aa r X i v : . [ m a t h . A T ] J u l SYMMETRY BREAKING AND LINK HOMOLOGIES II
NITU KITCHLOOA
BSTRACT . In the first part of this paper, we constructed a filtered U( r ) -equivariant stablehomotopy type called the spectrum of strict broken symmetries s B ( L ) of links L givenby closing a braid with r strands. Evaluating this filtered spectrum on suitable twisted U( r ) -equivariant cohomology theories gives rise to a spectral sequence of link invariantsthat converges to the cohomology of the limiting spectrum s B ∞ ( L ) . In this paper, we ap-ply Borel equivariant singular cohomology H ∗ U( r ) to our construction. We show that the E -term of the spectral sequence is isomorphic to an unreduced verision of triply-gradedlink homology. More precisely, we show that the E -term of the spectral sequence is iso-morphic to the Hochschild-homology complex of Soergel bimodules that was shown byM. Khovanov in [7] to compute triply-graded link homology. We also set up the theorythat allows for twisting equivariant cohomologies by adjoint-equivariant local systems on U( r ) . This allows us to twist Borel equivariant cohomology by a universal power series ofthe form ℘ ( x ) = P i ≥ b i x i with b i denoting formal variables. Based on computations, wespeculate that the specialization to ℘ ( x ) = x n gives rise to an E -term that is isomorphicto sl ( n ) -link homology. In other words, ℘ ( x ) = P i ≥ b i x i can be interpreted as the (dif-ferential of the) universal potential function with no linear term, in the language of matrixfactorizations [11]. C ONTENTS
1. Introduction 12. Borel equivariant cohomology of Broken symmetries and Soergel bimodules 53. A note on equivariant K-theory 134. Twistings, sl ( n ) -link homology, and cohomology operations 155. Appendix: Calculations in Borel equivariant cohomology 23References 331. I NTRODUCTION
In [4], we constructed a U( r ) -equivariant filtered homotopy type s B ( L ) , called the spec-trum of strict broken symmetries, which was an invariant of links L . The aim of this articleis to apply Borel equivariant singular cohomology H ∗ U( r ) to the above construction, and toinvoke the filtration to set up a spectral sequence that converges to the cohomology of thelimiting value s B ∞ ( L ) of the fitration. We also allow for twistings on Borel equivariantcohomology. The pages of the spectral sequence E q ( L ) for q ≥ recover important linkinvariants. In the untwisted case, we identify the E -term of our spectral sequence withan unreduced, integral form of triply-graded link homology [7]. Indeed, we show that he E -term of our spectral sequence can be identified with the complex of Hochschildhomologies of Bott-Samelson Soergel bimodules associated to braid words, so that thecohomology of the resulting complex computes triply graded link homology as shownby M. Khovanov in [7]. In the twisted case, we offer evidence to suggest that the E -termof the spectral sequence recovers sl ( n ) -link homology for any n (depending on the twist).Let us now recall the definition of the spectrum of strict broken symmetries s B ( L ) asdefined in [4], so as to apply the above construction. In order to make this definitionprecise, consider a braid element w ∈ Br( r ) , whose closure is the link L , and where Br( r ) stands for the braid group on r -strands. For the sake of exposition, in this introductionwe only consider the case of a positive braid that can be expressed in terms of positiveexponents of the elementary braids σ i for i < r . The general case will be described in latersections.Let I = { i , i , . . . , i k } denote an indexing sequence with i j < r , so that a positive braid w admits a presentation in terms of the fundamental generators of the r -stranded braidgroup Br( r ) , w = w I := σ i σ i . . . σ i k . Let T denote the standard maximal torus, and let G i denote the unitary form in the reductive Levi subgroup having roots ± α i . We consider G i as a two-sided T -space under the left(resp. right) multiplication.The equivariant U( r ) -spectrum of broken symmetries is defined as the (suspension) spec-trum corresponding to the U( r ) -space B ( w I ) B ( w I ) := U( r ) × T ( G i × T G i × T · · · × T G i k ) = U( r ) × T B T ( w I ) , with the T -action on B T ( w I ) := ( G i × T G i × T · · · × T G i k ) given by conjugation t [( g , g , · · · , g k − , g k )] := [( tg , g , · · · , g k − , g k t − )] . The U( r ) -stack U( r ) × T ( G i × T G i × T · · · × T G i k ) is equivalent to the stack of principal U( r ) -connections on the trivial U( r ) -bundle over S , endowed with a reduction of thestructure group to T at k distinct points, and so that the holonomy between successivepoints belongs the corresponding group of the form G i in terms of this reduction. Wedefine Definition. (Strict broken symmetries and their normalization), ( [4] definition 2.8)Let L denote a link described by the closure of a positive braid w ∈ Br( r ) with r -strands, and let w I be a presentation of w as w = σ i . . . σ i k . We first define the limiting U( r ) -spectrum s B ∞ ( w I ) of strict broken symmetries as the space that fits into a cofiber sequence of U( r ) -spaces: hocolim J ∈I B ( w J ) −→ B ( w I ) −→ s B ∞ ( w I ) . where I is the category of all proper subsets of I = { i , i , . . . , i k } .The spectrum s B ∞ ( w I ) admits a natural increasing filtration by spaces F t s B ( w I ) defined as thecofiber on restricting the above homotopy colimit to the full subcategories I t ⊆ I generated bysubsets of cardinality at least ( k − t ) , so that the lowest filtration is given by F s B ( w I ) = B ( w I ) .Define the spectrum of strict broken symmetries s B ( w I ) to be the filtered spectrum F t s B ( w I ) above. The normalized spectrum of strict broken symmetries of the link L is defined as s B ( L ) := Σ − k s B ( w I ) . n ([4] theorem 8.5) we proved the general form of the following result. Theorem.
As a function of links L , the filtered U( r ) -spectrum of strict broken symmetries s B ( L ) is well-defined up to quasi-equivalence ( [4] definition 3.4). In particular, the limiting equivariantstable homotopy type s B ∞ ( L ) is a well-defined link invariant in U( r ) -equivariant spectra (see [4] remark 3.5, and remark below). In Section 2 of this paper we will prove, and then generalize the following theorem toarbitrary braids
Theorem.
Given a link L given by the closure of a positive braid word w I of length k in r -strandsas above, one has a spectral sequence with E t,s = M J ∈I t / I t − H s U( r ) ( B ( w J )) ⇒ H s + t − k U( r ) ( s B ∞ ( L )) . The differential d is the canonical simplicial differential induced by the functor in ( [4] definition2.6). In addition, the terms E q ( L ) are invariants of the link L for all q ≥ . Remark.
As we showed in ( [4] theorem 2.12), the groups H ∗ U( r ) ( s B ∞ ( L )) are variants of Leehomology [12] . As such they only record the number of components of the link. However, thepages of the spectral sequence converging to it are non-trivial as we shall see below. Let us dig a little deeper into the E and E -terms of the spectral sequence above. Givenan indexing set I = { i , . . . , i k } , consider the Bott-Samelson variety defined as B t S ( w I ) := U( r ) × T ( G i × T G i × T · · · × T G i k /T ) = U( r ) × T B t S T ( w I ) , with the T -action on B t S T ( w I ) := G i × T G i × T · · · × T G i k /T given by the left action t [( g , g , · · · , g k − , g k )] := [( tg , g , · · · , g k − , g k )] . Notice that B t S ( w I ) supports two equivariant maps to U( r ) /T defined as π ( I ) : B t S ( w I ) −→ U( r ) /T, [( g, g , g , · · · , g k − , g k )] gT,τ ( I ) : B t S ( w I ) −→ U( r ) /T, [( g, g , g , · · · , g k − , g k )] gg . . . g k T. In particular, H ∗ U( r ) ( B t S ( w I )) is a bimodule over the T -equivariant cohomology of a pointH ∗ T , induced via the two maps respectively. As such, we will show in section 2 that thehomologies H ∗ U( r ) ( B t S ( w I )) are integral forms of Soergel bimodules (see [17], [6]). Theorem.
The H ∗ T -bimodule H ∗ U( r ) ( B t S ( w I )) is an integral form of the “Bott-Samelson” Soergelbimodule for the braid word w I . Furthermore, the Hochschild homology of this bimodule withcoefficients in H ∗ T is isomorphic to the cohomology of the space of broken symmetries H ∗ U( r ) ( B ( w I )) as an H ∗ T -module. Consequently, the E -term of the spectral sequence in the previous theorem isisomorphic to an unreduced integral form of triply-graded link homology as shown in [7, 8] , whosevalue on the unknot has the form Z [ x ] ⊗ Λ( y ) , with the degrees of x and y being and resp. Remark.
In ( [4] see remark 6.6), we showed that the filtered spectrum s B ( w I ) ∧ T S obtained bytaking the orbits under the right T -action is invariant under the Braid and inverse relations andrepresents a filtered equivariant homotopy type for the complex of Bott-Samelson Soergel bimodulesstudied by Rouquier in [20, 21] . n order to describe twistings of equivariant cohomology theories, let us start by notic-ing that the spaces of broken symmetries B ( w I ) admit a canonical equivariant map toconjugation action of U( r ) on itself ρ I : B ( w I ) −→ U( r ) , [( g, g , . . . , g k )] g ( g g . . . g k ) g − . Thinking of the co-domain U( r ) as the space of principal U( r ) -connections on the circle,we see that the map ρ I is the map that simply composes the holonomies of all the brokensymmetries along the circle. It follows that ρ I is compatible under inclusions of subsets J ⊆ I . We show in section 4 that this compatibility allows us to use equivariant localsystem on U( r ) to define twists θ of any suitable family E U( r ) of equivariant cohomologytheories. We also study the action of cohomology operations on these twisted theories.In section 4, we also describe a twist of Borel equivariant cohomology, by first extending itto a multiplicative equivariant theory H , so that H U( r ) is obtained from Borel equivariantsingular cohomology H U( r ) by adjoining formal variables b i in cohomological degree − i .Let ℘ ( x ) = P i ≥ b i x i be the formal power series with the coefficients being the variables b i with x having cohomological degree (so that the homogeneous degree of ℘ ( x ) is zero).We may use the above framework to describe a twist θ of H of Borel equivariant singularcohomology denoted by θ H ∗ . As before, this twist θ gives rise to a family of link homol-ogy theories defined as the E q -terms of the corresponding spectral sequence in twistedBorel equivariant singular cohomology for any q ≥ . For any q ≥ , the value of thesetheories on the unknot L is given by E , ∗ q ( L, θ ℘ ) = E , ∗ ( L, θ ℘ ) = Z J x K [ b , b , . . . ] h ℘ ( x ) i , where Z J x K [ b , b , . . . ] denotes graded power series in x with coefficients in the variables b i . In order to compute further examples we require the value of twisted Borel equivariantcohomology on spaces of broken symmetries. In section 4, we will describe the generalstructure of θ H ∗ U( r ) ( B ( w I )) .To give the reader a sense of the twisted Borel equivariant cohomologies of the space ofbroken symmetries, consider the special case of r = 2 (compare with [5] section 5). Let σ ∈ Br(2) be the unique generator, and let I = { , . . . , } ( k -terms), so that w I = σ k . Letus define the power series ℘ ( x, y ) in variables x, y in cohomolgical degree ℘ ( x, y ) = ℘ ( x ) − ℘ ( y ) x − y . Example. (Compare to [5]
Lemma 5.1)The twisted Borel equivariant cohomology groups of B ( σ k ) are trivial in odd degree. In evencohomological degree they are isomorphic to the result below. For k > , we have θ H ∗ U(2) ( B ( σ k )) = R [ δ , . . . , δ k ] h ˆ δ, δ j + ( x − y + 2 P i
For the Hopf-link L (the case k=2), the above E term is given by E , ∗ ( L, θ ) = Z J x, y K h x n , y n , S ( x, y ) i , E , ∗ ( L, θ ) = Z J x K h x n i , where S ( x, y ) denotes the symmetric sum x n − + x n − y + . . . + y n − . Example.
For the (2 , -torus knot L (the case k=3), the above E term is given by E , ∗ ( L, θ ) ∼ = x Z J x K h x n i , E , ∗ ( L, θ ) = Z J x, y K h x n , y n , S ( x, y ) , x − y i , E , ∗ ( L, θ ) = Z J x K h x n i . Based on these computations, and a comparison with sl ( n ) -link homology we conjecture Conjecture.
Under the algebraic specialization ℘ ( x ) = x n , the groups E ∗ , ∗ ( L, θ ) are isomorphicto sl ( n ) -link homology. This specialization has a topological lift i.e. it can be lifted to a map offiltered spectra. Consequently, we have a spectral sequence E ∗ , ∗ q ( L, θ ) , q ≥ . This topological liftalso suggests how one may resolve the conjecture, see remark 4.12.
