aa r X i v : . [ m a t h . A T ] J u l SYMMETRY BREAKING AND LINK HOMOLOGIES I
NITU KITCHLOOA
BSTRACT . Given a compact connected Lie group G endowed with root datum, and an el-ement w in the corresponding Artin braid group for G , we describe a filtered G -equivariantstable homotopy type, up to a notion of quasi-equivalence. We call this homotopy type Strict Broken Symmetries , s B ( w ) . As the name suggests, s B ( w ) is constructed from the stackof pincipal G -connections on a circle, whose holonomy is broken between consecutive sec-tors in a manner prescribed by a presentation of w . We show that s B ( w ) is independent ofthe choice of presentation of w , and also satisfies Markov type properties. Specializing tothe case of the unitary group G = U( r ) , these properties imply that s B ( w ) is an invariant ofthe link L obtained by closing the r -stranded braid w . As such, we denote it by s B ( L ) . Theconstruction of strict broken symmetries also allows us to incorporate twistings. Applyingsuitable U( r ) -equivariant (possibly twisted) cohomology theories E U( r ) to s B ( L ) gives riseto a spectral sequence of link invariants converging to E ∗ U( r ) ( s B ∞ ( L )) , where s B ∞ ( L ) isthe direct limit of the filtration. In [3, 4], we offer two examples of such theories. In the firstexample, we study a universal twist of Borel-equivariant singular cohomology H U( r ) . The E -term in this case appears to recover sl ( n ) link homologies for any value of n (dependingon the choice of specialization of the universal twist). We also show that Triply-graded linkhomology corresponds to the trivial twist. In the next example, we apply a version of anequivariant K-theory n K U( r ) known as Dominant K-theory, which can be interpreted astwisted U( r ) -equivariant K-theory built from level n representations of the loop group of U( r ) . In this case, the E -term recovers a deformation of sl ( n ) -link homology, and has theproperty that its value on the unknot is the Grothendieck group of level n -representationsof the loop group of U(1) , given by Z [ x ± ] / ( x n − . C ONTENTS
1. Introduction 22. (Strict) broken symmetries and the G -equivariant homotopy type 63. Some filtered algebra 114. Properties of strict broken symmetries: The Markov 1 property 135. Properties of strict broken symmetries: Braid invariance 146. Properties of strict broken symmetries: Inverse relation and reflexivity 197. Properties of strict broken symmetries: G = U( r ) and the Markov 2 property 238. The invariant s B ( L ) of links 269. The p -completion, Etal´e homotopy type and the Frobenius 28References 29 The author acknowledges support by the Simons Foundation and the Max Planck Institute for Math.during the development of these ideas. . I NTRODUCTION
The main result of this article is the construction of a filtered U( r ) -equivariant stable ho-motopy type s B ( L ) for links L that can be described as the closure of r -stranded braids,namely, elements of the braid group Br( r ) . We call this spectrum the spectrum of strictbroken symmetries because it is built from the stack of principal U( r ) -connections on a circlewith prescribed reductions of the structure group to the maximal torus at various pointson the circle. Even though we have invoked the category of equivariant spectra, for links L that can be expressed as the closure of a positive braid, our spectrum s B ( L ) can be de-scribed entirely by the geometry of an underlying U( r ) -equivariant space of strict brokensymmetries. For the convenience of non-experts, all the results in the introduction willbe formulated for links given by the closure of a positive braid, with the general resultfor arbitrary braids described in later sections. We also point out that several results inthis article will be shown to hold for arbitrary compact connected Lie groups G . We havechosen to highlight the case G = U( r ) in the introduction for the purposes of exposition.Before we proceed, let us say a few words about the category of G -spectra that will beused in this article. The main results of this article can be understood with very little back-ground on equivariant spectra. It is helpful to bear in mind that G -spectra may be seenas a natural localization of the category of G -spaces where one is allowed to desuspendby arbitrary finite dimensional G -representations. As with G -spaces, one may evaluate G -spectra on G -equivariant cohomology theories. Given a subgroup H < G , one has re-striction and induction functors defined respectively by considering a G -spectrum as an H -spectrum, or by inducing up an H -spectrum X to the G -spectrum G + ∧ H X . As onewould expect, the induction from from H -spectra to G -spectra is left adjoint to restriction.For those somewhat familiar with the language, by an equivariant G -spectrum we meanan equivariant spectrum indexed on a complete G -universe [11].The spectra we study in this article are filtered by a finite increasing filtration F t X . Theassociated graded object Gr t ( X ) of such a spectrum has a natural structure of a chaincomplex in the homotopy category of G -spectra. In particular, one may define an acyclicfiltered G -spectrum X so that the associated graded object Gr r ( X ) admits stable null ho-motopies. The notion of acyclicity allows us to define a notion of quasi-equivalence on ourcategory of filtered G -spectra by demanding that two filtered G -spectra are equivalent ifthey are connected by a zig-zag of maps each of whose fiber (or cofiber) is acyclic.Returning to the main application of this article, we show that a braid w on r -strandsgives rise to a filtered equivariant U( r ) -spectrum of strict broken symmetries, denoted by s B ( w ) , which is well defined up to quasi-equivalence. Before we get to the definition ofstrict broken symmetries, let us first offer a geometric description of the U( r ) -spectrumof broken symmetries. Consider a braid element w ∈ Br( r ) , where Br( r ) stands for thebraid group on r -strands. For the sake of exposition, consider the case of a positive braidthat can be expressed in terms of positive exponents of the elementary braids σ i for i < r .