The quotient criterion for syzygies in equivariant cohomology for elementary abelian 2 -group actions
TTHE QUOTIENT CRITERION FOR SYZYGIES INEQUIVARIANT COHOMOLOGY FOR ELEMENTARY ABELIAN -GROUP ACTIONS SERGIO CHAVES
Abstract.
Let G be a elementary abelian 2-group and X be a manifold witha locally standard action of G . We provide a criterion to determine the syzygyorder of the G -equivariant cohomology of X with coefficients over a field ofcharacteristic two using a complex associated to the cohomology of the facefiltration of the manifold with corners X { G . This result is the real version ofthe quotient criterion for locally standard torus actions developed in [16]. Introduction
Let G be a compact Lie group and X be a G -space. The G -equivariant coho-mology of X with coefficients over a field k is defined as the singular cohomology ofthe Borel construction [5] H ˚ G p X ; k q “ H ˚ p X G ; k q . It inherits a canonical modulestructure over the cohomology of the classifying space H ˚ p BG ; k q of G .If the restriction map H ˚ G p X ; k q Ñ H ˚ p X ; k q is surjective, the G -equivariantcohomology is a free module over H ˚ p BG ; k q as a consequence of the Leray-Hirschtheorem. In this case, we say that X is G -equivariantly formal over k . The con-verse also holds when extra assumptions over G are considered. For example, if G is connected, it is a direct consequence of the Eilenberg-Moore spectral sequenceassociated to the fibration X Ñ X G Ñ BG , and a discussion for nilpotent actionsof any G can be found in [2, § Q include smooth compact toric varieties and quasitoric manifolds [11], symplecticmanifolds with Hamiltonian torus actions [4] and G -spaces with vanishing odd ra-tional cohomology when G is connected. Moreover, when cohomology with rationalcoefficients is considered, the ring H ˚ p BG ; Q q becomes a polynomial ring in n gen-erators sitting in even degrees. In this case, a G -space X is equivariantly formal ifand only if its equivariant cohomology fits in a long exact sequence(1.1) 0 Ñ H ˚ G p X ; Q q Ñ F Ñ ¨ ¨ ¨ F n of free H ˚ p BG ; Q q -modules F j for 1 ď j ď n by the Hilbert Syzygy Theorem. Thisequivalence motivates the study of syzygies in equivariant cohomology started in[1] for torus actions.Recall that a finitely generated module M over a commutative ring R is a j -thsyzygy if there is an exact sequence(1.2) 0 Ñ M Ñ F Ñ ¨ ¨ ¨ Ñ F j Date : October 1, 2020. a r X i v : . [ m a t h . A T ] S e p CHAVES of free modules F k for 1 ď k ď j . In [1, Thm.5.7], the authors showed thatthe syzygy order of the equivariant cohomology of a space with a torus action isequivalent to the partial exactness of the Atiyah-Bredon with rational coefficients.This sequence was firstly discussed in [3] [6] and it is defined in the following way:Let T “ p S q n be a torus and X be a T -space. The filtration of X by its orbitdimensions X “ X T Ď X Ď ¨ ¨ ¨ Ď X n “ X induces a complex(1.3) 0 Ñ H ˚ T p X q Ñ H ˚ T p X T q Ñ H ˚` p X , X q Ñ ¨ ¨ ¨ Ñ H ˚` n p X n , X n ´ q that is referred nowadays as the Atiyah-Bredon sequence of the T -space X .The characterization of syzygies via the partial exactness of the sequence 1.3 canbe extended to any compact connected Lie group G over the rationals by restrictionof the action to a maximal torus T Ď G [15] and to any elementary abelian p -group G p over a field of characteristic p by transfer and restriction of the action underthe inclusion G p Ď T [2].Another remarkable characterization of syzygies in equivariant cohomology fortorus actions is the quotient criterion for locally standard actions developed in [16].Recall that these actions are modeled by the standard representation of T “ p S q n on C n . Such characterization is given as follows: for a locally standard smoothaction on a T -manifold X , the quotient space M “ X { T is a nice manifolds withcorners and the syzygy order of the T -equivariant cohomology of X it is determinedby the topology of filtration of M by its faces. In particular, this result recoversthe equivariant formality over Q of compact smooth toric varieties, torus manifoldsand quasi-toric manifolds.In this paper, we discuss locally standard torus actions modeled by the standardrepresentation of G “ p Z { Z q n on R n and cohomology with coefficients over a fieldof characteristic 2, that we refer as the “real version” of the torus case. We use thecharacterization of syzygies via the Atiyah-Bredon sequence for elementary abelian2-groups and the description of the Ext modules of the equivariant homology asthe cohomology of the Atiyah-Bredon complex as discussed in [2] analogously tothe torus actions case [16]. The main result of this document is the following.Let G “ p Z { Z q n be a elementary abelian 2-group and let X be a compactmanifold of dimension m ě n with a locally standard action of G . Then M “ X { G becomes a m -manifold with n -corners (Definition 3.1) and for any face P Ď M weconsider the complex B i p P q “ à Q Ď P rank Q “ i H ˚ p Q, B Q q with differential induced by the connecting homomorphism of the cohomologicallong exact sequence associated to the triple p Q, B Q, B Q zp P zB P qq . The syzygy or-der of the G -equivariant cohomology of X is determined by the vanishing of thecohomology of the complex B i p P q for any P in certain range as we state in thefollowing theorem. Theorem 1.1.
