Assignments for topological group actions
aa r X i v : . [ m a t h . A T ] S e p ASSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS
Oliver Goertsches Augustin-Liviu Mare
Fachbereich Mathematik und Informatik Dept. of Mathematics and Statisticsder Philipps-Universit¨at Marburg University of ReginaD-35032 Marburg, Germany Regina SK, S4S 0A2 Canada [email protected] [email protected]
Abstract.
A polynomial assignment for a continuous action of a compact torus T on atopological space X assigns to each p ∈ X a polynomial function on the Lie algebra of theisotropy group at p in such a way that a certain compatibility condition is satisfied. Thespace A T ( X ) of all polynomial assignments has a natural structure of an algebra over thepolynomial ring of Lie( T ). It is an equivariant homotopy invariant, canonically related tothe equivariant cohomology algebra. In this paper we prove various properties of A T ( X )such as Borel localization, a Chang-Skjelbred lemma, and a Goresky-Kottwitz-MacPhersonpresentation. In the special case of Hamiltonian torus actions on symplectic manifolds weprove a surjectivity criterion for the assignment equivariant Kirwan map corresponding toa circle in T . We then obtain a Tolman-Weitsman type presentation of the kernel of thismap.MSC: 55P91, 55N91, 53D20 Introduction
The notion of assignment associated to a torus action on a manifold was defined byGinzburg, Guillemin, and Karshon in [8], by means of a construction that takes into ac-count exclusively the orbit stratification and the relative position of the strata. They wereled to this construction while dealing with the existence problem of an abstract moment mapfor a given action. However, as the authors briefly mention, this new notion is susceptible tobe relevant for another important question in this area, namely, under which circumstancesis the equivariant cohomology algebra determined by the orbit stratification? Indeed, a fewyears later, Guillemin, Sabatini, and Zara have found in [14] a direct connection between theequivariant cohomology and a particular assignment space, which is called by them the alge-bra of polynomial assignments . Concretely, the connection is given by a ring homomorphism,which, for certain classes of actions, is injective and sometimes even bijective. For example,injectivity is achieved for equivariantly formal actions with isolated fixed points on com-pact manifolds and bijectivity for the sub-class of actions of Goresky-Kottwitz-MacPherson(GKM) type.This paper is based on the observation that the polynomial assignment algebra can bedefined for arbitrary continuous (torus) actions on topological spaces. More precisely, let X be a topological space and T a torus that acts on it. For any p ∈ X we denote by T p thecorresponding isotropy subgroup of T and by t p its Lie algebra (this will be referred to asthe infinitesimal isotropy at p ). Let also S ( t ∗ p ) be the algebra of polynomial functions on t p . Definition 1.1. A (polynomial) assignment for the T -action on X is a map A that assignsto each p ∈ X a polynomial A ( p ) ∈ S ( t ∗ p ) such that for any (connected) subtorus H ⊂ T themap A h on the fixed point set X H is locally constant. Here h is the Lie algebra of H and A h the map defined by A h ( p ) := A ( p ) | h , for all p ∈ X H . This looks different from the definition in [14] since, as already mentioned, the latterinvolves the orbit stratification. However, we will show in Section 3 below that for smoothactions on manifolds, there is no difference between the two notions.We denote by A T ( X ) the space of all assignments of the above type. It has an obviouscanonical structure of an S ( t ∗ )-algebra, which will be referred to as the assignment algebra of the torus action. It defines a functor from the category of topological T -spaces to thecategory of S ( t ∗ )-algebras; moreover, it is an equivariant homotopy invariant, see Section 2below. Our goal here is to present some results concerning A T ( X ) in the topological set-up.Direct connections with the equivariant cohomology algebra H ∗ T ( X ) in the spirit of [14] arealso discussed, although they are not of main interest for us. Polynomial assignments arestudied here in their own right.In fact, equivariant cohomology is rather relevant for us in an indirect way: that is, weconsider some results in this theory and prove assignment versions of them. In the firstpart we will consider the inclusion of the fixed point set X T into X along with the map A T ( X ) → A T ( X T ) induced by functoriality. After proving Borel type localization results,concerning the kernel and the cokernel of the aforementioned map, we obtain an assignmentversion of the GKM-theorem. It requires some extra assumptions on the (continuous) torusaction. Among others, we want the fixed point set X T to have only finitely many components,call them Z , . . . , Z n . Then the theorem says that A T ( X ) is isomorphic to the subspace of S ( t ∗ ) × · · · × S ( t ∗ ) ( n factors) consisting of tuples ( f , . . . , f n ) with the property that if Z i and Z j are contained in a connected component of some X H , where H ⊂ T is a codimensionone subtorus of Lie algebra h , then f i and f j are equal when restricted to h . The precisestatement can be found in Section 5.3. We emphasize that the result is purely topological.One class of torus actions for which it holds true is the one of equivariantly formal actions oncompact Hausdorff spaces with finitely many infinitesimal isotropies and finite dimensionalcohomology. We note that this is in the spirit of [8, Section 3.4]: a characterization of A T ( X ) similar to the one above is obtained there under the (more restrictive) hypothesesthat X is a manifold, the T -action is smooth, and the compatibility relations are assumedfor subtori H of arbitrary dimension.In the last section of the paper we consider the particular context of Hamiltonian torusactions on compact symplectic manifolds, which are prominent examples of equivariantlyformal, in general non-GKM, actions. More precisely, we study the assignment version ofthe equivariant Kirwan map. Recall that the Kirwan map is a basic instrument when dealing SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 3 with the cohomology of symplectic quotients. There is a rich literature devoted to this topic.We only mention here the seminal work [18] of Kirwan and the influential papers [11] byGoldin and [19] by Tolman and Weitsman, which are directly related to our interests. Tostate our result, let us denote by T the torus which acts and by M the compact symplecticmanifold which is acted on. Let also Q ⊂ T be a one-dimensional subtorus with Lie algebra q and Φ : M → q ∗ the moment map of the restricted Q -action. If µ ∈ q ∗ is a regular valueof Φ, the symplectic quotient M := Φ − ( µ ) /Q is a symplectic orbifold with a canonicalaction of T /Q . We first show that there is a well-defined equivariant assignment Kirwanmap κ A : A T ( M ) → A T/Q ( M ). In the case when M is a manifold, this was already noticedin [14]. Unlike its cohomological counterpart, κ A is in general not surjective. We prove thefollowing surjectivity criterion. Assume that for any connected component F of M T , theweights of the isotropy representation along F are linearly independent modulo collinearity(more precisely, after setting equal any two weights which are collinear, the resulting setmust be linearly independent). Then κ A is surjective. The proof uses ideas from Morsetheory for the moment map function Φ, which are applicable mainly due to the fact that A T is a topological, equivariant homotopy invariant. We also achieve a description of thekernel of κ A , which is the assignment version of a result previously obtained in cohomologyby Goldin [11]. As a consequence, concrete formulas for A T/Q ( M ) become available. Thedetails can be found in Section 8. Acknowledgements.
We wish to thank Silvia Sabatini for helpful comments. We alsothank the referee for carefully reading the manuscript and suggesting several improvements.2.
Basic facts
We start with a list of examples. The importance of the first one is rather historical. It isonly intended to remind that originally assignments arose from the study of moment maps,see [8].
Example 2.1.
Let (
M, ω ) be a symplectic manifold endowed with a Hamiltonian action ofa torus T . If Φ : M → t ∗ is a moment map, then A ( p ) := Φ( p ) | t p , p ∈ M , is a polynomialassignment.In the following situations the assignment algebra A T ( X ) can be obtained by direct cal-culations. Example 2.2. If X is the empty space acted on by a torus T , then there exists a uniquemap from X to S ( t ∗ ). This map does not change after multiplication with 0 ∈ S ( t ∗ ). Thus, A T ( ∅ ) = { } . Example 2.3.
In the case when a torus T acts trivially on a non-empty connected space X , then t p = t for all p ∈ X , hence any assignment is constant on X . This means that A T ( X ) = S ( t ∗ ). ASSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS
Example 2.4.
Let m, n > P ( m, n )is defined as the quotient ( C \ { (0 , } ) / ∼ , where( z , z ) ∼ ( λ m z , λ n z ) , ( z , z ) ∈ C \ { (0 , } , λ ∈ S . We consider the action of S on P ( m, n ), given by z. [ z : z ] := [ z k z : z ] , z ∈ S , where k > S is R . The action has two fixed points, p := [1 : 0] and p := [0 : 1]. The infinitesimal isotropy at any other point is { } . Inthis case, one can identify the symmetric algebra S ( t ∗ ) with the polynomial ring R [ u ]. Theassignments for our action are of the form A ( p ) = f , if p = p ,f , if p = p ,r, if p / ∈ { p , p } , where f , f ∈ R [ u ] and r ∈ R are such that f (0) = f (0) = r . Example 2.5.
