The equivariant cohomology for semidirect product actions
aa r X i v : . [ m a t h . A T ] S e p THE EQUIVARIANT COHOMOLOGY FOR SEMIDIRECTPRODUCT ACTIONS
SERGIO CHAVES
Abstract.
The rational Borel equivariant cohomology for actions of a com-pact connected Lie group is determined by restriction of the action to a maxi-mal torus. We show that a similar reduction holds for any compact Lie group G when there is a closed subgroup K such that the cohomology of the classi-fying space BK is free over the cohomology of BG for field coefficients. Westudy the particular case when G is a semi-direct product and K is its maxi-mal elementary abelian 2-subgroup for cohomology with coefficients in a fieldof characteristic two. This provides a different approach to investigate thesyzygy order of the equivariant cohomology of a space with a torus action anda compatible involution, and we relate this description with results for 2-torusactions. Introduction
Let G be a compact group and X be a finite G -CW complex. The G -equivariantcohomology of X with coefficients over a field k is defined as the singular cohomologyof the homotopy quotient H ˚ G p X ; k q : “ H ˚ p EG ˆ G X ; k q . It becomes canonicallya module over the cohomology of the classifying space H ˚ p BG ; k q (cohomologycoefficients will be omitted as long as there is no ambiguity). We say that X is G -equivariantly formal over k if the restriction map H ˚ G p X q Ñ H ˚ p X q is surjective.In this case, the Leray-Hirsch theorem implies that H ˚ G p X q is a free module over H ˚ p BG q .Freeness of the equivariant cohomology has been generalized to the study ofsyzygy modules. A detailed discussion of this topic was started by Allday-Franz-Puppe [3] for torus actions with cohomology over a field of characteristic zero.Recall that a finitely generated module M over a commutative ring R is a j -thsyzygy if there is an exact sequence(1.1) 0 Ñ M Ñ F Ñ ¨ ¨ ¨ Ñ F j of free modules F k for 1 ď k ď j . If R is a polynomial algebra in n variables over afield k , then the n -th syzygy modules correspond to the free ones as a consequenceof the Hilbert Syzygy theorem. In [3, Thm.5.7], the authors showed that the syzygyorder of the equivariant cohomology of a space with a torus action is equivalent tothe partial exactness of the Atiyah-Bredon sequence of such a space [5] [11]. Thissequence is defined in the following way: for a T -space X where T “ p S q n , thefiltration of X by the dimension of its orbits X “ X T Ď X Ď ¨ ¨ ¨ Ď X n “ X Date : October 1, 2020. induces a sequence0 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 . These results in equivariant cohomology for torus actions are key to extend thestudy of syzygies in equivariant cohomology for any compact connected Lie groupactions by considering the restriction of the action to a maximal tori [22], andelementary p -abelian groups actions by restriction and transfer of the action to atorus one [4].In this paper, we study a more general problem of characterizing syzygies inequivariant cohomology for compact Lie group actions in terms of the action toa suitable closed subgroup K Ď G , and considering coefficients over an arbitraryfield k . This generalizes the relation between compact connected Lie groups (resp. p -abelian groups) and their maximal tori in equivariant cohomology as discussedbefore. The methods developed in this paper also provide a new tool to recoverthese results.Let us consider a compact Lie group G and let K Ď G be a closed subgroup. Wedenote by W “ N G p K q{ K the Weyl group of K in G . Let us consider cohomologywith coefficients over a field k (that we omit it in our notation). Suppose that thecanonical map H ˚ p BK q Ñ H ˚ p G { K q arising from the fibration G { K Ñ BK Ñ BG is surjective and that there is an isomorphism of algebras H ˚ p BK q W – H ˚ p BG q .The following result provides a general reduction in equivariant cohomology. Theorem 1.1.
Let X be a G -space such that H ˚ p X q G “ H ˚ p X q , then W actson the K -equivariant cohomology of X and there is a natural isomorphism of H ˚ p BG q -algebras H ˚ G p X q – H ˚ K p X q W and a natural isomorphism of H ˚ p BK q -algebras H ˚ K p X q – H G p X q b H ˚ p BG q H ˚ p BK q . Observe that the condition of the map H ˚ p BK q Ñ H ˚ p G { K q being surjectiveallows us to describe the cohomology of the homogeneous spaces G { K in terms of thecohomology of the classifying space of G and K . This applies to the cohomology ofhomogeneous spaces of Lie groups [15], [29], [8] and for the equivariant cohomologyof Hamiltonian actions of non-abelian compact connected Lie groups in symplecticgeometry [7]. A pair of groups p G, K q satisfying this condition will be called a freeextension pair following the notation introduced in [7].As a consequence of Theorem 1.1, we can characterize syzygies in G -equivariantcohomology in terms of the K -equivariant cohomology. Corollary 1.2.
The module H ˚ G p X q is a j -th syzygy over H ˚ p BG q if and only if H ˚ K p X q is a j -th syzygy over H ˚ p BK q . With our methods, we recover in equivariant cohomology classical results forcompact connected Lie groups with cohomology over rational coefficients [30, Ch.III.Lem.4.12],or for finite groups with abelian p -Sylow subgroup with cohomology over a field ofcharacteristic p [12, Ch.III.Thm.10.3]. Moreover, they also allow us to study theequivariant cohomology for actions of semi-direct product of groups or groups fit-ting in a group extension. For example, actions of matrix groups, dihedral andsymmetric groups and torus action with compatible involutions. In particular, thelatter case has been of interest in symplectic geometry. Namely, let M be a symplec-tic manifold with a symplectic action of a torus T and an antisymplectic compatibleinvolution τ . The maximal elementary 2-abelian subgroup T of T acts on the fixed QUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS 3 point M τ and if M is T -equivariantly formal over Q , then M τ is T -equivariantlyformal over F [20],[6], [18].This situation motivates our study of the equivariant cohomology for semi-directproduct actions; in particular, we use it to approach the symplectic setting describedabove by considering the equivariant cohomology for actions of the group T ¸ Z { Z .We first prove a canonical description of the F -cohomology of the classifying space B p T ¸ Z { Z q as a tensor product of the cohomologies of BT and B Z { Z . Moreover,we describe the equivariant formality and syzygies in equivariant cohomology foractions of this group in terms of the maximal 2-elementary subgroup K and theinvariants under the action of the Weyl group W “ N G p K q{ K as stated in thefollowing result. Theorem 1.3.
Let G “ T ¸ Z { Z , let H be the maximal -elementary subgroup of G and W be the Weyl group of H in G . Then p G, H q is a free extension pair over F , and there is an isomorphism of algebras H ˚ p BG q – H ˚ p BH q W that extends toequivariant cohomology for any G -space X . In particular, H ˚ G p X q is a j -th syzygyover H ˚ p BG q if and only if H ˚ H p X q is a j -th syzygy over H ˚ p BH q . As consequences of this result, we recover the equivariant formality of the reallocus of conjugation spaces [26] and Hamiltonian torus actions on symplectic man-ifold when the cohomology of the space contains no 2-torsion.This document is organized as follows: In Section 2, we discuss free extensionpairs and the reduction of syzygies in equivariant cohomology for a pair of groupssatisfying this property. In Section 3, we approach torus actions and compatibleinvolutions by looking at the induced action of the semi-direct product of a torusand a 2-tori. Syzygies in equivariant cohomology for actions of such groups canbe reduced to 2-torus actions as discussed in Section 4. In Section 5, we discussa topological generalization of Hamiltonian actions on a symplectic manifold withan anti-symplectic compatible involution using the results discussed in the previoussections. Finally, in the last section of this document, we study a canonical semi-direct product action on the big polygon spaces to realize all possible syzygy ordersin equivariant cohomology analogous to the torus actions case [21] and we will relatethese results to the real big polygon spaces recovering some of the work discussedin [31].
Acknowledgments.
I would like to thank Matthias Franz for his collaborationand helpful discussions to develop this project. His suggestions about generalizingHamiltonian torus actions with an anti-symplectic involution in equivariant coho-mology was a fundamental pillar to obtain the results presented in this paper, aswell as the application on big polygon spaces. I am also grateful to Jeffrey Carlsonhis fruitful discussions. Finally, I want to thank Matthias Franz and F´elix BarilBoudreau for their comments and feedback on earlier versions of this document.2.
