Диссертация (1136188), страница 44
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Assume that X is T -equivariantly cellular. Then there is a degree-doubling map∗∗ΦtopX : ΩT (X) → M UT (X)which is a ring isomorphism.Proof. It follows from Lemma 3.6 and [22, Prop. 7.5] that there is a ring homo∗∗morphism ΦtopX : ΩT (X) → M UT (X).We now choose a sequence {(Vj , Uj )}j≥1 of good pairs for the T -action as in∼=Proposition 3.5. It follows from [22, Thm. 6.1] that ΩiT (X) −→ ΩiT (X, U) for Teach i ∈ Z. Since HT∗ (X, Z) = H ∗ X × EG , Z is torsion-free by Lemma 3.6,∼=Lemma 3.2 implies that M UTi (X) −→ M UTi (X, U). The theorem now follows fromProposition 3.5.
Let G be a connected reductive group with a maximal torus T and a Borelsubgroup B containing T . Since we have already noted that the flag variety G/B isT -equivariantly cellular, the following is an immediate consequence of Theorem 3.7.Corollary 3.8. There is a ring isomorphism∼=∗Φtop→ M UT∗ (G/B).G/B : ΩT (G/B) −For any character α ∈ Tb, let xα denote the first T -equivariant Chern class∈ S of the associated T -equivariant line bundle Lα on Spec (C) (see Subsection 2.1). Let Σ denote the root system of (G, T ), and W the Weyl group. Thefollowing description of Ω∗T (G/B) as a subring of S |W | follows immediately from∼=Corollary 3.8, [14, Thm.
3.1] and the isomorphism S −→ M U ∗ (BT ) ([26]).cT1 (Lα )400VALENTINA KIRITCHENKO, AMALENDU KRISHNATheorem 3.9. The inclusion ι : (G/B)T ,→ G/B of the fixed point locus inducesa ring isomorphismι∗ |W |Ω∗T (G/B) →(f)∈S|f≡f(modx)∀α∈Σ,∀w∈W.ww∈Wwswαα∼=A description of the kind obtained in Theorem 3.9 was first conceived in thepaper [13] of Goresky–Kottwitz–MacPherson, who showed that the equivariant singular cohomology of G/B can be described in such a way. Around the same time,Brion [5] showed that the T -equivariant Chow groups of G/B can also be describedin a similar way.
The results of Goresky–Kottwitz–MacPherson and Brion weresubsequently extended to a bigger class of equivariant cohomology theories suchas equivariant K-theory and equivariant complex cobordism of Kac–Moody flagvarieties by Harada–Henriques–Holm [14, Thm. 3.1]. Note that this description isdifferent from the Borel type description we obtain in the next sections.4. Equivariant complex cobordism of G/BIn this section, we continue working over the ground field C. Let G be a connected reductive group. We fix a maximal torus T of rank n and a Borel subgroupB containing T .
The Weyl group of G is denoted by W . In this section, we computethe equivariant complex cobordism ring M UT∗ (G/B) of the complete flag varietyG/B. For this description, we need the following special case of the Leray–Hirschtheorem for a multiplicative generalized cohomology theory.Theorem 4.1 (Leray–Hirsch). Let X be a (possibly infinite) CW-complex withpifinite skeleta and let F →− E−→ X be a fiber bundle such that the fiber F is a finiteCW-complex. Assume that there are elements {e1 , · · · , er } in M U ∗ (E) such that{f1 = i∗ (e1 ), . . . , fr = i∗ (er )} forms an L-basis of M U ∗ (F ) for each fiber F of thefiber bundle. Assume furthermore that H ∗ (X, Z) is torsion-free.
Then the mapΨ : M U ∗ (F ) ⊗L M U ∗ (X) → M U ∗ (E) XXΨfi ⊗ b i =p∗ (bi )ei1≤i≤r(4.1)1≤i≤ris an isomorphism of MU ∗ (X)-modules. In particular, MU ∗ (E) is a free MU ∗ (X)module with the basis {e1 , . . . , er }.This result is well known and can be found, for example, in [36, Thm. 15.47]and [20, Thm. 3.1].4.1.