Let n ≥ ξ , . . . , ξ n ∈ C the n -th roots of unity.Let X be the quotient space S × S / { ( ξ i , z ) ∼ ( ξ i , z ′ ) for all i = 1 , . . . , n and all z, z ′ ∈ S } .The action of S on S × S given by z. ( x , x ) := ( x , zx ) descends to X and has n fixedpoints, p , . . . , p n , which are represented by { ξ } × S , . . . , { ξ n } × S , respectively. At any p ∈ X which is not fixed by S , the isotropy is { } . Hence the assignments are this time A ( p ) = ( f i , if p = p i for some 1 ≤ i ≤ nr, if p = p i for all 1 ≤ i ≤ n, where f , . . . , f n ∈ R [ u ] and r ∈ R are such that f (0) = . . . = f n (0) = r . This is a slightlymodified version of an example from [20] (see Figure 4 in that paper). Example 2.6.
The torus T := S × S acts on the sphere S := { ( x , x ) ∈ C | | x | + | x | =1 } as follows: ( z , z ) . ( x , x ) := ( z x , z x ) . It is an easy exercise to show that, after identifying S ( t ∗ ) = R [ u , u ], the assignments ofthis action are of the following form: A (( x , x )) = f , if x = 0 ,f , if x = 0 ,r, if x = 0 and x = 0 , where f , f ∈ R [ u ] and r ∈ R are such that f (0) = f (0) = r .In what follows we will show that the assignment algebra of a torus action shares withequivariant cohomology two important properties: homotopy invariance and (a weak versionof) the Mayer-Vietoris sequence. SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 5 A T -equivariant map f : X → X ′ between two T -spaces X and X ′ induces an S ( t ∗ )-algebrahomomorphism f ∗ : A T ( X ′ ) → A T ( X ) between the corresponding assignment algebras: f ∗ ( A )( p ) := A ( f ( p )) | t p , for all p ∈ X. The map f ∗ ( A ) satisfies Definition 1.1. To verify this, take H ⊂ T and Y a component of X H . Then f ( Y ) is contained in a component of ( X ′ ) H , hence A ( f ( p )) | h is independent of p ∈ Y . Proposition 2.7.
Let f, g : X → X ′ be two T -equivariant maps which are homotopic toeach other through T -equivariant maps. Then f ∗ = g ∗ : A T ( X ′ ) → A T ( X ) .Proof. Let F : [0 , × X → X ′ be a T -equivariant homotopy from f to g . For everyassignment A ∈ A T ( X ′ ) and every point p ∈ X we have to show that f ∗ ( A )( p ) = g ∗ ( A )( p ),i.e., that A ( f ( p )) | t p = A ( g ( p )) | t p . But by equivariance, T p ⊂ T F ( t,p ) for all t ∈ [0 , t F ( t, p ) lies completely in a connected component of ( X ′ ) H , where H is theidentity component of T p . The claim thus follows from Definition 1.1. (cid:3) If X is a T -space and Y ⊂ X a T -invariant subspace, there is a restriction map A T ( X ) →A T ( Y ), A A | Y , which is the algebra homomorphism induced by the inclusion Y ֒ → X . Proposition 2.8.
Let X be a T -space and Y, Z ⊂ X two T -invariant subspaces, which areeither both open or both closed. Then the following sequence is exact: −→ A T ( Y ∪ Z ) −→ A T ( Y ) ⊕ A T ( Z ) −→ A T ( Y ∩ Z ) . Here A T ( Y ∪ Z ) → A T ( Y ) ⊕ A T ( Z ) is given by A ( A | Y , A | Z ) and A T ( Y ) ⊕ A T ( Z ) →A T ( Y ∩ Z ) by ( A, B ) A | Y ∩ Z − B | Y ∩ Z .Proof. The only nontrivial statement is that if B ∈ A T ( Y ) and C ∈ A T ( Z ) are such that B | Y ∩ Z = C | Y ∩ Z then there exists A ∈ A T ( Y ∪ Z ) such that A | Y = B and A | Z = C . Observefirst that the last two equations define A uniquely, as a map Y ∪ Z ∋ p A ( p ) ∈ S ( t ∗ p ). Itremains to show that A is an assignment on Y ∪ Z : that is, if H ⊂ T is a subtorus then A h is locally constant on ( Y ∪ Z ) H = Y H ∪ Z H . Case 1.
Both Y and Z are open in X . Take p ∈ Y H ∪ Z H , say p ∈ Y H . Then B h is constanton an open neighborhood of p in Y H , which is also a neighborhood of p in Y H ∪ Z H , since Y H is open in the latter union. Finally, by definition, A h and B h coincide on that neighborhood. Case 2.
Both Y and Z are closed in X . Again we take p ∈ Y H ∪ Z H . If p ∈ Y H \ Z H ,there exists an open neighborhood U of p in X such that B h is constant on U ∩ Y H and U ∩ Z H = ∅ . But then U ∩ ( Y H ∪ Z H ) = U ∩ Y H and on this set B h equals A h , hence thelatter is constant as well. If p ∈ Y H ∩ Z H , there exist open neighbourhoods U and U of p in X such that B h is constant on U ∩ Y H and C h is constant on U ∩ Z H . It follows that A h is constant on ( U ∩ U ) ∩ ( Y H ∪ Z H ). (cid:3) ASSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS Assignments for smooth group actions
In this section X is assumed to be a manifold and the T -action smooth. Polynomialassignments for such actions have been defined in [14] as follows (cf. also [8, Definition 3.7]).One first considers the corresponding infinitesimal orbit-type stratification of X . That is,the strata are connected components of spaces of the form Y := { p ∈ X | t p = h } , where h is an infinitesimal isotropy. Let us denote t Y := t p , where p ∈ Y . There is a partial order (cid:22) on the set of all strata given by Y (cid:22) Z if and only if Y ⊂ Z . Note that the last conditionimplies t Z ⊂ t Y . Definition 3.1. ([14, Definition 2.1]) A polynomial assignment for the T -action on X is afunction A that associates to each infinitesimal stratum Y a polynomial A ( Y ) ∈ S ( t ∗ Y ) suchthat if Y (cid:22) Z then A ( Z ) = A ( Y ) | t Z . The following proposition says that Definitions 1.1 and 3.1 are equivalent.
Proposition 3.2. (a) If A is like in Definition 1.1 then A is constant on each infinitesimalstratum Y . Let A ′ ( Y ) denote the common value of all A ( p ) , p ∈ Y . Then the map Y A ′ ( Y ) satisfies the requirement of Definition 3.1.(b) If A ′ is like in Definition 3.1 then the map A given by A ( p ) := A ′ ( Y ) , where p is inthe stratum Y , satisfies the requirement of Definition 1.1.Proof. (a) If Y is a stratum, then obviously Y ⊂ X T Y , where T Y ⊂ T is the connected Liesubgroup corresponding to t Y . This implies the claim.(b) Let H ⊂ T be a subtorus and p ∈ X H . Let U be a tubular neighborhood around T p ,i.e., an open neighborhood of
T p which is T -equivariantly diffeomorphic to T × T p ν p , where ν p is the normal space to T p at p .We claim that A ( q ) | h is independent of q ∈ U ∩ X H (this will imply that A h is locallyconstant on X H ). To prove this, we may assume that q ∈ ν p and q = 0. Since the T p -actionon ν p is linear, the infinitesimal stratum of q contains the half-line { xq | x > } in ν p .Denote the stratum of q by Y . Since the half-line mentioned above is contained in Y , wededuce that p ∈ Y . Hence the whole stratum of p is contained in Y , thus A ( q ) = A ( p ) | t q .But q ∈ X H , which implies that h ⊂ t q ⊂ t p and further that A ( q ) | h = A ( p ) | h . (cid:3) A Borel type localization theorem
Let X be a connected topological space acted on by a torus T . We assume throughoutthis section that the following assumption is fulfilled. Assumption 1.
The T -action on X has only finitely many infinitesimal isotropies.Recall that A T ( X ) has a canonical structure of an S ( t ∗ )-algebra. The goal here is to provean analogue of Borel’s localization theorem for equivariant cohomology, see for instance [13,Theorem C.20]. SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 7
Recall from Section 2 that if Y is a T -invariant subspace of X , then there is a naturalrestriction map A T ( X ) → A T ( Y ). Proposition 4.1.
If the T -action on X satisfies Assumption 1, then the kernel of the re-striction map r : A T ( X ) → A T ( X T ) is the S ( t ∗ ) -torsion submodule of A T ( X ) .Proof. We only need to show that any element in the kernel is S ( t ∗ )-torsion, since the otherinclusion is obvious (note that A T ( X T ) is a free S ( t ∗ )-module, see Example 2.3). Take A ∈ A T ( X ) such that r ( A ) = 0. Let h , . . . , h n be the infinitesimal isotropies of the T -action which are different from t . Pick a non-zero polynomial f in S ( t ∗ ) which vanishes on h ∪ · · · ∪ h n . If one multiplies by f any element of A T ( X \ X T ) one obtains zero. Considernow the map A T ( X ) → A T ( X T ) ⊕ A T ( X \ X T ), which is the direct sum of the restrictionmaps. This map is injective and maps f A to zero. Thus, f A = 0. (cid:3) Corollary 4.2.