2. B
OREL EQUIVARIANT COHOMOLOGY OF B ROKEN SYMMETRIES AND S OERGELBIMODULES
Before we start with the main results as stated in the previous section, let us first recall thegeneral formulation of the statements of the definitions and theorems for arbitrary links L . Let T = T r ⊆ U( r ) denote the standard maximal torus, and let Br( r ) denote the braidgroup generated by the standard braids σ i , ≤ i < r . The weights of T will be denoted by P i ≤ r Z h x i i , and will be identified with H T , so that the simple roots α i are expressed in thisbasis by x i − x i +1 . Let h i , ≤ i < r denote the co-roots, so that α j ( h i ) = a ij are the entries inthe Cartan matrix for U( r ) . In this article, we often require a basis of weights constructedout of dual co-roots. This basis { h ∗ i , ≤ i ≤ r } is defined in term of the generators x j as h ∗ i = X j ≤ i x j , in particular, we have h ∗ i ( h j ) = δ i,j for j < r. Identifying H T with H ( T ) via the transgression map, the elements { h ∗ i } (as well as theelements { x i } ) yield a Z -basis of H ( T, Z ) . The action of the Weyl group, whose generatorswe will also denote by σ i (since the group is clear from context), acts on a weight α by σ i α := α − α ( h i ) α i . Notice that by definition h ∗ r is the central character that is invariant under the Weyl group.Let G i ⊆ U( r ) denote the unitary form in the reductive Levi subgroup generated by theroots ± α i . Let ζ i denote the virtual G i representation ( g i − r R ) , where g i is the adjointrepresentation of G i denoted by Ad , and r R is the trivial representation of dimension r .Notice that the restriction of ζ i to T is isomorphic to the root space representation α i (as areal representation). Let S − ζ i denote the sphere spectrum for the virtual G i representation − ζ i defined as in ([4] definition 2.3). onsider a general indexing sequence for arbitrary braid words I := { ǫ i i , · · · , ǫ i k i k } ,where i j < r , and ǫ j = ± . Assume that w = w I := σ ǫ i i · · · σ ǫ ik i k . Recall the U( r ) -spectrumof broken symmetries ([4] definition 2.4), B ( w I ) := U( r ) + ∧ T B T ( w I ) , where B T ( w I ) := H i ∧ T . . . ∧ T H i k , and H i = S − ζ i ∧ G i + , if ǫ i = − , H i = G i + else . The T × T -action on H i is defined by demanding that an element ( t , t ) ∈ T × T actson S − ζ i ∧ G i + by smashing the action Ad ( t ) ∗ on S − ζ i with the standard T × T actionon G i + given by left (resp. right) multiplication. As before the T -action on B T ( w I ) is byconjugation on the first and last factor. It is clear that each bundle ζ i above represents a U( r ) -equivariant vector bundle over U( r ) × T ( G i × T · · · × T G i k ) := B ( w I + ) , where I + isthe indexing set obtained from I by replacing each ǫ j with . By construction B ( w I ) is thecorresponding equivariant Thom spectrum over B ( w I + ) B ( w I ) = B ( w I + ) − ζ I , where ζ I := M i j ∈ I | ǫ ij = − ζ i j . Definition 2.1. (The functor B ( w I ) , [4] defintion 2.6)Given a braid word w I , for I = { ǫ i i , · · · , ǫ i k i k } , let I denote the set of all subsets of I . Let usdefine a poset structure on I generated by demanding that nontrivial indecomposable morphisms J → K have the form where either J is obtained from K by dropping an entry i j ∈ K (i.e. anentry for which ǫ i j = 1 ), or that K is obtained from J by dropping an entry − i j (i.e an entry forwhich ǫ i j = − ). The construction B ( w J ) induces a functor from the category I to U( r ) -spectra.More precisely, given a nontrivial indecomposable morphism J → K obtained by dropping − i j from J , the induced map B ( w J ) → B ( w K ) is obtained by applying the map of ( [4] claim 2.5).Likewise, if J is obtained from K by dropping the factor i j , then the map B ( w J ) → B ( w K ) isdefined as the canonical inclusion. Definition 2.2. (Strict broken symmetries, [4] definitions 2.7, 2.8, 2.10)let I + ⊆ I denote the terminal object of I given by dropping all terms − i j from I (i.e. terms forwhich ǫ i j = − ). Define the poset category I to the subcategory of I given by removing I + . I = { J ∈ I , J = I + } The filtered U( r ) -spectrum F t s B ( w I ) of strict broken symmetries is defined via the cofiber se-quence of equivariant U( r ) -spectra hocolim J ∈I t B ( w J ) −→ B ( w I + ) −→ F t s B ( w I ) , where I t ⊆ I consisting of objects no more than t nontrivial composable morphisms away from I + . We define F s B ( w I ) = B ( w I + ) , and F k = ∗ for k < .The normalized spectrum of strict broken symmetries s B ( L ) is defined as s B ( L ) := Σ l ( w I ) s B ( w I )[ ̺ I ] , where Σ l ( w I ) denotes the suspension by l ( w I ) := l − ( w I ) − l + ( w I ) with l + ( w I ) being the numberof positive and l − ( w I ) being the number of negative exponents in the presentation w I for w interms of the generators σ i . Also, s B ( w I )[ ̺ I ] denotes the filtered spectrum s B ( w I ) with a shift inindexing given by F t s B ( w I )[ ̺ I ] := F t + ̺ I s B ( w I ) , with ̺ I being one-half the difference betweenthe cardinality of the set I , denoted by | I | , and the mimimal word length | w | , of w ∈ Br( r ) ̺ I = 12 ( | I | − | w | ) . ur goal now is to apply Borel equivariant cohomology H ∗ U( r ) to our construction. Definition 2.3. (Borel equivariant singular cohomology)Borel equivariant cohomology of a U( r ) -spectrum X is defined as the singular cohomology of theBorel construction X ∧ U( r ) EU( r ) + , where EU( r ) denotes the free contractible U( r ) -CW complex. Definition 2.4. (Schubert classes for Bott-Samelson varieties)Let I = { i , . . . , i k } , given any j ≤ k , let J j = { i , . . . , i j } . Define π ( j ) and τ ( j ) to be the maps π ( j ) : B t S ( w J j ) −→ B t S ( σ i j ) , [( g, g i , . . . , g i j )] [( gg · · · g i j − , g i j )] .τ ( j ) : B t S ( w J j ) −→ U( r ) /T, [( g, g i , . . . , g i j )] [ gg · · · g i j ] Notice that one has a pullback diagram with the vertical maps being canonical projections B t S ( w J j ) (cid:15) (cid:15) π ( j ) / / B t S ( σ i j ) (cid:15) (cid:15) B t S ( w J j − ) τ ( j − / / U( r ) /T. Since B t S ( w I ) projects canonically onto B t S ( w J j ) , we will use the same notation to denote themaps π ( j ) and τ ( j ) with domain B t S ( w I ) . Define δ j ∈ H r ) ( B t S ( w I )) as the pullback class π ( j ) ∗ ( δ ) , where δ ∈ H r ) ( B t S ( σ i j )) is the (unique) generator of H r ) ( B t S ( σ i j )) as a H ∗ T -modulethat satisfies δ = − α δ . Here i j = s , and α := x s − x s +1 denotes the character of the standardpositive simple root α s of U( r ) that corresponds to the subgroup G i j . In particular, we have δ j + [ α ] j δ j = 0 , where we define [ x ] j := τ ( j − ∗ ( x ) . Notice that B t S ( w I ) supports two equivariant maps to U( r ) /T defined as π ( I ) : B t S ( w I ) −→ U( r ) /T, [( g, g , g , · · · , g k − , g k )] gT,τ ( I ) : B t S ( w I ) −→ U( r ) /T, [( g, g , g , · · · , g k − , g k )] gg . . . g k T. These maps endow H ∗ U( r ) ( B t S ( w I )) with a bimodule structure over H ∗ T . Definition 2.5. (The redundancy set ν ( I ) and its complement ν ( I ) )Let I = { i , i , . . . , i k } denote a positive indexing sequence with each i j < r . Let ν ( I ) be the(unordered) set of integers s < r such that s occurs somewhere in I . Given s ∈ ν ( I ) , let ν ( s ) denote the number of times it occurs in I , and let I s ⊂ I denote the indexing subsequence I s = { i s , i s , . . . , i s ν ( s ) } of all elements i t ∈ I so that i t = s .Let W I ⊆ Σ r to be the subgroup of the Weyl group of U( r ) generated by the reflections σ s for s ∈ ν ( I ) . Let ν ( I ) be defined as the set of orbits of the canonical action of W I acting on { , . . . , r } .It is easy to see that ν ( I ) is in bijection with complement of the set I in { , . . . , r } , and hence thenotation. The maps and generators identified in definition 2.4 allow us to describe the equivariantcohomology of Bott-Samelson varieties and the space of broken symmetries heorem 2.6. Let I = { i , i , . . . , i k } denote a positive indexing sequence with i j < r . Then as aH ∗ T -algebra, we have H ∗ U( r ) ( B t S ( w I )) = H ∗ T [ δ , δ , . . . , δ k ] h δ j + [ α s ] j δ j , if i j ∈ I s . i . Moreover, for any weight α , the character α ∈ H T satisfies the following recursion relations [ α ] = α and [ α ] j = [ α ] j − + α ( h u ) δ j − , where i j − = u, with α ( h u ) denoting the value of the weight α evaluated on the coroot h u . Furthermore, thebehaviour under inclusions J ⊆ I , is given by setting all δ t = 0 for i t ∈ I/J . Theorem 2.7.
Let I = { i , i , . . . , i k } denote a positive indexing sequence with each i j < r .Then, for any s ∈ ν ( I ) , with the property that ν ( s ) = 1 , there exists a class β s ∈ H r ) ( B ( w I ) .Likewise, for any orbit l ∈ ν ( I ) , there exists a canonical class γ l ∈ H r ) ( B ( w I )) so that there ismap of H ∗ U( r ) ( B t S ( w I )) -modules where s and l range over the sets ν ( I ) and ν ( I ) respectively ̟ I : H ∗ U( r ) ( B t S ( w I )) ⊗ Λ( β s ) ⊗ Λ( γ l ) h ˆ δ s i −→ H ∗ U( r ) ( B ( w I )) , where ˆ δ s := X j ∈ I s δ j . If I is redundancy free, i.e. has the property that ν ( s ) = 1 for all s ∈ ν ( I ) , then the map ̟ I is anisomorphism. More generally, given any redundancy free subsequence I ⊆ I with the property ν ( I ) = ν ( I ) , then square free monomials in the generators δ i , i ∈ I/I generate a length filtrationof H ∗ U( r ) ( B ( w I )) so that the associated graded module Gr δ H ∗ U( r ) ( B ( w I )) is isomorphic to Gr δ H ∗ U( r ) ( B ( w I )) ∼ = H ∗ U( r ) ( B ( w I )) ⊗ H ∗ U( r ) ( B t S ( w I )) H ∗ U( r ) ( B t S ( w I )) , where we consider H ∗ U( r ) ( B t S ( w I )) as an H ∗ U( r ) ( B t S ( w I )) -module by identifying δ j , j ∈ I , withtheir namesakes δ j , j ∈ I . Naturality of the above description is seen as follows. Given an inclusion J ⊂ I , so that J = I −{ i j } , the induced map in equivariant cohomology is given by the setting theclass δ j to zero in the case ν ( J ) = ν ( I ) . If on the other hand, we have i j = s = ν ( I ) /ν ( J ) , thenthe restriction map is injective, and has the property that β s maps to the class [ α s ] j γ s , where γ s isdefined as γ l ( s ) , with l ( s ) ∈ ν ( J ) being the W J -orbit containing s . All other generators restrict totheir namesakes modulo the ideal generated by γ s . The proofs of theorems 2.6 and 2.7 are given in the Appendix. Instead, let us explore howthese results relate to Soergel bimodules and triply graded link homology.
Theorem 2.8.