Let k be a field of characteristic two and ď j ď n . H ˚ G p X ; k q isa j -th syzygy over H ˚ p BG ; k q if and only if for any face P of the manifold withcorners M “ X { G we have that H i p B ˚ p P qq “ for any i ą max p rank P ´ j, q . As consequence of this result, we immediately recover the equivariant formalityfor G -spaces whose orbit space and its faces are contractible; for example, the QUIVARIANT COHOMOLOGY FOR ELEMENTARY 2-ABELIAN GROUPS 3 real locus of quasitoric manifolds whose orbit space is a simple polytope. Thisalso provides a criterion to compute the syzygy order of the p Z { Z q n -manifoldsconstructed in [19], [21].This document is organized as follows: In section 2 we review equivariant coho-mology for elementary abelian 2-groups (or 2-tori) and provide a characterizationof syzygies in terms of decomposition of the subgroups of the 2-torus. In Section3 we review the concept of manifolds with corners and in Section 4 we discuss lo-cally standard 2-torus actions, provide a proof of the main results and discuss someconsequences of them. Acknowledgments.
I would like to thank Matthias Franz for his ideas and collab-oration on this work, as well as his comments on earlier versions of this document.I am also grateful to Christopher Allday, Matthias Franz and Volker Puppe forsharing an earlier copy of [2] with me. This work is based on the author’s doctoralthesis.2.
Remark on syzygies in equivariant cohomology for 2-torus actions
In this section, we review the characterization of syzygies in equivariant coho-mology for actions of a group G – p Z { Z q n isomorphic to a 2-torus of rank n andcohomology with coefficient over a field of characteristic two k . We will omit thecoefficient k in our notation for cohomology.We start by reviewing the construction of the Atiyah-Bredon sequence for 2-torus actions. The relation between syzygies in G -equivariant cohomology and theAtiyah-Bredon sequence, the G -equivariant homology and the equivariant Poincar´eduality has been developed in [2] where the authors generalize analogous resultsfrom the torus case [1]. Let G be a 2-torus of rank n and X be a G -space. The i -th G -equivariant skeleton of X is the space X i defined as the union of orbits of size atmost 2 i for ´ ď i ď n . The skeletons of X give rise to a filtration H “ X ´ Ď X Ď ¨ ¨ ¨ Ď X r “ X called the G -orbit filtration of X . This filtration induces a complex0 Ñ H ˚ G p X q Ñ H ˚ G p X q Ñ H ˚` G p X , X q Ñ ¨ ¨ ¨ Ñ H ˚` nG p X n , X n ´ q which is called the G -Atiyah-Bredon sequence of X and it will be denoted by AB ˚ G p X q . We will show a characterization of syzygies in terms of the exactnessof the Atiyah-Bredon sequence AB L p X K q for any decomposition G – K ˆ L analo-gously to [16, § G -equivariant cohomology is a module over the polynomialring H ˚ p BG q , we first review the following algebraic remark. Remark 2.1.
Let S be a polynomial ring over some field in n -variables of positivedegree, and let m be the maximal homogeneous ideal of S . For a graded finitelygenerated S -module M , the length of a maximal M -sequence of elements in m isdenoted by depth S M . It is related to the Ext functor via the formula depth S M “ min t k : Ext n ´ kS p M, S qu . See [13, Prop.A1.16] . On the other hand, the depth of M and the syzygy order of M are related as follows: M is a j -th syzygy over S if and only if for any primeideal p Ď S , depth S p M p ě min p j, dim S p q [7, § . Now we proceed to prove the following result.
CHAVES
Lemma 2.2.
Let X be a G -space. The G -equivariant cohomology H ˚ G p X q is a j -th syzygy over R if and only if for any decomposition G – K ˆ L into two tori K and L it holds that depth R L H ˚ L p X K q ě min p j, rank L q . Proof.