Assume that the T -action on X satisfies Assumption 1. Then the followingassertions are equivalent: (i) The set X T is not empty and the restriction map r : A T ( X ) → A T ( X T ) is injective. (ii) The assignment algebra A T ( X ) is S ( t ∗ ) -torsion free.Proof. The implication (ii) ⇒ (i) follows immediately from Proposition 4.1, see also Example2.2. It remains to justify (i) ⇒ (ii). But if X T = ∅ and r is injective, then A T ( X ) is asubmodule of A T ( X T ); since the latter is free, the former is torsion free. (cid:3) A class of examples which satisfy the two conditions in the corollary consists of smoothtorus actions on compact smooth manifolds that are equivariantly formal (see Section 5.1and Proposition 5.3 below).If T is a circle and the two conditions in the corollary hold true, then A T ( X ) is not onlytorsion free, but also free, because it is a submodule of A T ( X T ), which is free, and S ( t ∗ ) is aPID. In general, however, it is possible that A T ( X ) is torsion free but not free: see Example6.5.In the spirit of Borel’s localization theorem for equivariant cohomology, not only the kernelof r : A T ( X ) → A T ( X T ) is S ( t ∗ )-torsion, but also its cokernel: Proposition 4.3.
Consider an action of a torus T on a topological space X satisfyingAssumption 1. Then the cokernel of the map r : A T ( X ) → A T ( X T ) is S ( t ∗ ) -torsion.Proof. We have to show that for every assignment A ∈ A T ( X T ) there exists a polynomial f ∈ S ( t ∗ ) such that f A is in the image of r .Let f be any polynomial that vanishes on all proper (i.e., = t ) infinitesimal isotropies ofthe action. We define an assignment B on X by declaring B ( p ) = ( f A ( p ) ∈ S ( t ∗ ) , if p ∈ X T , if p / ∈ X T . ASSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS
This really defines an assignment. Let us check that it satisfies the condition in Definition1.1. If h is contained in a proper infinitesimal isotropy, then B h is identically zero; otherwise, X H = X T and for any p in a connected component of this space one has B h ( p ) = f | h A ( p ) | h ,which is constant on that component, since A is an assignment on X T . (cid:3) Corollary 4.4.
For an action of a torus T on a topological space satisfying Assumption 1,the restriction map r : A T ( X ) → A T ( X T ) is an isomorphism modulo torsion. Consequently,the rank of A T ( X ) over S ( t ∗ ) is equal to the number of connected components of X T . A Chang-Skjelbred lemma
Let X be a connected topological space acted on by the torus T . Consider the of the action, which is X := { p ∈ X | corank T p ≤ } . Besides Assumption 1 in the previous section, the following extra condition is needed here:
Assumption 2.
For any subtorus H ⊂ T , any component of X H has non-trivial intersectionwith X T and connected intersection with X .5.1. Example: equivariantly formal actions.
In this subsection we will show that As-sumption 2 is fulfilled if X is compact Hausdorff and the T -action is equivariantly formalin the sense that H ∗ T ( X ) is free relative to its canonical structure of H ∗ ( BT )-module. (Weconsider here ˇCech cohomology with real coefficients.) We first prove the following lemma. Lemma 5.1.
Assume that X is compact Hausdorff, the T -action on X is equivariantlyformal and dim H ∗ ( X ) < ∞ . Then for any subtorus H ⊂ T , the T -action on (any connectedcomponent of ) X H is equivariantly formal.Proof. Recall that if Y is any compact Hausdorff space acted on continuously by the torus T , then dim H ∗ ( Y T ) ≤ dim H ∗ ( Y ), with equality if and only if the T -action is equivariantlyformal (see e.g. [17, Corollary 2, p. 46]). In particular, the claim in the lemma is equivalentto dim H ∗ (( X H ) T ) = dim H ∗ ( X H ). But ( X H ) T = X T , and hencedim H ∗ ( X T ) ≤ dim H ∗ ( X H ) ≤ dim H ∗ ( X ) . By the aforementioned general result, we have dim H ∗ ( X T ) = dim H ∗ ( X ), hence the in-equalities above are both equalities. (cid:3) From [17, Corollary 1, p. 45] we deduce that under the assumptions in the lemma, anyconnected component of X H contains a T -fixed point. Furthermore, the 1-skeleton of thatcomponent is connected: this follows from [5, Proposition 2.5]. (Both these results are knownin the particular context of differentiable group actions on manifolds: see, e.g., [15, Theorem11.6.1] or [9, Lemma 3.1].) Example 5.2.
Not every action that satisfies Assumptions 1 and 2 is equivariantly formal.To see this, consider again Example 2.5: as noticed in [20], the action of S on X is notequivariantly formal; the reason is that it has finitely many fixed points and H ( X ) = R . SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 9
Another example is obtained by taking S , the (circular) subgroup of SU(3) which consistsof all diagonal matrices of the form Diag( z − , z − , z ), where z ∈ S . The action of S onSU(3) /S given by multiplication from the left is not equivariantly formal, see [4, Proposition8.9]. It satisfies Assumptions 1 and 2 in an obvious way. If one wants an action with anon-trivial 1-skeleton, take the direct product of the S -action above with itself: S × S onSU(3) /S × SU(3) /S . One can easily check that this is again not equivariantly formal andsatisfies Assumption 1. Assumption 2 only needs to be checked for H = { ( I, I ) } , H = { I } × S , and H = S × { I } and this can be done immediately (note that the 1-skeleton is theunion (cid:2) (SU(3) /S ) S × SU(3) /S (cid:3) ∪ (cid:2) SU(3) /S × (SU(3) /S ) S (cid:3) , which is a connected subspaceof SU(3) /S × SU(3) /S ).5.2. The Chang-Skjelbred “lemma”.
We will prove an assignment version of [5, Lemma2.3].
Proposition 5.3.
If Assumptions 1 and 2 are valid, then the restriction map r : A T ( X ) →A T ( X T ) is injective. Its image is the same as the image of r ′ : A T ( X ) → A T ( X T ) .Proof. We first show that r is injective. Take A ∈ A T ( X ) such that r ( A ) = 0. Take p ∈ X arbitrary. Denote by H the identity component of T p and by h its Lie algebra. ByAssumption 2, the connected component of X H through p contains a connected componentof X T . On the latter component A is identically zero, hence A h is identically zero on theformer component as well. This implies that A ( p ) = 0.For the second claim in the proposition, observe that one can factorize r as A T ( X ) → A T ( X ) r ′ → A T ( X T ) . Hence the image of r is contained in the image of r ′ . We now prove the other inclusion.We consider A ∈ A T ( X ) and construct B ∈ A T ( X ) such that r ( B ) = r ′ ( A ). It will beconvenient to use the following notation: if Z ⊂ X is a connected component of X T , then A ( Z ) := A ( z ) , for all z ∈ Z. Take p ∈ X , let H be the identity component of T p , set h := Lie( H ), and denote by Y theconnected component of X H that contains p . By Assumption 2, there exists a component Z of X T such that Z ⊂ Y . We set B ( p ) := A ( Z ) | h . We show that B is well defined (i.e., B ( p ) is independent of the choice of Z ) and an assign-ment as well. To this end, we take a subtorus G ⊂ T , G = T , a connected component Y of X G , two components Z and Z ′ of X T , both contained in Y , and show that(5.1) A ( Z ) | g = A ( Z ′ ) | g , where g := Lie( G ). Indeed, by Assumption 2, Y ∩ X is a connected subspace of X G . Hencethe function p A ( p ) | g is constant on Y ∩ X . Also note that Z and Z ′ are both containedin Y ∩ X . (cid:3) A GKM description of the assignment algebra.
Assumption 1 will be validthroughout this section. Let us consider all possible infinitesimal isotropies which havecodimension one in t ; say that they are g , . . . , g m . By definition, the 1-skeleton of the T -action on X is the union of all X g i := { p ∈ X | g i ⊂ t p } , 1 ≤ i ≤ m . The followingsupplementary requirement will be needed in this subsection: Assumption 3.
Each of the spaces X t and X g i , 1 ≤ i ≤ m , has finitely many connectedcomponents.For example, this condition is satisfied if X is compact Hausdorff with dim H ∗ ( X ) < ∞ ,see, e.g. [2, Corollary 3.10.2].Assumptions 1, 2, and 3 alone lead to a presentation of the algebra A T ( X ) which is similarto the one given by Goresky, Kottwitz, and MacPherson [12] for the equivariant cohomologyalgebra H ∗ T ( X ). Recall that the latter presentation requires several other assumptions: the T -action on X must be equivariantly formal, X T must be finite, and X must be a union of2-spheres (cf. also [15, Section 11.8]).We denote by Z , . . . , Z n the connected components of X T . Theorem 5.4.