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 braid group Br( r ) , w = w I := σ i σ i . . . σ i k . Let T , or T r (if we need to specify rank), be the standard maximaltorus, and let G i denote the unitary (block-diagonal) form in reductive Levi subgrouphaving roots ± α i . We consider G i as a two-sided T -space under the left(resp. right)multiplication. he equivariant U( r ) -spectrum of broken symmetries is defined as the (suspension) spec-trum corresponding to the U( r ) -space B ( w I ) that is induced up from a T -space B T ( 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 − )] . As mentioned above, the U( r ) -stack U( r ) × T ( G i × T G i × T · · · × T G i k ) is equivalent tothe stack of U( r ) -connections on a circle with k marked points, with the structure groupbeing reduced to T at the points, and symmetry being broken to G i along the i -th sector.One may heuristically relate the connections to links in the following way. Assume theexistence of a flow that sends a U( r ) -connection ∇ as above to a connection ∇ ∞ whose ho-lonomy along any sector between successive points is a permutation matrix that belongsto the corresponding block-diagonal subgroup. Now consider the parallel transport un-der ∇ ∞ of the standard orthonormal frame in C r . Projecting this frame onto a genericline bundle C × S ⊆ C r × S gives rise to the associated (geometric) link L ( ∇ ) in thetrivial line bundle ( C × S ) ⊂ R . In this context, connections whose holonomy preservesthe T -structure in any sector do not induce a braiding in that sector. In particular, torecover links with a prescribed braid presentation, one needs to factor out those connec-tions whose holonomy preserves the T -structure in any sector. The homotopical notionof the resulting object, where these redundant connections are factored out, is called theequivariant spectrum of strict broken symmetries . Definition. (Strict broken symmetries and their normalization)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 ) . In order for the normalized definition to make sense, one would require proving that theconstruction of s B ( L ) is independent (up to quasi-equivalence) of the braid presentation w I used to describe L . This comes down to checking the braid group relations, and thefirst and second Markov property. The first Markov property and the braid group rela-tions are established in sections 4 and 5-6 resp. Results of these sections in fact admit ageneralization to any compact connected Lie group G , and we work with that generalityin the first six sections. We have chosen to highlight the case G = U( r ) for the purposesof this introduction. he second Markov property imposes a stability condition on the construction, requiringthat it be invariant under the augmentation of w by the elementary braid σ r (or its inverse)so as to be seen as a braid in Br( r + 1) . This is equivalent to the observation that the link L is unchanged on adding an extra strand that is braided with the previous one. In provinginvariance under the second Markov property, we encounter a subtle point. Notice that s B ( L ) is induced up from a T r -spectrum we shall denote by s B T r ( L ) . Proving invarianceunder the second Markov property would therefore require showing that the U( r + 1) -spectrum obtained by considering L as the closure of wσ ± r is induced from s B T r ( L ) alongthe standard inclusion T r < U( r + 1) . This requirement is almost true but for a smallsubtlety. We show in section 7 that when L is seen as the closure of the ( r + 1) -strandedbraid wσ ± r , the corresponding U( r +1) -spectrum, s B ( L ) is induced up from s B T r ( L ) alonga different inclusion ∆ r : T r −→ U( r + 1) . This inclusion differs from the standard inclusionin the last entry. We proceed to resolve this issue in section 7 by inducing up to a largergroup. The upshot is that s B ( L ) is a link invariant. 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. In particular, the limiting equivariant stable homotopytype s B ∞ ( L ) is a well-defined link invariant in U( r ) -equivariant spectra (see remark 3.5). Wediscuss s B ∞ ( L ) below. An obvious way to obtain (group valued) link invariants from the filtered homotopy type s B ( L ) is to apply an equivariant cohomology and invoke the filtration to set up a spectralsequence. Let E G denote a family of equivariant cohomology theories indexed by thecollection G = U( r ) , with r ≥ , and naturally compatible under restriction E U( r ) ∼ = ι ∗ E U( r +1) , where ι : U( r ) −→ U( r + 1) . Therefore, given a family of equivariant cohomology theories E U( r ) as above that satisfiesthe conditions described in definition 8.6, the filtration of s B ( L ) described above doesindeed give rise to a spectral sequence that converges to E ∗ U( r ) ( s B ∞ ( L )) . The E -term ofthis spectral sequence is itself a link invariant, and is given by the cohomology of theassociated graded complex for the filtration of s B ( L ) . We have Theorem.
Assume that E U( r ) is a family of U( r ) -equivariant cohomology theories as above thatsatisfy the conditions of definition 8.6. Then, given a link L described as a closure of a positivebraid presentation w I on r -strands, one has a spectral sequence converging to E ∗ U( r ) ( s B ∞ ( L )) andwith E -term given by E t,s = M J ∈I t / I t − E s U( r ) ( B ( w J )) ⇒ E s + t − k U( r ) ( s B ∞ ( L )) . The differential d is the canonical simplicial differential induced by the functor described in defi-nition 2.6. In addition, the terms E q ( L ) are invariants of the link L for all q ≥ . In [4, 5], we will relate special cases of the above spectral sequence to various well-knownlink homology theories. The limiting spectrum s B ∞ ( L ) can actually be described explic-itly, and so we know exactly what the above spectral sequence converges to, yieldingimportant information about each stage E q ( L ) . To this point, in section 8 we will provea generalization of the following theorem for arbitrary compact connected Lie groups G ,and for braid words that are not necessarily positive. heorem. Given an indexing set I = { i , . . . , i k } , so that w I = σ i σ i . . . σ i k is braid word thatcloses to the link L . Let V I denote the representation of T given by a sum of root spaces V I = X j ≤ k w I j − ( α i j ) , where w I j − = σ i . . . σ i j − , w I = id, and where w I j − ( α i j ) denotes the root space for the root given by the w I j − translate of the simpleroot α i j . Then the U( r ) -equivariant homotopy type of s B ∞ ( L ) is given by the equivariant Thomspace (suitably desuspended) s B ∞ ( L ) = Σ − k U( r ) + ∧ T ( S V I ∧ T ( w ) + ) , where S V I denotes the one-point compactification of the T -representation V I , and T ( w ) denotesthe twisted conjugation action of T on itself t ( λ ) := w − twt − λ where w = σ i . . . σ i k , t ∈ T, λ ∈ T ( w ) . Remark.