Free extension pairs
Let G be a compact connected Lie group and T be a maximal torus in T . Therational equivariant cohomology of a G -space X is completely determined by theinduced action of T on X ; namely, there is an isomorphism of H ˚ p BT ; Q q -algebras(2.1) H ˚ T p X ; Q q – H ˚ G p X ; Q q b H ˚ p BG ; Q q H ˚ p BT ; Q q CHAVES and an isomorphism of H ˚ p BG ; Q q -algebras(2.2) H ˚ G p X ; Q q – H ˚ T p X ; Q q W where W “ N G p T q{ T is the Weyl group of T in G , and the action of W on T by conjugation extends to one on the Borel construction X T [28, Thm.2.2]. Suchan isomorphism also holds over Z if H ˚ p G ; Z q contains no torsion and over F p if H ˚ p G ; F p q contains no p -torsion as generalized in [10, Ch.VII]. These results fromLeray and Borel are discussed for principal G -bundles and can be adapted to theequivariant cohomology setting,The isomorphism (2.1) can be used to characterize free and torsion-free mod-ules in G -equivariant cohomology, and more generally, the syzygy modules for anycompact connected Lie group in terms of the restricted action to the maximal torus T in G [22, Prop.4.2]. The key ingredient is the fact that H ˚ p BT ; Q q becomes afree module over H ˚ p BG ; Q q via the map induced by the inclusion T Ñ G . Wegeneralize this situation by introducing the following definition. Definition 2.1.
Let G be a compact Lie group and K Ď G be a closed sub-group. The pair p G, K q has the free extension property over a field k if the map H ˚ p BK ; k q Ñ H ˚ p G { K ; k q is surjective. Observe that this is equivalent to the degeneracy at the E -term of the cohomo-logical Serre spectral sequence associated to the fibration G { K Ñ BK Ñ BG andthe trivial action of G on G { K in cohomology. In particular, H ˚ p BK ; k q becomesa finitely generated free H ˚ p BG ; k q -module by the Leray-Hirsch theorem. Recallthat we will often omit the coefficient field k in our notation for cohomology. Proposition 2.2.
Let K Ď H Ď G be a sequence of groups such that p G, H q and p H, K q are free extension pairs. The following statements are equivalent.(1) p G, K q is a free extension pair.(2) The action of G on H ˚ p G { K q is trivial.(3) The cohomological Serre spectral sequence arising from the fibration H { K Ñ G { K Ñ H { K degenerates at E .Proof. For any space X , let P X p t q denote the Poincar´e series of X with coefficientsin the field k . As p G, H q and p G, K q are free extension pairs, we get that P BK p t q “ P BG p t q P G { H p t q P H { K p t q and so H ˚ p BK q is a free module of rank b p G { H q b p H { K q .This implies that the cohomological Serre spectral sequence arisen from the fibration G { K Ñ BK Ñ BG degenerates at E and H ˚ p BK q – H ˚ p BG ; H ˚ p G { K qq – H ˚ p BG q b H ˚ p G { K q G as H ˚ p BG q -modules. Then P G { H p t q P H { K p t q “ P G { K p t q ifand only if G acts trivially on the cohomology of G { K . (cid:3) Proposition 2.3.
Let F Ñ E p ÝÑ B be a fibration such that the map H ˚ p E q Ñ H ˚ p F q is surjective. Let X be a connected space and f : X Ñ B be a continuousmap. Then in the pullback fibration F Ñ X f : “ f ˚ E q ÝÑ X , the map H ˚ p X f q Ñ H ˚ p F q is also surjective and there is an isomorphism of H ˚ p E q -modules H ˚ p X f q – H ˚ p X q b H ˚ p B q H ˚ p E q . QUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS 5
Proof.
The surjectivity of the map H ˚ p X f q Ñ H ˚ p F q follows from the commuta-tivity of the diagram F X f XF E B g f
We can choose an additive section α : H ˚ p F q Ñ H ˚ p E q of the surjective map i ˚ : H ˚ p E q Ñ H ˚ p F q that induces an isomorphism of H ˚ p B q -modules θ : H ˚ p B q b H ˚ p F q Ñ H ˚ p E q given by θ p a b t q “ p ˚ p a q α p t q using the Leray-Hirsch theorem.Similarly, the composite β “ g ˚ ˝ α is an additive section of j ˚ : H ˚ p X f q Ñ H ˚ p F q and there is an induced isomorphism of H ˚ p X q -modules φ : H ˚ p X q b H ˚ p F q Ñ H ˚ p X f q given by φ p b b t q “ q ˚ p b q β p t q . Under these choices, there is a commutativediagram H ˚ p B q b H ˚ p F q H ˚ p E q H ˚ p X q b H ˚ p F q H ˚ p X f q θf ˚ b id g ˚ φ Now we will show that the canonical map K : H ˚ p X q b H ˚ p B q H ˚ p E q Ñ H ˚ p X f q given by b b x ÞÑ q ˚ p a q g ˚ p x q is an isomorphism. It follows from the commutativediagram H ˚ p X q b H ˚ p F q H ˚ p X f q H ˚ p X q b H ˚ p B q H ˚ p B q b H ˚ p F q H ˚ p X q b H ˚ p B q H ˚ p E q H ˚ p X f q φ – id b θ K where all maps are isomorphisms. That the map K is one of H ˚ p E q -modulesfollows from naturality of the construction with respect to the map f : X Ñ B andconsidering the particular case id : B Ñ B . (cid:3) The following result is a particular case of the previous proposition.
Proposition 2.4.
Let p G, K q be a free extension pair over k and X be a G -space.There is a natural isomorphism of H ˚ p BK q -modules. (2.3) H ˚ K p X q – H ˚ G p X q b H ˚ p BG q H ˚ p BK q where X is a K -space by restriction of the G -action.Proof. The Borel constructions X K and X G sit in a pullback diagram X K X G BK BG where the horizontal maps are fibrations with fiber G { K . Since p G, K q is a free ex-tension pair, the map H ˚ p BK q Ñ H ˚ p G { K q is surjective and thus the isomorphism(2.3) follows by applying Proposition 2.3. CHAVES (cid:3)
This result allows us to describe the syzygies in G -equivariant cohomology interms of the K -equivariant cohomology analogously to the reduction from non-abelian compact connected Lie group actions to torus actions [22, Prop.4.2] as westate in the following result. Proposition 2.5.
Let p G, K q be a free extension pair and X be a G -space. Forany j ě , H ˚ G p X q is a j -th syzygy over H ˚ p BG q if and only if H ˚ K p X q is a j -thsyzygy over H ˚ p BK q .Proof. We use the characterization of syzygies via regular sequences [13, § R, S be rings such that S is a free finitely generated R -module. Let A be an S -algebra and B and R -algebra such that A – B b R S as S -modules. Then A is a j -th syzygy over S if and only if B is a j -th syzygy over R . The result then follows by combining these facts, the remark after Definition2.1 and Proposition 2.4. (cid:3) Besides the example where G is a compact connected Lie group and K “ T isthe maximal torus, we can also find free extension pairs in the following situations. Example 2.6.
Let k denote field of characteristic p and n ą p G, K q is a free extension pair. ‚ When G is a torus and K is the maximal elementary abelian p -subgroup of K [16, § ‚ When G is a finite abelian group and K is the subgroup isomorphic to theproduct of cyclic groups of order divisible by p in the elementary decomposition of G [12, Ch.III. § ‚ . Let p “
2. When G “ O p n q , SO p n q , U p n q or SU p n q and K is the maximalelementary abelian p -subgroup of G [30, Ch.III. § ‚ Let p “