If Assumptions 1, 2, and 3 are satisfied, then the image of A T ( X ) underthe injective algebra homomorphism r : A T ( X ) → A T ( X T ) is the subalgebra of A T ( X T ) = ⊕ nr =1 A T ( Z r ) = S ( t ∗ ) n which consists of all ( f , . . . , f n ) with the property that whenever Z r and Z s are contained in the same component of some X g i , ≤ i ≤ m , the difference f r − f s restricted to g i is identically zero.Proof. If A ∈ A T ( X ) then its restriction to X T is an n -tuple ( f , . . . , f n ) which obviouslysatisfies the conditions in the lemma. To prove the other inclusion, we start with ( f , . . . , f n )with the properties in the lemma. Consider the map A on X given by A ( p ) = ( f r , if p ∈ Z r f r | g i , if t p = g i and Z r ⊂ X g i . Note that, by hypothesis, the polynomial f r | g i does not depend on r with Z r ⊂ X g i . We nowshow that A is an assignment on X . Let H ⊂ T be a subtorus. If H = T then X H = X T and A is obviously constant on each component of the latter space. If H = T , we have X H = [ h ⊂ g i X g i . By Assumption 2, this is a connected space. We need to show that A h is constant on thisspace. Look at the connected components of the spaces X g i for all i ∈ { , . . . , m } with h ⊂ g i . The intersection of two such subspaces is empty or is a union of one or more Z r .Since X H is connected, for any of the two aforementioned components, say Y and Y ′ , thereexists a chain of components, Y , . . . , Y q , such that Y = Y , Y q = Y ′ and Y i ∩ Y i +1 = ∅ for all1 ≤ i ≤ q −
1. But A h is constant on each Y i , hence the values on Y and Y ′ are equal. Thus A is an assignment on X . Finally, by Proposition 5.3, A can be extended from X to anassignment on X . (cid:3) SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 11
The S -action described in Example 2.5 satisfies the hypotheses of the theorem. We havealready calculated A S ( X ) and the presentation we found is of GKM type. It is worthwhilenoting that no such presentation exists for H ∗ S ( X ). The reason is that, as already mentioned,the action has finitely many fixed points and H ( X ) = R .6. Relation to equivariant cohomology
An important class of assignments arise from equivariant cohomology. Let X be a compactHausdorff topological space. Recall that H ∗ T ( X ) = H ∗ ( E × T X ), where E = ET is the totalspace of the classifying bundle of T . (By H ∗ we mean ˇCech cohomology with real coefficients.)We will use the identification(6.1) H ∗ T ( T /H ) = S ( h ∗ ) , for any connected and closed subgroup H ⊂ T . Concretely, H ∗ T ( T /H ) = H ∗ ( E × T ( T /H )) = H ∗ ( E/H ) = H ∗ ( BH ) = S ( h ∗ ) . We now define γ X : H ∗ T ( X ) → A T ( X ) as follows: to α ∈ H ∗ T ( X ) corresponds the assignment A given by(6.2) A ( p ) := α | T p , p ∈ X. The right hand side represents the image of α under the map i ∗ p : H ∗ T ( X ) → H ∗ T ( T p ) inducedby the inclusion i p : T p ֒ → X (the identification (6.1) is taken into account). Proposition 6.1.
The map A defined by (6.2) is an assignment.Proof. We need to show that A satisfies the condition in Definition 1.1. Let H ⊂ T be aconnected and closed subgroup and Y ⊂ X a connected component of X H . For p ∈ Y , theinclusion i p : T p ֒ → X factorizes as T p ֒ → Y ֒ → X . Moreover, the map S ( t ∗ p ) → S ( h ∗ ) givenby restriction to h is actually the same as H ∗ T ( T p ) → H ∗ T ( T /H ) induced by a p : T /H → T p ⊂ Y , tH tp , for all t ∈ T (indeed, this is the only S ( t ∗ )-algebra homomorphism S ( t ∗ p ) → S ( h ∗ )). It is sufficient to show that the map a ∗ p : H ∗ T ( Y ) → H ∗ T ( T /H ) is independentof p ∈ Y . But the map E × T ( T /H ) → E × T Y induced by a p is given by [ e, H ] [ e, p ], forall e ∈ E . It can be factorized as: E × T ( T /H ) / / ' ' ❖❖❖❖❖❖❖❖❖❖❖ E × T YE × H Y rrrrrrrrrr where E × H Y → E × T Y is the canonical map induced by the inclusion H ⊂ T . Theleft-hand side map in the diagram is [ e, H ] [ e, p ], for all e ∈ E ; this map can also be expressed as: E × T ( T /H ) ∼ = (cid:15) (cid:15) / / E × H Y ∼ = (cid:15) (cid:15) E/H j p :[ e ] ([ e ] ,p ) / / ( E/H ) × Y Finally observe that j ∗ p : H ∗ (( E/H ) × Y ) → H ∗ ( E/H ) is independent of p ∈ Y : by theK¨unneth formula, H ∗ (( E/H ) × Y ) can be identified with H ∗ ( E/H ) ⊗ H ∗ ( Y ) and j ∗ p is theprojection of the latter space on H ∗ ( E/H ) ⊗ H ( Y ) ≃ H ∗ ( E/H ). (cid:3) Observe that both H ∗ T and A T are contravariant functors from the category of topologicalcompact Hausdorff T -spaces to the category of graded S ( t ∗ )-algebras. Proposition 6.2. γ is the only natural transformation between the two functors H ∗ T and A T .Proof. To prove that γ is a natural transformation we only have to verify that for everycontinuous T -equivariant map f : X → X ′ the diagram H ∗ T ( X ′ ) f ∗ / / γ X ′ (cid:15) (cid:15) H ∗ T ( X ) γ X (cid:15) (cid:15) A T ( X ′ ) f ∗ / / A T ( X )commutes. Take α ∈ H ∗ T ( X ′ ) and p ∈ X . Denote by i p : T p → X and i f ( p ) : T f ( p ) → X ′ the inclusion maps. Then the following diagram is commutative: T p f | Tp / / i p (cid:15) (cid:15) T f ( p ) i f ( p ) (cid:15) (cid:15) X f / / X ′ We thus have γ X ( f ∗ ( α ))( p ) = i ∗ p ( f ∗ ( α ))= ( f | T p ) ∗ ( i ∗ f ( p ) ( α )) = ( i f ( p ) ) ∗ ( α ) | t p = γ X ′ ( α )( f ( p )) | t p = f ∗ ( γ X ′ ( α ))( p ) . Here we have used that ( f | T p ) ∗ : H ∗ T ( T f ( p )) = S ( t ∗ f ( p ) ) → H ∗ T ( T p ) = S ( t ∗ p ) is just therestriction map induced by the inclusion t p ⊂ t f ( p ) .For the converse we let η denote any natural transformation between H ∗ T and A T . We fixan arbitrary compact Hausdorff T -space X and show that the S ( t ∗ )-algebra homomorphism η X : H ∗ T ( X ) → A T ( X ) necessarily coincides with γ X . As assignments are determined bytheir values at each point, we fix an arbitrary point p and consider again the inclusion SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 13 i p : T p → X . Then we have a commutative diagram H ∗ T ( X ) i ∗ p / / η X (cid:15) (cid:15) H ∗ T ( T p ) η Tp (cid:15) (cid:15) A T ( X ) i ∗ p / / A T ( T p )Both objects on the right are isomorphic to S ( t ∗ p ). We observe that the bottom horizontalmap is just evaluation at p . Moreover, the vertical map on the right is necessarily the unique S ( t ∗ )-algebra homomorphism S ( t ∗ p ) → S ( t ∗ p ), namely the identity. Thus, the commutativityof the diagram implies that η X = γ X . (cid:3) Remark 6.3.
Observe that by restriction to the even-dimensional part of the equivariantcohomology groups, γ induces a transformation between the functors H even T and A T . With thesame methods as in the proof of Proposition 6.2, one can show that γ | H even T is the only naturaltransformation between these two functors. Recall that another natural transformationbetween H even T and A T was introduced in [14], in the context of smooth T -manifolds, usingthe Cartan model of equivariant cohomology (cf. also Section 3 above). We deduce that thistransformation coincides with γ | H even T on compact smooth T -manifolds.As noted in [14, Section 4], if X is a compact manifold and the action of T on X is ofGKM type, then γ X is an isomorphism. Here are two non-smooth examples when γ X is anisomorphism. Example 6.4.
For the weighted projective plane P ( m, n ) already addressed in Example 2.4,the map γ : H ∗ S ( P ( m, n )) → A S ( P ( m, n )) is an isomorphism. This follows from the GKMpresentation of H ∗ S ( P ( m, n )). Example 6.5.