The proof given in [16, Prop.3.3] is purely algebraic and carries over to oursetting. It mainly uses the characterization of syzygies via depth as in Remark 2.1,and that one can restrict only to the prime ideals corresponding to those arisen asthe kernel of the restriction map H ˚ p BG q Ñ H ˚ p BK q for any subgroup K Ď G .Compare with [2, Lem.8.1]. (cid:3) The next proposition uses the G -equivariant homology of X and the equivariantextension of the Poincar´e duality for 2-torus actions. See [2] for a wider discussionon equivariant homology and the equivariant Poincar´e duality. Proposition 2.3.
Let X be a G -manifold and j ě . Then H ˚ p X q is a j -th syzygyif and only if H i p AB ˚ L p X K qq “ for any subgroup K occurring as an isotropysubgroup in X where L “ G { K and i ą max p rank L ´ j, q .Proof. The proof from the torus case [16, Cor.3.4] can be also adapted to oursetting. As discussed in [2, Thm.8.3], we may also assume that k “ F . Let K be a subgroup of G . Then X K is a closed submanifold of X by the tubularneighbourhood theorem. For a connected component Y Ď X K , there is a principalorbit G { G x where x P Y , so K Ď G x as subgroup. Set K “ G x . L “ G { K andwrite rank K “ rank K ` k for some integer k ě
0. Using Lemma 2.2 we getdepth R L H ˚ L p Y q ě depth R L H ˚ L p X K q“ depth R L H ˚ L p X K q ` k ě min p j, rank L q ` k ě min p j, rank L q . Following [2, Thm.8.9] and analogous to [1, Thm.4.8] for the torus case, we havethat the cohomology of the G -Atiyah-Bredon complex of X is isomorphic to theext of the equivariant homology of X ; namely, H i p AB ˚ G p X qq – Ext iR p H G ˚ p X q , R q for any i ě
0. This implies thatdepth R L H ˚ L p X K q “ min t i : Ext rank L ´ iR L p H ˚ L p X K q , R L q ‰ u“ min t i : H rank L ´ i p AB ˚ L p X K qq ‰ u by combining also the equivariant Poincar´e duality isomorphism H G ˚ p X q – H ˚ G p X q .Therefore, depth R L H ˚ L p X K q ě min p j, rank L q if and only if H i p AB ˚ L p X K qq “ i ą max p rank L ´ j, q . (cid:3) We finish this section with the following result (compare with [16, Prop.3.3]).
Proposition 2.4. If H ˚ G p X q is a j -th syzygy over R then so is H ˚ L p X K q over R L for any subgroup K Ď G and complementary subgroup L Ď G . Furthermore, L canbe canonically identified with the quotient G { K .Proof. Let K Ď G , Y “ X K and L be a complementary subgroup to K . We willshow that the condition of Lemma 2.2 holds for H ˚ L p Y q . Let K Ď L and choosea complementary subgroup L Ď L of K in L . Notice that K – K ˆ K is a QUIVARIANT COHOMOLOGY FOR ELEMENTARY 2-ABELIAN GROUPS 5 complementary subgroup of L in G , and we have that Y K “ p X K q K “ X K .Applying then Lemma 2.2 to the subgroup K Ď G , we get thatdepth R L,L H ˚ L p Y K q “ depth R L H ˚ L p X K q ě min p j, rank L q showing that H ˚ L p Y q is a j -th syzygy over R L again by Lemma 2.2. (cid:3) Remark on manifolds with corners
In this section we review the notion of manifolds with corners that generalizesthe concept of manifolds and manifolds with boundary in the classical setting. Theywere firstly developed in [9] and [12] in differential geometry and have been used intransformation groups on smooth manifolds [17], cobordism [18] and toric topology[8]. Recall that a topological manifold of dimension n is locally modeled by R n , amanifold with boundary is modeled by the half space r ,
8s ¸ R n ´ and a manifoldwith corners with be modeled by the intersection of (zero or more) half spaces in R n as we state in the following definition. Definition 3.1.
Let M be a paracompact Hausdorff space and m ě n ě beintegers. We say that M is an m -manifold with n -corners if M has an atlas tp U i , ϕ i qu where ϕ i : U i Ñ V i is a homeomorphism of U i onto an open subset V i or R m,n : “ r , n ˆ R m ´ n , and the map ϕ i ˝ ϕ ´ j : ϕ j p U i X U j q Ñ ϕ i p U i X U j q isthe restriction of a diffeomorphism between open sets in R m for all i, j . Even though we provide a general definition, in this document we will onlydiscuss compact spaces for simplicity. Examples of manifolds with corners includemanifolds, manifolds with boundary and convex simple polytopes. More interestingexamples of spaces that are manifolds with corners are given by the following figures:the teardrop and the eye-shaped space.
Figure 1.