Given a positive indexing sequence I = { i , . . . , i k } , the equivariant cohomologyH ∗ U( r ) ( B t S ( w I )) is isomorphic as bimodules to the “Bott-Samelson” Soergel bimodule [17] givenby a tensor product H ∗ U( r ) ( B t S ( σ i )) ⊗ H ∗ T . . . ⊗ H ∗ T H ∗ U( r ) ( B t S ( σ i k )) . Furthermore, each term H ∗ U( r ) ( B t S ( σ i )) is isomorphic to the bimoduleH ∗ U( r ) ( B t S ( σ i )) = H ∗ T ⊗ H ∗ Gi H ∗ T , where H ∗ G i denotes the G i -equivariant cohomology of a point. Finally, the equivariant cohomol-ogy H ∗ U( r ) ( B ( w I )) is isomorphic as H ∗ T -modules to the Hochschild homology of the bimoduleH ∗ U( r ) ( B t S ( w I )) with coefficients in the bimodule H ∗ T . roof. Let
EU( r ) -denote the free contractible U( r ) -complex that defines the universal prin-cipal G -bundle. Let us begin by noticing that one has a pullback diagram EU( r ) × T ( G i /T ) π (cid:15) (cid:15) τ / / EU( r ) /T π (cid:15) (cid:15) EU( r ) /T τ / / EU( r ) /G i . where τ is induced by the right action of G i on EU( r ) , and the map π by the projection G i /T −→ pt . Now all the spaces in the above diagram have torsion free, evenly gradedcohomology, and so we can invoke the Eilenberg-Moore spectral sequence [22] to calcu-late the cohomology of the pullback. The spectral sequence collapses to the above tensorproduct of the two copies of H ∗ T over H ∗ G i since H ∗ T is a free module over H ∗ G i .Notice also that H ∗ U( r ) ( B t S ( σ i )) is a free H ∗ T module under either the left or the right mod-ule structure as can be verified directly, or by using the fact that the Serre spectral se-quence of the fibration given by the map π collapses (since both the fiber and base haveevenly graded cohomology).Now consider the pullback diagram of definition 2.4. Inductively using freeness over H ∗ T ,and the Eilenberg-Moore spectral sequence we get the iterated tensor product decompo-sition for H ∗ U( r ) ( B t S ( w I )) as claimed in the statement of the theorem.It remains to identify H ∗ U( r ) ( B ( w I )) with the Hochschild homology of H ∗ U( r ) ( B t S ( w I )) withcoefficients in H ∗ T . For this, consider the principal T -fibration given by the canonicalprojection EU( r ) × U( r ) B ( w I ) −→ EU( r ) × U( r ) B t S ( w I ) . It is easy to see that this fibratrion is classified by the map
EU( r ) × U( r ) B t S ( w I ) π ( I ) × τ ( I ) −→ BT × BT −→ BT , where the first map is given by the product of the maps defined in definition 2.4 thatendow H ∗ U( r ) ( B ( w I )) with a bimodule sructure, and the second map represents the mapthat classifies the difference map T × T → T , ( s, t ) st − .We now consider the Serre spectral sequence for the fibration defined above. Let us pickgenerators { γ , . . . , γ r } ⊂ H ( T, Z ) so that γ i is identified with the dual co-root h ∗ i (definedat the beginning of this section) under the identification H ( T ) = H T . The E -term of theSerre spectral sequence computing H ∗ U( r ) ( B ( w I )) is given by Λ( γ , . . . , γ r ) ⊗ H ∗ U( r ) ( B t S ( w I )) , d ( γ k ) = π ( I ) ∗ ( h ∗ k ) − τ ( I ) ∗ ( h ∗ k ) . This complex is the standard Koszul complex that computes the Hochschild homology ofthe bimodule H ∗ U( r ) ( B t S ( w I )) with coefficients in H ∗ T . In order to prove the statement inthe theorem, it is sufficient to show that the spectral sequence collapses at E . In order toestablish this, we will proceed as follows.First we will filter the above complex so as to set up an (algebraic) spectral sequence thatconverges to the Hochschild homology. Then we will see that the E -term of that algebraicspectral sequence (i.e. the complex obtained after the first nontrivial differential) agreeswith the associated graded of the cohomology of H ∗ U( r ) ( B ( w I )) as established in theorem .7. It follows that there can be no further differentials, and that the Hochschild homologycomputed via the above Koszul complex agrees with H ∗ U( r ) ( B ( w I )) up to an associatedgraded quotient. However, theorem 2.7 shows that H ∗ U( r ) ( B ( w I )) is a free H ∗ T -module. Sothere are no H ∗ T module extension problems and we are done.It therefore remains to construct the algebraic spectral sequence mentioned above. Todo so, we pick a subset I ⊆ I as in theorem 2.7 so that I is redundancy free and that ν ( I ) = ν ( I ) . We then filter the above Koszul complex by the monomial-length filtration interms of square free monomials in δ j for j ∈ I/I . Now consider the Koszul differentials.By using the recursion relation in theorem 2.6, we see that the differential is given by d ( γ s ) = ˆ δ s = X j ∈ I s δ j , if s ∈ ν ( I ) , and d ( γ s ) = 0 if s / ∈ ν ( I ) . In the former case, the recursion also tells us that ˆ δ s + α s ˆ δ s = 0 up to lower filtrations (i.e.it is expressible in terms of square free monomials of length at least two). It follows that β s := γ s (ˆ δ s + α s ) is a cycle for s ∈ ν ( I ) at the E stage.It now follows easily that as a H ∗ U( r ) ( B t S ( w I )) -module, the E -term of this algebraic spec-tral sequence is isomorphic to an exterior algebra on the classes γ s for s ∈ ν ( I ) and β s for s ∈ ν ( I ) . The only relation is the relation ˆ δ s = 0 for all s ∈ ν ( I ) . This is precisely thestructure of the associated graded of H ∗ U( r ) ( B ( w I )) we described in theorem 2.7. Thereforethere can be no more differentials, and we are done with the proof of the theorem. (cid:3) Remark 2.9.
In ( [4] see remark 6.6), we pointed out that the filtered equivariant homotopy type s B ( w I ) ∧ T S is independent of the Braid and inversion relations, where we take the T -orbitsunder the right T -action on s B ( w I ) . In particular, it lifts the chain complexes of Bott-SamelsonSoergel bimodules studied by Rouquier in [20, 21] . It remains to establish the existence of the spectral sequence, whose E -term is isomorphicto an integral form of Hochschild homologies of Soergel bimodules. Theorem 2.10.
One has a spectral sequence with E -term given by E t,s = M J ∈I t / I t − H s U( r ) ( B ( w J )) ⇒ H s + t + l ( w I )U( r ) ( s B ∞ ( L )) . The differential d is the canonical simplicial differential induced by the functor in ( [4] definition2.6). In addition, the terms E q ( L ) are invariants of the link L for all q ≥ . Furthermore, the E -term of this spectral sequence is isomorphic to an unreduced, integral form of triply-gradedlink homology as defined in [7, 8] whose value on the unknot is H ∗ U(1) (U(1)) = Z [ x ] ⊗ Λ( y ) .Proof. By ([4] theorem 8.7), in order to prove the above theorem we need to verify two setsof conditions called the B and M2a/b-type conditions given in ([4] definition 8.6). TheM2a/b-type conditions pertains to the behaviour under the second Markov move andcomes down to verifying that the maps given in ([4] theorems 7.1 and 7.2) are both trivialon the level of the associated graded complex. This follows easily from the behavior ofthe cohomology of the equivariant spectra s B ( w I ) under inclusion of subsets J ⊆ I asdescribed in theorem 2.7. The B-type condition is the one flagged in ([4] remark 6.5). Weconsider it below. n [4], we considered a certain quasi-equivalence of filtered U( r ) -spectra (details will begiven momentarily) π m : s BS h ( i,j, ( w I ) −→ π ∗ m s BS h ( i,j,m +1) ( w I ) , where i, j are indices with < m < m i,j where m ij are the exponents in the Artin braidgroup for the Lie group G in question. In the case in hand, G = U( r ) and so the onlynontrivial exponents are m ij = 3 .Let Z m denote the fiber of π m . Then the relevant B-type condition demands that the fiberinclusion map on the associated graded Gr t Z m −→ Gr t s BS h ( i,j,m ) ( w I ) be surjective in Borel equivariant singular cohomology. The only indices that satisfy theabove parameters in the case of U( r ) are consecutive indices j = i + 1 < r and with m = 2 .Now the spectra Gr r s BS h ( i,i +1 , ( w I ) and Gr r s BS h ( i,i +1 , ( w I ) are coproducts of other U( r ) -spectra indexed on the same set, and so the relevant condition comes down to veri-fying a condition on the individual summands.In order to describe the above objects, let us recall the definition of broken Schubert spec-tra ([4] definition 5.6).Given indices i, j < r , let S h i,j,i denote the T × T -space given by the pullback diagram S h i,j,i (cid:15) (cid:15) ρ S h / / U( r ) (cid:15) (cid:15) X i,j,i / / U( r ) /T, where X i,j,i is the image if the following canonical map under group multiplication X i,j,i = Image of G i × T G j × T G i /T −→ U( r ) /T Notice that S h i,j,i is a T × T -invariant subspace of U( r ) , where T × T acts on G via left/rightmultiplication. Given any positive indexing sequence of the form I = { i , . . . , i k , i, j, i } ,we may construct the spectrum of broken Schubert spectra defined as the suspensionspectrum of the space BS h ( i,j, ( w I ) := U( r ) × T ( G i × T · · · × T G i k × T S h i,j,i ) , with the T -action on G i × T · · ·× T G i k × T S h i,j,i being endpoint conjugation as before. Sim-ilarly, we may define the broken Schubert spectra BS h ( i,j, ( w I ) . However, these spectraagree with the spectra of broken symmetries.We now prove a claim that is the heart of the argument that will feed directly into theproof of theorem 2.10 laim 2.11. Let i and j be indices with j = i + 1 < r . Given a positive sequence J = { i , . . . , i k } ,consider positive indexing sequences I ′ = { i , . . . , i k , i } , I ′′ = { i , . . . , i k , i, i } , I = { i , . . . , i k +3 } := { i , . . . , i k , i, j, i } so that J ⊂ I ′ and J ⊂ I ′′ ⊂ I in the obvious fashion. Then one has a diagram of cofiber sequencesof U( r ) -equivariant spectra, which is functorial in JZ I ′′ ι I ′′ / / f (cid:15) (cid:15) B ( w I ′′ ) g (cid:15) (cid:15) µ / / B ( w I ′ ) (cid:15) (cid:15) Z I ι I / / B ( w I ) / / BS h ( i,j, ( w I ) where µ : B ( w I ′′ ) −→ BS h ( w I ′ ) is defined by the multiplication in the last two factors. Further-more, the maps ι I and ι I ′′ are surjective in Borel equivariant singular cohomology.Proof. The existence of the commutative square in the right, and its functoriality in J follows from the definition of the spaces in question. Furthermore, by ([4] lemma 5.7), thespaces in the right square is a pushout, and consequently the map f is an equivalence. Itis clear that the map µ is split and so the map ι I ′′ is surjective in any cohomology theory.Let us pick the splitting h induced by the inclusion I ′ ⊂ I ′′ by including the index i as i k +3 in terms of the indexing sequence h : B ( w I ′ ) −→ B ( w I ′′ ) . From the choice of h , it follows that the image of the map ι I ′′ is isomorphic to the idealgenerated by the Schubert class δ k +1 in terms of theorem 2.7. We will now proceed toshow that the map g is surjective onto the cohomology of Z I ′′ , which we will identify asthe ideal generated by δ k +1 . We consider two cases. The first we consider is when theindex j belongs to the indexing set J . In that case, theorem 2.7 tells us that the map g issurjective and so obviously surjective onto the kernel of the map h .The only other case to consider is when j does not appear in the indexing sequence J . Inthis case, theorem 2.7 tells us that the image of g contains the submodule generated bythe class [ α j ] k +2 ∈ H ∗ U( r ) ( B t S ( w I ′′ )) . But the recursion relation in 2.6 gives us the relation [ α j ] k +2 = [ α j ] k +1 − δ i k +1 . Since [ α j ] k +1 belongs to the image of µ , we observe that the image of g in cohomologysurjects onto the ideal generated by δ i k +1 modulo the image of µ , which is what we wantedto prove. (cid:3) Let us now complete the proof of 2.10. As mentioned earlier, the proof follows once wehave verified the B-type condition of ([4] definition 8.6) which comes down to showingthat the map in ([4] claim 6.5) is injective. This is equivalent to claim 2.11 above withonly two cosmetic differences. The first involves working with Thom spectra instead.The Thom isomorphism theorem comes to our rescue here. The only other differenceinvolves working with the subsequence { i, j, i } anywhere in the sequence I and not justat the terminating three spots. This is again not an issue since we can invoke the firstMarkov property to move the subsequence to the end. (cid:3) . A NOTE ON EQUIVARIANT K- THEORY
Essentially all the results from the previous sections continue to hold with Borel equivari-ant singular cohomology replaced by equivariant K-theory. In particular, K ∗ U( r ) ( s B ∞ ( L )) is a link invariant that can be computed by means of a spectral sequence as in theorem2.10. Even though we do not verify it, it is likely that the E -term of this spectral se-quence is a variant of Hochschild homology of the K-theory based Soergel bimodulesK ∗ U( r ) ( B t S ( w I )) . Theorem.
