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Cohomology of GKM Fiber Bundles

COHOMOLOGY OF GKM FIBER BUNDLES
VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA Abstract. For GKM manifolds the equivariant cohomology ring of the manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant ?brations for which the total space and the base space are both GKM and derive a graph theoretical version of the Serre-Leray theorem. We also make some applications of this result to the equivariant cohomology theory of ?ag varieties. Keywords: Equivariant ?ber bundle, equivariant cohomology, GKM space, ?ag manifold MSC: 55R91, 05C25, 05E15, 55N91

arXiv:0806.3539v1 [math.CO] 22 Jun 2008

1. Introduction Let T be an n?dimensional torus and M a T ?manifold. We recall that the equivariant cohomology groups of M are de?ned as the usual cohomology groups of the quotient (M × E)/T , where E is the classifying bundle of the group T . In particular, if M1 and M2 are T ?manifolds and π : M1 → M2 is a T ?equivariant ?bration, one gets a ?bration (M1 × E)/T → (M2 × E)/T (1.2) and a Serre-Leray spectral sequence relating the equivariant cohomology groups of M1 and M2 . Moreover, the E2 ?term of this spectral sequence is the product H ? (F ) ? H ? ((E × M2 )/T ) (1.3) where F is the ?ber of the ?bration (1.2) and hence of the ?bration (1.1). Thus if this spectral sequence collapses one gets an isomorphism of additive cohomology
? ? HT (M1 ) ? H ? (F ) ? HT (M2 ) .

(1.1)

(1.4)

An alternative approach to computing the equivariant cohomology groups of a ? T ?manifold M is by Kirwan localization: Namely if HT (M ) is torsion-free, the restriction map ? ? HT (M ) → HT (M T ) ? is injective and hence to compute HT (M ) one is reduced to computing the image ? ? T T of HT (M ) in HT (M ). If M is ?nite, then
? HT (M T ) = p∈M T ? with one copy of S(t? ) ? HT (point) for each p ∈ M T , and determining where ? HT (M ) sits inside this sum is a challenging problem in combinatorics. However, one class of spaces for which this problem has a simple and elegant solution is a

S(t? ) ,

Date: Thursday, June 19, 2008.
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VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

class of spaces introduced by Goresky-Kottwitz-MacPherson in their seminal paper [GKM], and now known as GKM spaces. We recall that a space M , with M T ?nite, is a GKM space if, for every codimension one subtorus T ′ ? T , the connected ′ components of M T are either points or 2-spheres. To such a space one can attach a graph Γ = ΓM by decreeing that these twospheres with pre-assigned orientations are the oriented edges, EΓ , of Γ and that the points of M T are the vertices of Γ. It is easy to check that if S is one of these edge two-spheres, then S T consists of exactly two T ??xed points, p and q (the “North” and “South” poles of S) and one de?nes S to be the oriented edge of Γ joining q to p. Moreover we get from this data a map α : EΓ → Z? T of oriented edges of Γ into the weight lattice of T . This map assigns to the edge 2-sphere, S, with North pole p, the weight of the isotropy representation of T on the tangent space to S at p. This map is called the axial function of the graph Γ and the main result of [GKM] can be summarized as saying that there is an isomorphisms of rings ? ? (1.5) HT (M ) ? Hα (ΓM ) ,
? where Hα (ΓM ) is the cohomology ring of the pair (ΓM , α). (For the de?nition of this ring see Section 3.1 below.) Suppose that M1 and M2 are GKM spaces and π : M1 → M2 is a T ?equivariant ?bration. The purpose of this paper is to show how to reconcile the two descriptions ? of HT (M1 ) given above, the topological description (1.3) and the graph theoretical description (1.5). The complications involved in doing this stem from the fact that, on one hand, the generic ?ber, F , of the ?bration (1.1) is not intrinsically a T ?space and that, on the other hand, it is a T ?space, but in a lot of inequivalent ways. T Namely, for every p ∈ M2 F ? Fp = π ?1 (p) ;

however, for di?erent p’s the T ?structures on the di?erent Fp are usually di?erent. T Moreover, even if one identi?es F with Fp for some p ∈ M2 , the cohomology i groups that ?gure in (1.3) are not the equivariant cohomology groups HT (F ), but i just the standard cohomology groups H (F ). Hence if one wants to reproduce the isomorphism (1.3) graph theoretically, one has to ?nd, in some intrinsic way, a set of elements c1 , . . . , cN in HT (Fp ) which map bijectively onto a basis of H ? (F ) under ? ? the “forgetfulness” map HT (Fp ) → H ? (F ) and then show that HT (M1 ) is a free ? module over the ring HT (M2 ) with the ci ’s as generators. It turns out that there are a number of ways to construct these ci ’s; however, all constructions involve the same ingredient: The GKM structure on M2 gives us, for each path γ in the graph ΓM2 , a holonomy isomorphism
? ? Υγ : HT (Fp ) → HT (Fq )

(1.6)

where p and q are the end points of γ. The ci ’s are chosen so that, for all paths γ with initial and terminal point at p, we have Υγ (ci ) = ci . (1.7)

A few words about the contents of this paper. In Section 2 we review basic facts about GKM graphs and translate into graph theoretic language the notion of “?bration of GKM graphs.” (Fibrations of GKM graphs and GKM ?ber bundles

COHOMOLOGY OF GKM FIBER BUNDLES

3

were discussed in an earlier paper of ours; however our de?nitions here are slightly di?erent from and more natural than those in [GZ]. ) In Section 3 we review the de?nition of the cohomology ring of a GKM graph, describe what this cohomology ring looks like in the context of ?brations of GKM graphs, de?ne the holonomy isomorphism (1.6), show how to construct classes ci with the property (1.7) and prove a graph version of (1.4). (See Theorem 3.6 in Section 3.2.) In Section 4 we discuss a class of examples: generalized ?ag manifolds, which have been extensively studied in the combinatorics literature, but not from the perspective of this paper. Let G be a semisimple Lie group, B a Borel subgroup of G and P ? B a parabolic subgroup. In Section 4.1 we describe what the GKM graphs associated with the spaces G/B and G/P look like and in Sections 4.2-4.3 we discuss the ?bration of GKM graphs associated with the ?bration of T ?manifolds G/B → G/P and compute the group of holonomy automorphisms associated with this ?bration. In Section 5 we specialize to the case where G is one of the four classical simple Lie group types, An , Bn , Cn , or Dn , and give an explicit construction for these groups of classes, ci , satisfying (1.7). In the ?ag manifold examples in Sections 4-5, the holonomy action of the loop group of the base graph on the cohomology of the ?ber coincides with the natural Weyl group action on this cohomology, and in Section 6 we examine some implications of this fact. More explicitly, let G be a semisimple Lie group, T a maximal torus, B a Borel subgroup containing T , and W the Weyl group. We recall (see [GS]) that if M is a GKM space and the T ?action on M extends to a G?action, then
? ? HT (M ) = HG (M ) ?S(t? )W S(t? ) ,

and hence
? ? HG (M ) ? HT (M )W . ? In Section 6 we construct an explicit basis of HT (G/B) consisting of classes that are ? W ?invariant, hence of classes in HG (G/B). These invariant classes are obtained from the equivariant Schubert classes by averaging over the action of the Weyl group (“symmetrization”). Since W ?invariant classes on the ?ber are holonomy invariant, the symmetrized Schubert classes can be extended to global classes on the whole space, and those global classes have the property that their restrictions to each ?bers are generators for the equivariant cohomology of that ?ber. A notational remark: a torus which will reccur repeatedly in the examples is the (n ? 1)?dimensional torus n?1 T0 = T n /C ,

where T n = (S 1 )n is the standard n?dimensional torus, and C ? S 1 is the diagonal S 1 inside T n . We would like to thank Sue Tolman for her role in inspiring this work, to Ethan Bolker for helpful comments on an earlier version, and to Allen Knutson and Alex Postnikov for some very illuminating remarks concerning the de?nition of the invariant classes in the ?ag manifold case.

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VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

2. GKM Graphs 2.1. GKM graphs. Let Γ = (V, E) be a regular graph, where V is the set of vertices and E is the set of oriented edges. We will consider oriented edges, so each unoriented edge e joining vertices p and q will appear twice in E: once as (p, q) = p → q and the second time as (q, p) = q → p. When e is oriented from p to q, we will call p = i(e) the initial vertex of e, and q = t(e) the terminal vertex of e. For a vertex p, let Ep be the set of oriented edges with initial vertex p. De?nition 2.1. Let e = (p, q) be an edge of Γ, oriented from p to q. A connection along the edge e is a bijection ?e : Ep → Eq such that ?e (p, q) = (q, p). A connection on Γ is a family ? = (?e )e∈E of connections along the oriented edges of Γ, such that ?(q,p) = ??1 for every edge e = (p, q) of Γ. (p,q) De?nition 2.2. Let ? be a connection on Γ. A ??compatible axial function on Γ is a labeling α : E → t? of the oriented edges of Γ by elements of a linear space t? , satisfying the following conditions: (1) α(q, p) = ?α(p, q); (2) For every vertex p, the vectors {α(e) | e ∈ Ep } are mutually independent; (3) For every edge e = (p, q), and for every e′ ∈ Ep we have α(?e (e′ )) ? α(e′ ) = cα(e) , where c ∈ R is a scalar that depends on e and e′ . An axial function on Γ is a labeling α : E → t? that is a ??compatible axial function for some connection ? on Γ. De?nition 2.3. A GKM graph is a pair (Γ, α) consisting of a regular graph Γ and an axial function α : E → t? on Γ. Example 1 (The complete graph). Let Γ = Kn , the complete graph with n vertices. Let t? be an n?dimensional linear space and {x1 , . . . , xn } be a basis of t? . De?ne α : E → t? by α(i, j) = xi ? xj . If ?(i,j) : Ei → Ej sends (i, j) to (j, i) and (i, k) to (j, k) for k = i, j, then ? is a connection compatible with α. The image of α spans the (n ? 1)?dimensional subspace t? generated by α1 = x1 ? x2 , . . . , αn?1 = xn?1 ? xn . The GKM graph 0 (Kn , α) is the GKM graph of CP n?1 , the projective space of complex lines in Cn , n?1 with an action of the torus T0 induced by a linear action of T n on Cn . When n = 2, the graph Γ has two vertices, 1 and 2, joined by an edge. The oriented edge from 1 to 2 is labeled β = x1 ? x2 , and the oriented edge from 2 to 1 is labeled ?β = x2 ? x1 . The second condition in the de?nition of an axial function is automatically satis?ed. The complete graph K3 is shown in Figure 1(a). Example 2 (The permutahedron). Let Sn be the symmetric group of permutations of [n] = {1, . . . , n}, and let Γ = (Sn , E) be the graph with vertex set Sn . Two vertices are joined by an edge if the corresponding permutations di?er by a transposition. We will refer to Γ as Sn , and it will be clear from the context when Sn is the graph, the vertex set, or the group of permutations. For distinct a, b in [n], we denote by (a, b) the transposition that swaps a and b. If u ∈ Sn , then its neighbors are the permutations v = u(i, j) = (u(i), u(j))u. The last equality shows that two permutations that di?er by a transposition to the right (operating