Note that the structure group T of the above Thom spectrum can be reduced to a subtorus T w ⊆ T that is fixed by the Weyl element w that underlies w I . The torus T w is isomorphic toa product of rank one tori indexed by the components of L . More precisely, the factor correspondingto a particular component of L is the diagonal in the standard sub torus of T r indexed by thestrands belonging to that particular compoment. Since the cohomology of s B ∞ ( L ) (assumingThom isomorphism) depends only on the number of components of L , we may think of s B ∞ ( L ) as a stable lift of the Lee homology of L [10] . The spectrum s B ∞ ( L ) has a rich internal structure,which we study it in more detail in [5] . Let us point out an important piece of structure that is relevant to our framework. Noticethat each space of broken symmetries B ( w I ) admits a canonical map (given by composingthe holonomies along the sectors) to the stack of principal connections on a circle, whichis equivalent to the adjoint action of U( r ) -action on itself ρ I : B ( w I ) −→ U( r ) , [( g, g i , . . . , g i k )] g ( g i . . . g i k ) g − . These maps ρ I are clearly compatible under inclusions J ⊆ I . In particular, the spec-tra s B ( L ) can be endowed with a U( r ) -equivariant “local system” by pulling back U( r ) -equivariant local systems on U( r ) . We will use this structure in [3] to twist the equivariantcohomology theories E U( r ) considered above. More precisely, in [3, 4], we will study twoexamples of (twisted) cohomology theories and the corresponding spectral sequence. Thefirst example is given by Borel-equivariant singular cohomology H U( r ) . The second exam-ple is given by a version of an equivariant K-theory n K U( r ) known as Dominant K-theory,built from level n representations of the loop group of U( r ) .The author is indebted to M. Khovanov and L. Rozansky for their consistent interest andpatience during several discussions related to the subject. We thank D. Gepner, J. Morava,D. Rolfsen, J. Lurie and P. Wedrich for helpful discussions related to this article. The au-thor is also indebted to Vitaly Lorman, Apurva Nakade and Valentin Zakharevich for athorough reading of past work, especially with respect to the coherence properties en-countered in section 5. . (S TRICT ) BROKEN SYMMETRIES AND THE G - EQUIVARIANT HOMOTOPY TYPE
Let G be a compact connected Lie group of rank r , and semisimple rank r s ≤ r . Let usfix a root datum, and let T ⊂ G denote the maximal torus acted upon by the Weyl group W . Let σ i ∈ W for i ≤ r s , be the generators corresponding to the simple roots α i , where r s ≤ r is the semisimple rank of G . The Artin braid group for this root datum is a lift ofthe Coxeter presentation of W , and is defined by Br( G ) = h σ i , ≤ i ≤ r s , σ i σ j σ i . . . m ij terms = σ j σ i σ j . . . m ij terms i , where the integers m ij are determined by the entries of the Cartan matrix correspondingto the root datum. In the special case of G = U( r ) , the semisimple rank r s is r − and thebraid group is the classical braid group Br( r ) of braids with r strands.In this section we will construct a G -equivariant filtered stable homotopy type called the strict broken symmetries . All our G -spectra are genuine, by which we mean they are in-dexed on the complete G -universe. We first begin by defining the equivariant G -spectrumof broken symmetries given a presentation of an element w ∈ Br( G ) . Broken symmetrieswill then be assembled to construct strict broken symmetries. We begin with the defini-tions for positive braids. Definition 2.1. (Broken symmetries: Positive braids)Let I = { i , i , . . . , i k } denote an indexing sequence with i j ≤ r s , so that a positive braid w admitsa presentation in terms of the fundamental generators of Br( G ) , w = w I := σ i σ i . . . σ i k . Let G i denote the unitary form in the reductive Levi subgroup having roots ± α i . We consider G i as atwo-sided T -space under the canonical left(resp. right) multiplication.Define the equivariant G -spectrum of broken symmetries to be the suspension spectrum of B ( w I ) := G × T ( G i × T G i × T · · · × T G i k ) = G × T B T ( w I ) , where the T -action on B T ( w I ) := ( G i × T G i × T · · · × T G i k ) is given by endpoint conjugation t [( g , g , · · · , g k − , g k )] := [( tg , g , · · · , g k − , g k t − )] . Remark 2.2.
As was already mentioned in the introduction, let us point out again that the G -stack G × T ( G i × T G i × T · · ·× T G i k ) is equivalent to the stack of principal G -connections on the trivial G -bundle over S , endowed with a reduction of the structure group to T at k distinct markedpoints, and with the property that the holonomy along the i -th sector belongs to the subgroup G i in terms of this reduction. The stack of strict broken symmetries, which we will encounterlater, should be interpreted as the stack obtained from broken symmetries by factoring out thoseconnections whose holonomy preserves the T -structure in some sector. Let us now address the matter of braid elements w that admit a presentation with negativeexponents. Let ζ i denote the virtual G i representation ( g i − r R ) , where g i is the adjointrepresentation of G i , and r R is the trivial representation of dimension r (rank of G ). Noticethat the restriction of ζ i to T is isomorphic to the root space representation α i (as a realrepresentation). In fact, all results in the first six sections of this paper hold for G being the unitary form of an arbitrarysymmetrizable Kac-Moody group. efinition 2.3. (The dual Adjoint sphere spectrum)Let S − ζ i denote the sphere spectrum for the virtual real G i representation − ζ i . In what follows, wemay pick any model for this sphere. For instance, one may choose to define S − ζ i to be the dual of Σ − r S g i , and denote the left G i action on it by Ad ( g ) ∗ S − ζ i := Map( S g i , S r ) , Ad ( g ) ∗ ϕ := g ◦ ϕ ◦ Ad ( g − ) , ϕ ∈ Map( S g i , S r ) . Definition 2.4. (Broken symmetries: Arbitrary braids)Consider the more general indexing sequence expressed as I := { ǫ i i , · · · , ǫ i k i k } , where i j ≤ r s asbefore, and ǫ j = ± . Assume that w = w I := σ ǫ i i · · · σ ǫ ik i k . We define the equivariant G -spectrum B ( w I ) := G + ∧ 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 acts on S − ζ i ∧ G i + by smashing the action Ad ( t ) ∗ on S − ζ i with the standard T × T action on G i + given by left (resp.right) multiplication. As before the T -action on H i + ∧ T H i + ∧ T . . . ∧ T H i k is by conjugationon the first and last factor. Notice that B ( w I ) is an equivariant Thom spectrum over the stack ofbroken symmetries G × T ( G i × T · · · × T G i k ) . Our eventual goal is to study the naturality properties of the construction B ( w I ) in termsof subwords. In order to study this, we will require to make certain constructions knownas Pontrjagin-Thom constructions that require studying tubular neighborhoods. In orderto do so, let us fix a G -biinvariant metric on G . Claim 2.5.