One has a spectral sequence with E -term E t,s = M J ∈I t / I t − K s U( r ) ( B ( w J )) ⇒ K s + t + l ( w I )U( r ) ( s B ∞ ( L )) . The differential d is the canonical simplicial differential induced by the functor in ( [4] definition2.6). In addition, the terms E q ( L ) are invariants of the link L for all q ≥ . For the benefit of the reader, we explicitly describe the equivariant K-theory of the rel-evant objects below. In [5], we will dig deeper into equivariant K-theory by studyinga
Loop equivariant
K-theory, and show that the corresponding spectral sequence and theunderlying link invariant are closely related to sl ( n ) -link homology. In what follows, weset the representation ring of T as K T ( ∗ ) = Z [ x ± , . . . , x ± r ] in terms of the standard genera-tors, where we have used the abusive notation of expressing the multiplicative charactercorresponding to the linear character x i , also by the notation x i (as opposed to e x i ) .Before we begin with the description of the equivariant K-theory of the spaces B t S ( w I ) and B ( w I ) , let us point out an important piece of structure the spectral sequence based onequivariant K-theory admits. More precisely, we have the action of the Adams operations Ψ l on the spectral sequence. These operations are unstable operations , in that one may onlydefine them on the equivariant K-theory of spaces, and not spectra. As such, they act onthe spectral sequence for links L with a positive presentation. In order to extend themto spectra, one is required to invert the integer l . The action of Ψ l can be described in astraightforward way on the representation ring K T . More precisely, the operation Ψ l ismultiplicative and additive, and has the property Ψ l ( x i ) = x li for x i ∈ K T = Z [ x ± , . . . , x ± r ] . Example.
Let us consider the example of the Hopf-link L described by the presentation σ , with σ ∈ Br(2) being the generator. By ( [4] theorem 2.12), s B ∞ ( L ) is equivalent to the Thom spectrum Σ − (U(2) + ∧ T S α ⊕ α ) ∧ T + , where S α ⊕ α denotes the one point compactification of the root-space representation added to itsconjugate. Then the action of Ψ l on the Thom class Th in K ( s B ∞ ( L )) is given by Ψ l ( Th ) = S l ( α ) ( xy ) l − Th , where S l ( α ) = x l − + x l − y + . . . + xy l − + y l − , where we use x and y to denote the two standard generators x and x . The next definition and theorem describe the structure of the equivariant K-theory of B t S ( w I ) , and the following theorem describes that of B ( w I ) efinition. (Schubert classes in equivariant K-theory for Bott-Samelson varieties)Recall the pullback diagram 2.4. Define ∂ j ∈ K r ) ( B t S ( w I )) as the pullback class π ( j ) ∗ ( ∂ ) , where ∂ ∈ K r ) ( B t S ( σ i j )) is the (unique) generator of K r ) ( B t S ( σ i j )) as a K T -module that satisfies ∂ = ( e α s − ∂ . Here i j = s , and e α s := x s x − s +1 denotes the character of the standard positivesimple root α s of U( r ) that corresponds to the subgroup G i j . In particular, we have the relation ∂ j + (1 − [ e α s ] j ) ∂ j = 0 , where we define [ x ] j := τ ( j − ∗ ( x ) . We may now describe by the U( r ) -equivariant K-theory of B t S ( w I ) using these generators Theorem.
Let I = { i , i , . . . , i k } denote a positive indexing sequence with i j < r . Then as aK T -algebra, we haveK r ) ( B t S ( w I )) = K T [ ∂ , ∂ , . . . , ∂ k ] h ∂ j + (1 − [ e α s ] j ) ∂ j , if i j ∈ I s . i , K r ) ( B t S ( w I )) = 0 . Moreover, for any weight α , the character e α ∈ K T satisfies the following recursion relations [ e α ] = e α and [ e α ] j = [ e α ] j − + [ e α − e α − α ( h u ) α u − e α u ] j − ∂ j − , where i j − = u, with α ( h u ) denoting the value of the weight α evaluated on the coroot h u . Furthermore, thebehaviour under inclusions J ⊆ I , is given by using the above relations recursively, and settingall ∂ t = 0 for i t ∈ I/J . Theorem.
Let I = { i , i , . . . , i k } denote a positive indexing sequence with each i j < r . Then, forany s ∈ ν ( I ) , with the property that ν ( s ) = 1 , there exists a class β s ∈ K r ) ( B ( w I ) . Likewise,for any orbit l ∈ ν ( I ) , there exists a canonical class γ l ∈ K r ) ( B ( w I )) so that there is map ofK ∗ U( r ) ( B t S ( w I )) -modules where s and l range over the sets ν ( I ) and ν ( I ) respectively ̟ I : K ∗ U( r ) ( B t S ( w I )) ⊗ Λ( β s ) ⊗ Λ( γ l ) h ˆ ∂ s i −→ K ∗ U( r ) ( B ( w I )) , ˆ ∂ s := X j ∈ I s [ e h ∗ s − α s ] j ∂ j . If I is redundancy free, i.e. has the property that ν ( s ) = 1 for all s ∈ ν ( I ) , then the map ̟ I is anisomorphism. More generally, given any redundancy free subsequence I ⊆ I with the property ν ( I ) = ν ( I ) , then square free monomials in the generators ∂ i , i ∈ I/I generate a length filtrationof K ∗ U( r ) ( B ( w I )) so that the associated graded module Gr ∂ K ∗ U( r ) ( B ( w I )) is isomorphic to Gr ∂ K ∗ U( r ) ( B ( w I )) ∼ = K ∗ U( r ) ( B ( w I )) ⊗ K ∗ U( r ) ( B t S ( w I )) K ∗ U( r ) ( B t S ( w I )) , where we consider K ∗ U( r ) ( B t S ( w I )) as an K ∗ U( r ) ( B t S ( w I )) -module by identifying ∂ j , j ∈ I , withtheir namesakes ∂ j , j ∈ I . Naturality under inclusions J ⊂ I where J = I − { i j } is given by thesetting the class ∂ j to zero in the case ν ( J ) = ν ( I ) . If on the other hand, i j = s = ν ( I ) /ν ( J ) ,then the restriction is injective sending β s to the class ([1 − e α s ] j ) γ s . All other generators restrictto their namesakes modulo the ideal generated by γ s . . T WISTINGS , sl ( n ) - LINK HOMOLOGY , AND COHOMOLOGY OPERATIONS
In this section we would like to describe the framework that allows us to twist equivariantcohomology theories by local systems on U( r ) . Our prime example is Borel equivariantsinguar cohomology. In order to make this statement precise, we will first recall LU( r ) -equivariant lifts of our spaces of broken symmetries. Let us fix natural isomorphism U(1) ⋉ SU( r ) = U( r ) . In what follows, we take a “small model” for the loop group LU( r ) given by extending smooth loops on SU( r ) by the Lie group U(1) × Z . LU( r ) := (U(1) × Z ) ⋉ LSU( r ) , where we have identified U(1) × Z as a model of LU(1) . In ([5] section 3), we constructed asmall model A r of principal U( r ) -connections on the trivial U( r ) -bundle on S . We recall Definition 4.1. (The universal proper
LU( r ) -CW complex A r , [5] definition 3.9)Taking the small model for LU( r ) , there exists a proper LU( r ) -subspace A r of principal U( r ) -connections on S . The subgroup ΩU( r ) := ΩSU( r ) ⋊Z acts freely on A r so that there is principal ΩU( r ) -fibration defined as the “holonomy’ mapHol : A r −→ U( r ) . Furthermore, the induced space A r + ∧ ˜LU( r ) U( r ) + , along the evaluation map E : LU( r ) −→ U( r ) is equivalent to the conjugation action of U( r ) on itself. Now let I = { i , i , . . . , i k } denote an indexing sequence with i j < r . Let T denote thestandard maximal torus, and let G i denote the unitary (block-diagonal) form in the re-ductive Levi subgroup having roots ± α i . We consider G i as a two-sided T -space underthe canonical left(resp. right) multiplication. For the (positive) braid word w I , recall thespaces B ( w I ) of broken symmetries defined as B ( w I ) := U( r ) × T ( G i × T G i × T · · · × T G i k ) = U( r ) × T B T ( w I ) , with the T -action on B T ( w I ) := ( G i × T G i × T · · · × T G i k ) given by conjugation t [( g , g , · · · , g k − , g k )] := [( tg , g , · · · , g k − , g k t − )] . Definition 4.2. (Lifts of the spectra B ( w I ) to proper LU( r ) -CW specra, [5] definition 3.11)Given a positive braid word w I , let ρ I denote the canonical U( r ) -equivariant map on B ( w I ) in-duced by group multiplication on the factors and with values in the space U( r ) acting on itself byconjugation ρ I : B ( w I ) = U( r ) × T ( G i × T G i × T · · ·× T G i k ) −→ U( r ) , [( g, g i , . . . , g i k )] gg i . . . g i k g − . We define the space B ( w I ) E to be the pullback of the universal proper LU( r ) -CW complex A r along ρ I , and denote Hol I : B ( w I ) E −→ B ( w I ) to be the induced holonomy map. Note that wemay identify Hol I with B ( w I ) E + ∧ LU( r ) U( r ) + = B ( w I ) . For the diagonal inclusion ρ : U( r ) × T T −→ U( r ) , we obtain (U( r ) × T T ) E = LU( r ) × LT R r . iven an arbitrary indexing sequence I = { ǫ i i , · · · , ǫ i k i k } , we define the LU( r ) -equivariantspectrum of broken symmetries B ( w I ) E as the Thom spectrum of the pullback of − ζ I (section 2) B ( w I ) E := ( B ( w I + ) E ) − ζ I , where I + is the indexing sequence obtained from I by replacing each ǫ j with . The pullback isperformed along Hol I + : B ( w I + ) E −→ B ( w I + ) The above
LU( r ) -equivariant spectra give rise to a filtered object as before: Definition 4.3. ( LU( r ) -equivariant strict broken symmetries, [5] definitioin 4.4)let I + ⊆ I denote the terminal object of I given by dropping all terms − i j from I (i.e terms forwhich ǫ i j = − ). Define the poset category I to the subcategory of I given by removing I + . I = { J ∈ I , J = I + } We first define the equivariant
LU( r ) -spectrum s B ∞ ( w I ) E via the cofibration of LU( r ) -spectra hocolim J ∈I B ( w J ) E −→ B ( w I + ) E −→ s B ∞ ( w I ) E . We endow s B ∞ ( w I ) E with a natural filtration as ˜LU( r ) -spectra giving rise to the filtered spectrumof strict broken symmetries s B ( w I ) E as follows. The lowest filtration is defined as F s B ( w I ) E = B ( w I + ) E , and F k = ∗ , for k < . Higher filtrations F t for t > are defined as the cone on the restriction of π to the subcategory I t ⊆ I consisting of objects no more than t nontrivial composable morphisms away from I + . Inother words F t s B ( w I ) E is defined via the cofiber sequence hocolim J ∈I t B ( w J ) E −→ B ( w I + ) E −→ F t s B ( w I ) E . Now assume that E U( r ) is a family of U( r ) -equivariant associative (i.e. A ∞ ), and homotopycommutative cohomology theories. Let us further assume that one has a natural familyof maps ∆ r,s of associative cohomology theories for all pairs ( r, s ) so that r + s = t ∆ r,s : E U( r ) ∧ E U( s ) −→ ι ∗ r,s E U( r + s ) , where ι r,s : U( r ) × U( s ) −→ U( t ) that induce an equivalence under the restriction map ι r : U( r ) −→ U( r + 1)∆ r : E U( r ) ∼ = −→ ι ∗ r E U( r +) , where , ∆ r : E U( r ) id ∧ −→ E U( r ) ∧ E U(1) ∆ r, −→ ι ∗ r, E U( r +1) . Let us also assume that given any H ≤ U( r ) , a homotopy class x ∈ π H (E U( r ) ) := π (E U( r ) H ) is a unit if and only if the underlying class x ∈ π (E ♯ ) is a unit, where E ♯ denotes thespectrum E U( r ) with the U( r ) -action forgotten.Consider the subspace GL(E U( r ) ) ⊂ Ω ∞ E U( r ) consisting of components whose underlyingelement in the ring π (E ♯ ) is a unit. GL(E U( r ) ) is a group-like monoid with an U( r ) -action,so that its delooping BGL(E U( r ) ) admits an induced U( r ) -space structure. The technicalcondition on the homotopy classes imposed above insures that the group following com-pletion map GL(E U( r ) ) −→ Ω BGL(E U( r ) ) is an U( r ) -equivariant equivalence. efinition 4.4. (Twistings for the family of cohomology theories E )Given a family of sufficiently multiplicative cohomology theories E as above, a twisting θ for E isa family of equivariant homotopy classes θ r ∈ π Map U( r ) (U( r ) , BGL(E U( r ) )) , where U( r ) is seen as an U( r ) -space under conjugation. Furthermore, we demand that the follow-ing diagram commutes up to U( r ) × U( s ) -equivariant homotopy for all pairs r + s = t U( r ) × U( s ) ι r,s / / θ r × θ s (cid:15) (cid:15) U( t ) θ t (cid:15) (cid:15) BGL(E U( r ) ) × BGL(E U( s ) ) ∆ r,s / / BGL(E U( t ) ) . So, Ω θ r ∈ Map U( r ) (ΩU( r ) , GL(E U( r ) )) endows E U( r ) with an U( r ) -equivariant action of ΩU( r ) . Definition 4.5. (Twisted cohomology of the spectra of broken symmetries)Given a twisting θ for the family E , we define the twisted cohomology of the spectra B ( w I ) by θ E k U( r ) ( B ( w I )) := π Map U( r )E ( B ( w I ) E ∧ ΩU( r ) E U( r ) , Σ k E U( r ) ) , where B ( w I ) E ∧ ΩU( r ) E U( r ) denotes the coequalizer of the ΩU( r ) -action on B ( w I ) E , and the ΩU( r ) -action on E U( r ) along Ω θ r . The space Map U( r )E ( B ( w I ) E ∧ ΩU( r ) E U( r ) , Σ k E U( r ) ) denotes the spaceof U( r ) -equivariant E U( r ) -module maps from B ( w I ) E ∧ ΩU( r ) E U( r ) to Σ k E U( r ) . As a straightforward consequence of definition 4.3 we obtain
Theorem 4.6.