COHOMOLOGY OF GKM FIBER BUNDLES

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3 x2 ?x3 x2 ?x3 231

321 312

2 x1 ?x3 x1 ?x2 1

213 x1 ?x2

x1 ?x3 132 123

(a)

(b)

Figure 1. Complete graph K3 (a) and Permutahedron S3 (b) on positions) also di?er by a transposition to the left (operating on values). We denote the edge u → v = u(i, j) either by u?? v ?→
?(i,j)

or

u???? v, ?? ?→

(u(i),u(j))?

depending whether we want to emphasize right or left transpositions. Let t? be an n?dimensional linear space and {x1 , . . . , xn } be a basis of t? . Let α : E → t? be the axial function de?ned as follows. If u → v = u(i, j) is an oriented edge, with 1 i < j n, de?ne α(u, v) = xu(i) ? xu(j) . Note that α(u, v) is determined by the values changed from u to v. For an edge e = u → v = u(i, j), de?ne ?e : Eu → Ev by ?e (u, u(a, b)) = (v, v(a′ , b′ )) , where (a′ , b′ ) = (i, j)(a, b)(i, j). Then ? is a connection compatible with α. As in Example 1, the image of α spans the (n ? 1)?dimensional subspace t? generated by 0 α1 = x1 ? x2 , . . . , αn?1 = xn?1 ? xn . The GKM graph (Sn , α) is the GKM graph n?1 of Fn (C), the manifold of full ?ags in Cn , with an action of the torus T0 induced n n by a linear action of T on C . The permutahedron S3 is shown in Figure 1(b). As a general convention throughout this paper, the axial function α takes the same value on parallel edges with the same orientation. For example, α(123, 132) = α(231, 321) = x2 ? x3 . As we have seen in the examples above, the image of α may not generate the entire linear space t? . Let (Γ, α) be a GKM graph. For a vertex p, let hp = span{αe | e ∈ Ep } ? t? be the subspace of t? generated by the image of the axial function on edges with initial vertex p. If Γ is connected, then this subspace is the same for all vertices of Γ, and we will denote it by t? . We can co-restrict the axial function α : E → t? to 0 a function α0 : E → t? , and the resulting pair (Γ, α0 ) is also a GKM graph. 0 De?nition 2.4. An axial function α : E → t? is called e?ective if t? = t? . 0 In other words, an axial function α : E → t? is e?ective if and only if its image is not contained in a proper subspace of t? .

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VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

2.2. GKM subgraphs. Let (Γ, α) be a GKM graph, with Γ = (V, E) and axial function α : E → t? . Let ? be a connection compatible with α. Let Γ0 = (V0 , E0 ) be a subgraph of Γ, with V0 ? V and E0 ? E, such that, if e ∈ E is an edge with i(e), t(e) ∈ V0 , then e ∈ E0 . De?nition 2.5. A connected subgraph Γ0 is a ??GKM subgraph if for every edge e ∈ E0 , with i(e) = p and t(e) = q, we have ?e (Ep ∩ E0 ) = Eq ∩ E0 . The subgraph Γ0 is a GKM subgraph if it is a ??GKM subgraph for a connection ? compatible with α. In other words, Γ0 is a GKM subgraph if, for some connection ? compatible with the axial function α, the connection along edges of Γ0 send edges of Γ0 to edges of Γ0 and edges not in Γ0 to edges not in Γ0 . Then the connected subgraph Γ0 is regular, the restriction α0 of α to E0 is an axial function on Γ0 , and the connection ? induces a connection ?0 compatible with α0 . Therefore a GKM subgraph is naturally a GKM graph. Example 3. Let 2 m < n, and let Γ0 be the subgraph of Sn with vertex set k n ? m} . V0 = {u ∈ Sn | u(k) = k for all 1

Then Γ0 is a GKM subgraph of Sn , and can be identi?ed, as a graph, with Sm . Note, however, that the axial function on Γ0 is slightly di?erent from the standard axial function on Sm . This is not going to be a major issue, and we’ll see a bit later that Γ0 and Sm are isomorphic as GKM graphs. More generally, let I = {i1 , . . . , in?m } be an (n ? m)?element subset of [n], and ΓI the subgraph of Sn with vertices VI = {u ∈ Sn | u(k) = ik for all 1 k n ? m} . Then ΓI is a GKM subgraph of Sn , isomorphic, as a graph, with Sm . For example, if I = {2} ? [3], then ΓI is the S2 ? K2 graph with vertices p = 213 and q = 231 in S3 , and α(p, q) = x1 ? x3 . 2.3. Isomorphisms of GKM graphs. Let (Γ1 , α1 ) and (Γ2 , α2 ) be two GKM graphs, with Γ1 = (V1 , E1 ), α1 : E1 → t? and Γ2 = (V2 , E2 ), α2 : E2 → t? . 1 2 De?nition 2.6. An isomorphism of GKM graphs from (Γ1 , α1 ) to (Γ2 , α2 ) is a pair (Φ, Ψ), where (1) Φ : Γ1 → Γ2 is an isomorphism of graphs; (2) Ψ : t? → t? is an isomorphism of linear spaces; 1 2 (3) For every edge (p, q) of Γ1 we have α2 (Φ(p), Φ(q)) = Ψ ? α1 (p, q) . The ?rst condition implies that Φ induces a bijection from E1 to E2 , and the third condition can be restated as saying that the following diagram commutes: E1
α1 Φ

/ E2
α2

t? 1



Ψ

 / t? 2

As examples of isomorphisms of GKM graphs, we will discuss some of the graphs ΓI mentioned in the previous section.

COHOMOLOGY OF GKM FIBER BUNDLES

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Example 4. Let n 3, and let Sn be the GKM graph introduced in Example 2. For 1 k n, let Vk = {w ∈ Sn | w(1) = k} and let Γk be the GKM subgraph of Sn with vertex set Vk . For a permutation u ∈ Sn?1 , let u = u(1)u(2) . . . u(n?1)n ∈ Sn . (We use the one-line notation for permutations.) For 1 i < j n, let ci,j ∈ Sn be the cycle i → i+1 → · · · → j → i, and cj,i = c?1 the cycle j → j ?1 → · · · → i+1 → i → j. i,j Let ?k : Sn?1 → Γk be the isomorphism of graphs given by ?k (u) = ck,n u cn,1 . The cycle ck,n , operating on values, moves the value k to the last position and preserves the relative order of the values on the other positions. The cycle cn,1 , operating on positions, moves the value k from the last position to the ?rst and then shifts all the other positions to the right by one. The axial function on Γk takes values in t? = span(x1 , . . . , xn ) and the axial function on Sn?1 has values in the (n ? 1)?dimensional subspace span(x1 , . . . , xn?1 ). To identify Γk with Sn?1 , we need to have isomorphic target spaces for the axial functions, and we will simply consider the axial function on Sn?1 as taking values in t? . Let φk = ck,n : [n] → [n] and de?ne ψk : t? → t? as the linear extension of ψk (xi ) = xφk (i) , for 1 i n. If u → v is an edge of Sn?1 and v = u(i, j) with 1 i < j n ? 1, then α(?k (u), ?k (v)) = ψk (α(u, v)) hence (?k , ψk ) : Sn?1 → Γk is an isomorphism of GKM graphs. If n = 3 and k = 2, then V2 = {213, 231} and the axial function on Γ2 is α(213, 231) = x1 ? x3 . The graph isomorphism ?2 : S2 → Γ2 is ?2 (12) = 213 and ?2 (21) = 231. Then φ2 is the transposition (2, 3), hence the isomorphism ψ2 : t? → t? is given by ψ2 (x1 ) = x1 , ψ2 (x2 ) = x3 , and ψ2 (x3 ) = x2 , and α(?2 (12), ?2 (21)) = α(132, 312) = x1 ? x3 = ψ2 (x1 ? x2 ) = ψ2 ? α(12, 21) , which shows that ?2 and ψ2 are compatible. 2.4. GKM ?brations. We now introduce a special type of maps between GKM graphs. Let Γ and B be connected graphs and π : Γ → B be a morphism of graphs. Hence π is a map from the vertices of Γ to the vertices of B such that, if (p, q) is an edge of Γ, then either π(p) = π(q) or else (π(p), π(q)) is an edge of B. If (p, q) is an edge of Γ and π(p) = π(q), we will say that the edge (p, q) is vertical, and if (π(p), π(q)) is an edge of B, we will say that the edge (p, q) is horizontal. For a ⊥ vertex q of Γ, let Eq be the set of vertical edges with initial vertex q, and let Hq be ⊥ the set of horizontal edges with initial vertex q. Then Eq = Eq ∪ Hq and π induces canonically a map (dπ)q : Hq → (EB )π(q) from the horizontal edges at q to the edges of B with initial vertex π(q): if (q, q ′ ) ∈ Hq , then (dπ)q (q, q ′ ) = (π(q), π(q ′ )). De?nition 2.7. The morphism of graphs π : Γ → B is a ?bration of graphs 1 if for every vertex q of Γ, the map (dπ)q : Hq → (EB )π(q) is bijective. Fibrations have the unique lifting of paths property: Let π : Γ → B be a ?bration and (p0 , p1 ) an edge of B, and q0 ∈ π ?1 (p0 ) a point in the ?ber over p0 . Since (dπ)q0 : Hq0 → (EB )p0 is a bijection, there exists a unique edge (q0 , q1 ) such that (dπ)q0 (q0 , q1 ) = (p0 , p1 ). We will say that (q0 , q1 ) is the lift of (p0 , p1 ) at q0 . If γ is
1This is what we called submersion in [GZ]