The Pontrjagin-Thom construction along T ⊂ G i induces a canonical map of equi-variant T × T -spectra π i : S − ζ i ∧ G i + −→ T + , where the T × T -action on T is induced by left (rep. right) group multiplication.Proof. Let us describe the construction of the Pontrjagin-Thom construction along T ⊂ G i in some detail. First notice that the normal bundle of T in G i is canonically trivial (usingthe right T -translation of the complement of the Lie algebra of T , which we have denotedby ζ i . The conjugation action of T on this normal bundle can therefore be identified withthe standard root-space action of T on ζ i . Performing the Pontrjagin-Thom constructionamounts to collapsing a complement of a epsilon neighborhood of T (for some smallepsilon fixed throughout). Doing so gives rise to the map π i : S − ζ i ∧ G i + −→ S − ζ i ∧ S ζ i ∧ T + . We would like to identify S − ζ i ∧ S ζ i with S so as to identify the codomain with T + . Thisis clearly the case non-equivariantly but we must verify the required equivariance. Recallthe action of ( t , t ) ∈ T × T -action on S − ζ i ∧ G i + defined by smashing the action Ad ( t ) ∗ on S − ζ i with the standard T × T action on G i + given by left (resp. right) multiplication.Notice that t g i t = t g i t − t t = ( Ad ( t ) g i ) t t . In particular, performing the Pontrjagin-Thom construction on the inclusion T ⊂ G i turns the T × T -action on G i + into the expectedaction on S ζ i ∧ T + . It follows that the T × T -action on the product S − ζ i ∧ S ζ i is trivial, andamounts to group multiplication on the factor T + as we require. (cid:3) efinition 2.6. (The functor B ( w J ) )Given a braid word w I , for I = { ǫ i i , · · · , ǫ i k i k } , let I denote the set of all the 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 G -spectra. More precisely,given a nontrivial indecomposable morphism J → K obtained by dropping − i j from J , the in-duced map B ( w J ) → B ( w K ) is obtained by applying the map π i j of claim 2.5 in the correspondingfactor. Likewise, if J is obtained from K by dropping the factor i j , then the map B ( w J ) → B ( w K ) is defined as the canonical inclusion induced by the map T + → G i j + in the corresponding factor. We now define the equivariant G -spectrum of strict broken symmetries as follows Definition 2.7. (Limiting strict broken symmetries)let I + ⊆ I denote the terminal object of I given by dropping all terms − i j from I (i.e. termsfor which ǫ i j = − ). Define the poset category I to the subcategory of I given by removing theterminal object I + . I = { J ∈ I , J = I + } The equivariant G -spectrum s B ∞ ( w I ) is defined to be the cofiber of the canonical map: π : hocolim J ∈I B ( w J ) −→ B ( w I + ) . In other words, one has a cofiber sequence of equivariant G -spectra hocolim J ∈I B ( w J ) −→ B ( w I + ) −→ s B ∞ ( w I ) . Note that the above cofiber sequence can be induced from a cofiber sequence of T -spectra. Definition 2.8. (Filtration of strict broken symmetries via sub-posets I t ⊆ I )We endow s B ∞ ( w I ) with a natural filtration as G -spectra defined as follows. The lowest filtrationis defined as F s B ( w I ) = B ( w I + ) , 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, the filtered spectrum of strict broken symmetries F t s B ( w I ) is defined via the cofibersequence hocolim J ∈I t B ( w J ) −→ B ( w I + ) −→ F t s B ( w I ) . As before, note that F t s B ( w I ) = G + ∧ T F t s B T ( w I ) , with F t s B T ( w I ) defined just as above.Since I t ⊂ I t +1 , we obtain a canonical filtration of length k = | I | B ( w I + ) = F s B ( w I ) → F s B ( w I ) → F s B ( w I ) · · · → F k s B ( w I ) = s B ∞ ( w I ) . Remark 2.9.
It is straightforward to see that the associated graded of this filtration is given by thecofiber sequence F t − s B ( w I ) −→ F t s B ( w I ) −→ _ J ∈I t / I t − Σ t B ( w J ) . or the purposes of the rest of the article, it is helpful to normalize definition 2.7. Definition 2.10. (Normalized strict broken symmetries)Given an element w ∈ Br( G ) , we define the normalized G -spectrum of strict broken symmetries s B ( w ) := Σ l ( w I ) s B ( w I )[ ̺ I ] , where w I is any presentation of w , and l ( w I ) stands for the integer l − ( w I ) − l + ( w I ) with l + ( w I ) denoting the number of positive exponents and l − ( w I ) denoting the number of negative exponentsin the presentation w I for w in terms of the generators σ i . Here s B ( w I )[ ̺ I ] denotes the shift inindexing given by F t s B ( w I )[ ̺ I ] := F t + ̺ I s B ( w I ) where the integer ̺ I that induces the shift isgiven by one-half the difference between the cardinality of the set I , | I | , and the mimimal wordlength | w | , of w ∈ Br( G ) ̺ I = 12 ( | I | − | w | ) . Note that the notation for the normalization given above depends only on w ∈ Br( G ) , butmakes no reference to the presentation of w . For this definition to make sense, one needsto verify that the filtered spectrum s B ( w ) is independent of the presentation. That is infact the case, but the proof of that fact will take up the next several sections Theorem 2.11.
Given a braid element w ∈ Br( G ) , with two presentations w = w I = w I ′ ,then the filtered G -spectra Σ l ( w I ) s B ( w I )[ ̺ I ] and Σ l ( w I ′ ) s B ( w I ′ )[ ̺ I ′ ] are quasi-equivalent, wherequasi-equivalence is defined in definition 3.4. Before we move ahead to the next section that describes the notion of equivalence wework with, let us actually study the underlying homotopy type of the top filtration givenby Σ l ( w I ) s B ∞ ( w I ) . In fact, this homotopy type has a nice description in terms of a Thomspectrum. We have Theorem 2.12.
Given an indexing set I = { ǫ i , . . . , ǫ k i k } , Let V I denote the vitrual representa-tion of T given by a vitual sum of root spaces V I = X j ≤ k ǫ j w I j − ( α i j ) , where w I j − = σ i . . . σ i j − , w I = id, and where w I j − ( α i j ) denotes the root space for the root given by the w I j − translate of the simpleroot α i j . Let | V I | denote the virtual dimension of V I . Then the G -equivariant homotopy type of Σ l ( w I ) s B ∞ ( w I ) is given by the equivariant Thom spectrum Σ l ( w I ) s B ∞ ( w I ) = Σ −| V I | G + ∧ T ( S V I ∧ T ( w ) + ) , where S V I denotes the sphere spectrum for the virtual T -representation V I , and T ( w ) denotes thetwisted conjugation action of T on itself t ( λ ) := w − twt − λ where w = σ i . . . σ i k , t ∈ T, λ ∈ T ( w ) . Proof.