Consider X = Σ T , the unreduced suspension of the 2-torus T = T , withthe canonical T -action. The action has two fixed points and is free on their complement in X . Both H ∗ T ( X ) and A T ( X ) can be easily calculated by using the Mayer-Vietoris sequence,see Proposition 2.8. As it turns out, both algebras are isomorphic to the space { ( f , f ) ∈ S ( t ∗ ) × S ( t ∗ ) | f (0) = f (0) } (see also [6, Example 5.5]). This shows that the correspondingmap γ is again an isomorphism. It is shown in [1, Example 3.3] that H ∗ T ( X ) equipped withits canonical structure of S ( t ∗ )-module is torsion free but not free. Thus, the same can besaid about A T ( X ). Remark 6.6.
Observe that, in general, the assignment algebra can be defined “over Z ”:that is, in Definition 1.1, one considers A ( p ) ∈ S (( t p ) ∗ Z ), where ( t p ) ∗ Z is the weight latticeof t p , p ∈ X . It would be worthwhile investigating this new invariant, call it A T ( X ; Z ).The first natural question to be addressed is to which extent the Chang-Skjelbred lemmafor assignments, i.e. Proposition 5.3 above, can be extended over Z , in the same way as theusual Chang-Skjelbred lemma was extended for integral cohomology by Franz and Puppein [7, Corollary 2.2] and by Goertsches and Wiemeler in [10, Lemma 6.1]. After that, onecould start searching for extensions of Theorem 5.4. In this context, we point out that for general GKM actions, A T ( X ; Z ) is not isomorphic to H ∗ T ( X ; Z ). To see this, consider againExample 2.4, this time with m = n = 1; that is, we just look at the standard rotationaction of S on S , the spinning speed being k . Like in Example 2.4, A S ( S ; Z ) consists ofpairs f , f ∈ Z [ u ] with f − f divisible by u (independent of k ). However, H ∗ S ( S ; Z ) doesdepend on k : it consists of pairs f , f ∈ Z [ u ] with f − f divisible by ku . This exampleseems to indicate that, in general, the isomrphism between the assignment algebra over Z and the integral equivariant cohomology for GKM torus actions requires the supplementaryassumption that any of the weights at any fixed point is primitive, i.e., no integer multipleof another weight. 7. Locally free actions
In this section we will assume that X is a completely regular topological space. Thisassumption will allow us to use the Slice Theorem, cf., e.g., [3, Ch. II, Theorem 5.4], which isan essential ingredient for us. For example, any Hausdorff locally compact topological spaceis completely regular.Consider now a subtorus Q ⊂ T , whose induced action on X is locally free, i.e., the isotropy Q p is finite for all p ∈ X . The quotient T /Q acts canonically on
X/Q , as follows: tQ.Qp := Qtp , t ∈ T, p ∈ X . In this section we show that A T ( X ) is isomorphic to A T/Q ( X/Q ). In thecase when X is smooth and the Q -action is smooth and free, this result has been proved in[14, Section 8]: the isomorphism is constructed there explicitly by relating the stratificationsof X and X/Q . We adapted the approach from the aforementioned paper to our set-up.The main difference is that we use the pointwise definition of assignments. The two majorbenefits of this definition are that the result we will prove is purely topological, hence moregeneral, and that the whole construction involves only points rather than strata, and istherefore more transparent.Let π : X → X/Q be the canonical projection. We construct a map π ∗ : A T ( X ) →A T/Q ( X/Q ), as follows. Take p ∈ X . The isotropy group ( T /Q ) Qp is equal to T ( p ) /Q ,where T ( p ) := { t ∈ T | tp ∈ Qp } . The group T ( p ) acts transitively on Qp , thus the latter space is homeomorphic to both T ( p ) /T p and Q/Q p . Lemma 7.1.
The map T p /Q p → T ( p ) /Q given by the inclusion of T p into T ( p ) followed bythe canonical projection is a group isomorphism.Proof. The map is obviously injective. To prove surjectivity, observe that for any t ∈ T ( p )there exists g ∈ Q such that tp = gp , hence tQ is the image of g − tQ p . (cid:3) Denote by t p and t ( p ) the Lie algebras of T p and T ( p ) respectively. The differential atthe identity of the group isomorphism mentioned in Lemma 7.1 is a linear isomorphism, SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 15 whose inverse is ϕ p : t ( p ) / q → t p . Note that t ( p ) / q is the Lie algebra of ( T /Q ) Qp . Define π ∗ : A T ( X ) → A T/Q ( X/Q ),(7.1) π ∗ ( A )( Qp ) := ϕ ∗ p ( A ( p )) , for all p ∈ X. We need to show that the map π ∗ ( A ) satisfies the requirement of Definition 1.1. Consider asubtorus of T /Q , which is of the form
H/Q , where H is a subtorus of T with Q ⊂ H . Thefixed points of H/Q in X/Q are orbits Qp , with p ∈ X such that Hp = Qp . Let C be aconnected component of ( X/Q ) H/Q . Lemma 7.2. If π : X → X/Q is the canonical projection, then π − ( C ) is a connectedsubspace of X .Proof. Assume that π − ( C ) is a disjoint union of two non-empty open subspaces U and U .Both U and U are Q -invariant: if p ∈ U , then Qp is connected and Qp = ( U ∩ Qp ) ∪ ( U ∩ Qp ), the elements of the union being open subspaces of Qp . But then π ( U ) and π ( U )are disjoint as well. Since they are open in C , the latter space is not connected, which is acontradiction. (cid:3) Let us now denote by h and h p the Lie algebras of H and H p , respectively, where p ∈ X . Lemma 7.3.
The space h p is independent of p ∈ π − ( C ) .Proof. First, if p ∈ π − ( C ), then Qp = Hp , thus dim h p = dim h − dim q , which is independentof p . From the Slice Theorem, see [3, Ch. II, Theorem 5.4], any p ∈ π − ( C ) has an openneighborhood where all H -isotropy groups are contained in H p ; the Lie algebras of thesegroups are therefore all equal to h p . Since π − ( C ) is connected and h p is locally constantfor p ∈ π − ( C ), it is in fact globally constant. (cid:3) Set h ′ := h p , p ∈ π − ( C ). Let H ′ be the connected subgroup of T whose Lie algebra is h ′ (that is, the connected component of H p , with p as above). From the previous lemma, π − ( C )is contained in a connected component of X H ′ . For any p ∈ π − ( C ), the isomorphism ϕ p : t ( p ) / q → t p maps h / q to h ∩ t p = h ′ . Since A ( p ) | h ′ is independent of p in the aforementionedcomponent of X H ′ , it follows that π ∗ ( A )( Qp ) | h / q is independent of Qp ∈ C .We now have a well defined map π ∗ : A T ( X ) → A T/Q ( X/Q ). Recall that A T ( X ) has acanonical structure of an S ( t ∗ )-algebra. Via the canonical homomorphism S (( t / q ) ∗ ) → S ( t ∗ ), A T ( X ) can also be endowed with a structure of a S (( t / q ) ∗ )-algebra. One can verify that π ∗ is a homomorphism of S (( t / q ) ∗ )-algebras. Theorem 7.4.
The map π ∗ : A T ( X ) → A T/Q ( X/Q ) is an isomorphism of S (( t / q ) ∗ ) -algebras.Proof. We show how to construct σ : A T/Q ( X/Q ) → A T ( X ) which is inverse to π ∗ . To thisend we first consider for any p ∈ X the inverse of ϕ p , which is ψ p : t p → t ( p ) / q (the inclusionof t p into t ( p ) followed by the canonical projection). If B ∈ A T/Q ( X/Q ), then we define σ ( B )( p ) := ψ ∗ p ( B ( Qp )) , for all p ∈ X. We show that σ ( B ) satisfies the requirement of Definition 1.1. Take H ′ ⊂ T a subtorus withthe property that X H ′ = ∅ . Then H ′ ∩ Q is a finite group. Set H := H ′ · Q and note that itsLie algebra is h = h ′ ⊕ q . Let Y be a connected component of X H ′ . One can easily see that π ( Y ) is contained in (a component of) ( X/Q ) H/Q . For any p ∈ Y we have h ′ ⊂ t p , hence ψ p ( h ′ ) = h / q . Moreover, the map ψ p | h ′ : h ′ → h / q is independent of p ; if we denote this mapby ψ , we have σ ( B )( p ) | h ′ = ψ ∗ ( B ( Qp ) | h / q ) . Since B ( Qp ) | h / q is constant on any connected component of ( X/Q ) H/Q , the left-hand sideof the previous equation is constant on Y . At this point we conclude that the map σ : A T/Q ( X/Q ) → A T ( X ) is well defined. It only remains to observe that σ ◦ π ∗ and π ∗ ◦ σ areequal to the identity. (cid:3) Example 7.5.