Assume { θ E s U( r ) , r ≥ } is an INS-type θ -twisted equivariant cohomology theory(see [4] definition 8.6 for INS-type theories). Then given a link L described as a closure of a braidword w I on r -strands, one has a spectral sequence converging to the θ -twisted E -cohomology θ E ∗ U( r ) ( s B ∞ ( L )) and with E -term E t,s = M J ∈I t / I t − θ E s U( r ) ( B ( w J )) ⇒ θ E s + t + l ( w I )U( r ) ( s B ∞ ( L )) . The differential d is the canonical simplicial differential induced by the functor in ( [4] definition2.6). Furthermore, the terms E q ( L, θ ) are invariants of the link L for all q ≥ . Let us get to the main example of twisted spectra that interests us. Let H denote the(non-equivariant) Eilenberg-MacLane spectrum, and let BU denote the infinite complexGrassmannian classifying vector bundles of virtual dimension one, seen as an infiniteloop space under the direct sum of virtual vector bundles of dimension zero . The infinite loopstructure of BU induces the structure of a commutative ring spectrum on H ∧ BU + .The homotopy groups of H ∧ BU + are (by definition), the homology groups of BU , withan algebra structure induced by the h -space structure on BU . This algebra is a graded olynomial algebra on a collection of generators with one generator in each even degree.One may choose the Z -module spanned by these generators to be the image of ι ∗ : H ∗ (BU(1) , Z ) −→ H ∗ (BU , Z ) , where ι : BU(1) −→ BU is the canonical inclusion. In particular, we have π ∗ ( H ∧ BU + ) = H ∗ (BU) = Z [ b , b , . . . ] , | b i | = 2 i, where b i is the image of the fundamental generator of H i (BU(1)) under ι ∗ . We now definean equivariant theory H for which we subsequently construct a natural twist. Definition 4.7. (Borel-equivariant singular cohomology H with parameters)Define an equivariant cohomology theory H U( r ) by first constructing the naive U( r ) -equivarianttheory given by the Borel completion Map(EU( r ) , H ∧ BU + ) , and then inducing up from thetrivial to the complete universe. Definition 4.8. (A twist for H )Notice that BU acts on itself via direct sum of virtual bundles. This action extends to the equivari-ant loop map ℘ : Map(EU( r ) , BU) −→ GL( H U( r ) ) , B℘ : Map(EU( r ) , SU) −→ BGL( H U( r ) ) . Now let U( r ) be seen as a space with an U( r ) -action by conjugation. Consider the composite map λ r : EU( r ) × U( r ) U( r ) −→ EU × U U = LBU π U −→ U π SU −→ SU , where the map π U : LBU −→ U is the projection under the h-space splitting LBU = U × BU .Similarly, π SU : U −→ SU is the projection under the h-space splitting U = SU × U(1) . Let usdenote by ˇ λ r : U( r ) −→ Map(EU( r ) , SU)) the equivariant map given by the adjoint of the map λ r given above. Our twisting class θ := { θ r } is defined as the composite θ r = B℘ ◦ ˇ λ r : U( r ) −→ Map(EU( r ) , SU)) −→ BGL( H U( r ) ) . Since the maps π U and π SU are maps of h-spaces, and since this h-space structure is compatiblewith block-diagonal sums i r,s : U( r ) × U( s ) −→ U( t ) , it follows that the family θ := { θ r } servesas a twisting for H as required by definition 4.8. Claim 4.9.
Consider the map induced by θ on the level of fundamental groups π ( θ ) : Z = π U(1)) −→ π (BGL( H U(1) ) U(1) ) = π (GL( H U(1) ) U(1) ) ⊂ [BU(1) , H ∧ BU + ] . Then the image of ∈ Z under π ( θ ) is the canonical map ι : BU(1) −→ BU −→ H ∧ BU + . Al-ternatively, interpreting [BU(1) , H ∧ BU + ] as ( H ∧ BU + ) (BU(1)) = ( H ∧ BU + ) J x K , the imageof ∈ Z is given by the invertible power series ℘ ( x ) , where ℘ ( x ) := X i ≥ b i x i , b = 1 , so that ℘ ( x ) = X i ≥ b i x i . roof. Consider the map λ r for r = 1 λ : BU(1) × U(1) −→ EU × U U π −→ SU , where π = π SU π U in terms of the projections defined in definition 4.8. One may describe π explicitly as follows. First, we identify EU × U U as the direct limit of the spaces EU( r ) × U( r ) U( r ) . Consider an equivalence class [( F, A )] ∈ EU( r ) × U( r ) U( r ) , where F an r -frame in C ∞ , and A is a element in U( r ) . Given the pair ( F, A ) , one may construct a unitary matrixin U which is defined as the matrix A in the frame F , and as the identity matrix in thecomplement of the subspace spanned by F . Taking the limit as r grows, we obtain a map ˜ π : EU × U U −→ U , which is easily seen to be a map of h -spaces, with the h -space structure being induced bythe block-diagonal sum of matrices. The homomorphism π defined above is obtained byprojection to SU . The map λ given by restricting π to BU(1) × U(1) is easily seen to factorthrough a map (which we also denote by the same name) λ : Σ BU(1) −→ SU . The image of ∈ Z under π ( θ ) is precisely the map induced by the adjoint of λ . On theother hand, the geometric description of λ given above is well known (in the context ofBott periodicity) to be the map whose adjoint is ι (see [18] for instance). The proof of theclaim follows. (cid:3) An easy computation that one can make at this point is the twisted cohomology of thespectra of broken symmetries in the abelian case of
U(1) . Since B (1) = U(1) , we see that B (1) E = R , and that the twisted cohomology is a quotient of a graded power series whichis only non-trivial in odd degrees θ H ∗ +1U(1) (U(1)) = Z J x K [ b , b , . . . ] h ℘ ( x ) i , θ H ∗ U(1) (U(1)) = 0 . The extension to
U(1) × r is straightforward θ H ∗ + r U(1) × r (U(1) × r ) = Z J x , . . . , x r K [ b , b , . . . ] h ℘ ( x ) , . . . , ℘ ( x r ) i , θ H ∗ + r − × r (U(1) × r ) = 0 . The computation of the above result is identical to the proof of the corresponding state-ment for Dominant K-theory ([5] theorem 3.3 and corollary 3.4). Indeed, the calculation tocompute the twisted H -cohomology of any spectrum of broken symmetries is formallyidentical to that of the structure described for Dominant K-theory in [5].We first define the polynomials ℘ ( α ) analogous to the polynomials S ( α ) described in [5].Given a root α = x − y in U( r ) , define the formal expression ℘ ( α ) := ℘ ( x ) − ℘ ( y ) x − y . Recalling the terminology of theorems 2.6 and 2.7, we have heorem 4.10. Let I = { i , i , . . . , i k } denote a positive indexing sequence with each i j < r .Then the θ -twisted Borel equivariant cohomology of B ( w I ) is concentrated in degree r + 2 | ν ( I ) | (mod 2). If the sequence I = { i , i , . . . , i k } is redundancy free, i.e. has the property that ν ( s ) = 1 for all s ∈ ν ( I ) , then there is an isomorphism of ( H ∧ BU + ) ∗ U( r ) ( B t S ( w I )) -modules ̟ I : ( H ∧ BU + ) ∗ U( r ) ( B t S ( w I )) h ℘ ( α s ) , δ s , if s ∈ ν ( I ) , ℘ ( x l ) , if l ∈ ν ( I ) i ∼ = −→ θ H ∗ + r +2 | ν ( I ) | U( r ) ( B ( w I )) . More generally, given any positive indexing sequence I , multiplication with ˆ δ s := P j ∈ I s δ j istrivial in θ H ∗ U( r ) ( B ( w I )) for all s ∈ ν ( I ) , and given any redundancy free subsequence I ⊆ I with ν ( I ) = ν ( I ) , the square-free monomials in the generators δ i , i ∈ I/I generate a lengthfiltration of θ H ∗ U( r ) ( B ( w I )) so that the associated graded Gr δ θ H ∗ U( r ) ( B ( w I )) is isomorphic to Gr δ θ H ∗ U( r ) ( B ( w I )) ∼ = θ H ∗ U( r ) ( B ( w I )) ⊗ H ∗ U( r ) ( B t S ( w I )) H ∗ U( r ) ( B t S ( w I )) . In the above expression we consider H ∗ U( r ) ( B t S ( w I )) as an H ∗ U( r ) ( B t S ( w I )) -module by identifying δ j , j ∈ I with their namesakes δ j , j ∈ I . Given the inclusion J ⊂ I , so that J = I − { i t } , theinduced map in twisted cohomology is either given by the setting the class δ t to zero in the case ν ( J ) = ν ( I ) , or the injective map given by multiplication with [ α s ] t , if i t = s = ν ( I ) /ν ( J ) .Proof. The statement of the above theorem is formally the same as the that of Theorem5.4 (or theorem 3.17 in [5]). The proof will proceed in the same way, where we split theproof into two cases, with Case 1 addressing the first part of the theorem that deals withthe case when I is redundancy free. Case 2 addresses the second part of the theorem,and describes an inductive argument starting with any redundancy free susbset I ⊆ I with ν ( I ) = ν ( I ) , and sequentially adding indices. The proof of Case 2 is identical to thatgiven in the Appendix (Case 2 in 5.4, or Theorem 3.17 in [5]); so we only focus on Case 1.Given a redundancy free positive indexing sequence I , we invoke a spectral sequence ofmodules over ( H ∧ BU + ) ∗ U( r ) ( B ( w I )) converging to the twisted cohomology θ H ∗ U( r ) ( B ( w I )) after one single differential. This spectral sequence starts with the untwisted equivariantcohomology theory ( H ∧ BU + ) ∗ U( r ) ( B ( w I )) , which we completely understand by virtue oftheorem 2.7. To construct this spectral sequence, we filter H ∗ (BU + ) by powers of the idealgenerated by the elements b , b , . . . . This filtration is induced by a stably-split topologicalfiltration of BU + (see remark 4.12). This filtration induces a filtration of θ H ∗ U( r ) ( B ( w I )) whose associated quotient is ( H ∧ BU + ) ∗ U( r ) ( B ( w I )) , since the twisting is trivial under theassociated quotient. The first differential is determined by what it does on the class “1” d (1) = X s ∈ ν ( I ) , l ∈ ν ( I ) A s β s + B l γ l , for some coefficients A s and B l . Since ( H ∧ BU + ) ∗ U( r ) ( B ( w I )) injects into ( H ∧ BU + ) ∗ T r ( T r ) ,we know the differential using claim 4.9. It follows from our choice of generators β, γ (seeremark 5.5), that B l = ℘ ( x l ) , A s = ℘ ( α s ) . We may now compute the cohomology underthe first differential, and verify that it is a quotient of the ideal in ( H ∧ BU + ) ∗ U( r ) ( B ( w I )) generated by product of all the classes of the form β s and γ l . This ideal is easily seen to bea cyclic submodule for ( H ∧ BU + ) ∗ U( r ) ( B t S ( w I )) with a shift in degree given by r + 2 | ν ( I ) | ,and agrees with what we expect. By parity reasons, the spectral sequence collapses. (cid:3) he following theorem follows from theorem 4.6 once we verify that θ H ∗ U( r ) is an INS-type theory. This involves verifying the two sets of conditions called the B and M2a/b-type conditions given in ([4] definition 8.6). The B-type condition flagged in ([4] remark6.5) is verified as in the non-twisted case 2.11. The M2a/b-type conditions pertains tothe behaviour under the second Markov move and comes down to verifying that themap given in ([4] theorem 7.1) is trivial and that the map ([4] theorem 7.2) is injectiveon the level of the associated graded complex. This follows easily from the behavior ofthe cohomology of the equivariant spectra s B ( w I ) under inclusion of subsets J ⊆ I asdescribed in theorem 4.10 above. Theorem 4.11.