2. When G “ SU p q and H “ Q is the quaternionic group [1,Ex.2.10]. Proposition 2.7.
Let n ě and G n , K n be one of the following pair of groups(a) Let G n “ SO p n q and K n “ O p n q .(b) Let G n “ SU p n q and K n “ U p n q .(c) Let G n “ Sp p n q and K n “ Sp p n q ¸ Sp p q .There is an embedding of K n on G n ` such that p G n ` , K n q is a free extension pairover F in case (a) and over an arbitrary field in cases (b) and (c).Proof. Let Σ n “ R P n , C P n or H P n and L “ O p q , U p q or Sp p q in cases p a q , p b q or p c q respectively. There is a transitive action of G n ` on Σ n and thus the equivari-ant cohomology H ˚ G n ` p Σ n q is isomorphic to H ˚ p B p G n ` q x q . Using homogeneouscoordinates on Σ n , we see that the isotropy group of x “ r ¨ ¨ ¨ : 0 : 1 s is given by p G n ` q x – K n and thus the inclusion map p G n ` q x Ñ G n ` induces an embeddingof K n into G n ` . On the other hand, the restriction map H ˚ G n ` p Σ n q Ñ H ˚ p Σ n q is the restriction of the first characteristic class to a finite approximation whichshows that Σ n is G n ` -equivariantly formal. Combining both facts we have that p G n ` , K n q is a free extension pair and the cohomological Serre spectral sequenceassociated to the fibration Σ n Ñ BK n Ñ BG n ` degenerates at the E -page. QUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS 7
This argument can be generalized for any Grassmanian as follows. Let Σ n.k “ Gr p n, k q be the Grassmannian of k -dimensional planes in K n . There is a canonicaltransitive action of K n on Σ n,k and thus H ˚ K n p Σ n,k q – H ˚ p B p Σ n q X q for a chosen X P Gr p n, k q . If X “ x e , . . . , e k y , we see that p Σ n q X – S p G k ˆ G n ´ k q .There is a short split exact sequence0 Ñ S p G k ˆ G n ´ k q Ñ G k ˆ G n ´ k Ñ L Ñ H ˚ p BG k q b H ˚ p BG n ´ k q – H ˚ p BL q b H ˚ p BS p G k ˆ G n ´ k q and H ˚ p BS p G k ˆ G n ´ k qq – p H ˚ p BG k qb H ˚ p BG n ´ k qqb H ˚ p BL q k . Recall that H ˚ p Σ n,k q – H ˚ p G k q{ I n and so Σ n,k is K n -equivariantly formal; inparticular, this implies that p K n , S p G k ˆ G n ´ k qq is a free extension pair. (cid:3) Recall that for K Ď G a closed subgroup of a Lie group G , the Weyl group of K in G is defined as W “ N G p K q{ K where N G p K q denotes the normalizer of K in G . Theorem 2.8.
Let p G, K q be a free extension pair. Let W be the Weyl groupof K in G and suppose that there is an isomorphism of algebras H ˚ p BK q W – H ˚ p BG q . Then for any G -space X such that H ˚ p X q G “ H ˚ p X q , W acts on the K -equivariant cohomology of X and there is a natural isomorphism of H ˚ p BG q -modules H ˚ K p X q W – H ˚ G p X q .Proof. The map c g given by the conjugation of a chosen element g P G inducesthe identity map in the cohomology of H ˚ p BG q . This can be shown using Milnor’sjoin construction of BG . Moreover, this map induces a map X G Ñ X ˜ G where X is a ˜ G -space with the action of G induced by the map c g . Notice that the ˜ G -equivariant cohomology of X is isomorphic to its G -equivariant cohomology. Since G acts trivially on the cohomology of X , a (Serre) spectral sequence argument showsthat the map c g induces the identity on H ˚ G p X q . Therefore, there is a well definedaction of W on both H ˚ p BK q and H ˚ K p X q and the previous argument shows that H ˚ G p X q Ď H ˚ K p X q W . Moreover, the canonical map H ˚ G p X q b H ˚ p BG q H ˚ p BK q Ñ H ˚ K p X q of algebras is W -equivariant and an isomorphism by Proposition 2.4 andthus H ˚ K p X q – H ˚ G p X q W as H ˚ p BG q -modules since W acts trivially on H ˚ G p X q and H ˚ p BK q W – H ˚ p BG q by assumption. (cid:3) Observe that Propositions 2.4 and Theorem 2.8 can be summarized in Theorem1.1 as discussed at the beginning of this document. In the rest of this section, wediscuss the case when G is a semi-direct product; we first start with the followingresult. Proposition 2.9.
Let G and K be groups. Suppose that there is a subgroup N Ď G and a group homomorphism φ : K Ñ Aut p G q such that φ p k q| N is the identity forall k P K . Then p G, N q is a free extension pair if and only if p G ¸ φ K, N ˆ K q alsois.Proof. Under these assumptions, there are canonical isomorphisms H ˚ p B p N ˆ K qq – H ˚ p BN q b H ˚ p BK q and H ˚ pp G ¸ φ K q{p N ˆ K qq – H ˚ p G { N q . The com-mutative diagram H ˚ p BN q b H ˚ p BK q H ˚ pp G ¸ φ K q{p N ˆ K qq H ˚ p BN q H ˚ p G { N q CHAVES and the surjectivity of the vertical arrows implies that the top horizontal arrow issurjective if and only if the bottom also is. (cid:3)
It is not difficult to check from the definition of free extension pairs and the factthat the classifying space functor preserves finite products, that the product of twofree extension pairs is again a free extension pair.In the following remark we discuss a more general criterion for the productproperty of free extension pairs.
Remark 2.10.
Let p G, N q and p K, L q be two free extension pairs. Suppose thatthere is a group homomorphism φ : K Ñ Aut p G q such that φ k p N q Ď N then themap H ˚ p B p N ¸ L qq Ñ H ˚ p BN q is surjective if and only if p G ¸ ϕ K, N ¸ ϕ L q is afree extension pair. Let G “ N ¸ T be a semi-direct product group. The G -equivariant cohomologycan be computed stepwise as in the direct product case; namely, for a a G -space X , there is an isomorphism of k -algebras H ˚ G p X q – H ˚ K p X N q As a consequence ofProposition 2.9 and Remark 2.10, we can recover the free extension property for thematrix groups G n and K n of Proposition 2.7 since K n – G n ¸ K n { G n . Notice thatthe group L “ K n { G n is unique up to isomorphism for all n ą
1. We summarize itin the following corollary.
Corollary 2.11.
Let L “ K n { G n . The pairs p G n ` , L n q and p K n , L n q are freeextension pairs. In particular, for k “ F , L n can be replaced by the its maximalelementary abelian -subgroup. Torus actions and compatible involutions
In this section, we will consider cohomology with coefficients over k “ F (andwe will omit it in our notation). Let X be a space with an action of a torus T and let τ : X Ñ X be an involution. We say that τ is compatible if τ p g ¨ x q “ g ´ ¨ τ p x q for any g P T and x P X . Examples of such spaces appear naturallyas toric varieties in algebraic geometry, Hamiltonian torus actions on symplecticmanifolds and topological generalizations of these spaces as quasitoric manifolds,torus manifolds and moment angle complexes [17], [24], [14].The aim of this section is to describe the equivariant cohomology for any T -space X with a compatible involution τ . Firstly, notice that in this case there is awell defined action of the group G “ T ¸ τ Z { Z where Z { Z acts on T by inversion.Conversely, an action of G on X induces an action of T with a compatible involution τ on X . Therefore, the equivariant cohomology of a T -space with a compatibleinvolution can be approached by studying the G -equivariant cohomology of X .More generally, let m and n be positive integers. We will consider actions ofthe semidirect product group G “ T ¸ K where T “ p S q n , K “ p Z { Z q m and σ ¨ g “ g ´ for g P T and a generator σ P K . By our assumption on the base field k , the cohomology of the classifying spaces BT and BK are polynomial rings. Infact, we will show that under our assumptions there is a canonical isomorphism ofalgebras H ˚ p BG q – H ˚ p BT qb H ˚ p BK q and thus H ˚ p BG q is a polynomial algebrain p n ` m q -variables as we state in the following result. Theorem 3.1.
There is a unique graded algebra isomorphism H ˚ p BG q – H ˚ p BT qb H ˚ p BK q such that QUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS 9 ‚ The canonical map i ˚ : H ˚ p BG q Ñ H ˚ p BT q induced by the inclusion issurjective and ker p i ˚ q – p H ˚ p BK q ` q is the ideal generated by the positivedegree cohomology of BK . ‚ The canonical map p ˚ : H ˚ p BK q Ñ H ˚ p BG q induced by the projection isinjective and Coker p p ˚ q – H ˚ p BT q . ‚ There is an algebra homomorphism ϕ : H ˚ p BT q Ñ H ˚ p BG q , such that thecomposite i ˚ ˝ ϕ is the identity over H ˚ p BT q and Coker p ϕ q – H ˚ p BK q . ‚ The map j ˚ : H ˚ p BG q Ñ H ˚ p BK q induced by the inclusion j : t e u ˆ K Ñ G has kernel p H ˚ p BT q ` q and the composite j ˚ ˝ p ˚ is the identity over H ˚ p BK q .Proof. To compute the cohomology of BG , notice that the short exact sequence1 Ñ T Ñ G Ñ K Ñ BT Ñ BG Ñ BK.