Consider the action of T = S × S on S described in Example 2.6. Take S := { ( z, z ) | | z | = 1 } , which is a subgroup of T . It acts freely on X , thus A T ( S ) ≃A T/S ( S /S ) . We have S /S = C P = S and the T /S -action on it is equivalent to thecanonical “rotation” action of the circle S . Along with the presentation of A S ( S ) (seefor instance [14, Example 2.2]), these identifications lead readily again to the description of A T ( S ) given in Example 2.6. 8. Kirwan surjectivity
The assignment Kirwan map.
The following set-up is mentioned in [14, Section8.3]. One considers a symplectic manifold M equipped with a Hamiltonian action of a torus T as well as a subtorus Q ⊂ T , whose Lie algebra is q ⊂ t . The moment map of the Q -action is Φ : M → q ∗ . Let µ ∈ q ∗ be a regular value of this map. Then the actionof Q on the pre-image Φ − ( µ ) is locally free, hence the symplectic quotient M//Q ( µ ) :=Φ − ( µ ) /Q has a canonical structure of a symplectic orbifold. It also has a canonical actionof the torus T /Q . One way to obtain information about the equivariant cohomology algebra H ∗ T/Q ( M//Q ( µ )) is by identifying it with H ∗ T (Φ − ( µ )); the inclusion Φ − ( µ ) ֒ → M induces thealgebra homomorphism κ : H ∗ T ( M ) → H ∗ T (Φ − ( µ )). This is called the equivariant Kirwanmap and was first studied by Goldin in [11], inspired by Kirwan’s fundamental work [18].Relevant for our goal is the surjectivity of this map, which holds under the assumption thatΦ is a proper map (see [11, Theorem 1.2], cf. also [18] and [19]).A natural attempt is to obtain similar results about the assignment algebra of the T -actionon M . First, by Theorem 7.4, A T (Φ − ( µ )) ≃ A T/Q (Φ − ( µ ) /Q ). To complete the analogywith equivariant cohomology, one needs to understand whether the map κ A : A T ( M ) →A T (Φ − ( µ )) is surjective. We call κ A the assignment Kirwan map . We first give an examplewhich shows that, in general, κ A is not surjective. Example 8.1.
We consider the action of the torus T on C P given by( e πit , e πit ) . [ z : z : z : z ] = [ z : e πit z : e πit z : e πi ( t + t ) z ] . SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 17
The canonical identification of t ∗ with t = R leads to the following description of a momentmap:Φ : C P → R , Φ([ z : z : z : z ]) = 1 | z | + | z | + | z | + | z | (2 | z | + | z | , | z | + | z | ) . (As usual, C P is equipped with the Fubini-Study symplectic form.) The circle Q = { ( e πit , e πit ) | t ∈ R } ⊂ T acts on C P with moment map(8.1) Φ Q : C P → R , Φ Q ([ z : z : z : z ]) = 2 | z | + | z | + | z | | z | + | z | + | z | + | z | . The open subspace U := { [1 : z : z : z ] | z , z , z ∈ C } ⊂ C P is Q -invariant and themoment map is the restrictionΦ Q | U : U → R , Φ Q ([1 : z : z : z ]) = 2 | z | + | z | + | z | | z | + | z | + | z | . The pre-image Φ − Q (1) is clearly contained in U . Concretely, this space is just the unit sphere S in U = C . The induced T -action on S is given by( e πit , e πit ) . ( z , z , z ) = ( e πit z , e πit z , e πi ( t + t ) z ) , for all ( z , z , z ) ∈ C with | z | + | z | + | z | = 1. Thus the assignment algebra A T ( S )consists of triples ( f , f , f ), where f : { ( t , t ) ∈ R | t = 0 } → R ,f : { ( t , t ) ∈ R | t = 0 } → R ,f : { ( t , t ) ∈ R | t = − t } → R are polynomial functions with f (0 ,
0) = f (0 ,
0) = f (0 , A T ( U ) → A T ( S ) is not surjective (this implies that also κ A : A T ( C P ) → A T ( S ) is notsurjective, since it factorizes by the map above). This is because given ( f , f , f ) ∈ A T ( S ),one cannot always find f ∈ R [ t , t ] whose restrictions to the subspaces of equations t = 0, t = 0, and t = t are f , f , f , respectively. For instance, one can take f (0 , t ) = t , f ( t ,
0) = t , f ( t, − t ) = t . Assume there exists f ∈ R [ t , t ] with the aforementionedproperties. We may assume that f is homogeneous of degree one (otherwise, one can replaceit by its degree-one component). This means that f : R → R is a linear map. It must satisfy f (0 ,
1) = f (0 ,
1) = 1, f (1 ,
0) = f (1 ,
0) = 1, f (1 , −
1) = f (1 , −
1) = 1. This contradicts f (1 , −
1) = f (1 , − f (0 , Example 8.2. (We are grateful to the referee for kindly suggesting this example to us.)The surjectivity of κ A does hold in the following situation. Assume M is compact, the T -action has finitely many fixed points, and the weights of the isotropy representation atany fixed point are any three linearly independent. Furthermore, assume that the circle Q ⊂ T is generic, i.e. M Q = M T , and also that the Q -action on Φ − ( µ ) is free. This settingwas considered by Guillemin and Zara in [16], where they proved that under the aboveassumptions, the T /Q -action on Φ − ( µ ) /Q is GKM, see Theorem 1.5.1 in their paper. By Theorem 5.4 and Proposition 6.2, each of M and Φ − ( µ ) /Q has its assignment ring andequivariant cohomology ring isomorphic to each other. The surjectivity of the assignmentKirwan map then follows from the surjectivity of the genuine (cohomological) Kirwan map.8.2. A surjectivity criterion.
Let M be a compact symplectic manifold acted on by atorus T , the action being Hamiltonian. Inspired by Example 8.1, we make an assumptionwhich concerns the weights of the isotropy representation at fixed points. To formulate it,we first choose a Riemannian metric on M such that T acts isometrically on M . Let F be aconnected component of M T . For any p ∈ F , the normal space ν p F has a complex structurewhich is preserved by the T -action. Let α ,F , . . . , α m,F be the weights of the T -representationon ν p F (note that they must not be pairwise distinct). The number m is equal to half thecodimension of F in M and it may change from a connected component of M T to the other.The corresponding weight space decomposition is ν p F = m M i =1 C α i,F , where C α i,F is a copy of C acted on by T with weight α i,F . We say that two functions α, β ∈ t ∗ are equivalent, and denote this by α ∼ β , if α is a scalar multiple of β . The mainresult of this section is: Theorem 8.3.
Assume that for any connected component F of M T , the elements of the set { α ,F , . . . , α m,F } / ∼ are linearly independent. Then for any circle Q ⊂ T and any regularvalue µ of Φ : M → q ∗ , the map κ A : A T ( M ) → A T (Φ − ( µ )) is surjective. We need a preliminary result.
Lemma 8.4.
Let V be a real vector space of dimension n . Let also m be an integer with ≤ m ≤ n and β , . . . , β m some linearly independent elements of the dual space V ∗ . Finally,let V , . . . , V k be subspaces of V , each of them of the form ker β i ∩ · · · ∩ ker β i q , where i , . . . , i q ∈ { , . . . , m } . Assume that for each ≤ i ≤ k , f i is a polynomial in S ( V ∗ i ) such that f i | V i ∩ V j = f j | V i ∩ V j , for all ≤ i, j ≤ k . Then there exists f ∈ S ( V ∗ ) such that f | V i = f i , for all ≤ i ≤ k .Proof. The proof is by induction on n . For n = 1 the statement is trivially true. It nowfollows the induction step. For any 1 ≤ q ≤ m and any i , . . . , i q ∈ { , . . . , m } we constructa polynomial g i ,...,i q ∈ S ((ker β i ∩ · · · ∩ ker β i q ) ∗ ) such that: • if V i = ker β i ∩ · · · ∩ ker β i q then g i ,...,i q = f i ; • if { i ′ , . . . , i ′ r } ⊂ { i , . . . , i q } then g i ,...,i q = g i ′ ,...,i ′ r | ker β i ∩···∩ ker β iq .We proceed by recursion. First, for q = m : the intersection ker β ∩ · · · ∩ ker β m is equal toor contained in at least one V i ; we define g ,...,m as the restriction of f i to ker β ∩ · · · ∩ ker β m .Assume that we have constructed g on all intersections of at least q + 1 kernels. We wishto construct g i ,...,i q . If ker β i ∩ · · · ∩ ker β i q is equal to V i , for some 1 ≤ i ≤ k , we define g i ,...,i q := f i . Otherwise, we use the induction hypothesis to construct g i ,...,i q with prescribed SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 19 values on any intersection of the form ker β i ∩ ker β i ∩ · · · ∩ ker β i q (note that the space of allrestrictions β i | ker β i ∩···∩ ker β iq which are not identically zero consists of linearly independentelements of (ker β i ∩ · · · ∩ ker β i q ) ∗ ).We end up with polynomials g ∈ S ((ker β ) ∗ ) , . . . , g m ∈ S ((ker β m ) ∗ ) such that if V i =ker β i ∩ · · · ∩ ker β i q is contained in ker β j then f i = g j | V i . The goal is to construct f ∈ S ( V ∗ )such that f | ker β j = g j , for all 1 ≤ j ≤ m .Set W j = ker β j , 1 ≤ j ≤ m . We can find a basis w , . . . , w n of V such that W j =Span { w , . . . , w j − , w j +1 , . . . , w n } , 1 ≤ j ≤ m . If x , . . . , x n are the coordinates relative tothis basis, then W j is described by x j = 0 and g j is in R [ x , . . . , x j − , x j +1 , . . . , x n ]. For any J = { ≤ j < . . . < j k ≤ m } we denote by J c its complement in { , . . . , n } ; we also denoteby x J the vector in R n whose components are 0, except those of index j , . . . , j k , which are x j , . . . , x j k respectively. Set f := g + · · · + g m + X k ≥ ,I = { ≤ i <... A T ( M c + ǫ ) → A T ( M c − ǫ ) is surjective. Let k be theindex of C as a critical manifold of f . The negative spaces of the Hessian along C give riseto a vector bundle E → C , whose rank is k , such that E q is a subspace of ν q C , for all q ∈ C .By the equivariant Morse-Bott Lemma, see [21, Theorem 4.9], M c + ǫ is T -equivariantlyhomotopy equivalent to the space obtained from M c − ǫ by attaching the (closed) unit diskbundle D in E along its boundary S . Lemma 8.5.