Given a link L described as a closure of a braid word w I on r -strands, one has aspectral sequence converging to the θ -twisted H -cohomology θ H ∗ U( r ) ( s B ∞ ( L )) and with E t,s = M J ∈I t / I t − θ H s U( r ) ( B ( w J )) ⇒ θ H s + t + l ( w I )U( r ) ( s B ∞ ( L )) . The differential d is the canonical simplicial differential induced by the functor in ( [4] definition2.6). Furthermore, the terms E q ( L, θ ) are invariants of the link L for all q ≥ . Remark 4.12.
The spectrum Σ ∞ BU + has a A ∞ -multiplicative stable splitting, with associatedgraded object given by W k MU( k ) . This splitting induces a E -monoidal splitting of H ∧ BU + realizing the decomposition of Z [ b , b , . . . ] into homogeneous summands in the generators b i [3] .We define the H ∧ BU + -modules of associated quotients Q m ( H ∧ BU + ) = _ k ≤ m H ∧ MU( k ) , Q ( H ∧ BU + ) = H . Notice that the quotient Q ( H ∧ BU + ) is the augmented “summand of indecomposables” given by W i ≥ Σ i H h b i i . One is therefore allowed to specialize the filtered object θ H ∗ U( r ) ( s B ∞ ( L )) , by firstsmashing it (as modules over H ∧ BU + ) with Q ( H ∧ BU + ) , and then pinching out all summandsother than H h b i and H h b n i . Our computations suggest that E -term of the spectral sequence forthis (specialized) filtered object is essentially an augmented version of sl ( n ) -link homology. It islikely that one may be able to compare the spectral sequence used in the proof of theorem 4.10 aboveand the one described in [19] converging to sl ( n ) -link homology, to prove our conjecture. Let us now inquire into the framework that describes the structure of cohomology opera-tions on the twisted theories. In principle, cohomology operations may “act” on the actualtwist θ , and so one must work with universal theories where this action is incorporatedinto the coefficients. Let us describe this by means of two important examples. Example 4.13. (The positive Witt algebra)Recall that the equivariant theory H was defined as a Borel equivariant completion ofH ∧ BU + . By Thom isomorphism, it is easy to see that H ∧ BU + ∼ = H ∧ MU , where MU denotes the (non-equivariant) complex cobordism spectrum. The theory H ∧ MU is uni-versal in the sense that its homotopy Z [ b , b , . . . ] represents (as functors from rings to sets)the data given by a formal group law F , endowed with an isomorphism e F ( x ) from the dditive group law to F . In other words, the generating function e F ( x ) is the “universalexponential” e F ( x ) := X i ≥ b i x i +1 , b = 1 , note that e F ( x ) can be identified with x + x ℘ ( x ) . Consider the map η : H ∧ MU −→ H ∧ MU ∧ MU that includes MU into the right factor.As before, the theory H ∧ MU ∧ MU also has a universal description in that its homotopy Z [ b i , s j , i, j ≥ represents a pair of formal group laws ( F, G ) , and a pair of isomorphisms e F ( x ) := X i ≥ b i x i +1 , f ( x ) := X i ≥ s i x i +1 , b = s = 1 with e F ( x ) being the universal exponential for F , and f ( x ) being an isomorphism from F to G . As such, the map η represents the composite isomorphism f ( e F ( x )) from theadditive group law to G , i.e. the exponential for G .Invoking Thom isomorphism, the map η represents a map of multiplicative cohomologytheories η : H ∧ BU + −→ H ∧ BU + ∧ BU + . Performing the Borel equivariant completion along η gives rise to a map of equivarianttheories endowed with twisting classes θ and ˜ θ respectively η : H −→ ˜ H , η ( θ ) := ˜ θ. Using the description in terms of formal group laws gives us an explicit description of thepower series ˜ ℘ ( x ) for the equivariant theory ˜ H . By the definition of η , we have x + x ˜ ℘ ( x ) = η ( x + x ℘ ( x )) = X i ≥ η ( b i ) x i +1 = X i ≥ s i ( X j ≥ b j x j ) i +1 , b = s = 1 . We claim that the map of twisted theories η incorporates the data that amounts to (amongother things) a representation of the positive Witt algebra. To see this, let us consider thefinite-type Z -dual W of Z [ s , s , . . . ] . In other words W is a free Z -module on the duals ofthe monomials in s i . Then it follows that the map η turns W into an algebra of operatorson H ∗ U(1) (the so-called Landweber-Novikov algebra [16]). To get a sense of this algebra,let L m denote the primitive element in W given by L m ( s m ) = 1 , and defined to evaluatetrivially on all other monomials in s j . By the above formula for η , we get a formula forthe action of the operator L m on the class y := x + x ℘ ( x ) L m ( y ) := L m ( η ( x + x ℘ ( x ))) = ( X i ≥ b i x i +1 ) m +1 = y m +1 ∂∂y ( y ) . This suggests that the sub algebra of W generated by the elements L m is isomorphic tothe positive Witt algebra. This is indeed the case. We expect that this action is related tothe results of [9]. Example 4.14. (The even Steenrod Algebra)In the previous example, we studied the Borel equivariant completion H of H ∧ MU . Letus now define the equivariant theory H ( F p ) given by the Borel equivariant completion f H ( F p ) ∧ MU , where p is a prime, and H ( F p ) denotes the mod p Eilenberg-MacLanespectrum. Since we have a map of multiplicative theories ζ : H ∧ MU −→ H ( F p ) ∧ MU , induced by the canonical maps on the respective factors, the map ζ induces a map ofequivariant cohomology theories, each endowed with a twist ζ : H −→ H ( F p ) ζ ( θ ) := θ p Notice that π ∗ ( H ( F p ) ∧ MU) = F p [ b , b . . . ] induced by ζ . As before, we consider the mapgiven by including H ( F p ) as the left factor η A : H ( F p ) ∧ MU −→ H ( F p ) ∧ H ( F p ) ∧ MU inducing a map η A : H ( F p ) −→ A ( F p ) , η A ( θ p ) := θ A , where A ( F p ) is the equivariant theory obtained on completing H ( F p ) ∧ H ( F p ) ∧ MU . Thehomotopy of H ( F p ) ∧ H ( F p ) ∧ MU is of the form π ∗ ( H ( F p ) ∧ H ( F p ) ∧ MU) = π ∗ ( H ( F p ) ∧ H ( F p )) [ b , b , . . . ] , where the homotopy groups of the spectrum H ( F p ) ∧ H ( F p ) is none other than the dualmod- p Steenrod algebra. As such π ∗ ( H ( F p ) ∧ H ( F p )) is an algebra on free variables ξ i for i ≥ in homological degree p i − , as well as a family of free variables in odd degree.As in the previous example, the subalgebra of π ∗ ( H ( F p ) ∧ H ( F p ) ∧ MU) generated by theclasses ξ i and b j represents the data given by a pair of isomorphisms a ( x ) and e F ( x ) offormal group laws over F p , where e F ( x ) is as before, and a ( x ) is an automorphism of theadditive formal group law [1, 23] a ( x ) = X i ≥ ξ i x p i , ξ = 1 . As in the previous example, η A represents composition of isomorphisms. We may apply η A to H ( F p ) ∗ U(1) and describe η A applied to the class x + x ℘ ( x ) by a formula x + x ℘ A ( x ) = η A ( x + x ℘ ( x )) = X i ≥ η A ( b i ) x i = X i ≥ b i ( X j ≥ ξ j x p j ) i , ξ = b = 1 . The reader may verify that this formula describes a co-action of the dual even Steenrodalgebra on H ( F p ) ∗ U(1) extending the standard co-action on Borel-equivariant cohomologyH ( F p ) ∗ U(1) = F p [ x ] .5. A PPENDIX : C
ALCULATIONS IN B OREL EQUIVARIANT COHOMOLOGY
The task we aim to achieve in the Appendix is to compute the Borel equivariant cohomol-ogy of spaces of the form B ( w I ) for some positive indexing sequence I . The answer willbe expressed in terms of the Borel equivariant cohomology of the Bott-Samelson spaces B t S ( w I ) . And so we will begin with the structure of the latter.Consider the Bott-Samelson variety B t S ( σ i ) = U( r ) × T ( G i /T ) , where σ i ∈ Br( r ) is thestandard braid for ≤ i < r . Then we have Theorem 5.1. H ∗ U( r ) ( B t S ( σ i )) is a rank two free module over H ∗ T , generated by classes { , δ } ,where δ ∈ H r ) ( B t S ( σ i )) is uniquely defined by the property δ + αδ = 0 , where α ∈ H T is thecharacter x i − x i +1 corresponding to the root α . roof. The proof of theorem 5.1 is classical. Consider the two T -fixed points of G i /T giventhe the cosets T /T and σ i T /T . The normal bundle of σ i T in G i /T is isomorphic to the T -representation with linear character given by the dual α of α . The two fixed points T /T and σ i T /T give rise to two sections s and s σ i respectively of the bundle U( r ) × T ( G i /T ) −→ U( r ) /T. Pinching off the section s gives rise to a cofiber sequence which splits into short exactseqences in equivariant cohomology −→ H ∗ T (Σ α ) f ∗ −→ H U( r ) ( B t S ( σ i )) s ∗ −→ H ∗ T −→ . Let λ ∈ H U( r ) ( B t S ( σ i )) be the class given by the image to the Thom class of the representa-tion α under the map f ∗ . We see from from the above sequence that { , λ } are H ∗ T -modulegenerators of H ∗ U( r ) ( B t S ( σ i )) . Now consider the map induced by the inclusion of fixedpoints s ⊔ s σ i : (U( r ) /T ) ⊔ (U( r ) /T ) −→ B t S ( σ i ) . The above short exact sequence shows that λ is uniquely determined by the fact thatit restricts trivially along s ∗ (by construction) and restricts to the element − α along s ∗ σ i since − α is the Euler class of the representation α . The generator ∈ H r ) ( B t S ( σ i )) clearly restricts to ∈ H T along both fixed points. It now follows that any element inH ∗ U( r ) ( B t S ( σ i )) is uniquely determined by its restrictions along these two fixed points.Applying this observation to λ yields the relation λ + αλ = 0 . Our generator δ is simply defined as λ . It is straightforward to see that δ is the uniqueclass with the property that δ + α δ = 0 . (cid:3) Remark 5.2.
Consider the map ρ i : U( r ) × T ( G i /T ) −→ U( r ) /T, [( g, g i T )] gg i T. Then, given γ ∈ H ∗ U( r ) (U( r ) /T )) = H ∗ T , we my ask to express the element ρ ∗ i ( γ ) in terms ofour generators { , δ } . This is easily done by using the restrictions along the two fixed points.Expressing ρ ∗ i ( γ ) as ρ ∗ i ( γ ) = a + b δ, we may restrict along s ∗ to deduce that a = γ . Then restricting along s ∗ σ i says that b = γ − σ ( γ ) α . Let us now recall the definition 2.4 of Bott-Samelson varieties and their Schubert classes.Given I = { i , . . . , i k } , and any j ≤ k , let J j = { i , . . . , i j } . Define π ( j ) and τ ( j ) to be themaps π ( j ) : B t S ( w J j ) −→ B t S ( σ i j ) , [( g, g i , . . . , g i j )] [( gg · · · g i j − , g i j )] .τ ( j ) : B t S ( w J j ) −→ U( r ) /T, [( g, g i , . . . , g i j )] [ gg · · · g i j ] Notice that one has a pullback diagram with vertical maps being canonical projections B t S ( w J j ) (cid:15) (cid:15) π ( j ) / / B t S ( σ i j ) (cid:15) (cid:15) B t S ( w J j − ) τ ( j − / / U( r ) /T. ince B t S ( w I ) projects canonically onto B t S ( w J j ) , we will use the same notation to denotethe maps π ( j ) and τ ( j ) with domain B t S ( w I ) . Define δ j ∈ H r ) ( B t S ( w I )) as the pullbackclass π ( j ) ∗ ( δ ) , where δ ∈ H r ) ( B t S ( σ i j )) is the (unique) generator of H r ) ( B t S ( σ i j )) as aH ∗ T -module that satisfies δ = − α δ . Here i j = s , and α := x s − x s +1 denotes the characterof the standard positive simple root α s of U( r ) that corresponds to the subgroup G i j . Inparticular, we have the relation δ j + [ α ] j δ j = 0 , where we define [ x ] j := τ ( j − ∗ ( x ) . Recalling definition 2.5, we have
Theorem 5.3.