Recall the definition of s B ∞ ( w I ) via the cofiber sequence hocolim J ∈I B ( w J ) −→ B ( w I + ) −→ s B ∞ ( w I ) , here I + is the terminal element of the set I as defined in 2.6, and I denotes the sub-category of I consisting of all objects besides I + . Now recall that the spectrum B ( w J ) isdefined by B ( w J ) := G + ∧ T B T ( w J ) , where B T ( w J ) := H i j ∧ T . . . ∧ T H i js , and H i = S − ζ i ∧ G i + , if ǫ i = − , H i = G i + else . Consider the pushout category P with three objects { , , ∞} , and two morphisms ema-nating from to the other objects. Now consider the suspension of the category I thatindexes the diagram for s B ∞ ( w I ) . One may define this suspension as the quotient ofthe k -fold product category P I (with objects identified with functions on I with valuesin the object set of P ) taking the value ∞ on any element of I is identified with a dis-tinguished object (also called ∞ ). The explicit identification of the above quotient withthe suspension of I is given by defining the function ϕ ∈ P I corresponding to the sub-set J = { ǫ i j i j , . . . ǫ i js i j s } ⊆ I as ϕ ( ǫ s i s ) = 1 if ǫ s i s ∈ J and ǫ s < or ǫ s i s / ∈ J . Define ϕ ( ǫ s i s ) = 0 otherwise. Since P I is a product of pushout categories, and the functor de-scribing s B ∞ ( w I ) decomposes as a smash product of functors, we may express the ho-motopy colimit as the smash product s B ∞ ( w I ) = G + ∧ T P ( H i ) ∧ T . . . ∧ T P ( H i k ) , where P ( H i ) are the pushout diagrams constructed from the elementary morphisms de-scribed in definition 2.6. More precisely, if ǫ i = − , then we have S − ζ i ∧ G i + (cid:15) (cid:15) / / T + (cid:15) (cid:15) ∗ / / P ( H i ) . Since S ζ i ∧ T + is obtained from G i by pinching out the T × T -subspace given by theleft coset σ i T (see proof of claim 2.5), the above pushout is equivalent to the spectrum Σ S − ζ i ∧ σ i T + . On the other hand, if ǫ i = 1 , then the pushout diagram is given by T + (cid:15) (cid:15) / / G i + (cid:15) (cid:15) ∗ / / P ( H i ) . which is equalent to the spectrum S ζ i ∧ σ i T + , again using the proof of claim 2.5. Since the T -representation ζ i is isomorphic to α i , we may smash them together to obtain s B ∞ ( w I ) = Σ m G + ∧ T ( S ǫ ζ i ∧ σ i T + ) ∧ T . . . ∧ T ( S ǫ k ζ ik ∧ σ i k T + ) , where m is the number of negative exponents. Now suspending by a sphere of dimension l ( w I ) , and collecting all the terms σ i T + is easily seen to yield the result we seek to prove. (cid:3) . S OME FILTERED ALGEBRA
Before we address the the particular properties enjoyed by s B ( w I ) , let us digress brieflyinto the theory of filtered G -spectra so as to define a lax notion of equivalence of filteredequivariant G -spectra that would be relevant for our purposes.By a bounded-below filtered G -spectrum X := { F t X } , we mean a filtered G -spectrumwith the filtration being the trivial spectrum below some fixed integer n . We have a col-lection of cofiber sequences: · · · F t X → F t +1 X → F t +1 X/F t X → Σ F t X · · · which assemble to a collection of maps ∂ t : F t +1 X/F t X −→ Σ F t X −→ Σ( F t X/F t − X ) . Furthermore, ∂ t − ◦ ∂ t is null homotopic for each i . In particular one obtains a bounded-below graded chain complex in the homotopy category of G -spectra associated to X . Definition 3.1. (The associated graded and the shift functor for filtered spectra)The associated graded chain complex of a bounded-below filtered G -spectrum X is defined as { Gr t ( X ) } := { Σ − t ( F t X/F t − ) , ∂ t − } . Given a bounded below filtered G -spectrum X , we define the shifted spectrum X [ ̺ ] as F t X [ ̺ ] := F t + ̺ X. Remark 3.2.
Notice that desuspension and shift together amount to a reindexing of the associatedgraded complex. In other words, we have { Gr t (Σ − ̺ X [ ̺ ]) } = { Gr t + ̺ ( X ) } . Example 3.3.
The associated graded spectrum associated to s B ( w I ) is given by Gr t ( s B ( w I )) = _ J ∈I t / I t − B ( w J ) , with ∂ t being induced by a signed sum along nontrivial indecomposable morphisms. Definition 3.4. (Acyclicity and quasi-equivalence of filtered G -spectra)A filtered G -spectrum X is said to be acyclic if the associated graded complex Gr t ( X ) admits a“null chain homotopy” h t for all t ≥ , whose graded commutator with ∂ is an equivalence h t : Gr t ( X ) −→ Gr t +1 ( X ) , ∂ t ◦ h t + h t − ◦ ∂ t − : Gr t ( X ) ≃ −→ Gr t ( X ) . A map of filtered G -spectra ρ : X −→ Y is defined as a collection of maps ρ t : F t X −→ F t Y ,compatible with the filtration. The map ρ is said to be an elementary quasi-equivalence if thefiltered spectrum defined by the fibers (or cofibers) of ρ t , is acyclic. Two filtered G -spectra are saidto be quasi-equivalent if they are connected by a zig-zag of elementary quasi-equivalences. We willrefer to usual levelwise equivalences as honest equivalences. Remark 3.5.
Notice that the definitions imply that if two filtered spectra X and Y are quasi-equivalent, then their limiting G -spectra X ∞ := hocolim t F t X and Y ∞ := hocolim t F t Y respec-tively are G -equivariantly homotopy equivalent. The converse need not be true as can be easilyseen. laim 3.6. Let ρ : X −→ Y be an elementary quasi-equivalence of filtered G -spectra. Given a G -equivariant cohomology theory E G so that the map of cochain complexes induced by ρρ ∗ : E ∗ G (Gr t ( Y )) −→ E ∗ G (Gr t ( X )) , is either injective for all t , or surjective for all t . Then ρ ∗ is a quasi-isomorphism.Proof. The proof is straightforward. We prove it under the injectivity assumption, thesurjective case is analogous. Assuming injectivity, we notice that there is a short exactsequence of cochain complexes → E ∗ G (Gr t ( Y )) ρ ∗ −→ E ∗ G (Gr t ( X )) −→ E ∗ G (Gr t ( Z )) → , where Z := { F t Z } is the fiber of ρ . By definition of acyclicity, the cochain complex E ∗ G (Gr t ( Z )) is acyclic. The long exact sequence in cohomology therefore implies that ρ ∗ isa quasi-isomorphism. (cid:3) It is easy to see why the requirement of injectivity or surjectivity in the above claim is nec-essary. For example, given a filtered spectrum X , consider the canonical map of filteredspectra given by the shift that maps each filtrate into the next s : X −→ X [1] , F t X −→ F t +1 X. It is straightforward to see that s is an elementary quasi-equivalence. However, the in-duced map s ∗ is trivial on the associated graded complex in any cohomology theory E G .Indeed, this example is universal in a suitable sense in describing what happens in thecase of elementary quasi-equivalences which are trivial on the associated graded object. Claim 3.7.