Examples of manifold with corners: Teardrop and Eye-shaped figureFor any m -manifolds with n -corners M , the boundary B M becomes a p m ´ q -manifold with p n ´ q -corners. Moreover, we can filter M in the following way. Forany z “ p x, y q P R m,n , let c z be the number of zero coordinates of x in r , n . If M is an m -manifold with n -corners, then c x is well-defined for any x P M . We saythat F is a facet of M if F is the closure of a connected component of the subspace M “ t x P M : c x “ u . Notice that F is an p m ´ q -dimensional submanifold withboundary of B M and Ť F facet F “ B M . Moreover, any finite intersection of facets Ş ki “ F i is either empty, or a disjoint union of submanifold of M of codimension k .Analogously, a face of M of codimension k is defined as the closure of a connectedcomponent of the subspace M k “ t x P M : c x “ k u . CHAVES
Remark 3.2.
Any manifold with corners become a filtered space by setting X i “ Ť k ď i M n ´ i , so X i consists of all faces of codimension at least n ´ i . In particular, X “ M n and X n “ M . For example, a face of R m,n “ r , n ˆ R m ´ n of codimension k is the subspace A I “ tp x, y q P R m,n : x i “ i R I u for some I Ď t , . . . , n u , | I | “ k . Definition 3.3.
Let M be a manfold with corners. We say that M is a nicemanifold with corners if for any face P of M there are exactly k -facets F , Ď , F k such that P is a connected component of the intersection Ş ki “ F i . For example, in the spaces from Figure 3, we have that the teardrop is a 2-manifold with 2-corners that is not nice but the eye-shaped figure it is a nice2-manifolds with 2-corners. For our interests, we will focus on nice manifolds withcorners as they generalize the notion of simple polytopes in toric topology as theywill show up as the orbit space of locally standard actions as discussed in thefollowing section.4.
The quotient criterion for locally standard 2-torus actions
In this section we discuss locally standard 2-torus actions on manifolds as theirquotients will be nice manifolds with corners and then we will prove the quotientcriterion for these particular actions analogously to [16] for locally standard torusactions. Recall that we are considering cohomology with coefficients over a field k of characteristic two and we will omit it in our notation.We start by reviewing the standard action on R m . Let G be a 2 torus of rank n with n ď m . The standard action of G on R m is defined as follows: Identifying Z { Z “ t˘ u , we have a canonical action of G on R m given by p g , . . . , g n q ¨ p x , . . . , x n , x n ` , . . . , x m q “ p g x , . . . , g n x n , x n ` , . . . , x m q and thus the quotient space R m { G – R m,n “ r , n ˆ R m ´ n is a manifold withcorners. This leads to the following definition. Definition 4.1.
Let G be a -torus of rank n and X be a G -manifold of dimension m with m ě n . A G -standard chart p U, ϕ q of x P X is a G -invariant open neigh-bourhood U of x in X and a G -equivariant homeomorphism ϕ : U Ñ V on some G -invariant open set V Ď R m (with the standard action defined above). We saythat X is a locally standard G -manifold (or that the G -action is locally standard)if X has an atlas tp U i , ϕ i qu consisting of standard charts such that the change ofcoordinates ϕ i ˝ ϕ ´ j : ϕ j p U i X U j q Ñ ϕ i p U i X U j q is a G -equivariant diffeomorphismfor all i, j . Under these assumptions, one can check that the quotient space M “ X { G becomes an m -manifold with n -corners. Let π : X Ñ X { G denote the quotientmap. For any subspace A Ď X { G we write π ´ p A q “ X A ; however, we will identify X G with its image in X { G .Let P be a face of M . Notice that all points in X lying over the interior of P have a common isotropy group G P Ď G . We denote by G ˚ P “ G { G P whichis isomorphic to a complementary 2-torus of G P in G . Also, we write rank P “ rank G ˚ P . Observe that X P “ π ´ p P q is a connected component of X G P and theset X P zB P “ X P z X B P is the open subset of X P where G ˚ P acts freely. This impliesthat QUIVARIANT COHOMOLOGY FOR ELEMENTARY 2-ABELIAN GROUPS 7 (4.1) H G ˚ P p X P , X B P q – H ˚ p P, B P q . For two faces
P, Q of X { G we write P Ď Q if P Ď Q and rank Q “ rank P ` F Ď X { G , then X F is a closed submanifold of X of codimension 1 and wedenote by N F its normal bundle. For any face P Ď F , consider E F.P the vectorbundle over X P zB P obtained as the pullback of N F under the inclusion of X P zB P on X F . Under the inclusion p X P , X B P q Ñ p E F,P , E F,P q , the equivariant Thomclass of E F,P induces a class in the equivariant cohomology e F,P P H G p X P , X B P q .Finally, we denote by t F,P P H ˚ p BG p q the restriction of e F,P under the inclusion p pt, Hq Ñ p X P , X B P q . Notice that t F,P is the equivariant Euler class of the G p -equivariant line bundle over a point which corresponds to the generator of H ˚ p BG P q in H p BG P q . Remark 4.2.