Equivariant complex cobordism of G/BIn what follows, we assume all spaces to be pointed and let pX : X → pt bethe structure map. Let M U ∗ (BT ) = M UT∗ (pt) denote the coefficient ring of theT -equivariant complex cobordism. It is well known ([26]) that M U ∗ (BT ) is isomorphic to S(T ) (which is denoted by S in this text). The isomorphism sendsa character χ of T to the first Chern class of the T -equivariant line bundle Lχon BT . Note that each character χ of T also gives rise to the G-equivariant lineEQUIVARIANT COBORDISM OF FLAG AND SYMMETRIC VARIETIES401Bbundle Lχ := G × Lχ on G/B. We will also use that M U ∗ (BT ) = M U ∗ (BB ) isG∗isomorphic to M UG(G/B) since G/B × EG = EG /B and we can choose EG = EB .For any finite CW-complex X with a G-action, consider the fiber bundleBiGπXG/B −−→ X × EG −−X→ X × EG ,where iX is the inclusion of the fiber at the base point.
Put i = iX and π = πXwhen X is the base point. This gives rise to the following commutative diagram:M U ∗ (BG )π ∗ //M U ∗ (BT )p∗G,Xi∗ //M U ∗ (G/B)p∗T,X∗M UG(X)∗πX// M U ∗ (X)T(4.2)// M U ∗ (G/B).i∗XRecall that the torsion index of G is defined as the smallest positive integer tGsuch that tG times the class of a point in H 2d (G/B, Z) (where d = dim(G/B))belongs to the subring of H ∗ (G/B, Z) generated by the first Chern classes of linebundles Lχ (e.g., tG = 1 for G = GLn ; see [39] for computations of tG for othergroups).
If G is simply connected then this subring is generated by H 2 (G/B, Z).For the rest of this section, an abelian group A will actually mean its extensionA ⊗Z R, where R = Z[t−1G ]. In particular, all the cohomology and the cobordismgroups will be considered with coefficients in R.We shall use the following key fact to prove the main result of this section.∗Lemma 4.2.
The homomorphism i∗ : M UG(G/B) → M U ∗ (G/B) is surjectiveover the ring R.∗Proof. Since M UG(G/B) ' M U ∗ (BT ) ' S, the image of i∗ is the subring of∗M U (G/B) generated by the first Chern classes of line bundles Lχ . To prove surjectivity of i∗ , we have to show that M U ∗ (G/B) is generated by the first Chernclasses.Since G/B is cellular, the cobordism ring M U ∗ (G/B) is a free L-module.Choose a basis {ew }w∈W in M U ∗ (G/B) such that all ew are homogeneous (e.g.,take resolutions of the closures of cells). Consider the homomorphismϕ : M U ∗ (G/B) → M U ∗ (G/B) ⊗L R.Since H ∗ (G/B, R) is torsion free, we have the isomorphism M U ∗ (G/B) ⊗L R 'H (G/B, R). Note that H ∗ (G/B, R) is generated by the first Chern classes bydefinition of the torsion index, and the homomorphism ϕ takes the Chern classes tothe Chern classes.
Hence, there exist homogeneous polynomials {%w }w∈W , where%w ∈ Sym(Tb) ⊗ R ⊂ S such that ϕ(ew ) = ϕ(i∗ (%w )). Then the set of cobordismclasses {i∗ (%w )}w∈W is a basis over L in M U ∗ (G/B, R). Indeed, consider thetransition matrix A from the basis {ew }w∈W to this set (order ew and %w so thattheir degrees decrease). The elements of A are homogeneous elements of L and∗402VALENTINA KIRITCHENKO, AMALENDU KRISHNAA ⊗L R is the identity matrix.