Let ρ : X −→ Y be an elementary quasi-equivalence of filtered G -spectra, then thereexists a filtered G -spectrum P ρ endowed with elementary quasi-equivalences ι Y : Y −→ P ρ , ι X : X [1] −→ P ρ , with ι X ◦ s ≃ ι Y ◦ ρ. In particular, P ρ furnishes a quasi-equivalence between Y and X [1] . Furthermore, if the map ofthe associated graded cochain complexes induced by ρρ ∗ : E ∗ G (Gr t ( Y )) −→ E ∗ G (Gr t ( X )) , is trivial in E G -cohomology, then both maps ι ∗ Y and ι ∗ X are quasi-isomorphisms on the associatedgraded complex.Proof. Let us define the filtered G -spectrum by defining { F t P ρ } as the homotopy pushout F t X ρ (cid:15) (cid:15) s / / F t X [1] ι X (cid:15) (cid:15) F t Y ι Y / / F t P ρ . By construction, the fiber of ι X is the fiber of ρ , which is acyclic. Similarly, the fiber of ι Y is the fiber of s , which is also acyclic. Hence, both ι X and ι Y are elementary quasi-equivalences. Now assume that ρ is trivial in E G -cohomology. Then it is easy to see fromcomparing the long-exact sequences in cohomology for the two rows (resp. columns),that ι ∗ X (resp. ι ∗ Y ) is surjective on the associated graded complex. By claim 3.6, it followsthat they are quasi-isomorphisms. (cid:3) . P ROPERTIES OF STRICT BROKEN SYMMETRIES : T HE M ARKOV PROPERTY
Beginning with this section we shall prove various helpful properties of the G -spectrumof strict broken symmetries. To begin with, we will prove a Markov 1 type result whichessentially says that s B ( w I ) is equivalent to the spectrum s B ( w I ) , where I is the sequenceobtained by cyclicly permuting I . More precisely Definition 4.1. (The permuted poset I )Given an indexing sequence I = { ǫ i , ǫ i , . . . , ǫ k i k } , we define the sequence I = { ǫ i , ǫ i , . . . , ǫ k i k , ǫ i } . Notice that given a subset J ∈ I , the image of J under this permutation is denoted by J ∈ I .This gives rise to an isomorphism of the posets τ : 2 I −→ I which restricts to an isomorphism τ : I −→ I , where I is the poset of all non-terminal objects in I . With the above definition in place, we prove the Markov 1 property:
Theorem 4.2.
The functors B ( w J ) and B ( w J ) ◦ τ are equivalent. In particular, τ induces alevelwise (honest) equivalence of filtered G -spectra τ t : F t s B ( w I ) ≃ −→ F t s B ( w I ) , t ≥ . Proof.
We require a natural equivalence between the G -spectra B ( w J ) and B ( w J ) . Let J = { ǫ i j i j , . . . , ǫ i js i j s } be an element in I , so that J is defined as follows J = { ǫ i j − i j − , ǫ i j − i j − , . . . , ǫ i js − i j s − } ⊆ I, if j > , and J = { ǫ i j − i j − , . . . , ǫ i js − i j s − , ǫ i i } ⊆ I, if j = 1 . Recall from definition 2.4 that B ( w J ) : G + ∧ T ( H i j ∧ T H i j ∧ T . . . ∧ T H i js ) , where H i = S − ζ i ∧ G i + , if ǫ i = − , H i = G i + else . Let us first consider the case of a positive braid w I , so that H i j = G i j + for all j . In thatcase, we define τ on the underlying topological space by τ [( g, g i j , . . . , g i js )] = [( g, g i j − , . . . , g i js − )] , if j > , and τ [( g, g i j , . . . , g i js )] = [( gg i , g i j − , . . . , g i js − , g i )] , if j = 1 . It is easy to see that this map is well defined. The map defined above extends (by per-muting the equivariant spheres) to the equivariant vector bundle obtained by smashingthis space with the equivivariant spheres of the form S − ζ i . In other words, one may re-place G i j by H i j in the above description to obtain a map covering the space level mapdescribed above. This defines the natural equivalence of functors we seek τ : B ( w J ) −→ B ( w J ) . (cid:3) . P ROPERTIES OF STRICT BROKEN SYMMETRIES : B
RAID INVARIANCE
We now move towards showing that s B ( w I ) depends only on the braid element w and noton the presentation I used to express it. This property requires proving two results. Thefirst result would require showing that s B ( w I ) is invariant under the braid relations, andthe second result would require us to show invariance under the inverse relation, namelythat one may contract successive terms in I of the form { . . . , − i, i, . . . } or { . . . , i, − i, . . . } without changing the equivariant homotopy type up to quasi-equivalence.We begin our goal by first proving the following theorem on braid invariance: Theorem 5.1.
For a fixed pair of indices ( i, j ) , let I ( i,j ) be an arbitrary sequence that containsthe braid sequence O ( i,j ) := { i, j, i, j, . . . } with m ij -terms as a subsequence of consecutive terms.Define I ( j,i ) to be the sequence obtained by replacing the braid subsequence with its counterpart { j, i, j, i . . . } with m ij -terms. Then the filtered G -spectra s B ( w I ( i,j ) ) and s B ( w I ( j,i ) ) are con-nected by a sequence of zig-zags of elementary quasi-equivalences. In particular, they are quasi-equivalent. We will only consider the nontrivial case where m i,j > . The case when m i,j = 2 isstraightforward since the (left/right) T × T -spaces G i × T G j and G j × T G i can both beidentified with the same space, namely the group generated by G i and G j . The proof oftheorem 5.1 for m i,j > is fairly technical, and packed with several constructions andcorresponding definitions. The reader interested in the bigger picture may safely ignorethe rest of the section. For those who choose the path of most resistance, the generalargument of the proof can be outlined as follows.We will introduce two sequences of filtered G -spectra s BS h ( i,j,m ) ( w I ) and s BS h ( j,i,m ) ( w I ) resp. for ≤ m ≤ m ij called strict broken Schubert spectra. The spectra in either sequencewill be shown to belong to the same quasi-equivalence class by a sequence of zig-zagsof elementary quasi-equivalences. Moreover, by construction, the sequences will beginwith s B ( w I ( i,j ) ) and s B ( w I ( j,i ) ) respectively, and with both sequences ending with theexact same spectrum, allowing us to deduce the quasi-equilance between s B ( w I ( i,j ) ) and s B ( w I ( j,i ) ) . In other words, the beginning and ending terms of the sequences are s BS h ( i,j, ( w I ) = s B ( w I ( i,j ) ) , s BS h ( j,i, ( w I ) = s B ( w I ( j,i ) ) ,s BS h ( i,j,m ij ) ( w I ) = s BS h ( j,i,m ij ) ( w I ) . As with strict broken symmetries, the filtered spectra s BS h ( i,j,m ) ( w I ) and s BS h ( j,i,m ) ( w I ) will be defined via a homotopy colimit of two functors BS h ( i,j,m ) ( w J ) and BS h ( j,i,m ) ( w J ) (resp.) defined using a poset category J ( i,j ) m . Before we do that, we require Definition 5.2. (Schubert varieties and their lifts)Let O ( i,j ) denote the indexing sequence { i, j, i, . . . } ( m ij -terms). Let K ⊆ O ( i,j ) be any subset K = { k , k , . . . , k q } . The Schubert variety X K is defined as the image under group multiplication X K = Image of G k × T · · · × T ( G k s /T ) −→ G/T. efine S h K ⊂ G be the T × T -invariant subspace to be the pullback (where T × T acts on G vialeft/right multiplication) S h K (cid:15) (cid:15) / / G (cid:15) (cid:15) X K / / G/T.