For any face P of X { G , consider the k -algebra R P “ H ˚ p BG P q .Then P is a connected component of the intersection Ş P Ď F Ď X { G F and thus X P is a connected component of the intersection Ş P Ď F Ď X { G X F . Moreover, for anypoint x P X P zB P , there is an isomorphism G P – ś P Ď F Ď X { G G F by looking to astandard chart of x in X . This implies that t F,P is a basis for the vector space H p BG P q which extends to an isomorphism of algebras R P – k r t F,P : P Ď F Ď X { G s If P Ď Q , G Q Ď G P and we have a canonical map ρ P Q : R P Ñ R Q . It followsfrom the naturality of the Euler class and the above remark that ρ P Q p t F,P q “ t F,Q if Q Ď F and 0 otherwise. Now we will proceed to prove the following lemma. Proposition 4.3.
Let P be a face in X { T .(i) The composition φ P : H ˚ p P, B P q – ÝÑ H ˚ G ˚ P p X P , X B P q Ñ H ˚ G p X P , X B P q induces a map ψ P : H ˚ p P, B P q b R P Ñ H ˚ G p X P , X B P q which is an isomor-phism of graded vector spaces.(ii) If P Ď Q the following diagram H ˚ p P, B P q b R P H ˚ G p X P , X B P q H ˚` p Q, B Q q b R Q H ˚` G p X Q , X B Q q δ b ρ PQ ψ P δψ Q is commutative where δ is the connecting homomorphism arisen from the co-homology long exact sequence of the triple p Q, B Q, B Q zp P zB P qq .Proof. To prove the first claim, notice that the map φ P : H ˚ p P, B P q – ÝÑ H ˚ G ˚ P p X P , X B P q Ñ H ˚ G p X P , X B P q is the composite of the isomorphism (4.1) and the map in equivariant cohomologyinduced by the canonical projection G Ñ G ˚ P “ G { G P . Suppose that F , . . . , F k are the facets containing P . Using (i), we can define a map ψ P : H ˚ p P, B P q b R P Ñ H ˚ G p X P , X B P q CHAVES by setting ψ P p α b t m F ,P ¨ ¨ ¨ t m k F k ,P q “ φ P p α q e m F ,P ¨ ¨ ¨ e m k F k ,P . On the other hand, wehave an isomorphism of algebras ρ : H ˚ G p X P , X B P q Ñ R P b H G ˚ P p X P , X B P q Ñ H ˚ p P, B P q b R P by choosing a splitting of G “ G P ˆ G ˚ P . In particular, for e F,P P H p X P , X B P q , wehave that ρ p e F,P q P H p P, B P q b p R P q ‘ H p P, B P q b p R P q – p R P q ‘ H p P, B P q .As t F,P is the restriction of e F,P to R P we have then that ρ p e F,P q “ t F,P ` a F forsome a F P H p P, B P q . Using this computation we get that for α P H ˚ p P, B P q itholds that ρ ˝ ψ P p α b t m F ,P ¨ ¨ ¨ t m k F k ,P q “ ρ p φ P p α q e m F ,P ¨ ¨ ¨ e m k F k ,P q“ p α b q ρ p e F ,P q m ¨ ¨ ¨ ρ p e F k ,P q m k “ p α b qp b t F ,P ` a F b q m ¨ ¨ ¨ p b t F k ,P ` b a F k q m k “ α b p t m F ,P ¨ ¨ ¨ t m k F k ,P q ` S where S consists of sum of terms in H ˚ p P, B P q b H ˚ p R P q whose elements in thesecond factor are of degree lower than m ` ¨ ¨ ¨ ` m k ; therefore, we obtained that ρ ˝ ψ P is bijective and so is ψ P .Finally, to prove (iii), we need to check that δψ P p α b t m F ,P ¨ ¨ ¨ t m k F k ,P q “ ψ Q p δ p α qb ρ P Q p t m F ,P b t m k F k ,P qq . As the maps φ P , φ Q arise from natural constructions, theycommute with δ . Furthermore, since ρ P Q p t F,P q is either t F,Q if Q Ď F or zerootherwise, we only need to prove that δ p βe F,P q is either δ p β q e F,Q if Q Ď F or zerootherwise. Recall that δ arises from the connecting homomorphism δ : H ˚ p P, B P q – H ˚ pB Q, B Q zp P zB P qq Ñ H ˚ p Q, B Q q which induces the map δ : H ˚ G p X P , X B P q – H ˚ G p X B Q , X B Q zp P zB Q q q Ñ H ˚` G p X Q , X B Q q . In the case P Ď Q , by the Thom isomorphism theorem we have isomorphisms H ˚´ G p X P zB P q – H ˚ G p X P , X B P q and H ˚´ G p X Q zB Q q – H ˚ G p X Q , X B Q q induced bythe multiplication by e F,P and e F,Q respectively. As both e F,P and e F,Q are re-strictions of the equivariant Euler class of the normal bundle N F , we have that δ p βe F,P q “ δ p β q e F,Q . In the second case, we have that e F,P is then the restric-tion of the Euler class of the normal bundle of X P in X Q as P Ď Q . By theThom-Gysin exact sequence we have that δ vanishes precisely in the multiples of e F,P . (cid:3) For a face P of X { G , the filtration by its faces leads to an spectral sequencewith E -term given by E p,q “ à Q Ď P rank Q “ i H p ` q p Q, B Q q ñ H ˚ p P q the columns of this spectral sequence give rise to a complex that will be denotedby B i p P q . This complex will be related to the Atiyah-Bredon sequence discussedat the beginning of this section and it will provide a criterion to the syzygies in G -equivariant cohomology as it is shown in the following theorem which is analogousto [16, Thm.1.3] for the torus case. Theorem 4.4.