By degree arguments, it follows that the matrix Ais upper-triangular and the diagonal elements are equal to 1, so A is invertible.Hence, M U ∗ (G/B) has a basis consisting of polynomials in the first Chernclasses and the homomorphism i∗ is surjective over R. To compute M UT∗ (G/B) and M UT∗ (X), we can now apply the same strategy asin the cohomology case (see, e.g., [4, Prop. 1])∗By Lemma 4.2, we can choose polynomials {%w }w∈W in MUG(G/B) ' MU ∗ (BT )∗∗= S such that {i (%w )}w∈W form an L-basis in M U (G/B). Set %w,X = p∗T,X (%w )for each w ∈ W .
Define L-linear mapss : M U ∗ (G/B) → S, sX : M U ∗ (G/B) → M UT∗ (X)s (i∗ (%w )) = %w and sX (i∗ (%w )) = %w,X .(4.3)Note that maps iX and i are W -equivariant. In particular, the map s is alsoW -equivariant.Lemma 4.3. Let X be a finite CW-complex with a G-action such that HT∗ (X, R)is torsion-free.∗(i) The map M U ∗ (G/B)⊗L M UG(X) → M UT∗ (X) which sends (b, x) to sX (b)·∗∗πX (x) is an isomorphism of M UG(X)-modules.
In particular, M UT∗ (X) is∗a free M UG (X)-module with the basis {%w,X }w∈W .∗∗(ii) The map S × M UG(X) → M UT∗ (X) which sends (a, x) to p∗T,X (a) · πX(x)yields an isomorphism of graded L-algebras∼=∗Ψtop→ M UT∗ (X).X : S ⊗M U ∗ (BG ) M UG (X) −(4.4)∗Proof. It follows from our assumption and [16, Prop. 2.1(i)] that HG(X, R) is∗∗∗torsion-free. Since i = iX ◦ pT,X , we conclude from the above construction thati∗ (%w ) = i∗X p∗T,X (%w ) = i∗X (%w,X ). Since {i∗ (%w )}w∈W form an L-basis ofM U ∗ (G/B), the first statement now follows immediately by applying Theorem 4.1iBπGXto the fiber bundle G/B −−→ X × EG −−X→ X × EG .∗To prove the second statement, we first notice that M UG(X) → M UT∗ (X) is a∗∗map of M U (= L)-algebras and so is the map S → M UT (X).
In particular, beingthe product of these two maps, (4.4) is a morphism of L-algebras. Moreover, itfollows from the first part of the lemma that S ∼= M U ∗ (BT ) is a free M U ∗ (BG )∗∗module with basis {%w }w∈W and M UT (X) is a free M UG(X)-module with basistop{%w,X }w∈W . In particular, ΨX takes the basis elements %w ⊗ 1 onto the basiselements %w,X . Hence, it is an algebra isomorphism. We now compute M U ∗ (BG ).Proposition 4.4. The natural map M U ∗ (BG ) → (M U ∗ (BT ))phism of R-algebras.Wis an isomor-Proof. Note that in the proof of Lemma 4.2, we can choose %w0 = 1 (here w0 isthe longest length element of the Weyl group).
Then applying Theorem 4.1 to theiπfiber bundle G/B →− BT −→ BG (as in the proof of Lemma 4.3 for X = pt), we getΨ(1 ⊗ b) = Ψ(i∗ (%w0 ) ⊗ b) = π ∗ (b)%w0 = π ∗ (b) for any b ∈ M U ∗ (BG ),(4.5)EQUIVARIANT COBORDISM OF FLAG AND SYMMETRIC VARIETIES403where Ψ is as in (4.1). In particular, π ∗ is the composite map1⊗idΨπ ∗ : M U ∗ (BG ) −−−→ M U ∗ (G/B) ⊗L M U ∗ (BG ) −→ M U ∗ (BT ).(4.6)Hence to prove the proposition, it suffices to show using Theorem 4.1 that theWmap 1 ⊗ id induces an isomorphism M U ∗ (BG ) → (M U ∗ (G/B) ⊗L M U ∗ (BG ))over R.1⊗idWe first show that the map M U ∗ (BG ) −−−→ M U ∗ (G/B) ⊗L M U ∗ (BG ) is splitinjective.
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