Remark 5.3.
Notice that S h K depends only on the reduced sequence for K , namely the se-quence obtained by contracting all sequentially repreated elements. For instance S h K for thesubsequence K = { i, j, j, i } ⊂ O ( i,j ) := { i, j, i, j, i } agrees with S h K red , where K red = { i, j, i } . Definition 5.4. (Poset of reduced sequences)Let O ( i,j ) denote the indexing sequence { i, j, i, . . . } ( m ij -terms). Let O ( i,j ) red denote the quotientof the poset of all subsets of O ( i,j ) under the equivalence relation that identifies two indexingsequences if they have the same reduced sequence (see remark 5.3 above). For any < p < m ij ,we have exactly two elements of O ( i,j ) red given by reduced sequences of length p , namely ( ijij . . . ) or ( jiji . . . ) . There are two more additional sequences given by the empty sequence and the sequence ( ijij . . . ) with m ij -terms. There is a unique nontrivial morphism from one sequence into a strictlylonger sequence.For ≤ m ≤ m ij , let O ( i,j,m ) ⊆ O ( i,j ) denote the indexing sequence containing the last m -terms,and let O ( i,j,m ) red ⊆ O ( i,j ) red denote sub-poset of sequences in the equivalence class of those subsets in O ( i,j,m ) . In particular, O ( i,j,m ) red has m elements. Let us briefly explore sequences of categories that contain the above posets.
Definition 5.5. (Indexing sequences containing posets of reduces sequences)Consider an indexing sequence I ( i,j ) that contains the subsequence O ( i,j ) , so that we have I ( i,j ) = { ǫ i , . . . , ǫ l i l , O ( i,j ) , ǫ l + m ij +1 i l + m ij +1 , . . . , ǫ k i k } = I ( i,j,m ij ) a O ( i,j ) , where we define I ( i,j,m ij ) = { ǫ i , . . . , ǫ l i l , ǫ l + m ij +1 i l + m ij +1 , . . . , ǫ k i k } . Similarly, we define I ( i,j,m ) by the presentation I ( i,j,m ) = { ǫ i , . . . , ǫ l + m ij − m i l + m ij − m , ǫ l + m ij +1 i l + m ij +1 , . . . , ǫ k i k } , so that I ( i,j ) = I ( i,j,m ) ` O ( i,j,m ) .Define the poset categories J ( i,j ) m = 2 I ( i,j,m ) × O ( i,j,m ) red , and I m = J ( i,j ) m /J ( m, +) , where J ( m, +) isthe terminal object. Notice that for m < m ij there is a projection functor π m : J ( i,j ) m −→ J ( i,j ) m +1 thanks to the definition of these categories as a quotient of I ( i,j ) . There is also an inclusion functor ι m : J ( i,j ) m +1 −→ J ( i,j ) m that sends any object of the form ( ˜ J, ˜ K ) , for which ˜ K also belongs to O ( i,j,m ) red ⊂ O ( i,j,m +1) red , over to the object ( ˜ J, ˜ K ) . It sends any object of the form ( ˜ J, ˜ K ) , for which ˜ K / ∈ O ( i,j,m ) red (there are two such objects ˜ K ), over to ( J, K ) , where J = ˜ J ∪ { i l + m ij − m } , and K isobtained from ˜ K by dropping the first term. Notice that π m ◦ ι m is the identify functor of J ( i,j ) m +1 . We finally get to the definition of the family of filtered G -spectra s BS h ( i,j,m ) ( w I ) . efinition 5.6. (Strict broken Schubert spectra)Define functors of broken Schubert spectra BS h ( i,j,m ) ( w ( J,K ) ) on the category J ( i,j ) m as follows: BS h ( i,j,m ) ( w ( J,K ) ) = G + ∧ T ( H i j ∧ T · · · ∧ T H i jq ∧ T S h K + ∧ T H i jq +1 ∧ T · · · ∧ T H i js ) , where J ∈ I ( i,j,m ) and K ∈ O ( i,j,m ) red . We also recall our convention that H i = S − ζ i ∧ G i + if ǫ i = − , and H i = G i + if ǫ i = 1 . As in the case of strict broken symmetries, we define the strictbroken Schubert spectra as the filtered equivariant G -spectra F t s BS h ( i,j,m ) ( w I ) by the cofibersequence hocolim ( J,K ) ∈I tm BS h ( i,j,m ) ( w ( J,K ) ) −→ BS h ( i,j,m ) ( w J ( m, +) ) −→ F t s BS h ( i,j,m ) ( w I ) , where I tm we recall is the sub poset of elements in I m that are no more than t non-trivial de-composable morphisms away from the terminal object J ( m, +) . Notice that the filtered spectrum s BS h ( i,j, ( w I ) agrees with s B ( w I ( i,j ) ) , and that we have s BS h ( i,j,m ij ) ( w I ) = s BS h ( j,i,m ij ) ( w I ) since the functor BS h ( i,j,m ij ) ( w ( J,K ) ) is the same in both cases. As mentioned previously, we will presently show that all the filtered spectra of the type s BS h ( i,j,m ) ( w I ) are in the same quasi-equivalence class. Of course, the same will be truefor ( i, j ) replaced by ( j, i ) . That would constitute the proof of theorem 5.1 as indicatedabove.However, we need a preliminary lemma that will help us compare pointwise fibers alonga map of homotopy colimits. Lemma 5.7.
