evidently , the mappings on the diagram (8) do no t dep end on the elements n ∈ N ( H ), but they depend o n the element g ∈ G . The a ction o f the gr oup N ( H ) ov er 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, consider ing the max imalit y o f the gro up H , the action N ( H ) /H × M − → M is free and, by hypothesis, there is no more fixed of the action o f the subgr oup H in the total space of the bundle ξ , i.e. N ( H ) acts quasi-freely ov er the base and has normal stationar y subgroup H . Definition 3 If the c ondition (7) holds, we wil l say that the gr oup G acts quasi- fr e ely over the bund le ξ with (non-normal) stationary sub gr oup H . As w e will s ee in theorem 3, for classifying purp oses, it is enough to consider bundles with normal stationary subgro up. Let X ( ρ ) b e the canonica l mo del for the representation ρ : H − → GL ( F ) with action of the g roup N ( H ). Define a cano nical mo del X ( ρ g ) for the representa- tion ρ g : g H g − 1 s g − 1 − → H ρ − → GL ( F ) , s g ( n ) = g ng − 1 . The action of the gr oup N ( g H g − 1 ) over X ( ρ g ) is defined using the homomorphism of right g H g − 1 -mo dules u g : g H g − 1 s g − 1 − → H u − → N ( H ) s g − → N ( g H g − 1 ) by the formula (2). Let GX ( ρ ) := G [ g ] ∈ G/ N ( H ) X ( ρ g ) i.e. if l H l − 1 = g H g − 1 , then the spaces X ( ρ g ) and X ( ρ l ) coincide. This notation will b e clear after the next lemma. Lemma 4 The gr oup G acts over the sp ac e GX ( ρ ) qu a si-fr e ely with (non- normal) stationary sub gr oup H and, under this action, the sp ac e GX ( ρ ) c o- incides with the orbit of the su bsp ac e X ( ρ ) . In p articular, we have the r elations N ( H ) ( X ( ρ )) = X ( ρ ) and ( GX ( ρ )) gH g − 1 = N ( g H g − 1 ) /g H g − 1 . 8 Pro of. The action G × GX ( ρ ) → GX ( ρ ) is defined in the following wa y: for a fixed g ∈ G define the mapping g : X ( ρ ) − → X ( ρ g ) as s g × Id : N ( H ) 0 × F − → N ( g H g − 1 ) 0 × F ( N ( H ) 0 = N ( H ) /H ) and, if l H l − 1 = g H g − 1 , then the mapping l : X ( ρ ) − → X ( ρ l ) is c ho sen to make the diagramm X ( ρ g ) s − 1 g × Id − → X ( ρ ) k   y l − 1 g X ( ρ l ) l − 1 − → X ( ρ ) . (9) commutativ e, i.e. l = ( s g × Id ) ◦ ( g − 1 l ) where the mapping g − 1 l : X ( ρ ) − → X ( ρ ) = X ( ρ g − 1 l ) (10) is the canonica l left transla tion by the element g − 1 l ∈ N ( H ). Corollary 3 Ther e is an isomorphism g : Aut N ( H ) ( X ( ρ )) ≈ − → Aut N ( gH g − 1 ) ( X ( ρ g )) (11) that dep ends only on the class [ g ] ∈ G/ N ( H ) . Pro of. W e ha ve a diagram (8) for ξ = GX ( ρ ). Suc h a dia gram always induces an isomorphis m Aut N ( H ) ( X ( ρ )) ≈ − → Aut N ( gH g − 1 ) ( X ( ρ g )) by the rule A 7→ g A g − 1 and, if l ∈ [ g ] ∈ G/ N ( H ) then l − 1 g ∈ N ( H ) commutes with A ∈ Aut N ( H ) ( X ( ρ )). Therefore g A g − 1 = g ( g − 1 l )( l − 1 g ) A g − 1 = g ( g − 1 l ) A ( l − 1 g ) g − 1 = l A l − 1 . Definition 4 The sp ac e GX ( ρ ) is c al le d t he c anonic al mo del for the c ase when the sub gr oup H < G is not normal. Lemma 5 Aut G ( GX ( ρ )) ≈ Aut N ( H ) ( X ( ρ )) (12) 9 Pro of. By definition, a n elemen t of the group Aut G ( X ) is an equiv ariant mapping A a such that the pair ( A a , a ) defines the commutativ e diag ram X A a − → X   y   y G/H a − → G/H , that commutes with the ca no nical action, i.e. the mapping a ∈ Aut G ( G/H ) satisfies the condition a ∈ Aut G ( G/H ) ≈ N ( H ) / H, a [ g ] = [ g a ] , [ g ] ∈ N ( H ) /H. Therefore, A a = ( A a [ g ]) [ g ] ∈ N ( H ) /H ∈ Aut N ( H ) ( X ( ρ )). The v alue of the op erators ( A a [ g ]) [ g ] ∈ G/H can b e calculated in terms o f the op erator A a [1] as in lemma 2 from [1, p. 9]. Denote by g V ect G ( M , ρ ) the category o f vector bundles with quasi-fr e e ac tio n of the group G ov er the base M . Theorem 3 g V ect G ( M , ρ ) ≈ V ect N ( H ) ( M H , ρ ) . Pro of. F rom lemma 3 follows that the bundles ξ [1] and ξ [ g ] equiv ariantly isomorphic and are given by mappings M gH g − 1 / N ( g H g − 1 ) 0 − → B Aut N ( gH g − 1 ) ( X ( ρ g )) , and M H / N ( H ) 0 − → B Aut N ( H ) ( X ( ρ )) , that can be put in the commutativ e diagra m M H / N ( H ) 0 − → B Aut N ( H ) ( X ( ρ ))   y ¯ g   y ¯ g M gH g − 1 / N ( g H g − 1 ) 0 − → B Aut N ( gH g − 1 ) ( X ( ρ g )) . Here, g : ξ H − → ξ gH g − 1 is the a ction o ver the bundle ξ . The arrow on the right side is induced by the isomo r phism (11) a nd do es not dep end on the element g ∈ [ g ] ∈ G/ N ( H ). References [1] Mishc henko A.S., Mor ales M el´ endez, Quitzeh. Description of G -bu nd les over G -sp ac es with qu a s i-fr e e pr op er action of discr ete gr oup [2] Luk e G., Mishchenk o A. S., V e ctor Bund les And Their Applic ations. Kluw er Academic Publishers Group (Netherla nds), 199 8. ISBN: 978 07923515 4 2 [3] P . Co nner, E. Floyd. Differ ent i able p erio dic maps. 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