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VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

a path p0 → p1 → · · · → pm in B and q0 ∈ π ?1 (p0 ) is a point in the ?ber over p0 , then we can uniquely lift γ to a path γ(q0 ) = q0 → q1 → · · · → qm in Γ starting at q0 , by successively lifting the edges of γ. Example 5. For n 3, consider the graph morphism π : Sn → Kn from the permutahedron Sn to the complete graph Kn , sending a permutation u ∈ Sn to π(u) = u(1). For i ∈ [n], the ?ber Fi = π ?1 (i) consists of permutations of [n] that have i in the ?rst position. Let q ∈ Fi . The vertical edges at q are the edges that join q with vertices of the form q ′ = (h, k)q, where i ∈ {h, k}. The horizontal edges at q join q with vertices of the form q ′ = (i, j)q for j = i, and (dπ)q (q, q ′ ) = (i, j). Then (dπ)q is a bijection from the horizontal edges at q and the edges of Kn with initial vertex i = π(q). Hence π is a ?bration. The case n = 3 is shown in Figure 2. If γ is the cycle 1 → 2 → 3 → 1 in K3 , then the lift of γ at 123 is the path γ : 123 → 213 → 312 → 132 in S3 .
1 π? (3)

3 x2 ?x3 π 2 x1 ?x2 1 x1 ?x3 231 213

321 x2 ?x3 312

x1 ?x3 132 123
1 π? (1)

1 π? (2) x1 ?x2

Figure 2. Fibration S3 → K3 We will now add the GKM package to a ?bration, and de?ne GKM ?brations. Let (Γ, α) and (B, αB ) be two GKM graphs, with axial functions α : E → t? and αB : EB → t? taking values in the same linear space t? . Let ? and ?B be connections on Γ and B, compatible with α and αB , respectively. De?nition 2.8. A map π : (Γ, α) → (B, αB ) is a (?, ?B )?GKM ?bration if it satis?es the following conditions: (1) π is a ?bration of graphs; (2) If e is an edge of B and e is any lift of e, then α(e) = αB (e); (3) Along every edge e of Γ, the connection ? sends horizontal edges into horizontal edges and vertical edges into vertical edges; (4) The restriction of ? to horizontal edges is compatible with ?B , in the following sense: Let e = (p, q) be an edge of B and e = (p′ , q ′ ) the lift of e at p ′ . Let e′ ∈ Ep and e′′ = (?B )e (e′ ) ∈ Eq . If e′ is the lift of e′ at p ′ and e′′ is the lift of e′′ at q ′ , then (?)e (e′ ) = e′′ . e

COHOMOLOGY OF GKM FIBER BUNDLES

9

A map π : (Γ, α) → (B, αB ) is a GKM ?bration if it is a (?, ?B )?GKM ?bration for some connections ? and ?B . Example 6. The ?bration π : Sn → Kn , π(u) = u(1) introduced in Example 5 is a GKM ?bration for the GKM structures on Kn and Sn constructed in Examples 1 and 2: Let q ∈ Fi and q ′ = (i, j)q ∈ Fj . Then (q, q ′ ) is the lift at q of the edge (i, j) of Kn , and α(q, q ′ ) = xi ? xj = αi,j . Moreover, π is compatible with the connections on Sn and Kn , hence π is a GKM ?bration. 2.5. Fiber bundles. Let π : Γ → B be a ?bration of graphs. For a vertex p of B, let Vp = π ?1 (p) ? V and let Γp be the subgraph of Γ with vertex set Vp . For every edge (p, q) of B, de?ne a map Φp,q : Vp → Vq as follows. For p′ ∈ Vp , de?ne Φp,q (p′ ) = q ′ , where (p′ , q ′ ) is the lift of (p, q) at p′ . It is easy to see that Φp,q is bijective, with inverse Φq,p . What is not true, in general, is that Φp,q is an isomorphism of graphs from Γp to Γq . Example 7. Let Γ be the regular 3?valent graph consisting of two quadrilaterals (p1 , p2 , p3 , p4 ) and (q1 , q3 , q2 , q4 ) joined by edges (pi , qi ) for i=1,2,3,4. (See Figure 3.) p3 p2 p4 p1 p q q4 q1 q3 q2

Figure 3. Twisted ?bration Let B be a graph with two vertices p and q joined by an edge. Let π : Γ → B be the morphism of graphs π(pi ) = p and π(qi ) = q for i = 1, 2, 3, 4. Then π is a ?bration and Φp,q (pi ) = qi for i = 1, 2, 3, 4. However, (p1 , p2 ) is an edge in Γp , but (q1 , q2 ) is not an edge in Γq . While the ?bers Γp and Γq are isomorphic as graphs, the map Φp,q is not an isomorphism. We will be interested in ?brations for which the ?bers are canonically isomorphic. De?nition 2.9. A ?bration π : Γ → B is a ?ber bundle 2 if for every edge (p, q) of B, the maps Φp,q : Γp → Γq are morphisms of graphs. If π : Γ → B is a ?ber bundle, then Φp,q is bijective, and both Φp,q : Γp → Γq and Φ?1 = Φq,p : Γq → Γp are morphisms of graphs. Therefore the maps Φp,q are p,q isomorphisms of graphs. The simplest example of a ?ber bundle is the projection of a direct product of graphs onto one of its factors, π : Γ = B × F → B. We will call such ?ber bundles trivial bundles.
2This is what we called ?bration in [GZ]

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VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

Example. More interesting is the ?bration π : Sn → Kn in Example 5. For a permutation u ∈ Sn we de?ned π(u) = u(1) and Γi = π ?1 (i). For an edge (i, j) of Kn , the map Φi,j : Γi → Γj is given by Φi,j (q) = (i, j)q. If (q, q ′ ) is an edge in Γi , then q ′ = (h, k)q, where i ∈ {h, k}. Then Φi,j (q ′ ) = (i, j)(h, k)q = (h′ , k ′ )(i, j)q = (h′ , k ′ )Φi,j (q) , where (h′ , k ′ ) = (i, j)(h, k)(i, j). Then (h′ , k ′ )(j) = j, so j ∈ {h′ , k ′ }, and hence (Φi,j (q), Φi,j (q ′ )) is an edge in Γj . Therefore Φi,j is a morphism of graphs from Γi to Γj , and since this is valid for all edges (i, j) of Kn , we conclude that π : Sn → Kn is a ?ber bundle. If π : Γ → B is a ?ber bundle and B is connected, then all the ?bers are isomorphic. Let F be a graph isomorphic to all the ?bers. Then for each vertex p of B, there is an isomorphism of graphs ?p : F → Γp = π ?1 (p). For every edge (p, p ′ ) of B, let ?p,p ′ : F → F be the automorphism of F given by ?p,p ′ = ??1 ? Φp,p ′ ? ?p . p′ Therefore for every ?ber bundle over B with typical ?ber F , one has a family (?e )e of graph automorphisms of F , with one automorphism for every oriented edge e = (p, p ′ ) of B, and such that ?p ′ ,p = ??1 ′ . p,p Example. For the ?ber bundle π : Sn → Kn , the ?ber Γi = π ?1 (i) over a vertex i of Kn , is isomorphic, as a graph, with Sn?1 , and an isomorphism ?i : Sn?1 → Γi has been constructed in Example 4. For 1 i < j n the graph automorphism of Sn?1 associated to the edge (j, i) of Kn is ?j,i = ??1 ?Φj,i ??j , given by multiplying i a permutation in Sn?1 to the left by the restriction of c?1 (i, j)cj,n = c?1 = cj?1,i i,n i,j?1 to Sn?1 . In particular, ?i,i+1 = ?i+1,i is the identity, and ?i,i+2 = ?i+2,i is the left multiplication by the transposition (i, i + 1). 2.6. GKM ?ber bundles. Let (Γ, α) and (B, αB ) be two GKM graphs, with the same target space t? for the axial functions α and αB . De?nition 2.10. A map π : (Γ, α) → (B, αB ) is a GKM ?ber bundle if (1) π is a GKM ?bration; (2) π is a ?ber bundle; (3) For every oriented edge (p, q) of B, there exists a linear automorphism Ψp,q of t? such that Υp,q = (Φp,q , Ψp,q ) : (Γp , α) → (Γq , α) is an isomorphism of GKM graphs and such that Υ?1 = Υq,p . p,q If π : (Γ, α) → (B, αB ) is a GKM ?bration, then for each p ∈ B, the ?ber (Γp , α) is a GKM subgraph of (Γ, α). A GKM ?ber bundle is essentially a ?ber bundle for which the transition maps from one ?ber to another are isomorphisms not just of graphs, but of GKM graphs. Since Φ?1 = Φq,p , the condition Υ?1 = Υq,p is equivalent to Ψ?1 = Ψq,p . If the p,q p,q p,q axial function on the ?ber is e?ective, then the linear isomorphism Ψp,q is uniquely determined, and it satis?es the condition Ψq,p = Ψ?1 . If the axial function is p,q not e?ective, then we can still choose linear isomorphisms Ψp,q and Ψq,p such that Ψq,p = Ψ?1 . p,q 2.6.1. Transition isomorphisms. We can be more speci?c about the transition isomorphisms Ψp,q . Proposition 2.11. Let (p, q) be an edge of B and let Ψp,q : t? → t? . Let t? and t? p q be the subspaces of t? generated by the images of the axial function α on vertical edges above p and q, respectively. Then