Let X be a G -manifold with a locally standard action of a -torus G . Then H ˚ G p X q is a j -th syzygy over H ˚ p BG ; k q if and only if for any face P of the manifold with corners M “ X { G we have that H i p B ˚ p P qq “ for any i ą max p rank P ´ j, q QUIVARIANT COHOMOLOGY FOR ELEMENTARY 2-ABELIAN GROUPS 9
Proof.
Let Q be a face of X { G . We define the element t Q “ ś Q Ď F Ď X { G t F,Q P R Q . These elements induce an isomorphism of vector spaces R Q – À Q Ď P R P t P .On the other hand, by Proposition 4.3 there is an isomorphism of vector spaces H ˚ p Q, B Q q b R Q Ñ H ˚ G p X Q , X B Q q compatible with the differentials. We have thenan isomorphism(4.2) à Q :rank Q “ i H ˚ p Q, B Q q b R Q – à Q :rank Q “ i H ˚ G p X Q , X B Q q . Noticing that the i -th equivariant skeleton of X is given by X i “ ď P rank P “ i X P “ ď P rank P “ i ` X B P , we see that the last term of (4.2) is the i -th term of the Atiyah-Bredon sequence AB iG p X q and so there is an isomorphism (with an appropriatedegree shift) à Q rank Q “ i à PQ Ď P H ˚ p Q, B Q q b R P t P – à P Ď X { G B i p P q b R P t P – AB iG p X q compatible with the differentials. Therefore, H i p AB ˚ G p X qq “ À P Ď X { G H i p B ˚ p P qqb R P t P .From Proposistion 2.3, we have that H ˚ G p X q is a j -th syzygy if and only if H i p AB G ˚ P p X P qq “ P and i ą max p rank P ´ j, q . The isomorphismabove shows that this condition is equivalent to the vanishing of H i p B ˚ p P qq for all P and i ą max p rank P ´ j, q . (cid:3) We will use this criterion to construct syzygies in G -equivariant cohomology for2-torus actions. The dimension of a manifold with a locally standard action of a2-torus is constrained to the rank of the torus. In fact, if G “ p Z { Z q r and X is a G -manifold with a locally standard action of G and X G ‰ H , then dim X ě r . Infact, if the action is locally standard, then X G is a submanifold of codimension atleast r and there can not be any fixed points if dim X ă r . Example 4.5. If X is a manifold with a locally standard action of Z { Z , then theorbit space M “ X { G is a manifold with boundary. Conversely, any manifold withboundary can be realized as the orbit space of the manifold X “ p M \ M q{B M with the involution induced by the map M \ M Ñ M \ M that swaps factors.The action is locally standard on X as it can be seen as the reflection along thehyperplane where B M lies and so X G “ B M .Theorem 4.4 translates in this case on the statement that X is G -equivariantlyformal if and only if the map H ˚ pB M q Ñ H ˚` p M, B M q is surjective, or equiv-alently, the map H ˚ p M q Ñ H ˚ pB M q induced by the inclusion is injective. Forexample, if M “ S ˆ r , s is a cylinder, then the map H ˚ p M q Ñ H ˚ pB M q isinjective and so the manifold X is G -equivariantly formal. Then X is homeomor-phic to the torus S ˆ S and the involution is given by the axis reflection on one S -factor.On the other hand, M does not need to be orientable; for example, if M isthe Mobius strip, then M can be realized as the orbit space of a Klein bottle X . Moreover, the map induced in cohomology by the inclusion B M Ñ M is the zeromap and thus Theorem 4.4 implies that H ˚ G p X q is not equivariantly formal.Let G “ Z { Z ˆ Z { Z and let M be a nice manifold with corners locally diffeo-morphic to R n ˆ r , . Then the face lattice of M consists of facets F (of rank 1)and faces P of rank 0. Suppose that X is a p n ` q -manifold with a locally standardaction of G . From Theorem 4.4 we have the following cases. ‚ H ˚ G p X q is a 2-nd syzygy (or equivariantly formal) if and only if for anyfacet F the map H ˚ pB F q Ñ H ˚` p F, B F q is surjective and the sequence À P H ˚ p P q Ñ À F H ˚` p F, B F q Ñ H ˚` p M, B M q Ñ ‚ H ˚ G p X q is a 1-st syzygy if and only if for any facet F the map H ˚ pB F q Ñ H ˚` p F, B F q is surjective and the sequence À P H ˚ p P q Ñ À F H ˚` p F, B F q Ñ H ˚` p M, B M q Ñ Example 4.6.