The restriction map A T ( D ) → A T ( S ) is surjective.Proof. Take A ∈ A T ( S ). By Proposition 2.7, A T ( D ) ≃ A T ( C ), hence our surjectivity state-ment amounts to showing that there exists B ∈ A T ( C ) such that for any p ∈ C and any v inthe fiber S p one has A ( v ) = B ( p ) | t v . To this end, we consider the infinitesimal stratificationof C , whose elements are X , . . . , X n , with isotropy algebras k , . . . , k n respectively, such that if X a ⊂ X b then b ≤ a . For a ∈ { , . . . , n } , the weights of the (isotropy) k a -representationon T p M are independent of p ∈ X a . This representation leaves both T p C and E p invari-ant. Denote the weights of the k a -representation on E p by γ a, , . . . , γ a,ℓ , where ℓ is half therank of E . If F is a connected component of M T which is contained in X a , the functions γ a, , . . . , γ a,ℓ are restrictions to k a of certain weights of the isotropy representation along νF (more precisely, the weights of the T -representation on E q ⊂ ν q F , where q ∈ F ). Consider k a, := ker γ a, , . . . , k a,ℓ := ker γ a,ℓ , which are subspaces of k a . Note that the functions γ a, , . . . , γ a,ℓ may be pairwise proportionalor even equal and consequently the spaces above are not necessarily distinct. For any i ∈{ , . . . , ℓ } the spaces { v ∈ E p | x.v = γ a,i ( x ) v, for all x ∈ k a } with p ∈ X a give rise to asplitting of E | X a as a direct sum of T -equivariant subbundles. The vectors in the intersectionof this subbundle with S form a connected subspace of S where the infinitesimal isotropy is k a,i ; hence these vectors are all mapped to the same polynomial g a,i ∈ S ( k ∗ a,i ). The idea is touse induction on a ∈ { , . . . , n } to construct f a ∈ S ( k ∗ a ) such that:(i) if X a ⊂ X b then f a | k b = f b ;(ii) f a | k a,i = g a,i .(After performing this construction, we define B as the map which assigns to each stratum X a the polynomial f a .)Let us first take a = 1. The corresponding X is the regular stratum of the action. Onlycondition (ii) needs to be satisfied. To justify that f ∈ S ( k ∗ ) with these properties exists,pick F a component of C T . As already pointed out, γ , , . . . , γ ,ℓ are restrictions to k of someweights of the T -representation along νF . But k is an intersection of kernels of weights ofthe same representation, hence k , , . . . , k ,ℓ are of the same type. One applies Lemma 8.4 for V = t and β , . . . , β m the weights of the T -representation along νF (modulo the equivalencerelation mentioned in Theorem 8.3, these weights are linearly independent). One uses that A is an assignment on S . It follows that there exists a polynomial in S ( t ∗ ) whose restrictionto k ,i is g ,i , for all i = 1 , . . . , ℓ . By restricting this polynomial to k one obtains the desired f .It now follows the induction step. That is, assuming that f , . . . , f a − are known, we showhow to construct f a . First note that if X a ⊂ X b and a = b then b < a and hence f b isknown. Pick F a connected component of M T contained in X a . Then k a , k a, , . . . , k a,ℓ are allsubspaces of t that can be obtained by intersecting kernels of weights of the T -representationalong νF ; the same can be said about k b , whenever X a ⊂ X b , since F is then contained in X b . One uses again Lemma 8.4. The compatibility conditions that need to be checked areof three types:1. if X a ⊂ X b and X a ⊂ X b ′ then f b | k b ∩ k b ′ = f b ′ | k b ∩ k b ′ ;2. if X a ⊂ X b and i ∈ { , . . . , ℓ } , then f b | k b ∩ k a,i = g a,i | k b ∩ k a,i ;3. if i, i ′ ∈ { , . . . , ℓ } then g a,i | k a,i ∩ k a,i ′ = g a,i ′ | k a,i ∩ k a,i ′ . SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 21
To justify 1, pick again F a connected component of M T contained in X a . Pick q ∈ F andconsider the weight space decomposition of ν q F (the normal space to F in C ). Then k b ∩ k b ′ isan infinitesimal isotropy of vectors/points in ν q F that are also in a tubular neighbourhood of F in C . Moreover, this Lie algebra is the infinitesimal isotropy of a stratum, say X c , whoseclosure contains q , as well as points in X b and points in X b ′ . Thus X b ⊂ X c and similarly X b ′ ⊂ X c . By the induction hypothesis, both f b | k b ∩ k b ′ and f b ′ | k b ∩ k b ′ are equal to f c . For 2, onetakes into account that for any p ∈ X b , the k b -representation on E p has the same weights.If p ∈ X a , these weights are the restrictions to k b of the weights of the k a -representation on E p , which are γ a, , . . . , γ a,ℓ . The kernels of the restrictions are just k b ∩ k a,i , where 1 ≤ i ≤ ℓ .The connected component of C k b which contains X b is a submanifold of C . For any p in thissubmanifold one considers { v ∈ E p | x.v = γ a,i ( x ) v, for all x ∈ k b } and obtains in this way avector bundle. Take v in the intersection of S with the fiber over p and v ′ in the intersectionof S with the fiber over p ′ , where p ∈ X a and p ′ ∈ X b . One can join p and p ′ by a path in C k b , then one can lift it and get a path from v to v ′ in the vector bundle intersected with S .Since A is an assignment on S , the image of v ′ under A is g a,i | k b ∩ k a,i . Property 2 now followsfrom the induction hypothesis. As about 3, it is a direct consequence of the fact that A is anassignment on S . By Lemma 8.4, conditions 1, 2 and 3 imply that there exists a polynomialin S ( t ∗ ) which satisfies conditions (i) and (ii) above. One defines f a as the restriction to k a of this polynomial. (cid:3) Theorem 8.3 now follows from the following lemma.
Lemma 8.6.
The restriction map A T ( M c + ǫ ) → A T ( M c − ǫ ) is surjective.Proof. We identify M c + ǫ = M c − ǫ ∪ S D . The result follows readily from the Mayer-Vietorissequence (see Proposition 2.8) for the spaces M c − ǫ and D , which are closed in M c + ǫ andwhose intersection is S . (cid:3) Remark 8.7.
The assumption that f − ( c ) contains only one critical manifold has been usedin the proof above. By dropping it, no essential changes are necessary. Indeed, if the criticalmanifolds in f − ( c ) are C i , . . . , C i r , then, by the equivariant Morse-Bott Lemma (see [21,Theorem 4.9]), M c + ǫ is obtained from M c − ǫ by attaching some disk bundles D , . . . , D r over C i , . . . , C i r along their boundaries S , . . . , S r . Like in Lemma 8.5, the map A T ( D i ) → A T ( S i )is surjective, for all 1 ≤ i ≤ r . One uses again an obvious Mayer-Vietoris argument. Example 8.8.
As a direct application of Theorem 8.3 one can show that if M equippedwith a T -action is a toric manifold, then for any circle Q ⊂ T , the assignment Kirwan mapcorresponding to any regular value µ of the Q -moment map is surjective. In the case when Q acts freely on the preimage Φ − ( µ ), this is just a special case of Example 8.2. Example 8.9.