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11

(1) Ψp,q (t? ) = t? . p q (2) There exists a linear map cp,q : t? → R such that for x ∈ t? we have Ψp,q (x) = x + cp,q (x)αp,q . and the map cp,q is uniquely de?ned on t? . p (3) αp,q is an eigenvector of Ψp,q with eigenvalue λp,q = 1+cp,q (αp,q ). Moreover, λp,q λq,p = 1. (4) The functions cp,q and cq,p are related by cp,q = λp,q cq,p Proof. Let (p ′ , p ′′ ) be an edge of the ?ber Γp , and let (q ′ , q ′′ ) the corresponding edge of Γq , with q ′ = Φp,q (p ′ ) and q ′′ = Φp,q (p ′′ ). Then αq′ ,q′′ = Ψp,q (αp′ ,p′′ ) . Since t? is generated by the vectors αp′ ,p′′ and t? is generated by the vectors αq′ ,q′′ , p q it follows immediately that Ψp,q (t? ) = t? . p q The compatibility condition along the edge (p′ , q ′ ) implies that αq′ ,q′′ ? αp′ ,p′′ is a multiple of αp′ ,q′ = αp,q , hence there exists a constant c = c(αp′ ,p′′ ) such that Ψp,q (αp ′ ,p ′′ ) = αp ′ ,p ′′ + c(αp′ ,p′′ )αp,q . Therefore, there exists a unique function c : t? → R such that p Ψp,q (x) = x + c(x)αp,q , for all x ∈ t? . The linearity of c follows from the linearity of Ψp,q . If t? = t? , then p p we de?ne cp,q = c, and if not, we de?ne cp,q as a linear extension of c to all t? . Then Ψp,q (αp,q ) = αp,q + cp,q (αp,q )αp,q = λp,q αp,q , and therefore αp,q = Ψp,q (Ψq,p (αp,q )) = λp,q λq,p αp,q , and since αp,q = 0, it follows that λp,q λq,p = 1. Moreover, from Ψp,q (Ψq,p (x)) = x for all x ∈ t? we get, after a straightforward computation, that cp,q (x) = λp,q cq,p (x). In other words, the restriction of the linear isomorphism Ψp,q to the e?ective part of the restriction of α to Γp is a translation in the direction of αp,q . 2.6.2. The holonomy subgroup. For a path γ : p0 → p1 → · · · → pm?1 → pm in B from p0 to pm , let Υγ = Υpm?1 ,pm ? · · · ? Υp0 ,p1 : (Γp0 , α) → (Γpm , α) be the GKM graph isomorphism given by the composition of the transition maps. Let p ∈ B be a vertex, and let ?(p) be the set of all loops in B that start and end at p. If γ ∈ ?(p) is a loop based at p, then Υγ is an automorphism of the GKM graph (Γp , α). Let Kp = {Υγ | γ ∈ ?(p)} ? Aut(Γp , α) . Then Kp is a subgroup of Aut(Γp , α), and we will call it the holonomy subgroup of the ?ber Γp . If π : (Γ, α) → (B, αB ) is a GKM ?ber bundle and B is connected, then all the ?bers are isomorphic as GKM graphs. Let (F, αF ) be a GKM graph isomorphic to all the ?bers, with αF : EF → t? . Then for each vertex p of B, there is an isomorphism of GKM graphs ρp = (?p , ψp ) : (F, αF ) → (Γp , α). For every edge

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VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

(p, p ′ ) of B, let ρp,p ′ = (?p,p ′ , ψp,p ′ ) : (F, αF ) → (F, αF ) be the automorphism of (F, αF ) given by ?p,p ′ = ??1 ? Φp,p ′ ? ?p p′
?1 ψp,p ′ = ψp ′ ? Ψp,p ′ ? ψp

Therefore for every GKM ?ber bundle over (B, αB ) with typical ?ber (F, αF ), one has a family (ρe )e = (?e , ψe )e of automorphisms of (F, αF ), with one automorphism for every oriented edge e = (p, p ′ ) of B, and such that ρp ′ ,p = ρ?1 ′ . More general, p,p if γ is any path in B then the composition of the transition maps along the edges of γ de?nes an automorphism ργ = (?γ , ψγ ) of (F, αF ). Let p be a vertex of B and Kp = {ργ | γ ∈ ?(p)} ? Aut(F, αF ) . Then Kp is a subgroup of Aut(F, αF ). If p, p ′ are two vertices of B, then the subgroups Kp and Kp ′ are conjugated by ργ , where γ is any path in B connecting p and p ′ . Example 8. We have already shown that π : Sn → Kn , π(u) = u(1) is a GKM ?bration and a ?ber bundle. Recall that for i = j, the isomorphism of graphs Φi,j : Γi → Γj is Φi,j (u) = (i, j)u. De?ne Ψi,j : t? → t? as the linear extension of Ψi,j (xk ) = x(i,j)k , for 1 k n. Then Υi,j = (Φi,j , Ψi,j ) : (Γi , α) → (Γj , α) is an ?1 isomorphisms of GKM graphs, and Υi,j = Υj,i , so π : Sn → Kn is a GKM ?ber bundle. Each ?ber is isomorphic, as a GKM graph, with Sn?1 . An explicit isomorphism (?i , ψi ) of GKM graphs between Sn?1 and Γi has been constructed in Example 4. If 1 i < j n, then ρj,i = (?j,i , ψj,i ) : Sn?1 → Sn?1 , where ?j,i (u) = cj?1,i u and ψj,i : t? → t? is the linear extension of ψj,i (xk ) = xcj?1,i (k) , for 1 k n. 3. Cohomology of GKM Fiber Bundles 3.1. Cohomology of GKM graphs. Let (Γ, α) be a GKM graph, with Γ = (V, E) a regular graph and α : E → t? an axial function. Let S(t? ) be the symmetric algebra of t? ; if {x1 , . . . , xn } is a basis of t? , then S(t? ) ? R[x1 , . . . , xn ]. De?nition 3.1. A cohomology class on (Γ, α) is a map ω : V → S(t? ) such that for every edge e = (p, q) of Γ, we have ω(q) ≡ ω(p) (mod αe ) . (3.1) The compatibility condition (3.1) means that ω(q) ? ω(p) = αe f , for some element f ∈ S(t? ), and is equivalent to ω(q) = ω(p) on ker(αe ). If ω and τ are cohomology classes, then ω + τ and ωτ are also cohomology classes.
? De?nition 3.2. The cohomology ring of (Γ, α), denoted by Hα (Γ), is the subring ? of Maps(V, t ) consisting of all the cohomology classes. ? Moreover, Hα (Γ) is a graded ring, with the grading induced by S(t? ). We say ? that ω ∈ Hα (Γ) is a class of degree k if for every p ∈ V , the polynomial ω(p) ∈ Sk (t? ) k is homogeneous of degree k. If Hα (Γ) is the space of classes of degree k, then ? Hα (Γ) = k 0 ? ? If ω ∈ and h ∈ S(t ), then hω ∈ Hα (Γ), hence Hα (Γ) is an S(t? )?module; ? it is in fact a graded S(t )?module. ? Hα (Γ) ? k Hα (Γ) .

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13

Remark 3.3. If M is a GKM manifold and Γ = ΓM is its GKM graph, then odd 2k k HT (M ) = 0 and HT (M ) ? Hα (Γ). Let (Γ0 , α) be a GKM subgraph of (Γ, α). If f : V → S(t? ) is a cohomology class on Γ, then the restriction of f to V0 is a cohomology class on Γ0 . Therefore the ? ? inclusion i : (Γ, α) → (Γ0 , α) induces a ring morphism i? : Hα (Γ) → Hα (Γ0 ). In general this morphism is not surjective. If ρ = (?, ψ) : (Γ1 , α1 ) → (Γ2 , α2 ) is an isomorphism of GKM graphs, de?ne ρ? : Maps(V2 , S(t? )) → Maps(V1 , S(t? )) by (ρ? (f ))(p) = ψ ?1 (f (?(p))) , for p ∈ V1 , where ψ ?1 : S(t? ) → S(t? ) is the algebra isomorphism extending the linear isomorphism ψ ?1 : t? → t? . Then ρ? is a ring isomorphism and (ρ? )?1 = (ρ?1 )? . Proposition 3.4. If ρ = (?, ψ) : (Γ1 , α1 ) → (Γ2 , α2 ) is an isomorphism of GKM ? ? graphs, then ρ? : Hα2 (Γ2 ) → Hα1 (Γ1 ) is a ring isomorphism. Proof. All we have to show is that if f : V2 → S(t? ) satis?es the compatibility conditions on (Γ2 , α2 ), then ρ? f : V1 → S(t? ) satis?es the compatibility conditions on (Γ1 , α1 ). ? Let f ∈ Hα2 (Γ2 ). Let (p, p ′ ) be an edge of Γ1 . Then (?(p), ?(p ′ )) is an edge of Γ2 and α2 (?(p), ?(p ′ )) = ψ(α1 (p, p ′ )) . Then (ρ? f )(p ′ ) ? (ρ? f )(p) = ψ ?1 (f (?(p ′ ))) ? ψ ?1 (f (?(p))) = ψ ?1 (f (?(p ′ )) ? f (?(p))) .
? Since f ∈ Hα2 (Γ2 ), there exists Q ∈ S(t? ) such that

f (?(p ′ )) ? f (?(p)) = α2 (?(p ′ ), ?(p))Q and then (ρ? f )(p ′ ) ? (ρ? f )(p) = α1 (p, p ′ )ψ ?1 (Q) . ? Therefore ρ? f ∈ Hα1 (Γ1 ). Note that ρ? is a ring isomorphism, but not an isomorphism of S(t? )?modules, unless ψ : t? → t? is the identity map. 3.2. Cohomology of GKM ?ber bundles. Let π : (Γ, α) → (B, αB ) be a GKM ?ber bundle, with typical ?ber (F, αF ). The main goal of this paper is to describe the relationship between the cohomology ring of the total space (Γ, α) and the cohomology rings of the base (B, αB ) and the ?ber (F, αF ). Let f : VB → S(t? ) be a cohomology class on the base (B, αB ), and de?ne the pull-back π ? (f ) : VΓ → S(t? ) by π ? (f )(q) = f (π(q)). Then π ? (f ) is a cohomology ? ? class on (Γ, α), and π de?nes an injective morphism of rings π ? : HαB (B) → Hα (Γ). ? ? In particular, Hα (Γ) is an HαB (B)?module.
? ? De?nition 3.5. A cohomology class h ∈ Hα (Γ) is called basic if h ∈ π ? (HαB (B)). ? ? ? ? ? Let (Hα (Γ))bas = π ? (HαB (B)) ? Hα (Γ). Then (Hα (Γ))bas is a subring of Hα (Γ), ? and is isomorphic to HαB (B).