Let M be the 1-simplex tp x , x q P R , x ` x ď , x , x ě u .The manifold X “ ˜ X { „ can be taken as the real projective space R P as shownin the following figure.Since M and its faces are contractible it is easy to check that the G -equivariantcohomology of X is a free module by looking at the complex B ˚ p P q describedabove. Similarly, the action of G on X can be represented as the reflection alongthe main diagonals on the square. Therefore, X G consists of 3 points and thus b p X q “ b p X G q “ M “ tp u, z q P pr ,
8s ˆ R q : | z i | ` | u i | “ , u ` u ` u “ u and i “ , ,
3, where R ` denotes the non-negative real numbers. Then M isa smooth manifold with corners locally diffeomorphic to r , ˆ R . The pro-jection M Ñ p R q of the first component induces a homeomorphism between M and the subspace of p R q consisting of these triples p u , u , u q such thatmax t| u | , | u | , | u |u ď u ` u ` u “
0. The latter space describes theconfiguration of triangles (including degenerate triangle) in R with sides of lengthat most 1. Therefore, M is homeomorphic to the intersection of a 6-dimensional ballwith a linear subspace of codimension 2 and thus M is topologically a 4-dimensionalball. In particular, B M – S and H ˚ p M, B M q – r H ˚ p S q . Now we will look at theface decomposition of M . QUIVARIANT COHOMOLOGY FOR ELEMENTARY 2-ABELIAN GROUPS 11 ‚ M has exactly one face P of rank zero. Namely, it is given by those elements p u, z q P M such that z i “
0, and then u i P S for all i . Since one of the u s entries depends on the other two, P can be identified with the manifold P “ tp u , u , u q P p S q : u ` u ` u “ u “ tp x, y q P S ˆ S : | x ´ y | “ u P then is the configuration space of equilateral triangles in R with onevertex in the origin and two over the circle. Each of these configurationsis determined by a rotation of any of the pairs p , e iπ { q or p , e ´ iπ { q . Inparticular, this implies that P – S \ S . Thus we have that H p P q – H p P q “ k ‘ k and it is zero in any other degree. ‚ M has three faces of rank 1. Namely, Q , Q and Q where Q ij consistsof the pairs p u, z q P M such that z i “ z j “
0. We identify Q ij with themanifold with boundary Q “ tp x, y q P S ˆ S : | x ´ y | ď u In terms of configuration spaces, this consists of isosceles triangles withone vertex in the origin, two over the circle and whose base is of length atmost 1 (Here we allow the degenerate triangle). We can show that there isa homeomorphism Q – S ˆ I given by a rotation of the pairs p , e itπ { q P Q where ´ ď t ď
1. Computing the relative cohomology H ˚ p Q, P q of thecylinder relative to the boundary we see that H p Q, P q – H p Q, P q – k and it is zero in other degrees. ‚ M has three facets (of rank 2). Namely, F , F , F where F i consists of thepairs p u, z q P M such that z i “
0. We identify F i with the manifold withcorners F “ tp x, y q P S ˆ D : | x ´ y | ď u This space describes the configuration of triangles with one side of length1, and two of length at most 1. Each of these configurations is determinedby a rotation of the pairs p , se itπ { q P F where 0 ď s ď ´ ď t ď F is homeomorphic to S ˆ I ˆ I – S ˆ D . Looking at the relativecohomology H ˚ p F, B F q of the solid torus with respect to its boundary (thetorus) we find that H p F, B F q – H p F, B F q – k and it is zero in otherdegrees.The face lattice of M is then PQ Q Q F F F M Consider the manifold X “ tp z, u q P p R ˆ R q : | z i | ` | u i | “ , u ` u ` u “ u with the locally standard action of G on X given by multiplication on the variables z i . Then X is G -locally standard manifold and X { G – M . We will see that the G -equivariant cohomology of X is a first syzygy but not a second syzygy usingTheorem 4.4. It is a first syzygy as the maps H ˚ p P q Ñ H ˚` p Q, P q H ˚ p Q j , P q ‘ H ˚ p Q k , P q Ñ H ˚` p F i , B F i q , à i “ H ˚ p F i , B F i q Ñ H ˚` p M, B M q are surjective as it can be seen by using the explicit computation of these groupsmentioned above. On the other hand, H ˚ G p X q is not a second syzygy as the sequence à j “ H ˚ p Q j , B Q j q Ñ à i “ H ˚` p F i , B F i q Ñ H ˚` p M, B M q Ñ H p B ˚ p M qq ‰