Let T be an n -dimensional torus. Pick some weights α , . . . , α m ∈ t ∗ Z andconsider first the induced actions of T on C P , then the induced diagonal action on C P ×· · · × C P ( m factors). The action has 2 m fixed points, which are m -tuples of the type( p ± , . . . , p ± ), where p + = [1 : 0] and p − = [0 : 1]. The corresponding isotropy weights at any such point are ± α , . . . , ± α m . If the elements of { α , . . . , α m } / ∼ are linearly independent,then for any circle Q ⊂ T and any regular value µ of the moment map Φ Q , the assignmentmoment map A T ( C P × · · · × C P ) → A T (Φ − Q ( µ )) is surjective. This happens for instanceif the weights are all equal, i.e. α = . . . = α m . Note that in this case, the rings H ∗ T ( C P ×· · · × C P ) and A T ( C P × · · · × C P ) are not isomorphic, cf. e.g. [14, Example 7.4]. Thus,unlike the previous example, the surjectivity of the assignment Kirwan map is not a directconsequence of the surjectivity of the genuine Kirwan map. Remark 8.10.
In Theorem 8.3 it is essential to make the linearly independence assumptionalong all connected components of M T . If one takes for instance Example 8.1, the isotropyweights at [0 : 1 : 0 : 0] are − t , t − t ) , and t − t . They are linearly independent modulothe equivalence relation in Theorem 8.3. Nonetheless, we have seen that the corresponding κ A is not surjective. As expected, there is at least one other fixed point where the assumptionis not satisfied: for example, at [1 : 0 : 0 : 0] the weights are 2 t , t , and t + t .8.3. The kernel of the Kirwan map.
As before, M is a compact symplectic manifoldequipped with a Hamiltonian action of a torus T . Let Q ⊂ T be again a circle. Recallthe identification q ∗ = R , made by means of an inner product on t . Let Φ : M → R be the moment map of the Q -action. Under the assumption that 0 is a regular value ofthe latter map, we describe the kernel of κ A : A T ( M ) → A T (Φ − (0)). Our description issimilar in spirit to the one given by Tolman and Weitsman [19] in the context of equivariantcohomology. Theorem 8.11. If is a regular value of Φ : M → R , then the kernel of κ A : A T ( M ) →A T (Φ − (0)) is equal to the direct sum K + ⊕ K − , where K ± consist of all A ∈ A T ( M ) with the property that A ( F ) = 0 for all connected components F of M T with Φ( F ) > (resp. Φ( F ) < ).Proof. We first show that if A ∈ K + then A ( q ) = 0 for all q ∈ Φ − (0). To this end,let G denote the identity component of the isotropy group T q and let C be the connectedcomponent of q in M G . This is a T -invariant symplectic submanifold of M . The map Φ | C is not constant, since the action of Q on C is non-trivial (recall that the action of Q onΦ − (0) is locally free and q ∈ Φ − (0)). Observe now that 0 is in the interior of the linesegment Φ( C ): otherwise q would be an extremal point of Φ | C , hence a critical point, whichis impossible, since q is not Q -fixed (again because the action of Q on Φ − (0) is locally free).That is, Φ( C ) is an interval [ a, b ], where a < < b . We claim that Φ − ( b ) ∩ C containspoints that are T -fixed. (The reason is that Φ( C ) is obtained from the image of C underΦ T : C → t ∗ by projecting it orthogonally onto the line q ∗ ; but Φ T ( C ) is a polytope whosevertices are of the form Φ T ( F ), where F is a connected component of C T .) Thus there exists p ∈ C T with Φ( p ) = b >
0. We have A ( p ) = 0 and hence A ( q ) = A ( p ) | Lie( G ) = 0.Similarly, if A ∈ K − then A ( q ) = 0 for all q ∈ Φ − (0). We have proved that K + ⊕ K − ⊂ ker κ A . SSIGNMENTS FOR TOPOLOGICAL GROUP ACTIONS 23
The next goal is to prove the other inclusion. Take A in A T ( M ) whose restriction toΦ − (0) is identically 0. Consider the map A − on the set of all connected components of M T with values in S ( t ∗ ) given by A − ( F ) := ( , if Φ( F ) < A ( F ) , if Φ( F ) > . We show that A − extends to an assignment on M . By Theorem 5.4, we need to check thatif g ⊂ t is a codimension-one isotropy subalgebra and F , F are connected components of M T contained in the same connected component of M g , then A − ( F ) − A − ( F ) vanishes on g . This is certainly true if Φ( F ) and Φ( F ) have the same sign. Let us now assume thatΦ( F ) < < Φ( F ). The connected component of M g mentioned above contains at least onepoint q with Φ( q ) = 0. Since q is not T -fixed, the isotropy algebra t q is equal to g . We thushave A − ( F ) | g = A ( F ) | g = A ( q ) = 0, which shows that A − ( F ) is equal to A − ( F ) on g .Similarly, take the map A + on the set of all connected components of M T with values in S ( t ∗ ), given by A + ( F ) := ( A ( F ) , if Φ( F ) < , if Φ( F ) > . In the same way as before, A + extends to an assignment on M . We obviously have A = A + + A − , A + ∈ K + , and A − ∈ K − . (cid:3) Example 8.12.
The building stone of our example is the “rotation” action of S on thesphere S = C P , which is z. [ z : z ] := [ zz : z ] , z ∈ S , [ z : z ] ∈ C P . To describe a moment map it will be convenient to identify C P with the unit 2-sphere in R : the height function h : C P → R is a moment map. The critical points are q + := [1 : 0]and q − := [0 : 1], the North pole and the South pole on the sphere; that is, h ( q + ) = 1 and h ( q − ) = −
1. The actual example we will be looking at is the action of T = S × S on M := C P × C P × C P given by( z , z ) . ( q , q , q ) := ( z .q , z .q , z .q ) . Let t = R × R be the Lie algebra of T . A moment map of the above action is M → R × R , ( q , q , q ) ( h ( q ) + h ( q ) , h ( q )). Inside T we choose the diagonal circle ∆( S ) = { ( z, z ) | z ∈ S } . By restriction to this subgroup one obtains the diagonal action of S on C P × C P × C P , whose moment map is Φ : M → R , Φ( q , q , q ) = h ( q ) + h ( q ) + h ( q ).The critical points are the S -fixed points, that is, ( q ± , q ± , q ± ), eight points altogether.Consequently, the singular values are − , − , , and 3. In particular, 0 is a regular value.Denote by M = Φ − (0) / ∆( S ) the symplectic quotient at 0 and set T := T / ∆( S ). By themethod described in this section we can calculate A T ( M ), as follows. First, Theorem 5.4allows us to describe A T ( M ). Concretely, M T consists again of the eight points ( q ± , q ± , q ± ). We label them as follows: p := ( q − , q − , q − ) , p := ( q + , q − , q − ) , p := ( q − , q + , q − ) , p := ( q − , q − , q + ) ,p := ( q + , q + , q − ) , p := ( q + , q − , q + ) , p := ( q − , q + , q + ) , p := ( q + , q + , q + ) . The one-codimensional isotropies of the T -action are { e } × S and S × { e } . The fixedpoint sets of these two subgroups are C P × C P × { q ± } and { q ± } × { q ± } × C P . Byidentifying S (( t ) ∗ ) = R [ u , u ], we deduce that A T ( M ) consists of all 8-tuples ( f , . . . , f )where f i ∈ A T ( { p i } ) = R [ u , u ], 1 ≤ i ≤
8, such that: f − f , f − f , f − f , and f − f are divisible by u ; f − f , f − f , f − f , f − f , f − f , and f − f are divisible by u . A basis of A T ( M ) over R [ u , u ] consists of: A := (1 , , , , , , , A := (0 , u , , , , u , , A := (0 , , u , , , , u , A := (0 , , , u , , u , u , u ); A := (0 , , , , u , , , u ); A := (0 , , , , , u u , , A := (0 , , , , , , u u , A := (0 , , , , , , , u u ) . The weights of the T isotropy action at any fixed point, regarded as vectors of t , are( ± , , ( ± , , and (0 , ± p , p , p , and p to positive numbers and p , p , p , and p to negative numbers.Consequently, K + is spanned over R [ u , u ] by: B := ( u , , , u , , , , B := (0 , u u , , , , , , B := (0 , , u u , , , , , B := (0 , , , u u , , , , K − is spanned by A , A , A , and A . Theorems 8.3 and 8.11 imply that A T ( M ) ≃ A T ( M ) / ( K + ⊕ K − ) , by an isomorphism of S ( t ∗ )-algebras. Recall that t = ( R ⊕ R ) / ∆( R ), hence S ( t ∗ ) can beidentified with the subring R [ u − u ] of R [ u , u ]. In conclusion, A T ( M ) ≃ Span R [ u ,u ] ( A , . . . , A ) / Span R [ u ,u ] ( B , . . . , B , A , . . . , A ) , the quotient in the right hand side being regarded as an R [ u − u ]-algebra. A basis of A T ( M ) as a module over R [ u − u ] consists of the classes of A , A , A , and A . They area generating system since for any i = 1 , , ,
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