Theorem 3.6. Let π : (Γ, α) → (B, αB ) be a GKM ?ber bundle, and let c1 , . . . , cm be cohomology classes on Γ such that, for every p ∈ B, the restrictions of these ? classes to the ?ber Γp = π ?1 (p) form a basis of Hα (Γp ) over S(t? ). Then, as ? ? ? HαB (B)?modules, Hα (Γ) is isomorphic to the free HαB (B)?module on c1 , . . . , cm .

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VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

? Proof. Every linear combination of c1 , . . . , cm with coe?cients in HαB (B) is clearly a cohomology class on Γ. If such a combination is the zero class, then m

βk (p)ck (p ′ ) = 0
k=1

for every p ∈ B and p ∈ Γp . Since the restrictions of c1 , . . . , cm to Γp are independent, it follows that βk (p) = 0 for every k = 1, . . . , m and p ∈ B, hence the classes ? β1 , . . . , βm are zero. Therefore the free HαB (B)?module on c1 , . . . , cm is included ? in Hα (Γ). We will now prove the opposite inclusion. ? Let c ∈ Hα (Γ) be a cohomology class on Γ. For p ∈ B, the restriction of c to the ?ber Γp is a cohomology class on Γp . Since the restrictions of c1 , . . . , cm to Γp generate the cohomology of Γp , there exist polynomials β1 (p), . . . , βm (p) in S(t? ) such that, for every p ′ ∈ Γp , we have
m



c(p ′ ) =
k=1

βk (p)ck (p ′ ) .

We will show that the maps βk : B → S(t? ) are in fact cohomology classes on B. Let e = p → q be an edge of B, with associated weight αe = αpq ∈ t? . Let p ′ ∈ Γp , and let q ′ ∈ Γq such that p ′ → q ′ is the lift of p → q. Then α(p ′ , q ′ ) = α(p, q) = αe .
m

c(q ′ ) ? c(p ′ ) =
k=1 m

(βk (q)ck (q ′ ) ? βk (p)ck (p ′ ))
m

=
k=1

(βk (q) ? βk (p))ck (p ′ ) +
k=1

βk (q)(ck (q ′ ) ? ck (p ′ )) .

Since c, c1 , . . . , cm are classes on Γ, the di?erences c(q ′ ) ? c(p ′ ), ck (q ′ ) ? ck (p ′ ) are multiples of αe , for all k = 1, . . . , m. Therefore, for all p ′ ∈ Γp ,
m

(βk (q) ? βk (p))ck (p ′ ) = αe η(p ′ ) ,
k=1

where η(p ′ ) ∈ S(t? ). We will show that η : Γp → S(t? ) is a cohomology class on Γp . If p ′ and p ′′ are vertices in Γp , joined by an edge p ′ p ′′ , then
m

(βk (q) ? βk (p))(ck (p ′′ ) ? ck (p ′ )) = αe (η(p ′′ ) ? η(p ′ )) .
k=1

Each ck is a cohomology class on Γ, so ck (p ′′ ) ? ck (p ′ ) is a multiple of α(p ′ , p ′′ ), for all k = 1, . . . , m. Then αe (η(p ′′ ) ? η(p ′ )) is also a multiple of α(p ′ , p ′′ ). But αe and α(p ′ , p ′′ ) point in di?erent directions as vectors, so, as linear polynomials, they are relatively prime. Therefore η(p ′′ ) ? η(p ′ ) must be a multiple of α(p ′ , p ′′ ). Therefore η is a cohomology class on Γp . The restrictions of c1 , . . . , cm are a basis for the cohomology ring of Γp , hence there exist polynomials Q1 , . . . , Qm ∈ S(t? ) such that
m

η(p ′ ) =
k=1

Qk ck (p ′ ) .

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Then

m

(βk (q) ? βk (p) ? Qk αe )ck = 0
k=1

on the ?ber Γp . Since the classes c1 , . . . , cm restrict to linearly independent classes on ?bers, it follows that βk (q) ? βk (p) = Qk αe ,
? which implies that βk ∈ HαB (B). Therefore every cohomology class on Γ can be ? written as a linear combination of classes c1 , . . . , cm , with coe?cients in HαB (B).

3.3. Invariant classes. The main purpose of this section is to describe a method of constructing global classes c1 , . . . , cm with the properties required by Theorem 3.6. Let π : (Γ, α) → (B, αB ) be a GKM ?ber bundle, with typical ?ber (F, αF ). Let p be a ?xed vertex of B and let ρp = (?p , ψp ) : (F, αF ) → (Γp , α) be a GKM isomorphism from F to the ?ber above p. For a loop γ ∈ ?(p), let ργ = (?γ , ψγ ) be the GKM automorphism of (F, αF ) determined by γ. Let K = Kp be the holonomy ? subgroup of Aut(F, αF ) generated by all automorphisms ργ , and let f ∈ (HαF (F ))K be a cohomology class on the ?ber, invariant under all the automorphisms in K. ? Then fp = (ρ?1 )? (f ) ∈ Hα (Γp ) is a class on the ?ber over p, invariant under all the p automorphisms in Kp ? Aut(Γp , α). Moreover, if t? is the subspace of t? generated p by the values of α on the edges of Γp , then, for q ∈ Γp , we have fp (q) ∈ S(t? ). p We will extend the class fp from the ?ber Γp to the entire total space Γ. Let p ′ be another vertex of B, and let γ be a path in B from p ′ to p. Let ? ? ? Υγ : Hα (Γp ) → Hα (Γp ′ ) be the ring isomorphism induced by the GKM graph isomorphism Υγ : (Γp ′ , α) → (Γp , α). Lemma 3.7. If γ1 and γ2 are two paths in B from p ′ to p, then Υ?1 (fp ) = Υ?2 (fp ). γ γ Proof. Let γ ∈ ?(p) be the loop in B obtained by following the edges of γ1 , in reverse order, from p to p′ , then the edges of γ2 from p′ to p. Then Υγ = Υγ2 Υ?1 , γ1 hence Υ? = (Υγ2 Υ?1 )? = (Υ?1 )?1 Υ?2 . Since Υ? ∈ Kp and fp is Kp ?invariant, we γ1 γ γ γ γ have Υ? (fp ) = fp and therefore Υ?1 (fp ) = Υ?2 (fp ). γ γ γ
? We de?ne fp′ = Υ? (fp ) ∈ Hα (Γp′ ), where γ is any path in B from p′ to p. If γ ? is the subspace of t generated by the values of α on the edges of Γp′ , then for every q ′ ∈ Γp′ we have fp′ (q ′ ) ∈ S(t?′ ). p

t ?′ p

Proposition 3.8. Let c = cf,p : VΓ → S(t? ) be de?ned by c|Γq = fq for all q ∈ B. ? Then c ∈ Hα (Γ). Proof. Since the restrictions of c to ?bers are classes on ?bers, it su?ces to show that c satis?es the compatibility conditions along horizontal edges. Let (q1 , q2 ) be a horizontal edge of Γ and let e = (p1 , p2 ) be the corresponding edge of B. Then c(q2 ) ? c(q1 ) = fp2 (q2 ) ? fp1 (q1 ) = Ψe (fp1 (q1 )) ? fp1 (q1 ) . Since Ψe (x) = x + c(x)αe on t?1 , it follows that, for every Q ∈ S(t?1 ), the di?erence p p Ψe (Q)?Q is a multiple of αe . For Q = fp1 (q1 ) we get that c(q2 )?c(q1 ) is a multiple of αe = α(q1 , q2 ), and therefore c satis?es the compatibility condition imposed by the horizontal edge (q1 , q2 ).

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Note that c depends not only on the class f on the typical ?ber F , but also on the point p where we start realizing f on Γ. The choice of p is limited by the fact that f has to be invariant under the subgroup Kp determined by p. ? Suppose that the S(t? )?module HαF (F ) has a basis {f1 , . . . , fm }, consisting of classes invariant under Kp , for some p ∈ B. Let cj = cfj ,p , for j = 1, . . . , m. Then the classes c1 , . . . , cm have the property that their restrictions to each ?ber form a basis for the cohomology of the ?ber. 4. Flag Manifolds as GKM Fiber Bundles 4.1. GKM graphs of ?ag manifolds. Let g a semisimple Lie algebra, h ? g a Cartan subalgebra, and t ? h a compact real form. Let g=h⊕
α∈?



be the Cartan decomposition of g, where ? ? t? is the set of roots. Let ?+ be a choice of positive roots and ?0 = {α1 , . . . , αn } ? ? be the corresponding simple roots. For α ∈ ?, let sα : t? → t? be the re?ection generated by α, and, for 1 i n, let si = sαi be the re?ection generated by the simple root αi . Let S ? ?0 be a subset of simple roots, and let S ? ?+ be the set of positive roots that can be written as linear combinations of roots in S. Let WS = sα | α ∈ S be the group generated by the re?ections corresponding to the roots in S . Then WS is generated by re?ections corresponding to the simple roots in S. When S = ?0 , the corresponding group W = W?0 is the Weyl group of g and WS ? W is a subgroup of W for every S ? ?0 . As a general convention, we will use Greek letters α, β for roots and for axial functions (whose values are, in this case, roots, and it will be clear from the context whether α is a root or an axial function), and Roman letters u, v, w, for elements of the Weyl group W . Then wβ is the element of t? obtained by applying w ∈ W to β ∈ t? , and wsβ is the element of the Weyl group obtained by multiplying w ∈ W with the re?ection sβ ∈ W corresponding to the root β. Then wsβ = swβ w , hence two elements that di?er by a re?ection to the left also di?er by a re?ection to the right. If α ∈ S and β ∈ ?+ \ S , then sα β = β ? nβ,α α, with nβ,α ∈ Z. The positive root β is a linear combination, with positive coe?cients, of simple roots and with at least one strictly positive coe?cient in front of a simple root not in S. Since α is a linear combination of roots in S, it follows that sα β has at least one strictly positive coe?cient, in front of a simple root not in S, hence sα β ∈ ?+ \ S . Since WS is generated by the s′ s with α ∈ S , it follows that ?+ \ S is WS ?invariant. α Let w ∈ W and let w = sβ1 · · · sβm be a decomposition of w into re?ections. If α ∈ S and β ∈ ?+ \ S then sβ sα = sα ssα β , and sα β ∈ ?+ \ S . We can therefore push all the re?ections coming from roots in ′ ′ ′ ′ S to the left, and get that w = usβ1 . . . sβk with u ∈ WS and β1 , . . . , βk ∈ ?+ \ S .