Observe that the action of π p BK q on H ˚ p BT q is induced by the action of K on T and hence on BT . In fact, each generator σ P K , induces an action on the integralcohomology H ˚ p BT ; Z q given by multiplication by -1 and thus trivial in cohomologyover F . Therefore, the E -term of the Serre spectral sequence associated to thefibration 3.1 is given by E p,q – H p p BK ; H q p BT qq ñ H ˚ p BG q and then we have an isomorphism of algebras E – H ˚ p BK q b H ˚ p BT q We will show that this spectral sequence degenerates at this term by induction on m “ rank K . Let us assume then that m “
1. By degree reasons, the only possiblenon-zero differential d is determined by d : E , Ñ E , . Choose generators x i P H p BT q for 1 ď i ď n and t P H p BK q . Under these identifications, we havethat d p x i q “ α i t with either α i “ α i “ T T ¸ Z { Z {
21 1 ¸ Z { Z { ✲ ✲✲✻ ✲✻ ✻ induces a map of spectral sequences E p,qs Ñ r E p,qs , where r E – H ˚ p B Z { q is the r E page of the spectral sequence associated to the bottom exact sequence. This impliesthat d “ r d “ r E ˚ degenerates at the E -term. We have then an isomorphismof H ˚ p BK q -modules H ˚ p BK q b H ˚ p BT q – H ˚ p BG q . On the other hand, since H ˚ p BT q is a finitely generated polynomial algebra,we can choose a multiplicative section ˜ ϕ : H ˚ p BT q Ñ H ˚ p BG q of the surjective map of H ˚ p BG q Ñ H ˚ p BT q induced by the inclusion map. Therefore, such amap together with the canonical map p ˚ : H ˚ p BK q Ñ H ˚ p BG q gives rise to anisomorphism of graded H ˚ p BK q -algebras˜ θ : H ˚ p BK q b H ˚ p BT q Ñ H ˚ p BG q given by ˜ θ p α b β q “ p ˚ p α q ˜ ϕ p β q .Under this isomorphism, the canonical map induced by the inclusion j : B p ˆ K q Ñ BG might satisfy j p x i q “ w for some 1 ď i ď n . If that is the case, thenwe consider the section ϕ p x i q “ ˜ ϕ p x i q ` w if ˜ ϕ p x i q “ w and ϕ p x i q “ ˜ ϕ p x i q if˜ ϕ p x i q “
0. As discussed before, it induces an isomorphism of algebras θ : H ˚ p BK q b H ˚ p BT q Ñ H ˚ p BG q ;furthermore, such a section is unique since it is determined by the condition j ˚ ϕ “
0, and thus it makes the isomorphism θ unique as well. Therefore, thecomposite j ˚ θ : H ˚ p BK q b H ˚ p BT q Ñ H ˚ p BG q Ñ H ˚ p BK q has kernel H ˚ p BT q ` . Now notice that the composite H ˚ p BT q ϕ ÝÑ H ˚ p BG q i ˚ ÝÑ H ˚ p BT q where i ˚ is induced by the inclusion T Ñ G is the identity on H ˚ p BT q since i ˚ p w q “ ϕ was constructed as a section of this map. This implies that themaps H ˚ p BT q ϕ ÝÑ H ˚ p BG q and H ˚ p BG q i ˚ ÝÑ H ˚ p BT q coincide with the canonical inclusion and restriction respectively. Using a similarargument over the composite H ˚ p BK q p ˚ ÝÝÑ H ˚ p BG q j ˚ ÝÑ H ˚ p BK q , which is theidentity over H ˚ p BK q , we conclude that the map H ˚ p B Z { q p ˚ ÝÝÑ H ˚ p BG q haskernel H ˚ p BT q ` .The inductive argument follows in a similar fashion, noticing that G – p T ¸ ˆ K q¸ Z { Z where rank p K q “ m ´ B p T ¸ ˆ K q Ñ BG Ñ B Z { Z . (cid:3) Now we will study the algebraic properties of the G -equivariant cohomologyas a module over H ˚ p BG q . Notice that for any G -space X , there is an inducedinvolution τ on the space X T ; moreover, the Borel constructions X G and p X T q τ are homotopic. Using this remark we prove the following result. Proposition 3.2.
Let X be a G -space and assume that X is T -equivariantly formal.Then X is G -equivariantly formal if and only if the Borel construction X T is G { T -equivariantly formal.Proof. Write G “ T ¸ K where K “ x τ y – G { T . Firstly, let us suppose that X is G -equivariantly formal, by Theorem 3.1 and the above remark we get isomorphisms H ˚ τ p X T q – H ˚ G p X q – H ˚ p BG q b H ˚ p X q– H ˚ p BK q b H ˚ p BT q b H ˚ p X q– H ˚ p BK q b H ˚ p X T q QUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS 11 and so X T is K -equivariantly formal. Reversing the above sequence of isomor-phisms, the converse of the statement holds. However, we need to be careful withthe H ˚ p BG q -module structure of H ˚ G p X q and the H ˚ p BK q -module structure of H ˚ K p X T q . From the diagram H ˚ G p X q H ˚ K p X T q H ˚ p BG q H ˚ τ p BT q H ˚ p Bτ q H ˚ p Bτ q –– and the canonical isomorphism H ˚ p BG q – H ˚ p Bτ q b H ˚ p BT q constructed in The-orem 3.1, the H ˚ p Bτ q -module structure on H ˚ τ p X T q coincides with the restrictionof the H ˚ p BG q -module structure on H ˚ G p X q to the action of those elements of theform α b P H ˚ p Bτ q b H ˚ p BT q – H ˚ p BG q . (cid:3) Now we will apply this theorem to the conjugation spaces introduced by Haussmann-Holm-Puppe [26]; among these spaces, complex Grassmannian, toric manifolds,polygon spaces and some symplectic manifolds fit. A conjugation space X satisfiesthat H odd p X q “ T -equivariantly formal for anyaction a torus T . They also satisfy the following property [26, Thm.7.5]. Theorem 3.3.
Let X be a conjugation space with conjugation τ . Suppose thata torus T acts on X and that the action is compatible with τ . Then X T is aconjugation space where the conjugation on X T is the one induced by τ . This theorem shows that X T is τ -equivariantly formal. Then immediately fromTheorem 3.2 we obtain the following result. Corollary 3.4.
Let X be a T -space which is also a conjugation space with a com-patible involution τ . Then X is G -equivariantly formal. Reduction to 2-torus actions
In this section, we will use the results from section 2 on free extension pairs tostudy the equivariant cohomology for torus actions and compatible involutions byreducing to the maximal elementary 2-abelian subgroup (or 2-torus). Let H “ T ˆ K – p Z { Z q n ` m denote the maximal 2-torus subgroup in G “ T ¸ K where T ď T is the maximal 2-torus subgroup in T . Since p S , Z { Z q is a free extension pair over F , by Proposition 2.9, Remark 2.10 and Theorem 3.1 it follows that p G, H q is afree extension pair as well. Let us choose generators H ˚ p BT q – F r x , ¨ ¨ ¨ .x n s and H ˚ p BH q – F r t , . . . , t n , y , . . . , y n s so that H ˚ p BT q – F r t , . . . , t n s and the mapinduced by the inclusion T Ñ T maps x i to t i for all 1 ď i ď n . We now computeexplicitly the module structure of H ˚ p BH q over H ˚ p BG q as stated in the followinglemma. Lemma 4.1.
The map i ˚ : H ˚ p BG q Ñ H ˚ p BH q induced by the inclusion i : H Ñ G is given by i ˚ p c i q “ t i ` t i p w ` ¨ ¨ ¨ w m q for all ď i ď n and i ˚ p w j q “ w j forall ď j ď m . Proof.