Let I = { i , i , . . . , i k } denote a positive indexing sequence with i j < r . Then as aH ∗ T -algebra, we have H ∗ U( r ) ( B t S ( w I )) = H ∗ T [ δ , δ , . . . , δ k ] h δ j + [ α s ] j δ j , if i j ∈ I s . i . Moreover, for any weight α , the character α ∈ H T satisfies the following recursion relations [ α ] = α and [ α ] j = [ α ] j − + α ( h u ) δ j − , where i j − = u, with α ( h u ) denoting the value of the weight α evaluated on the coroot h u . Furthermore, thebehaviour under inclusions J ⊆ I , is given by setting all δ t = 0 for i t ∈ I/J .Proof.
The proof of the above theorem is a simple induction argument. Let I = { i , . . . , i k } as above. Assume that the classes { δ i , . . . , δ i k } have been constructed as in definition 2.4.Consider the indexing subsequence J k − = { i , . . . , i k − } . One has a bundle with fiber G i k /T B t S ( w I ) −→ B t S ( w J k − ) , [( g, g i , . . . , g i k )] [( g, g i , . . . , g i k − )] . As before, the above fibration supports two sections s J and s J,σ k induced by the two cosets { T /T, σ i k T /T } ⊂ G i k /T . Furthermore, using definition 2.4 one has a diagram of cofibersequences induced by the inclusion of the section s J B t S ( w J k − ) s J / / τ ( k − (cid:15) (cid:15) B t S ( w I ) / / π ( k ) (cid:15) (cid:15) Σ α ik B t S ( w J k − ) τ ( k − (cid:15) (cid:15) U( r ) /T s / / B t S ( w i k ) / / Σ α ik U( r ) /T. By induction, the classes { δ i , . . . , δ i k − } restrict to generators of H ∗ U( r ) ( B t S ( w J k − )) . Soby degree reasons, we see that the above diagram gives rise to a diagram of short exactsequences in equiariant cohomology. In particular, we see that H ∗ U( r ) ( B t S ( w I )) is a freemodule of rank two on H ∗ U( r ) ( B t S ( w J k − )) generated by the classes { , λ k } , where λ k is theimage of the Thom class in H ∗ U( r ) (Σ α ik B t S ( w J k − )) . By theorem 5.1, and using naturality,we see that this class is δ i k . This completes the induction argument. It remains to provethe recursive relation formula. This follow from unraveling the formula in remark 5.2. (cid:3) We now move to the the proof of theorem 2.7. The proof is fairly long and technical andso before we begin with the proof, let us briefly outline the main steps in the argument. e begin an induction argument with I being the empty set, for which we know theresult. In order to proceed with the induction argument, we first recall the space B T ( w I ) = G i × T G i × T · · · × T G i k . Even though we have considered B T ( w I ) as a T -space under conjugation, let us observethat B T ( w I ) extends to a T × T -space, with an element ( t , t ) acting via ( t , t )[( g , . . . , g k )] := [( t g , g , . . . , g k − , g k t )] . As such, we have the decomposition B ( w I ) = U( r ) × T ( G i × T G i × T · · · × T G i k ) = U( r ) × T ( B T ( w J ) × T B T ( w i k )) with the outer T -action being the conjugation action on both factors, and the inner T -action being the right, left actions on the two factors respectively.Then we consider two cases. The first case is when the index i k does not occur in J . Inthis case, the last factor may be altered to obtain a direct product of T -spaces. This allowsus to compute the equivariant cohomology of B T ( w I ) . The second case we consider iswhen i k occurs in J . In this case, we collect all the factors B T ( w i t ) so that i t = i k , and setup a spectral sequence converging to H ∗ U( r ) ( B ( w I )) , whose E -term is informed by the in-duction assumption. By degree reasons, this spectral sequence collapses at E confirmingthe induction step. Theorem 5.4.
Let I = { i , i , . . . , i k } denote a positive indexing sequence with each i j < r .Then, for any s ∈ ν ( I ) , with the property that ν ( s ) = 1 , there exists a class β s ∈ H r ) ( B ( w I ) .Likewise, for any orbit l ∈ ν ( I ) , there exists a canonical class γ l ∈ H r ) ( B ( w I )) so that there ismap of H ∗ U( r ) ( B t S ( w I )) -modules where s and l range over the sets ν ( I ) and ν ( I ) respectively ̟ I : H ∗ U( r ) ( B t S ( w I )) ⊗ Λ( β s ) ⊗ Λ( γ l ) h ˆ δ s i −→ H ∗ U( r ) ( B ( w I )) , where ˆ δ s := X j ∈ I s δ j . If I is redundancy free, i.e. has the property that ν ( s ) = 1 for all s ∈ ν ( I ) , then the map ̟ I is anisomorphism. More generally, given any redundancy free subsequence I ⊆ I with the property ν ( I ) = ν ( I ) , then square free monomials in the generators δ i , i ∈ I/I generate a length filtrationof H ∗ U( r ) ( B ( w I )) so that the associated graded module Gr δ H ∗ U( r ) ( B ( w I )) is isomorphic to Gr δ H ∗ U( r ) ( B ( w I )) ∼ = H ∗ U( r ) ( B ( w I )) ⊗ H ∗ U( r ) ( B t S ( w I )) H ∗ U( r ) ( B t S ( w I )) , where we consider H ∗ U( r ) ( B t S ( w I )) as an H ∗ U( r ) ( B t S ( w I )) -module by identifying δ j , j ∈ I , withtheir namesakes δ j , j ∈ I . Naturality of the above description is seen as follows. Given an inclusion J ⊂ I , so that J = I −{ i j } , the induced map in equivariant cohomology is given by the setting theclass δ j to zero in the case ν ( J ) = ν ( I ) . If on the other hand, we have i j = s = ν ( I ) /ν ( J ) , thenthe restriction map is injective, and has the property that β s maps to the class [ α s ] j γ s , where γ s isdefined as γ l ( s ) , with l ( s ) ∈ ν ( J ) being the W J -orbit containing s . All other generators restrict totheir namesakes modulo the ideal generated by γ s .Proof. Before the we begin the actual proof, let us comment on the part of the theoremthat is straightforward to prove. Namely, we will construct the classes γ l , as well as pointout why the classes ˆ δ s yield relations in cohomology. We begin with the relations. In rder to see this, notice that the proof of theorem 2.8 shows that the left and right H ∗ T -module structures must agree on H ∗ U( r ) ( B ( w I )) . Therefore the class τ ( I ) ∗ ( h ∗ s ) − π ( I ) ∗ ( h ∗ s ) ∈ H ∗ U( r ) ( B t S ( w I )) must give rise to a relation in cohomology, where τ ( I ) and π ( I ) were themaps defined in definition 2.4. From the recurrence relation given in theorem 5.3, we maywrite the above difference as the class τ ( I ) ∗ ( h ∗ s ) − π ( I ) ∗ ( h ∗ s ) = X j ∈ I s δ j := ˆ δ s . This shows that the classes ˆ δ s yield relations. Let us now define the classes γ l for anyorbit l ∈ ν ( I ) . Given an orbit l , let us pick an index i ∈ l arbitrarily. Let e denote thecardinality of the set ν ( I ) , so that one has a standard sub torus T e ⊆ T r corresponding tothese chosen indices. Now, let G I denote the subgroup generated by the subgroups G s for s ∈ ν ( I ) . In particular, we have a decomposition of the form G I = L I ⋊ T e where L I isthe semi-simple factor in G I . It follows that the maximal torus T r can be factored as T r = T I × T e , where T I is the maximal torus of G I generated by all the rank-one tori of the co-roots exp( i R h s ) for s ∈ ν ( I ) . In particular, we have a split fibration ET × T ( L I,i × T I . . . × T I L I,i k ) −→ ET × T ( B ( w I )) −→ T e , where L I,i := L I ∩ G i . The classes γ l that we seek are defined by pulling back the standardgenerators of the cohomology of T e . Restricting these classes to H (ET × T T ) , one caneasily check that that the restriction of the classes γ l can be expressed explicitly in termsof the standard generators { x i } of H (ET × T T ) as the class P i ∈ l x i . Case I
We now begin the main body of the proof. We will prove theorem 5.4 by induction on thelength of the indexing sequence. We split the induction into two cases and begin with thecase ν ( J ) ( ν ( I ) . Let I be the positive indexing sequence I = { i , . . . , i k } . By the use ofthe first Markov property, we may work with the special case where J is obtained from I by dropping the last index so that J = { i , . . . , i k − } . Let us express the block-diagonalgroup G i k ⊆ U( r ) as the group T r − × U(2) by reordering the standard basis of C r ifnecessary. We consider the new basis of the diagonal torus T r ⊂ U( r ) of the form T × T ,where T is the standard basis of the diagonal maximal torus in U(2) ⊂ G i k , and T is therank ( r − -torus endowed with the coroot basis exp( i R h u ) , u = i k , where h u denotes theco-root corresponding to the simple root α u . Let us endow H ( T , Z ) with the standardbasis { x, y } , and consider the projection map T r = T × T −→ T . We will denote by γ the pullback of the generator x under this projection map. It isstraightforward to check that γ = γ l ( i k ) , where l ( i k ) ∈ ν ( J ) is the W J -orbit containing i k .Notice also that h γ, h i k i = 1 , where we have identified the coroot h i k with an element inthe lattice H ( T r , Z ) . It follows that γ restricts to a generator of H (∆ , Z ) , where ∆ ⊂ T r isthe standard maximal torus of the semi simple factor in G i k , which can be identified with SU(2) . We shall denote this generator of H (∆ , Z ) also by γ . aving chosen the above basis, notice that we may express any block-diagonal group G i j for i j = i k as the form G ,i j ⋊ T , where G ,i j has maximal torus T . In particular, we havea decomposition as topological-spaces B T ( w J ) = G i × T G i . . . × T G i k − = ( G ,i × T G ,i . . . × T G ,i k − ) × T := B T ( w J ) × T . Equivariantly, the above decomposition results in a product decomposition of T -spaces B T ( w I ) = ( G ,i × T G ,i . . . × T G ,i k − ) × U(2) = B T ( w J ) × U(2) with T -acting by conjugation on both factors.Now consider the space B T ( w J ) = B T ( w J ) × T above. Note that each group G ,i j for j = k , belongs to a canonical subgroup L ⊂ U( r ) given by the semi-simple factor inthe group corresponding to the Dynkin diagram obtained by deleting the node i k in theDynkin diagram for U( r ) . Since L is a (product) of special unitary groups of strictlylower rank, we see that EU( r ) × U( r ) B ( w J ) is equivalent to a product EU( r ) × U( r ) B ( w J ) ∼ = (EU( r ) × U( r ) B ( w J )) × (EU( r ) × U( r ) B ( w J )) , where U( r ) × U( r ) ⊂ U( r ) is a non-trivial block decomposition of U( r ) . In particular, theinduction hypothesis applies allowing us to verify the structure of H ∗ U( r ) ( B ( w J )) . Next,let us consider the pullback diagram of fibrations, with the vertical fibers being the space ET × T ( B T ( w J )) ET × T ( B T ( w J )) (cid:15) (cid:15) / / ET × T ( B T ( w I )) (cid:15) (cid:15) ET × T T / / ET × T U(2) . Since the left hand side fibration is trivial, the corresponding Serre spectral sequence col-lapses. We will now proceed to verify that the map ET × T T −→ ET × T U(2) is injec-tive in cohomology (implying that the Serre spectral sequence for the right hand fibrationalso collapses), and identify its image. Indeed, this is a special case of the statement weare hoping to prove. A simple matter of identifying the image of the generators willthen verify the induction hypothesis for H ∗ T ( B T ( w I )) = H ∗ U( r ) ( B ( w I )) . That verification isstraightforward, and will be left to the reader.It therefore remains to verify the special case of U(2) above. Notice that we have a com-mutative diagram induced by the determinant map along the vertical direction ET × T T det (cid:15) (cid:15) / / ET × T U(2) det (cid:15) (cid:15) S / / S . with the map on fibers being being given by the inclusion ET × T ∆ −→ ET × T SU(2) ,where ∆ ⊂ SU(2) is the standard (skew) diagonal inclusion of the maximal torus.As before, we may reduce our task to verifying the injectivity of ET × T ∆ −→ ET × T SU(2) .For this, recall one has a homotopy decomposition of