COHOMOLOGY OF GKM FIBER BUNDLES

17

We can also push all the re?ections coming from roots in S to the right, and get ′′ ′′ ′′ ′′ that w = sβ1 . . . sβk u with u ∈ WS and β1 , . . . , βk ∈ ?+ \ S . Let W/WS = {vWS | v ∈ W } be the set of right cosets. We will now brie?y describe the GKM structure of W/WS ; more details of this construction are available in [GHZ]. Let π : W → W/WS be the projection, and, for v ∈ W , let [v] = π(v). Let Γ(W/WS ) be the graph with vertices W/WS and with vertices [w], [v] ∈ W/WS joined by an edge if and only if [v] = [wsβ ] for some β ∈ ?+ \ S . The edge e = ([w] → [wsβ ] = [swβ w]) is labeled by αe = α([w], [wsβ ]) = wβ. The connection ?e along the edge e = ([w], [wsβ ]) sends the edge ([w], [wsβ ′ ]) to the edge ([wsβ ], [wsβ sβ ′ ]). Then (Γ(W/WS ), α) is a GKM graph. When S = ? we have WS = {1} and W/WS = W . Let Γ(W ) be the corresponding GKM graph. Two vertices w, v ∈ W are joined by an edge in Γ(W ) if and only if w?1 v = sβ for some β ∈ ?+ (or, equivalently, if v = wsβ = swβ w), and the edge w → wsβ = swβ w is labeled by wβ. If S ? ?0 , let Γ(WS ) be the GKM subgraph of ΓW with vertex set WS . Two vertices w, v ∈ WS are joined by an edge in Γ(WS ) if and only if w?1 v = sβ for some β ∈ S , or, equivalently, if v = wsβ = swβ w.
4321 3421 4312

34
4132

3241

24 23
3214 4123

14 13
2314 2143 2134 1234 1423

12
1243

Figure 4. GKM ?ber bundle S4 → J(4, 2) For example, if ? is the root system of type A3 and S = {α1 , α3 }, then W = S4 , the group of permutations of [4], and WS = S2 × S2 is the subgroup generated by the commuting transpositions s1 = (1, 2) and s3 = (3, 4). Then W/WS is identi?ed with the set of 2-element subsets of [4] by ω → {u(1), u(2)}, where u is the unique representative of the class ω with u(1) < u(2) and u(3) < u(4). The corresponding graph is the Johnson graph J(4, 2), and it is the GKM graph of the Grassmannian manifold G2 (C4 ) of 2-dimensional (complex) subspaces in C4 , with an 3 action of T0 induced by the standard linear action of T 4 on C4 . The GKM ?bration

18

VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

S4 → J(4, 2) is the combinatorial description of the ?bration F4 (C) → G2 (C4 ) that sends a complete ?ag in F4 (C) to its two dimensional component. Figure 4 shows the graphical representation of this ?bration, with the ?bers being the squares. (The internal edges of S4 have been omitted.) 4.2. GKM ?ber bundles over partial ?ags. Proposition 4.1. The projection π : W → W/WS is a GKM ?ber bundle with typical ?ber WS . Proof. For a vertex w ∈ W , the vertical edges at w are the edges
⊥ Ew = {(w, wsβ ) | β ∈ S } ,

and the horizontal edges are Hw = {(w, wsβ ) | β ∈ ?+ \ S } , hence (dπ)w : Hw → E[w] , de?ned by (dπ)w (w, wsβ ) = ([w], [wsβ ]) is a bijection. Let v ∈ π ?1 ([w]). Then v = wu, for some u ∈ WS . The lift of the edge ([w], [wsβ ]) at v is the edge (v, swβ v) = (v, vsv?1 wβ ) = (v, vsu?1 β ) and since u?1 β ∈ ?+ \ S , we have α(v, vsu?1 β ) = vu?1 β = wβ = α([w], [wsβ ]) . The connections on W and W/WS are compatible, hence π : W → W/WS is a GKM ?bration. For [w] ∈ W/WS , the ?ber Γ[w] above [w] is Γ[w] = {wu | u ∈ WS } ? W . Each edge of W/WS is of the form ([w], [v]), with w, v ∈ W and v = wsβ for some β ∈ ?+ \ S . The transition map from the ?ber Γ[w] to the ?ber Γ[v] is Φ[w],[v] (q) = swβ q for all q ∈ Γ[w] . If Ψ[w],[v] : t? → t? is given by Ψ[w],[v] (α) = swβ α, then (Φ[w],[v] , Ψ[w],[v]) : Γ[w] → Γ[v] is an isomorphism of GKM graphs and Ψ?1 = Ψ[v],[w]. Hence π is a GKM ?ber bundle. [w],[v] Let v ∈ π ?1 ([w]), and de?ne ?v : WS → Γ[w] , ?v (u) = vu and ψv : t? → t? , ψv (β) = vβ. Then (?v , ψv ) : WS → Γ[w] is an isomorphism of GKM graphs, hence the typical ?ber of π is WS . 4.3. Holonomy subgroup. In this section we determine the holonomy subgroup of Aut(WS , α) determined by loops in the base W/WS . Let w ∈ WS , let Φw : WS → WS , Φw (u) = wu, and Ψw : t? → t? , Ψw (β) = wβ. Then Υw = (Φw , Ψw ) : Γ(WS ) → Γ(WS ) is a GKM automorphism. Moreover, the map Υ : WS → Aut(WS , α), Υ(w) = Υw is an injective morphism of groups and we identify Υ(WS ) with WS and consider that WS acts on Γ(WS ). Proposition 4.2. The subgroup of Aut(WS , α) determined by loops in the base W/WS is Υ(WS ). Proof. Let v0 ∈ W be a ?xed element of the Weyl group and Γ[v0 ] = π ?1 ([v]) ? W be the ?ber through v0 . Let ([v0 ], [v1 ]) be an edge of W/WS , with v1 = v0 sβ for β ∈ ?+ \ S . Then Φ[v0 ],[v1 ] (v0 u) = sv0 β v0 u = v0 sβ u = v1 u for every u ∈ WS . Let γ ∈ ?([v0 ]) be a loop in W/WS based at [v0 ], given by [v0 ] → [v1 ] → · · · → [vm?1 ] → [vm ] = [v0 ] ,

COHOMOLOGY OF GKM FIBER BUNDLES

19

where vk = vk?1 βk , with βk ∈ ?+ \ S for every k = 1, . . . , m. Then the induced map Φγ : Γ[v0 ] → Γ[v0 ] is given by Φγ = Φ[vm?1 ,vm ] ? · · · ? Φ[v0 ,v1 ] , and Φγ (v0 u) = vm u = v0 sβ1 · · · sβm u. As a map from WS to WS ,
?1 Φγ (u) = v0 vm u ,

where v0 → v1 → · · · → vm is the lift of γ at v0 . Since v0 and vm are in the same ?1 ?ber, it follows that w = v0 vm ∈ WS and Φγ = Φw . Similarly,
?1 Ψγ (β) = v0 vm β = Ψw (β)

for all β ∈ t? and the automorphism of WS induced by the loop γ is Υγ = Υw . We will now show that for every v ∈ WS , there exists a loop γ in W/WS such that Υv = Υγ . Since WS is generated by si for αi ∈ S, it su?ces to construct such loops for Υsi for αi ∈ S. Let αi ∈ S ?0 . The Weyl group W acts transitively on ?, hence there exists w ∈ W such that wαi ∈ ?+ \ S . Let w = uv be a decomposition of w such that u ∈ WS and v = sβ1 · · · sβm with β1 , . . . , βm ∈ ?+ \ S . Then u?1 wαi ∈ ?+ \ S . Consider the path γ in W/WS that starts with [v0 ] → [v0 sβm ] → · · · → [v0 sβm · · · sβ1 ] = [v0 v ?1 ] , continues with [v0 v ?1 ] → [v0 v ?1 su?1 wαi ] → [v0 v ?1 su?1 wαi sβ1 ] → [v0 v ?1 su?1 wαi sβ1 sβ2 ] , and ends with [v0 v ?1 su?1 wαi sβ1 sβ2 ] → · · · → [v0 v ?1 su?1 wαi sβ1 sβ2 · · · sβm ] = [v0 v ?1 su?1 wαi v] . The lift of γ at v0 ends at v ?1 su?1 wαi v = v0 si , hence Υγ = Υv?1 v0 si = Υsi ,
0

and the subgroup of Aut(WS , α) determined by loops in W/WS is Υ(WS ). 5. Fibrations of Classical Groups In this section we consider the GKM bundle W → W/WS when S = ?0 \ {α1 }, where ?0 is the set of simple roots for a classical root system and α1 is one of the endpoint roots in the Dynkin diagram. 2) is ?0 = {α1 , . . . , αn }, 5.1. Type A. The set of simple roots of An (for n where αi = xi ? xi+1 , for i = 1, . . . , n. The set of positive roots is ?+ = {xi ? xj | 1 S = {xi ? xj | 2 ?+ \ S = {βj | βj = x1 ? xj , 2 Let ω1 =[id] ωj =[sβj ] , for 2 j n+1. j i<j i<j n + 1} n + 1} , j n} . and xi ? xj = αi + . . . + αj?1 . If S = {α2 , . . . , αn }, then is the set of positive roots for a root system of type An?1 , and n + 1} = {α1 + · · · + αj | 1