By theorem 3.1 we can assume that m “ n “ H ˚ p BG q – F r x, w s and H ˚ p BH q – F r t, w s . Notice that the statement i ˚ p w q “ w is clear as itfollows from the map induced by the inclusion of Z { Z into the second factor of S ¸ Z { Z which factors through H ˚ p BH q , Now write i ˚ p x q “ αt ` βtw ` γw for α, β, γ P F . As before, the inclusion of Z { Z into the first and second factor of G show that α “ γ “ β , we consider the inclusionof G into SO p q by identifying G with O p q as in Proposition 2.7 . Recall that H ˚ p BSO p qq – F r ω , ω s where | ω i | “ i for i “ , H Ñ SO p q induces the map φ : H ˚ p BSO p qq Ñ H ˚ p BH q given by φ p ω q “ t ` tw ` w and φ p ω q “ t w ` wt . Since φ factors through i ˚ , this implies that β “ i ˚ p x q “ t ` tw . (cid:3) In Theorem 3.1 we showed that the cohomology of H ˚ p BG q behaves as thecohomology of the classifying space of the direct product T ˆ K . However, Lemma4.1 implies that their cohomology as modules over the Steenrod algebra are not. Remark 4.2.
The mod2-cohomology of the classifying spaces BG and B p T ˆ K q is isomorphic as F -algebras but not as modules over the Steenrod algebra.Proof. As before, we may assume n “ m “
1. For x P H p BG q generator, write Sq p x q “ αxw ` βw for α, β P F . By naturality of the Steenrod operations, wehave that i ˚ p Sq p x qq “ Sq p i ˚ p x qq where i ˚ is the map induced by the inclusion H Ñ G . Therefore, α p t w ` tw q ` βw “ Sq p t ` tw q “ t w ` wt by Lemma4.1 and so α “ , β “
0. On the other hand, a similar argument applied to theinclusion j : H Ñ T ˆ K shows that Sq p x q “ j ˚ p x q “ t . (cid:3) Notice that for m ě
2, there is a group isomorphism G – pp S q n ¸p Z { Z q m ´ q¸ Z { Z where the action of the last factor of Z { Z is trivial on p Z { Z q m ´ . Therefore,as in the proof of Theorem 3.1, for the results of this section we may assume that m “ S be any commutative ring and consider the polynomial ring S r a, b s . Let z “ a ` ab P S r a, b s . Since a “ z ´ ab , one can check that(4.1) S r a, b s{ S r z, b s “ aS r z, b s “ t p p z, b q P S r z, b s : a | p p z, b qu . We will use this fact to prove the following proposition.
Proposition 4.3. H ˚ p BH q is a free module of rank n over H ˚ p BG q ; moreover, itis freely generated by the elements of the form t ǫ t ǫ ¨ ¨ ¨ t ǫ n n where ǫ i P t , u for all i ,and the canonical multiplicative structure of H ˚ p BH q as an F -algebra induces an H ˚ p BG q -algebra structure completely determined by multiplication of the elementsof this basis.Proof. Let R “ k r c , ¨ ¨ ¨ , c n , w s – H ˚ p BH q , M “ k r t , . . . , t n , w s – H ˚ p BG q andset y i “ t i ` t i w P M for i “ , . . . , n . Recall that M is an R -module by extendingthe action c i ¨ “ y i and w ¨ “ w . Consider the filtration of M by the R -submodules(4.2) F “ k r y , y , . . . , y n , w s Ď F Ď ¨ ¨ ¨ Ď F n “ M QUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS 13 where F i “ k r t , . . . , t i , y i ` , . . . y n , w s for i “ , . . . n ´
1. We will prove byinduction that F i is a free R -module of rank 2 i for i “ , . . . , n . The statement for i “ i ą
0, set S i “ k r t , . . . , t i ´ , y i ` , . . . , y n s and notice that(4.3) F i { F i ´ – S i r t i , w s{ S i r y i , w s – t i S r y i , w s “ t i F i ´ as in (4.1) and thus it is a free R -module by induction. This implies that the shortexact sequence of R -modules(4.4) 0 Ñ F i ´ Ñ F i Ñ F i { F i ´ Ñ F i – F i ´ ‘ F i { F i ´ . Finally, we have then that F i is a free R -module of rank 2 i ´ ` i ´ “ i by induction again. The claim about the basiselements follows also by iterating (4.1) and (4.3) in the decomposition(4.5) F k “ F ‘ F { F ‘ ¨ ¨ ¨ ‘ F k { F k ´ for all k “ , . . . , n . (cid:3) Combining Propositions 2.5 and 4.3 we can state the following result.
Corollary 4.4.
Let X be a G -space. H ˚ G p X q is a j -th syzygy over H ˚ p BG q if andonly if H ˚ H p X q is a j -th syzygy over H ˚ p BH q . (cid:3) Proposition 2.4 shows that the H -equivariant cohomology of X is determinedby the G -equivariant cohomology of X . As in the case for compact connected Liegroups and their maximal torus for rational coefficients, we can also describe the G -equivariant cohomology of X in terms of the Weyl invariants of the H -equivariantcohomology of X . Recall that the Weyl group of H in G is defined as the quotient W “ N G p H q{ H where N G p H q denotes the normalizer of H in G . We first proofthe following proposition. Proposition 4.5.
Let W “ N G p H q{ H be the Weyl group of H in G . Then W –p Z { q n and there is an isomorphism of algebras H ˚ p BG q – H ˚ p BH q W where theaction on the cohomology of H ˚ p BH q is induced by the conjugation action of W on H .Proof. Write H “ xp g , e q , . . . , p g n , e q , p , τ qy where g i “ i -th factor S of T .We claim that N G p H q – p Z { q n ¸ Z { Z where p Z { q n “ x θ , . . . , θ n y is generated byelements θ i “ g i and Z { Z acts on Z { p g, σ q P G where g P T and σ P x τ y , p g, σ q commutes with every element in H of the form p g i , e q and so we only need to look at the conjugation of the element p , e q P H by p g, σ q . Namely, if p g, σ q P N G p H q we have that p g, σ qp , τ qp g σ , σ q “ p g , τ q P H andthus we get g P x θ , . . . , θ n y . This implies that W – p Z { Z q n is generated by thecosets p θ i , e q H for i “ , . . . , n .Recall that for any topological group the map induced in the cohomology ofthe classifying space by the conjugation of a fixed element is the identity map[2, Ch.II Thm.1.9] and so i ˚ p H ˚ p BG qq Ď H ˚ p BH q W . It only remains to checkthe reverse inclusion to finish the proof. We now compute the induced action on the cohomology of H ˚ p BH q . Choose a decomposition H ˚ p BH q – k r t , . . . , t n , w s where the variables t i are dual to the generators g i and w is to τ in F r H s . For a fixed i P t , . . . , n u , notice that any p θ i , e q H P W acts trivially on the generators p g j , e q P H ; on the other hand, we have that p θ i , e q H ¨ p , τ q “ p θ i , e qp , τ qp θ i g i , τ q “ p g i , τ q This implies that the induced map ϕ i by the action of p θ i , e q H on the cohomologyring F r t , . . . , t n , w s is given by ϕ i p t j q “ t j for j ‰ i , ϕ i p t i q “ t i ` w and ϕ i p w q “ w .This follows, for instance, from a group cohomology argument.By Proposition 4.3, consider an element P “ ř I P Λ P I t I P H ˚ p BH q W where P I P H ˚ p BG q are uniquely determined. We will show that P I “ I p k q ‰ ď k ď n . Let I P Λ be such that I p k q ‰
0, then ϕ k p P I t I q “ P I t I ` wP I t I k where I k p j q “ I p j q if j ‰ k and I k p k q “
0. Under this notation, we have that ϕ k p t I k q “ t I k and then the equation P “ ϕ k p P q implies that P I k ` wP I “ P I k andso P I “ (cid:3) Actually, the isomorphism of Proposition 4.5 can be extended to a naturalisomorphism in equivariant cohomology as we state in the following consequence oftheorem 2.8
Corollary 4.6.
Let X be a G -space, H the maximal -torus in G and W the Weylgroup of H in G . Suppose that G acts trivially on the cohomology of X . Then thereis a natural isomorphism of H ˚ p BG q -algebras H ˚ G p X q – H ˚ H p X q W induced by the inclusion H Ñ G . Equivariant Cohomology for the real locus
Let M be a symplectic manifold with an action of a torus T . A consequenceof the work of Frankel [20], Atiyah [6] and Kirwan [27] in equivariant cohomologyfor Hamiltonian torus actions is that the action on M is Hamiltonian if and only if M is T -equivariantly formal. Moreover, if M admits a compatible anti-symplecticcompatible involution τ , the real locus M τ inherits a canonical action of T and M τ is T -equivariantly formal as shown in [18], and extended later in [9]. This canbe summarized in the following result. Theorem 5.1.