SU(2) as T -spaces under conjugtion SU(2) / ∆ (cid:15) (cid:15) / / ∗ (cid:15) (cid:15) ∗ / / SU(2) , where ∆ ⊂ SU(2) stands for the standard (skew) diagonal maximal torus. A similarhomotopy pushout exists for ∆ as a T -space under conjugation S (cid:15) (cid:15) / / ∗ (cid:15) (cid:15) ∗ / / ∆ , so that the inclusion ι : ∆ ⊂ SU(2) is induced by a map of diagrams with ι : S ⊂ SU(2) / ∆ being given the the inclusion of the two ∆ -fixed points. In particular, we have a mapbetween the following two pushouts induced by ι . BT ` BT (cid:15) (cid:15) / / BT (cid:15) (cid:15) ET × T (SU(2) / ∆) / / (cid:15) (cid:15) BT (cid:15) (cid:15) BT / / ET × T ∆ BT / / ET × T SU(2) . The Mayer-Vietoris sequences for each pushout shows thatH ∗ T (ET × T ∆) = H ∗ T ⊗ Λ( γ ) , H ∗ T (ET × T (SU(2) / ∆)) = H ∗ T ⊗ Λ( β ) , where β restricts to αγ using the proof of therem 5.1.This completes the proof of the induction statement for Case I. Case II
We now move to the second case. As before, by using the first Markov property, we mayassume that J is obtained from I by dropping the last index i k . We further assume that theindex i k appears somewhere in the indexing sequence J . In other words, we consider thecase when ν ( I ) = ν ( J ) . Assume that s = i k , so that ν ( s ) > and let I s = { i s , i s , . . . , i k } .Similar to the earlier decomposition, we begin with the following decomposition of T -spaces B T ( w I ) = ( G i × T . . . × T G i s × T . . . × T G i s × T . . . × T G i k ) . We may idenitfy G s with T r − × U(2) , as before so that the above decomposition may bewritten as B T ( w I ) = ( G i × T . . . × ∆ SU(2) × ∆ . . . × ∆ SU(2) × ∆ . . . × ∆ SU(2)) , with the factor SU(2) occuring at the spots { i s , . . . , i k } , and ∆ corresponds to the skewdiagonal maximal torus of SU(2) .Our strategy is to start with the the indexing subsequence J = I/ { i s , . . . i k } obtainedby removing all but one copy of SU(2) above, and to sequentially insert the others in a anner that allows us to prove theorem 5.4 for each augmented sequence by an inductionargument. More precisely, we consider the family of sequences J r +10 := J r ∪ { i s r +1 } , ≤ r ≤ ν ( s ) − , J = J , J ν ( s )0 = I. Case I allows us to begin our induction argument by confirming the statement of theorem5.4 for B ( w J ) assuming that we knew the statement for all indexing sequences for which s / ∈ ν ( I ) . In other words, our proof happens one index s at a time. We now proceed withinduction by constructing a principal ∆ -bundle of T -spaces given by η r : ˜ B ( w J r − ) −→ B ( w J r ) , where ˜ B ( w J r − ) is defined as the space ˜ B ( w J r − ) = U( r ) × T ( G i × T . . . × ∆ SU(2) × . . . × T G i k ) , where the left-right ∆ × ∆ -orbits at the s r -st spot is replaced by a single two-sided ∆ -orbitdefined via the following action of λ ∈ ∆ on U( r ) × T ( G i × T . . . × SU(2) × . . . × T G i k ) λ ( g, . . . , g i sr − , g i sr , g i sr +1 , . . . ) := ( g, . . . , g i sr − λ − , λ g i sr λ − , λg i sr +1 , . . . ) , if r < ν ( s ) − λ ( g, g i , . . . , g i k − , g i k ) := ( gλ − , λ g i , . . . , g i k − λ − , λ g i k λ − ) , if r = ν ( s ) − . Our strategy is to first understand the equivariant cohomology of ˜ B ( w J r − ) and then usea spectral sequence to compute the equivariant cohomology of the quotient of the (free) ∆ -action on it, which we have identified with the space B ( w J r ) . Applying the homotopypushout result [2] to the s r -th factor in the expression for ˜ B ( w J r − ) gives rise to a givesrise to a pushout diagrams of U( r ) -spaces fibering over the space B ( w J r − ) B ( w J r − ) ˜ × (SU(2) / ∆) (cid:15) (cid:15) / / B ( w J r − ) (cid:15) (cid:15) B ( w J r − ) / / ˜ B ( w J r − ) , where B ( w J r − ) ˜ × (SU(2) / ∆) denotes a fibration over B ( w J r − ) , with structure group ∆ ,and with fiber being the left ∆ -space SU(2) / ∆ . As before, we may compare the abovepushout with the following pushout diagram over B ( w J r − ) B T ( w J r − ) × ( S ` S ) (cid:15) (cid:15) / / B T ( w J r − ) (cid:15) (cid:15) B T ( w J r − ) / / ( B T ( w J r − ) × ∆) . As in Case I, we can compare the Mayer-Vietoris sequences to deduce that the equivariantcohomologies have the following form for degree one and degree three classes τ and ∂ τ respectively H ∗ U( r ) ( B ( w J r − ) × ∆) = H ∗ U( r ) ( B ( w J r − )) ⊗ Λ( τ ) , H ∗ U( r ) ( ˜ B ( w J r − )) = H ∗ U( r ) ( B ( w J r − )) ⊗ Λ( ∂ τ ) , where ∂ τ restricts to the class − [ α s ] s r ⊗ τ . otice that both the above cohomologies are comodules over the exterior coalgebra givenby the cohomology of Λ( τ ) = H ∗ (∆) , with coaction induced by the principal ∆ -action µµ ∗ : H ∗ U( r ) ( B ( w J r − ) × ∆) −→ H ∗ U( r ) (( B ( w J r − ) × ∆) × ∆) , and identifying the left hand side with H ∗ U( r ) ( B ( w J r − )) ⊗ Λ( τ ) , and the right hand sidewith H ∗ U( r ) ( B ( w J r − )) ⊗ Λ( τ ) ⊗ Λ( τ ) . Our first order of business is to show that the coaction µ ∗ is standard. In other words, let us observe that this coaction is given by µ ∗ ( A ⊗ ( a + bτ )) = A ⊗ ( a ⊗ bτ ⊗ b ⊗ τ ) . In order to establish this, it is sufficient to show that the right action of ∆ on B ( w J r − ) istrivial in cohomology, where we recall that this action is given by λ [( g, . . . , g i sr − , g i sr , g i sr +1 , . . . )] := [( g, . . . , g i sr − λ, g i sr +1 , . . . )] Notice that this action is identical to the following action where we keep conjugatingelements to the left till we reach the spot s r − [( g, . . . , g i sr − λ, λ − g i sr − +1 λ, λ − g i sr − +2 λ . . . , λ − g i sr − λ, g i sr +1 , . . . )] Clearly, one may express the above as a product of several individual actions of ∆ , withthe left most action being by translation under ∆ at the spot s r − , and the others to itsright being by conjugation. It is sufficient to show that each of these actions are trivial incohomology. To see this, let us consider the conjugation action first. Since all cohomologygroups are torsion free, it is sufficient to show that this action is trivial when composedwith the degree two self map of ∆ . But since the identity component of the automorphismgroup of G i is SO(3) , the image of ∆ composed with the degree two map is null homo-topic. Next, consider the left most action by translation with ∆ at the spot s r − . However,the group at this spot is SU(2)) according to the definition of B ( w J r − ) . It follows thattranslation factors through the group SU(2) , which is simply connected. As before, theaction is trivial in cohomology.Having established the fact that the action of ∆ is standard, it follows easily that thecomodule structure on H ∗ U( r ) ( ˜ B ( w J r − )) = H ∗ U( r ) ( B ( w J r − )) ⊗ Λ( ∂ τ ) is determied by thediagonal on ∂ τ , where it is given by µ ∗ ( ∂ τ ) = ∂ τ ⊗ − [ α s ] s r ⊗ τ, where [ α s ] s r is in H ∗ U( r ) ( B ( w J r − )) . The above computation will feed into the Rothenberg-Steenrod spectral sequence, whichis a cohomologically graded, multiplicative spectral sequence that computes the coho-mology of principal quotients. This spectral sequence is built from the co-action and has E p,q = CoTor p,q Λ( τ • ) ( H ∗ U( r ) ( ˜ B ( w J r − )) , Z ) ⇒ H p + q U( r ) ( B ( w J r )) . Since all our comodules are finite type free Z -modules, we may work with the Z -dual ofH ∗ U( r ) ( ˜ B ( w J r − )) , which is a module over the exterior algebra Λ ∗ generated by classes dualto τ . In particular, we may cast the above spectral sequence asExt p,q Λ ∗ ( H ∗ U( r ) ( ˜ B ( w J r − )) ∗ , Z ) ⇒ H p + q U( r ) ( B ( w J r )) . We now use the standard Koszul resolution to compute the above Ext groups to see thatthey are computed by the complexH ∗ U( r ) ( B ( w J r − )) ⊗ Z [ X ] ⊗ Λ( ∂ τ ) , d ( ∂ τ ) = − [ α s ] s r X. y induction, multiplication by nonzero elements of H ∗ T is injective. It follows using thefirst Markov move that multiplication with − [ α ] i s is also injective. We therefore have E = H ∗ U( r ) ( B ( w J r − ))[ X ] h [ α s ] s r X i , where the class X is in bidegree (1 , . We now show that there can be no other dif-ferentials. By multiplicativity of this spectral sequence, differentials can only exist if d connects the classes of type β with an integral multiple of X . To show that such adifferential cannot happen, we may restrict to the subspace B ( w I rs ) ⊆ B ( w J r ) , where I rs = { i s , . . . , i s r } . In this special case, notice that B ( w I r − s ) is a retract of B ( w I rs ) (viagroup multiplication on the last two factors). Proceeding with the argument as above inthis special case, we see that there are no differentials eminating from the vertical axis.In particular, X cannot be a target. By naturallity, we have shown that the spectral se-quence above converging to H ∗ U( r ) ( B ( w J r )) collapses at E . It is easy to see that this class X is represented by the pullback of δ i sr .Furthermore, let us notice that the E -term notwithstanding, the [ α s ] s r torsion does notexist in H ∗ U( r ) ( B ( w J r )) since it gets resolved by the extension δ i sr + [ α s ] s r δ i sr = 0 . We con-clude that H ∗ U( r ) ( B ( w J r )) has a basis of { , δ i sr } as a H ∗ U( r ) ( B ( w J r − )) -module, and up toan associated graded object. This establishes the induction step and proves the theorem.Notice that our answer is expressed in terms of the equivariant cohomology of B ( w J ) instead of any redundancy free subsequence I ⊂ I as stated in the theorem. However, aswe mentioned earlier, one may repeat the above argument one index at a time to recoverthe precise statement of the theorem.Notice also that the above construction was natural in the subset { i s , . . . i k } . In otherwords, given any inclusion I ′ ⊂ I so that ν ( I ′ ) = ν ( I ) , the induced map in equivariantcohomology is given by setting the generators δ j to zero for j ∈ I/I ′ . This completesthe proof the second case. Along with the first case, this almost completes the proofof theorem 5.4. The only thing remaining to show is the naturality properties of ourcomputation.Given an inclusion J ⊂ I , so that J = I − { i j } , and ν ( J ) = ν ( I ) , we can invoke Case 2 toshow that the induced map in equivariant cohomology is given by the setting the class δ j to zero. The situation when i j = s = ν ( I ) /ν ( J ) is explained in the following remark. (cid:3) Remark 5.5.
Consider the restriction map H ∗ U( r ) ( B ( w I )) −→ H ∗ (ET × T T ) . The kernel of thismap is given by the ideal generated by the elements δ i . In particular, the restriction map is injectiveif ν ( s ) = 1 for all s ∈ ν ( I ) , namely if I is redundancy free. The discussion preceeding Case 1 inthe proof of theorem 5.4 described the restriction applied to the class γ l for any orbit l ∈ ν ( I ) γ l X i ∈ l x i . In Case 1, we also constructed the class β s ∈ H r ) ( B ( w I )) with the property that the restric-tion of β s to H r ) ( B ( w J )) for any J = I − { i j } , and with i j = s = ν ( I ) /ν ( J ) , is given by [ α s ] j γ s . Unlike the classes γ l , there is no canonical choice for β s unless the indexing sequence I as the property that ν ( s ) = 1 . It follows from the definition of these classes that the restrictionH r ) ( B ( w I )) −→ H (ET × T T ) applied to (any choice of) the class β s is given by the formula β s X i ≤ s, i ∈ l ( s ) α s x i , where l ( s ) denotes the W I -orbit that contains the index s . Now consider the one-step restriction r ( I, J ) : H ∗ U( r ) ( B ( w I )) −→ H ∗ U( r ) ( B ( w J )) , where J ⊂ I , so that J = I − { i j } , and i j = s = ν ( I ) /ν ( J ) . The we know from Case 1 thatthe generator β s restricts to [ α s ] j γ s . On restricting all the way to H (ET × T T ) and using therestriction formulas above, we see that r ( I, J ) has the property that it sends any other generator γ l or β t for t = s , to its namesake in H ∗ U( r ) ( B ( w J )) , modulo terms that involve γ s , and classes δ i . . R EFERENCES
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