20

VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

Then W/WS = {ω1 , . . . , ωn+1 }, and the graph structure of W/WS is that of a complete graph with n + 1 vertices. If τ : W/WS → t? is given by τ (ωi ) = xi for all i = 1, . . . , n + 1, then the axial function α on W/WS is given by α(ωi , ωj ) = τ (ωi ) ? τ (ωj ) = xi ? xj .
1 Then τ ∈ Hα (W/WS ) is a class of degree 1. Using a Vandermonde determinant argument, one can easily show that the classes {1, τ, . . . , τ n } are linearly indepen? dent over S(t? ), and in fact form a basis of the free S(t? )?module Hα (W/WS ). The Weyl group W is isomorphic with the symmetric group Sn+1 , acting on roots by w · (xi ? xj ) = xw(i) ? xw(j) . The simple re?ection si acts as the transposition (i, i + 1), and, more general, the re?ection generated by the root xi ?xj acts as the transposition (i, j). The subgroup WS is the subgroup of W = Sn+1 consisting of the permutations that ?x the element 1. With the identi?cation W/WS ? Kn+1 , the projection π : W → W/WS is the map π : Sn+1 → Kn+1 , π(w) = w(1). Let n > 1 and, for I = [i1 , . . . , in?1 ], let cI : Sn → S(t? ) be given by
n?1 n?1 cI (u) = u · (xi1 · · · xn?1 ) = xi1 xi2 · · · xu(n?1) . 1 u(1) u(2)

i

i

Then cI is an invariant class on Sn , and we will construct a basis of the S(t? )?module ? Hα (Sn ) consisting of classes of the type cI for speci?c indices I. Consider the GKM ?ber bundle π : S3 → K3 , π(u) = u(1). The ?ber π ?1 (3) is canonically isomorphic to S2 , and since S2 ? K2 , the cohomology of S2 is a free S(t? )?module with a basis given by the invariant classes c[0] and c[1] . The invariant class c[0] on this ?ber is extended, using transition maps between ?bers to the constant class c[0,0] ≡ 1 on the total space. The invariant class c[1] extends to the class c[0,1] . The cohomology of the base K3 is generated, over S(t? ), by 1, τ , and τ 2 , and these classes lift to basic classes c[0,0] , c[1,0] , and c[2,0] on S3 . Theorem 3.6 implies that the cohomology of S3 is a free S(t? )?module, with a basis given by {cI | I = [i1 , i2 ], 0 i1 2, 0 i2 1} . Their values on W (A2 ) = S3 are given in Table 1. c[0,0] 1 1 1 1 1 1 c[0,1] x2 x1 x3 x3 x1 x2 c[1,0] x1 x2 x1 x2 x3 x3 c[1,1] x1 x2 x2 x1 x1 x3 x2 x3 x3 x1 x3 x2 c[2,0] x2 1 x2 2 x2 1 x2 2 x2 3 x2 3 c[2,1] x2 x2 1 x2 x1 2 x2 x3 1 x2 x3 2 x2 x1 3 x2 x2 3

123 213 132 231 312 321

Table 1. Invariant classes on W (A2 )

Repeating the procedure further, we get the following result.
? Theorem 5.1. A basis of the S(t? )?module Hα (Sn ) is given by the invariant classes

B(An?1 ) = {cI | I = [i1 , . . . , in?1 ], 0

i1

n?1, 0

i2

n?2, . . . , 0

in?1

1} .

COHOMOLOGY OF GKM FIBER BUNDLES

21

5.2. Type B. The set of simple roots of Bn (for n 2) is ?0 = {α1 , . . . , αn }, where αi = xi ? xi+1 , for i = 1, . . . , n ? 1 and αn = xn . The set of positive roots is ?+ = {xi | 1 If S = {α2 , . . . , αn }, then S = {xi | 2 i n} ∪ {xi ± xj | 2 i<j j n} n} . is the set of positive roots for a root system of type Bn?1 , and
± ?+ \ S = {β1 = x1 } ∪ {βj = x1 ? xj | 2

i

n} ∪ {xi ± xj | 1

i<j

n} .

Let
+ ω1 =[id] ,
j

? ω1 = [sβ1 ]

+ ωj =[sβ + ] = [sx1 ?xj ] for 2 ? ωj =[sβ ? ] = [sx1 +xj ] for 2
j

j j

n n.

+ ? + ? Then W/WS = {ω1 , ω1 , . . . , ωn , ωn }, and the graph structure of W/WS is that ? of a complete graph with 2n vertices. If τ is the map τ : W/WS → t? , τ (ωj ) = ?xj , with 1 j n and ? ∈ {+, ?}, then the axial function α is given by ? α(ωi i , ωj j ) = ? ? τ (ωi i ) ? τ (ωj j ) ??i ?i 1 )) 2 (τ (ωi ) ? τ (ωi ?

for 1 for 1

i=j n i n.

Note that although W/WS and K2n are isomorphic as graphs, they are not isomorphic as GKM graphs. One way to see that is to notice that
+ ? ? ? ? + α(ω1 , ω1 ) + α(ω1 , ω2 ) + α(ω2 , ω1 ) = ?x1 = 0 .

Nevertheless, as in the K2n case, the set of classes {1, τ, . . . , τ 2n?1 } is a basis for ? the free S(t? )?module Hα (W/WS ).
? ω2

s2 s1 s2 + π ?1 (ω1 ) π
x2

? π ?1 (ω2 )

s2 s1 s2 s1 s2 s1 s2 = s2 s1 s2 s1
? π ?1 (ω1 )

x1 +x2
? ω1 + ω1

x1

x1 ?x2

x2
+ ω2

id
x1 ?x2

s1 s2 s1
x1 +x2

s1 x 1 s1 s2 + π ?1 (ω2 ) Figure 5. Fibration of B2

An alternative description of the Weyl group W is that of the group of signed permutations (u, ?), with u ∈ Sn and ? = (?1 , . . . , ?n ), ?j = ±1. The element (u, ?) is represented as (?1 u(1), . . . , ?n u(n)). Then sxi is just a change of the sign of xi , sxi ?xj corresponds to the transposition (i, j), with no sign changes, and sxi +xj corresponds to the transposition (i, j) with both signs changed. In particular, id is the identity permutation with no sign

22

VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

changes, sβ1 is the identity permutation with the sign of 1 changed, s+j is the β transposition (1, j) with no sign changes, and sβ ? is the transposition (1, j) with j sign changes for 1 and j. In general, if u ∈ Sn and ? = (?1 , . . . , ?n ) ∈ Zn , then 2 the element w = (u, ?) ∈ W acts by (u, ?) · xk = ?k xu(k) . Then W/WS can be ? identi?ed with {±1, ±2, . . . , ±n} by ωj → ?j, and the projection π : W → W/WS is π((u, ?)) = ?1 u(1). For I = [i1 , . . . , in?1 ], let cI : W → S(t? ) be given by cI ((u, ?)) = (?1 xu(1) )i1 · · · (?n?1 xu(n?1) )in?1 .
? Then cI ∈ (Hα (W ))W is an invariant class, and we will construct a basis of the free ? ? S(t )?module Hα (W ) consisting of classes of the type cI , for speci?c indices I. When n = 2, the ?ber over 2 is π ?1 (2) = {(2, 1), (2, ?1)} and is identi?ed with ? WS = S2 = {1, ?1}. A basis for Hα (WS ) is given by the invariant classes {c[0] , c[1] }, where c[0] ≡ 1 and c[1] (1) = x1 , c[1] (?1) = ?x1 . These classes are extended to the invariant classes c[0,0] and c[0,1] on W . The classes 1, τ , τ 2 , and τ 3 on the base lift to the basic classes c[0,0] , c[1,0] , c[2,0] , ? and c[3,0] on W . Then a basis for the free S(t? )?module Hα (W ) is

B(B2 ) = {cI | I = [i1 , i2 ], , 0

i1

3, 0

i2

1} .

The values of these classes on the elements of W (B2 ) are shown in Table 2. Repeating the procedure further, we get the following result. c[0,0] 1 1 1 1 1 1 1 1 c[0,1] x2 ?x2 ?x2 x2 x1 ?x1 ?x1 x1 c[1,0] x1 x1 ?x1 ?x1 x2 x2 ?x2 ?x2 c[1,1] x1 x2 ?x1 x2 x1 x2 ?x1 x2 x1 x2 ?x1 x2 x1 x2 ?x1 x2 c[2,0] x2 1 x2 1 x2 1 ?x2 1 x2 2 x2 2 x2 2 x2 2 c[2,1] x2 x2 1 ?x2 x2 1 ?x2 x2 1 x2 x2 1 x2 x1 2 ?x2 x1 2 ?x2 x1 2 x2 x1 2 c[3,0] x3 1 x3 1 ?x3 1 ?x3 1 x3 2 x3 2 ?x3 2 ?x3 2 c[3,1] x3 x2 1 ?x3 x2 1 x3 x2 1 ?x3 x2 1 x3 x1 2 ?x3 x1 2 x3 x1 2 ?x3 x1 2

(1, 2) (1, ?2) (?1, ?2) (?1, 2) (2, 1) (2, ?1) (?2, ?1) (?2, 1)

Table 2. Invariant classes on W (B2 )

? Theorem 5.2. A basis of the S(t? )?module Hα (W (Bn )) is given by the invariant classes

B(Bn ) = {cI | I = [i1 , . . . , in ], 0

i1

2n ? 1, 0

i2

2n ? 3, . . . , 0

in

1} .