Let M be a symplectic manifold with a symplectic action of a torus T and a compatible anti-symplectic involution τ . If M is T -equivariantly formal,the real locus M τ is T -equivariantly formal. In this section we generalize the notion of spaces with a torus action and acompatible involution to a large class of groups, and we study the equivariant coho-mology for the fixed point subspace under the compatible involution to generalizeTheorem 5.1 into a topological setting. We first introduce the following definitionmotivated by the case when X is a complex variety and the involution is inducedby the complex conjugation. Definition 5.2.
Let X be a space with involution τ . The real locus of X is definedas the fixed point subspace X τ . Let G be a compact group, X be a G -space and τ X be an involution on X . Wesay that τ X is a compatible involution of X if there is a group homomorphism τ G : G Ñ G such that τ G “ id and τ X p g ¨ x q “ τ G p g q ¨ τ X p x q for any g P G and QUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS 15 x P X . The condition of compatibility is equivalent to an action of the group G τ “ G ¸ τ Z { Z on X . To simplify our notation, the involutions τ X and τ G willbe both referred as τ , and their domain can be inferred from the context. Noticethat the subgroup G τ of τ -fixed points of G acts on the real locus X τ . Definition 5.3.
Let H be a τ -invariant subgroup of G . We say that p G, H q is a τ -free extension if both p G, H q and p G τ , H τ q are free extensions. Notice that if τ acts trivially on H , then X L is the real locus of X . We proceedto prove the following result. Theorem 5.4.
Let G be a compact group and let X be a G -space with a compatibleinvolution involution τ . Suppose that there is a τ -invariant -torus H in G suchthat p G, H q is a τ -free extension. For any splitting H τ – H τ ˆ L and for anyinteger j ě , if H ˚ G τ p X q is a j -th syzygy over H ˚ p BG τ q , then so is H ˚ G τ p X L q asa module over H ˚ p BG τ q .Proof. As H is a 2-torus, p G, H q is a free extension if and only if p G τ , H τ q is. Infact, it follows from the commutativity of the diagram H ˚ p BH τ q H ˚ p G τ { H τ q H ˚ p BH q H ˚ p G { H q where the map H ˚ p G τ { H τ q Ñ H ˚ p G { H q is an isomorphism and H ˚ p BH τ q Ñ H ˚ p BH q is surjective. If X is a j -th syzygy over H ˚ p BG τ q , it follows from Propo-sition 2.4 that it also is a j -th syzygy as a module over H ˚ p BH τ q – H ˚ p B p H ˆ τ qq .We can use now the tools for syzygies for 2-torus actions discussed in [16] In fact,from Theorem [16, Thm.2.1] applied to the subgroup L Ď H τ , we obtain that H ˚ H τ { L p X L q – H ˚ H p X L q is a j -th syzygy over H ˚ p B p H τ { L qq – H ˚ p BH τ q . Finally,as p G, H q is a τ -free extension pair, from Proposition 2.5 we get that X is also a j -th syzygy over H ˚ p BG τ q . (cid:3) This theorem applies, for instance, to the groups G “ T ¸ p Z { Z q n for any n ě H is the maximal 2-torus in G . It also applies to SO p n q with the canonical τ -action that makes the isomorphism SO p n q ¸ τ Z { Z – O p n q . In this case, H isthe maximal 2-torus in SO p n q . In particular, we have a generalization of Theorem5.1 given by the following result. Theorem 5.5.
Let G “ T ¸ Z { Z and X be a G -space. If H ˚ G p X q is a j -th syzygyover H ˚ p BG q , then so is H ˚ T p X τ q as a module over H ˚ p BT q . In particular, if X is G -equivariantly formal, then the real locus X τ is T -equivariantly formal. Example 5.6.
Let X be a T -space. Suppose X is also a conjugation space with acompatible conjugation τ . Then from Theorem 5.5 and corollary 3.4 we have thatthe real locus X τ is T -equivariantly formal.The assumptions of Theorem 5.5 cannot be weakened. For example, If X is a G -space such that it is simultaneously T -equivariantly formal and τ -equivariantlyformal, it is not necessarily true that X is G -equivariantly formal or that its reallocus X τ is T -equivariantly formal as the next example shows. Example 5.7.
Let X “ tp u, z q P C ˆ R : | u | ` | z | “ u “ S , let T “ S act on X by g ¨ p u, z q “ p gu, z q ; more precisely, by scalar multiplication in the first factor.Let τ be the involution τ p u, z q “ p ¯ u, ´ z q which is compatible with the torus action.Notice that X T “ tp , q , p , ´ qu – S and X τ “ tp´ , q , p , qu – S . Therefore,the action of T on X τ is the multiplication by ˘ T -space.This implies that its T -equivariant cohomology is not free over H ˚ p BT q . On theother hand, H ˚ T p X q is a free H ˚ p BT q -module since X and X T have the same Bettisum.One of the main issues of this example is that X G “ H . Even requiring X G ‰ H ,a counterexample can be found and its construction will be motivated by [32, Sec.5]. First we recall the following construction of topological spaces. Definition 5.8.
Let f : X Ñ Y be a G -map between G -spaces X and Y . Themapping cylinder is defined as the G -space M f “ p X ˆ r , sq \ Y { „ where p x, q „ f p x q , with the action given by g ¨ p x, t q “ p gx, t q for p x, t q P X ˆ r , s and the actionon Y . Notice that it is well defined at the points of the form p x, q since f is a G -map. The space M f is G -homotopic to Y and therefore H ˚ p M f q – H ˚ p Y q . Also, thefixed point subspace p M f q G – M f G where f G : X G Ñ Y G . Now let g : X Ñ Z be a G -map and M g the corresponding mapping cylinder. Then the space M f,g “ M f Y X ˆt u M g has cohomology groups fitting in the long exact sequence0 Ñ H p M f,g q Ñ H p Y q ‘ H p Z q Ñ H p X q Ñ H p M f,g q Ñ ¨ ¨ ¨ following from the Mayer-Vietoris long exact sequence. Moreover, M f,g becomes a G -space and p M f,g q G – M f G ,g G . In particular, we have Proposition 5.9.
Let m, n, r be distinct positive integers, h : S m Ñ S n a mapbetween spheres and consider f “ h ˆ id : S m ˆ S r Ñ S n ˆ S r and g : S m ˆ S r Ñ S m the projection on the first factor. Then H ˚ p M f,g q is free over Z { Z where a copy of Z { Z happens in degrees , n, m ` r ` , n ` r and it is zero otherwise. In particular, b p M f,g q “ . Using this construction, we have the following proposition.
Proposition 5.10.
There is a topological space M with an action of a torus T and a compatible involution τ such that M G ‰ H , M is T -equivariantly formaland Z { Z -equivariantly formal, but the real locus M τ is not T -equivariantly formalwith respect to the induced action of the -torus T Ď T on M τ . (cid:3) Proof.