0. In fact, the complex B ˚ p M q takes the form k Ñ k Ñ Ñ ˚ “
1. The map k Ñ k is given by p a, b, c q “ p a ` b, a ` c, b ` c q which is ofrank 2 and then H p B ˚ p M qq ‰ X with an action of G “ p Z { Z q suchthat the equivariant cohomology H ˚ G p X q is torsion-free but not free as H ˚ p BG q -module. The manifold X realizes the smallest possible dimension where a manifoldwith a locally standard action of a 2-torus G whose equivariant cohomology istorsion-free but not free exists. As we previously discussed, if rank G ď G “
3. On the other hand, if the dimension of amanifold is the same as the rank of the torus, its G -equivariant cohomology is freeif and only if it is torsion-free [16, § X ą References [1] C. Allday, M. Franz, and V. Puppe. Equivariant cohomology, syzygies and orbit structure.
Transactions of the American Mathematical Society , 366(12):6567–6589, 2014.[2] C. Allday, M. Franz, and V. Puppe. Syzygies in equivariant cohomology in positive charac-teristic.
ArXiv preprint, arxiv:2007.00496 , 2020.[3] M. F. Atiyah.
Elliptic operators and compact groups , volume 401. Springer, 1974.[4] M. F. Atiyah. Convexity and commuting hamiltonians.
Bulletin of the London MathematicalSociety , 14(1):1–15, 1982.[5] A. Borel. Seminar on transformation groups.
Bull. Amer. Math. Soc , 67:450–454, 1961.[6] G. E. Bredon. The free part of a torus action and related numerical equalities.
Duke Mathe-matical Journal , 41(4):843–854, 1974.[7] W. Bruns and U. Vetter.
Determinantal rings , volume 1327. Springer, 2006.
QUIVARIANT COHOMOLOGY FOR ELEMENTARY 2-ABELIAN GROUPS 13 [8] V. M. Buchstaber and T. E. Panov.
Toric topology , volume 204. American MathematicalSoc., 2015.[9] J. Cerf. Topologie de certains espaces de plongements.
Bulletin de la Soci´et´e Math´ematiquede France , 89:227–380, 1961.[10] S. Chaves. The equivariant cohomology for semidirect product actions.
ArXiv preprint,arxiv:2009.08526 , 2020.[11] M. W. Davis, T. Januszkiewicz, et al. Convex polytopes, coxeter orbifolds and torus actions.
Duke Mathematical Journal , 62(2):417–451, 1991.[12] A. Douady. Vari´et´es `a bord anguleux et voisinages tubulaires.
S´eminaire Henri Cartan , 14:1–11, 1961.[13] D. Eisenbud.
The geometry of syzygies: a second course in algebraic geometry and commu-tative algebra , volume 229. Springer Science & Business Media, 2005.[14] M. Franz. Big polygon spaces.
International Mathematics Research Notices , 2015(24):13379–13405, 2015.[15] M. Franz. Syzygies in equivariant cohomology for non-abelian lie groups.
ConfigurationSpaces , pages 325–360, 2016.[16] M. Franz. A quotient criterion for syzygies in equivariant cohomology.
TransformationGroups , 22(4):933–965, 2017.[17] K. J¨anich. On the classification of regular O p n q -manifolds in terms of their orbit bundles. Proceedings of the Conference on Transformation Groups , 1968.[18] G. Laures. On cobordism of manifolds with corners.
Transactions of the American Mathe-matical Society , 352(12):5667–5688, 2000.[19] Z. L¨u and M. Masuda. Equivariant classification of 2-torus manifolds.
ArXiv preprint,arXiv:0802.2313 , 2008.[20] V. Puppe. Equivariant cohomology of p Z q r -manifolds and syzygies. Fundamenta Mathemat-icae , pages 1–20, 2018.[21] L. Yu. On the constructions of free and locally standard Z { Osaka Journal of Mathematics , 49(1):167–193, 2012.(Sergio Chaves)
Department of Mathematics, Western University, London, ON. Canada
E-mail address ::