2) is ?0 = {α1 , . . . , αn }, 5.3. Type C. The set of simple roots of Cn (for n where αi = xi ? xi+1 , for i = 1, . . . , n ? 1 and αn = 2xn . The set of positive roots is ?+ = {2xi | 1 i n} ∪ {xi ± xj | 1 i < j n} . If S = {α2 , . . . , αn }, then S = {2xi | 2 i n} ∪ {xi ± xj | 2 i<j j n} n} . is the set of positive roots for a root system of type Cn?1 , and
± ?+ \ S = {β1 = 2x1 } ∪ {βj = x1 ? xj | 2

COHOMOLOGY OF GKM FIBER BUNDLES

23

Let
+ ω1 =[id] ,
j

? ω1 = [sβ1 ]

+ ωj =[sβ + ] = [sx1 ?xj ] for 2 ? ωj =[sβ ? ] = [sx1 +xj ] for 2
j

j j

n n.

This is essentially the same as the type B case, and W (Cn ) ? W (Bn ) is the group + ? + ? of signed permutations of n letters. Then W/WS = {ω1 , ω1 , . . . , ωn , ωn }, and the graph structure of W/WS is that of a complete graph with 2n vertices. The axial function on W/WS is given by
? ? α(ωi i , ωj j ) = τ (ωi i ) ? τ (ωj j ) , ? ?

hence W/WS is isomorphic, as a GKM graph, with the complete graph K2n . Then ? ? Hα (W (Cn )) ? Hα (W (Bn )), with B(Cn ) = B(Bn ) as a basis consisting of invariant classes. 5.4. Type D. The set of simple roots of Dn (for n 3) is ?0 = {α1 , . . . , αn }, where αi = xi ? xi+1 , for i = 1, . . . , n ? 1 and αn = xn?1 + xn . The set of positive roots is ?+ = {xi ? xj | 1 If S = {α2 , . . . , αn }, then S = {xi ? xj | 2 i<j n} ∪ {xi + xj | 2 i<j n} . If n 4, then S is the set of positive roots for a root system of type Dn?1 and if n = 3, then S is the set of positive roots of A1 × A1 . In both cases
+ ?+ \ S = {βi = x1 ? xi | 2

i<j

n} ∪ {xi + xj | 1

i<j

n} .

i

? n} ∪ {βi = x1 + xi | 2

i

n} .

Let
+ ω1 = [id] ? ω1 = [sβ ? sβ + ] = [sβ + sβ ? ] = [sβ + sβ ? ] for all 2
j j j j i i

i, j

n

+ ωi = [sβ + ] for all 2
i

i i

n n

? ωi

= [sβ ? ] for all 2
i

+ ? + ? Then W/WS = {ω1 , ω1 , . . . , ωn , ωn } and the graph structure of W/WS is that + ? n of the complete n?partite graph K2 ([Di, p. 15]), with partition classes {ωi , ωi } ? ? for 1 i n. If τ : W/WS → t is given by τ (ωi ) = ?xi , where ? ∈ {+, ?}, then the axial function α on W/WS is ? ? α(ωi i , ωj j ) = τ (ωi i ) ? τ (ωj j ) = ?i xi ? ?j xj . ? Then Hα (W/WS ) is a free S(t? )?module, with a basis given by 1, τ , . . . , τ 2n?2 , and η = x1 · · · xn τ ?1 . An alternative description of the Weyl group W is that of the group of signed permutations (u, ?) with an even number of sign changes. Then sxi ?xj corresponds to the transposition (i, j), with no sign changes, and sxi +xj corresponds to the transposition (i, j) with both signs changed. In particular, id is the identity permutation with no sign changes, s+j is the transposition (1, j) with no sign changes, β sβ ? is the transposition (1, j) with sign changes for 1 and j, and σβ + σβ ? is the
j j j

?

?

24

VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

identity permutation with the sign changes for 1 and j. In general, if u ∈ Sn and ? = (?1 , . . . , ?n ) ∈ Zn with ?1 · · · ?n = 1, then the element w = (u, ?) ∈ W acts 2 by (u, ?) · xk = ?k xu(k) . Then W/WS can be identi?ed with {±1, ±2, . . . , ±n} by ? ωi → ?i, and the projection π : W → W/WS is π((u, ?)) = ?1 u(1). For I = [i1 , . . . , in?1 ], let cI : W → S(t? ) be given by cI ((u, ?)) = (?1 xu(1) )i1 · · · (?n?1 xu(n?1) )in?1 .
? Then cI ∈ (Hα (W ))W is an invariant class, and we will construct a basis of the free ? ? S(t )?module Hα (W ) consisting of classes of the type cI , for speci?c indices I. 3 When n = 3, the ?ber π ?1 (3) of the GKM ?ber bundle π : D3 → K2 is

π ?1 (3) = {(3, 1, 2), (3, 2, 1), (3, ?2, ?1), (3, ?1, ?2)} and is identi?ed with WS = S2 × S2 = {(1, 2), (2, 1), (?2, ?1), (?1, ?2)}. Then ? Hα (WS ) is generated by the WS ?invariant classes {cI | I ∈ P2 }, where P2 = {[0, 0], [1, 0], [2, 0], [0, 1]} .
3 The classes 1, τ , τ , τ , τ 4 , η on K2 lift to the basic classes c[0,0,0] , c[1,0,0] , c[2,0,0] , ? c[3,0,0] , c[4,0,0] , and c[0,1,1] . Then a basis for the free S(t? )?module Hα (W ) is 2 3

{cI | I = [i1 , i2 , i3 ] ∈ P3 } , where P3 is the set of triples [i1 , i2 , i3 ] ∈ Z3 0 , such that i1 i2 i3 = 0 and either i1 4, i2 2, i3 1 or [i1 , i2 , i3 ] = [0, 1, 2] or [0, 3, 1]. Repeating this process further, we get the following general result.
? Theorem 5.3. A basis of the S(t? )?module Hα (Dn ) is given by

B(Dn ) = {cI | I ∈ Pn } , where P2 = {[0, 0], [1, 0], [2, 0], [0, 1]} and Pn is de?ned inductively as the set of indices [i1 , . . . , in ] such that either ? 0 i1 2n ? 2 and [i2 , . . . , in ] ∈ Pn?1 , or ? i1 = 0 and [i2 ? 1, . . . , in ? 1] ∈ Pn?1 . By a result of Chevalley ([Hu, Section 3.6]), for a root system R, the symmetric algebra S(t? ) is a free module over the ring of invariants S(t? )W . The results of this section show that for every v ∈ W , {cI (v) | cI ∈ B(R)} is a basis of this free module. 6. Symmetrization of Schubert Classes In Section 5 we constructed invariant classes for classical groups by iterating the GKM ?ber bundle construction. In this section we state a di?erent method of constructing invariant classes. The proofs will be included in a separate paper. ? For every u ∈ W , there exists a unique class τu ∈ Hα (W ), called the equivariant Schubert class of u, that satis?es the following conditions: (1) τu is homogeneous of degree ?(u), where ?(u) is the length of u; (2) τu is supported on {v |u v}, where is the strong Bruhat order, and

COHOMOLOGY OF GKM FIBER BUNDLES

25

(3) τu is normalized by the condition τu (u) =
? Hα (W );

{β | β ? 0, u?1 β ? 0}

The set {τu | u ∈ W } of equivariant Schubert classes is a basis of the S(t? )?module ? however, these classes are not invariant under the action of W on Hα (W ). ? sym ? For f ∈ Hα (W ) we de?ne the W ?invariant class f : W → S(t ) by 1 fw , f sym = |W |
w∈W w ?

where the permuted class f : W → S(t ) is given by f w (u) = w?1 · f (wu), u ∈ W . Theorem 6.1. Let u, v, w ∈ W . Then τu (ab) =
v
Lu

τuv?1 (a)τv (b)
w τuv?1 (w?1 )τv , v
Lu

τu =
w τu

=
v
Lu

(?1)?(vu

?1

)

τvu?1 (w?1 )τv .
L

where

L

is the left weak order, de?ned by v

u ?? ?(uv ?1 ) = ?(u) ? ?(v).

sym Theorem 6.2. For u ∈ W , let τu be the symmetrization of τu . If sym τu = v∈W

au,v τv ,

(6.1)

then (1) The matrix (au,v )u,v is lower triangular with respect to the left weak order on W : au,v = 0 ?? v L u . (2) The entries on the diagonal are all 1: au,u = 1 for all u ∈ W . (3) The entry au,v is, after multiplication by |W |, a homogeneous polynomial of degree ?(u) ? ?(v) in the negative simple roots, with non-negative integer coe?cients: |W |au,v ∈ Z
?(u)??(v) [?α1 , . . . , ?αn ] 0

.

As a consequence we get the following construction of a basis consisting of invariant classes.
sym Theorem 6.3. The set {τu | u ∈ W } of symmetrized classes is a basis of the ? ? S(t )?module Hα (W ).

References
[Bi] [BH] [Di] Billey, Sara. Kostant polynomials and the cohomology ring for G/B. Duke Math. J. 96, no. 1, 205–224, 1999 Billey, Sara, and Mark Haiman. Schubert Polynomials for the Classical Groups. Journal of the AMS, 8, No 2, 443–482, 1995. Diestel, Reinhard. Graph Theory. Graduate Texts in Mathematics 173, Springer-Verlag, New York, 2000.

26

VICTOR GUILLEMIN, SILVIA SABATINI, AND CATALIN ZARA

Humphreys, James. Re?ection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, 1990. [GHZ] Guillemin, Victor, Tara Holm, and Catalin Zara. A GKM description of the equivariant cohomology ring of a homogeneous space. J. Algebraic Combin. 23, no. 1, 21–41, 2006. [GKM] Goresky, Mark, Robert Kottwitz, and Robert MacPherson. Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131, no. 1, 25–83, 1998. [GS] Guillemin, Victor and Shlomo Sternberg. Supersymmetry and equivariant de Rham theory. Mathematics Past and Present. Springer-Verlag, Berlin, 1999. [GZ] Guillemin, Victor and Catalin Zara. 1-skeleta, Betti numbers, and equivariant cohomology. Duke Math. J. 107, no. 2, 283–349, 2001 Department of Mathematics, MIT Department of Mathematics, MIT Department of Mathematics, University of Massachusetts Boston

[Hu]


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