Let X “ S , Y “ S and h : X Ñ Y be the Hopf map, which can bewritten as h p u, z q “ p u ¯ z, | u | ´ | z | q . Here S is seen as the unit sphere in C and S as the unit sphere in C ˆ R . Let T “ S act on S and S as the complexmultiplication in the first component respectively, and τ be the involution on S and S given by the complex conjugation in the first component respectively. Then τ iscompatible with the torus action and X T – S , X τ – S , Y T – S and Y τ – S .Now let Z “ S be the unit sphere in C , let T act on Z by multiplication in thefirst component and τ be the involution on Z given by the complex conjugation inthe first component, and multiplication by ´ Z T – S and Z τ – S ; moreover, the action of the 2-torus T Ď T on Z τ isfree. QUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS 17
Let M “ M f,g be the construction of Proposition 5.9. We have that the Bettisums b p M q “ b p M T q “ b p M τ q “ M is T -equivariantly formal but M τ is not T -equivariantly formal since b pp M τ q T q “ ă b p M τ q . (cid:3) Actions on big polygon spaces
Big polygon spaces provide remarkable examples for the study of torus equivari-ant cohomology since their equivariant cohomology is not free over the cohomologyof the classifying space of the torus but realize all other possible syzygy order. Theywill also allow us to realize all possible syzygies in G -equivariant cohomology. Thesespaces where introduced in [21] where their non-equivariant and T -equivariant coho-mology was determined and an upper bound for their syzygy order was conjectured.This was proved later in [23]. They generalize chain spaces and polygon spaces stud-ied in different contexts in [19] and [25] for instance. The real analogous of thesespaces is also studied in [31] for the case of 2-torus actions and cohomology with F -coefficients. Before discussing these spaces, we will review the construction of acohomology class that will be useful for the results of this section. We use the equi-variant homology for 2-torus actions; for its construction and properties we followthe work in [4].Let G be a 2-torus, M be a closed G -manifold of dimension m and N Ď M be aclosed G -invariant submanifold of M of dimension n . Let j G ˚ : H G ˚ p N q Ñ H G ˚ p M q denote the map induced in equivariant homology by the inclusion j : N Ñ M ; sim-ilarly, j ˚ G : H ˚ G p M q Ñ H ˚ G p N q denotes the map induced in equivariant cohomology.As M and N satisfy Poincar´e duality (cohomology with coefficients over a fieldof characteristic two), there are isomorphism P D M : H ˚ G p M q Ñ H Gm ´˚ p M q and P D N : H ˚ G p N q Ñ H Gn ´˚ p N q . Consider the composite map ν N,M : H ˚ G p N q P D N ÝÝÝÑ H Gn ´˚ p N q j G ˚ ÝÝÑ H Gn ´˚ p M q P D ´ M ÝÝÝÝÑ H m ´ n `˚ G p M q j ˚ G ÝÑ H m ´ n `˚ G p N q . Under this construction, we introduce the following definition.
Definition 6.1.
The G -equivariant Euler class of N with respect to M denotedby e G p N Ď M q is defined as the cohomology class ν N,M p q P H m ´ nG p N q . Consider the following example (compare with [21, Lem.4.2]).
Example 6.2.
Let G “ G ˆ G be a 2-torus of rank 2 where G “ t , g u , G “t , τ u . Let G act on C where g acts as the multiplication by ´ τ as the complexconjugation. Let x, w denote the generators of H ˚ p BG q and H ˚ p BG q dual to g and τ respectively. Then the equivariant Euler class e G p Ď C q “ αx ` βxw ` γw P H p BG q . Let K “ t , s u and t be the generator of H ˚ p BK q . Consider the followingcases ‚ Let s act on C in the same fashion as g and let j : K Ñ G be the mapsending s to g , then the induced map in cohomology is given by j ˚ p x q “ t and j ˚ p w q “
0. Notice that e K p Ď C q “ t since g acts non-trivially inboth components of C “ R ‘ R and the Euler class is multiplicative. Fromthe naturality of the Euler class we get αt “ j ˚ p e G p Ď C qq “ e G p Ď C q “ t ; therefore, α “ ‚ Let s act on C in the same fashion as τ . As before, the map j : K Ñ G sending s to τ induces the map in cohomology mapping x to 0 and w to t . In this case, e K p Ď C q “ τ acts trivially on one real factor of C .Therefore, by naturality, we obtain γ “ ‚ Finally, let s act on C as gτ , and j : K Ñ G sends s to p g, τ q and, incohomology, both x, w are sent to t . Since s acts trivially on one real factorof C , e K p Ď C q “ t ` βt “ j ˚ p e G p Ď C qq “ e K p Ď C q “
0. Therefore, β “ e G p Ď C q “ x p x ` w q . Definition 6.3.
Let a, b, n be positive integers and M “ p S a ` b ´ q n Ď p C a ˆ C b q n .Let ℓ “ p l , . . . , l n q P R n be such that l i ą for all i and such that it cannot be splitas the sum of two vectors ℓ and ℓ of equal sum. The big polygon space is definedas X a,b p ℓ q “ p u, z q P M : n ÿ i “ l i u i “ + The space X a,b p ℓ q inherits an action of an n -dimensional torus T by componen-twise complex multiplication on the variables z i . In this case, X a,b p ℓ q becomes acompact orientable T -manifold of dimension p a ` b ´ q n ´ a and its equivariantdiffeomorphic type depends on ℓ [21, Lem.2.1]. Moreover, the complex conjugationon M induces a compatible involution τ on X a,b p ℓ q and its real locus is the real bigpolygon space.The first property of the G -equivariant cohomology of X a,b p ℓ q is that it is notfree over the cohomology of H ˚ p BG q as we show in the following result. Proposition 6.4. H ˚ G p X q is not free over H ˚ p BG q . In fact, H ˚ G p X q is not a j -thsyzygy for j ě p n ` q{ . As a consequence of Theorem 5.5, we can recover a theorem of Puppe [31,Thm.1.2] that bounds the syzygy order of the real big polygon spaces.
Corollary 6.5.
The equivariant cohomology of the real big polygon space X τ underthe action of the -torus T is a j -th syzygy j ď p n ` q{ .Proof. By Corollary 4.4, it is enough to restrict to the action of the maximal 2-torus H of G . Let us denote by X the big polygon space to simplify notation. On theone hand, the integer cohomology of X is free and its Betti sum is b p X q “ n [21,Prop.3.3]. On the other hand, when a ą
2, the F -cohomology of the fixed pointsubspace X H is isomorphic to a quotient of an exterior algebra on n -generators bya non-trivial ideal [19, Prop. 4.2 ] and so b p X G q ă n . The same bound holdswhen a “ X H – S ˆ X T { SO p q and the computation of the Bettisum. This shows that b p X H q ă b p X q and thus X is not G -equivariantly formal bythe Betti sum criterion for 2-torus actions [33, Prop.III.4.16]. The last assertion ofthe Corollary follows from the fact that X is a compact manifold and it satisfiesPoincar´e duality over F -cohomology. (cid:3) To bound the syzygy order of the G -equivariant cohomology of the big poly-gon spaces we use a similar approach as in the torus case. Firstly, we need toexplicitly describe the generators of the non-equivariant and equivariant homol-ogy of the spaces M and M z X . Namely, for any subset J Ď t , , . . . , n u , write J c “ t , . . . , n uz J and J Y j “ J Y t j u , and define ℓ p J q “ ř j P J l J . We say that J is short if ℓ p J q ă ℓ p J c q . We also define the manifolds V J “ tp u, z q P M : @ j R J p u j , z j q “ ˚u QUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS 19 W J “ tp u, z q P M : @ j, k R J, u j “ u k , z j “ z k “ u where ˚ P S a ` b ´ X p C a ˆ t uq is a chosen base point. Notice that V J is homeo-morphic to a product of | J | spheres of dimension d and W J – V J ˆ S a ´ . Thesehomeomorphisms imply that V J Ď W J , dim V J “ | J | d and dim W J “ | J | d `p a ´ q .Let r V J s , r W J s be the respective homological orientation classes of V J and W J and r V J s H , r W J s H their equivariant lifting. Then H ˚ p M q is free with basis tr V J s : J Ď t , . . . , n uu and H ˚ p M z X q is free with basis tr V J s , r W J s : J short u [21, Lem.3.2].Analogously to [21, Lem. 4.5, Prop.4.6] we have the following description in H -equivariant cohomology. Proposition 6.6.
Let ι : M z X Ñ M be the inclusion map.(i) H H ˚ p M q is a free H ˚ p BH q -module with basis tr V J s H , J Ď t , . . . , n uu .(ii) H H ˚ p M z X q is a free H ˚ p BH q -module with basis tr V J s H , r W J s H , J short u .(iii) ι H ˚ pr V J s H q “ r V J s H and ι H ˚ pr W J s H q “ ř j R J w bj p w j ` w q b r V J Y j s H .Proof. Notice that b p M q “ b p M H q as M H – p S a ´ q n and so M is H -equivariantlyformal. Moreover, we obtain that the restriction map H H ˚ p M q Ñ H ˚ p M q which is the edge homomorphism of the homological spectral sequence with E -termgiven by E “ H ˚ p M q b H ˚ p BH q and converging to H H ˚ p M q is surjective since thebasic elements r V J s have a lifting in H ˚ H p M q . Therefore, as in the Leray-HirschTheorem, the spectral sequence collapses and so tr V J s H , J Ď t , . . . , n uu is a basisof H H ˚ p M q over H ˚ p BH q , thus proving p i q . The proof of p ii q follows in a similarfashion.To prove p iii q we will use the H -equivariant Euler class to compute explicitlythe map ι H ˚ on the generators of H H ˚ p M z X q .Let K “ K ˆ K where K “ t , g u , K “ t , τ u , g denotes the action inducedby multiplication by ´ τ the complex conjugation in C , and let x, w denote thecanonical generators of H ˚ p BK q and H ˚ p BK q dual to the generators of K and K respectively. Similarly to 6.2, we get e K p S K Ď S q “ x b p x ` w q b , or equivalently, r S K s K “ x b p x ` w q b r S s K .Now the proof for the case of the torus action on the big polygon space found in[21, Lem.4.5] can be imitated in our situation to show that p iii q holds. Firstly, theidentity i H ˚ pr V J s H q “ r V J s H follows from the naturality of the equivariant homology,that is, from the commutative diagram H H ˚ p V J q H H ˚ p M z X q H H ˚ p M q ι H ˚ . To compute i H ˚ pr W J sq , we need to “enlarge” the acting group. For J Ď r n s ,define τ J to be the involution on M given by the complex conjugation on thevariables u j : j P J and z j : j P J , and write σ J “ τ J c . Set H J “ T ˆ τ J ˆ σ J and H Ñ H J the map induced by the identity on T and the map which sends τ to p τ J , σ J q . Thus we get a map H ˚ p BH J q “ k r t , . . . , t n , w τ , w σ s Ñ H ˚ p BH q “ k r t , . . . , t n , w s sending w τ and w σ to w which is the identity in the other variables.Moreover, we have maps in equivariant homology H H J ˚ p M q Ñ H H ˚ p M q Notice that the H J -action on M induces an action of H on M ; such an actioncoincides with the initial action of H on M described at the beginning of thesection. Also, we have similar restriction maps for the H J -invariant submanifolds X, M z X Ď M .Set ˜ M “ M X p C a ˆ t uq n – p S a ´ q n . For J Ď r n s , let ∆ J be the inclusion of S a ´ into the factors j P J of ˜ M . Notice that there is a homeomorphism W J – V J ˆ ∆ J c ; moreover, such homeomorphism yields to an equivariant decomposition H J “ p K J ˆ τ J qˆ p K J c ˆ σ J q where K J Ď T is the 2-subtorus of non-trivial factorsin the position j P J . Therefore, by the K¨unneth theorem in equivariant homologyfor 2-torus actions (compare with [21, Prop.4.1]) we have that r W J s H J “ r V J s K J ˆ τ J ˆ r ∆ J c s K Jc ˆ σ J By naturality of the Euler class, we have(6.1) i H J ˚ pr W J s H J q “ i K J ˆ τ J ˚ pr V J s K J ˆ τ J q ˆ i K Jc ˆ σ J ˚ pr ∆ J c s K Jc ˆ σ J q As above, it is straightforward to check that i K J ˆ τ J ˚ pr V J s K J ˆ τ J q “ r V J s K J ˆ τ J ,so it only remains to compute the last term of (6.1). Without loss of generalitywe can assume that J “ H , so ∆ J c “ ∆ is the diagonal of ˜ M , σ J “ τ , τ J istrivial and H J “ H . So we need to compute i H ˚ pr ∆ s H q . Since in H ˚ p ˜ M q wehave that r ∆ s “ ř nj “ r ∆ j s and ˜ M is H -equivariantly formal, we have then inequivariant homology that r ∆ s H “ ř nj “ r ∆ j s H . Consider the inclusion K Ñ H into the j -th factor of T and denote the image by K j . This map induces incohomology an identification of x with t j . Observe that ∆ j “ V T j “ V K j j and thus r ∆ j s H “ r V K j j s K j ˆ τ . We obtain by naturality of the Euler class and the abovecomputation that r ∆ j s H “ t bj p t j ` w q b r V j s H . Finally this implies that(6.2) i H ˚ pr ∆ s H q “ n ÿ j “ t bj p t j ` w q b r V j s H For the general case, using this computation, for any J we have again by (6.1) that i H J ˚ pr W J s H J q “ i K J ˆ τ J ˚ pr V J s K J ˆ τ J q ˆ i K Jc ˆ σ J ˚ pr ∆ J c s K Jc ˆ σ J q“ r V J s K J ˆ τ J ˆ ÿ j R J t bj p t j ` w σ q b r V j s K Jc ˆ σ J “ ÿ j R J t bj p t j ` w σ q b r V J Yt j u s H J The computation for the H -equivariant cohomology follows by naturality and usingthe restriction map H ˚ p BH J q Ñ H ˚ p BH q which sends w σ to w . (cid:3) Let R “ H ˚ p BH q “ F r t , . . . , t n , w s and write y j “ t j p t j ` w q . Let n “ m ` ℓ “ p , . . . , q . We will use the Koszul resolution of L “ R {p y b , . . . , y bn q to QUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS 21 identify the H -equivariant cohomology of the big polygon space X “ X a,b p ℓ q withthe Koszul syzygies appearing in such resolution. The assumption on ℓ is made sofor J Ď r n s is short if and only if ℓ p J q ă m .The following result follows from the analogous case of equivariant cohomologyfor torus actions on the big polygon spaces [21, §
5] and only an outline of the proofwill be presented.
Theorem 6.7.
Let n “ m ` , m ě . The G -equivariant cohomology of theequilateral big polygon space X “ p u, z q P p S a ` b ´ q n : n ÿ i “ u i “ + is an m -th syzygy but not an p m ` q -st syzygy.Proof. Let ι : M z X Ñ M be the inclusion and let ι H ˚ be the induced map inequivariant homology. For simplicity set d “ a ` b ´
1; The equivariant Poincar´e-Alexander-Lefschetz duality [4, Thm.7.6] implies that there is a short exact sequence(6.3) 0 Ñ Coker i H ˚ r nd s Ñ H ˚ H p X q Ñ ker i H ˚ r nd ´ s Ñ . From Proposition 6.6 we have that H ˚ H p M z X q – à | J |ď m p R ¨ r V J s H ‘ R ¨ r W J s H q and H ˚ H p M q – ‘ J Ďr n s R ¨ r V J s H as R -modules. By Proposition 6.6 (iii), the Kernel ofthe map ι H ˚ : à | J |ď m R ¨ r V J s H ‘ à | J |ă m R ¨ r W J s H Ñ H H ˚ p M q is the free R -submodule of H H ˚ p M z X q generated by the elements r W J s H ´ ř j P J y bj r V J Y j s H where | J | ă m since ι H ˚ pr V J s H q “ r V J s H and ι H ˚ pr W J s H q “ ř j R J t bj p t j ` w q b r V J Y j s H .On the other hand, the map ι H ˚ : à | J |“ m R ¨ r W J s H Ñ à | J |“ m ` R ¨ r V J s Ď H H ˚ p M q can be identified with the map d m ` in the Koszul resolution of L “ R {p y b , . . . , y bn q described above whose kernel is the Koszul syzygy K m ` . So we obtain thatker p ι H ˚ q – à | J |ă m R r´| J | d ´ ¯ d s ‘ K m ` r´ md ´ ¯ d ` s The degree shifts follows from the fact that dim W J “ | J | d ` ¯ d and dim V J “ | J | d and the convention that the Koszul syzygies are generated in degree 0.Similarly, we can see that im p ι H ˚ q – À | J |ď m R ¨ r V J s H ‘ im p d m ` q and thusCoker p ι H ˚ q “ H H ˚ p M q{ im p ι H ˚ q – à | J |ą m ` R ¨ r V J s H ‘ Coker p d m ` q . Notice that from the Koszul resolution it follows that Coker p d m ` q – im p d m ` q “ K m the m -th Koszul syzygy of L . Summarizing, we obtained thatCoker p ι H ˚ q – à | J |ą m ` R r´| J | d s ‘ K m r´p m ` q d s . and thus both ker p ι H ˚ q and Coker p ι H ˚ q are m -th syzygies. To finish the proof, it isenough to show that the sequence (6.3) splits. This will follow from [31, Lem.3.12] and using that the singular Cartan model as a free R -model for the G -equivariantcohomology. (cid:3) Finally, from 5.5 we can obtain the syzygy order of the equilateral real bigpolygon spaces recovering also one of the main results in [31, Thm.1.2].
Corollary 6.8.
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Department of Mathematics, Western University, London, ON. Canada
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