Assume K ⊆ O ( i,j ) is a subsequence so that K = { k , k , . . . , k q } , with k m +1 = k m ,where k m is the counterpart of k m . So for instance if k m = i , then k m = j and vice versa. Assume X and Y are T × T -spaces so that X is free as a right T -space. Then the following diagram is anhonest pushout of equivariant T × T -spaces X × T G k × T S h K × T Y (cid:15) (cid:15) / / X × T G k × T S h K ′ × T Y (cid:15) (cid:15) X × T S h K × T Y / / X × T S h K ′′ × T Y, where K ′ is defined as the set { k , k , k , . . . , k q } , and K ′′ = { k , k , k , k , . . . , k q } . All maps inthe above diagram are given by the canonical maps. The vertical maps being induced by multipli-cation in G , and the horizontal ones being the standard inclusion induced by K ⊂ K ′ ⊂ K ′′ .Proof. Using the left freeness of X as a right T -space, we see that the above diagram fibersover the following diagram, with fiber Y : X × T G k × T X K (cid:15) (cid:15) / / X × T G k × T X K ′ (cid:15) (cid:15) X × T X K / / X × T X K ′′ . gain, using the freeness of X as a right T -space, we see that the above diagram itselffibers over X/T , with fiber being the diagram: G k × T X K (cid:15) (cid:15) / / G k × T X K ′ (cid:15) (cid:15) X K / / X K ′′ . It is therefore enough to prove that the diagram shown above is an honest pushout. Thisis essentially an application of the Bruhat decomposition theorem [9]. The Bruhat decom-position theorem says that there is a canonical (left) T -equivariant CW decomposition ofthe Schubert varieties of the form X K . Furthermore, the (open) cells are a product (undergroup multiplication) of the 2-cells of the form C k , where C k denotes any lift of the space ( G k /T ) − ( T /T ) to G k .Let us make the Bruhat decomposition precise in the case of interest to us. Let K red de-note the reduced set corresponding to K . Recall from remark 5.3, that K red is obtainedfrom K by contracting all repeated indices. Let n be the cardinality of K red . Bruhat de-composition then gives us a cellular decomposition of X K with the top cell given by thealternating product C k × C k × C k × · · · n -terms. Lower dimensional open cells aregiven by alternating products of the form C i × C j × C i × · · · or C j × C i × C j × · · · with p terms for ≤ p < n . In particular, K ′ is obtained from K by adding two more cellsgiven by C k × C k × C k × · · · n -terms, and the cell C k × C k × C k × · · · ( n + 1) -terms.The space G k × T X K ′ is therefore obtained from G k × T X K by adding yet another twocells C k × C k × C k × C k × · · · ( n + 1) -terms, and the cell C k × C k × C k × C k × · · · ( n + 2) -terms. It follows that the cofiber of the inclusion of G k × T X K ⊂ G k × T X K ′ has a T -invariant CW decomposition with four cells given by C k × C k × C k × · · · n -terms, thecell C k × C k × C k × · · · ( n + 1) -terms, the cell C k × C k × C k × C k × · · · ( n + 1) -terms,and the cell C k × C k × C k × C k × · · · ( n + 2) -terms. These are precisely the four cells thatbuild X K ′′ from X K . In particular, the cofiber of the horizontal maps in the above diagramare mapped homeomorphic under the vertical map . This is equivalent to saying that thediagram is a pushout. (cid:3) Remark 5.8.
Notice that the left vertical map in the diagram for lemma 5.7 splits at T × T -spaces,hence we have the equality in the category of G -spectra ( X × T G k × T S h K × T Y ) + = F ∨ ( X × T S h K × T Y ) + , where F is the equivariant G -spectrum given by the fiber of (either) vertical map. It is easy to verifythat the statement of lemma 5.7, and the above splitting also holds if we replace each corner of thecommutative diagram of 5.7 by the Thom spectrum of a bundle ζ pulled back from the pushout X × T S h K ′′ × T Y . Our next step is to show that s BS h ( i,j,m ) ( w I ) and s BS h ( i,j,m +1) ( w I ) are quasi-equivalent.We do that by means of a zig-zag of elementary quasi-equivalences induced by the func-tors π m and ι m of definition 5.5. For m < m i,j , consider maps π m : s BS h ( i,j,m ) ( w I ) −→ π ∗ m s BS h ( i,j,m +1) ( w I ) ←− s BS h ( i,j,m +1) ( w I ) : ι m , where π m : s BS h ( i,j,m ) ( w I ) −→ π ∗ m s BS h ( i,j,m +1) ( w I ) is induced by the natural transforma-tion of functors between BS h ( i,j,m ) ( w ( J,K ) ) and π ∗ m BS h ( i,j,m +1) ( w ( J,K ) ) = BS h ( i,j,m +1) ( w π m ( J,K ) ) , nd the map ι m : s BS h ( i,j,m +1) ( w I ) = ι ∗ m π ∗ m s BS h ( i,j,m +1) ( w I ) −→ π ∗ m s BS h ( i,j,m +1) ( w I ) is in-duced by the functor ι m of definition 5.5.We begin by analyzing the map π m . Let Z m denote the filtered G -spectrum representingthe fiber of π m . Consider the fibration induced by π m on the level of associated graded Gr t ( Z m ) −→ _ ( J,K ) ∈I tm / I t − m BS h ( i,j,m ) ( w ( J,K ) ) Gr( π m ) −→ _ ( J,K ) ∈I tm / I t − m BS h ( i,j,m +1) ( w π m ( J,K ) ) . Our goal is to show that Z m is acyclic. Notice that for any object ( J, K ) , for which i l + m ij − m / ∈ J , the map Gr( π m )) : BS h ( i,j,m ) ( w ( J,K ) ) −→ BS h ( i,j,m +1) ( w π m ( J,K ) ) is an equivalence. Hence, the fiber of Gr( π m ) is detected on objects ( J, K ) for which i l + m ij − m ∈ J . We decompose such objects into two types. The first type of objects arethose for which the first term of K is i l + m ij − m , and the second type for which the firstterm is not i l + m ij − m . Consider the boundary map ∂ : Gr t ( Z m ) −→ Gr t − ( Z m ) on objectsof the first type. We see that precisely one component of this boundary maps to an objectof the second type, and on that component, it is equivalent to the map on the homotopyfibers of vertical maps in a diagram of the form described in lemma 5.7 and remark 5.8.Since these maps are cellular, lemma 5.7 implies that the map ∂ on this component givesrise to an equivalence. One therefore has a retraction to ∂ on objects of the second type,giving rise to a stable chain homotopy.It remains to show that ι m : s BS h ( i,j,m +1) ( w I ) −→ π ∗ m s BS h ( i,j,m +1) ( w I ) is also a quasi-equivalence. For this, let F t W m denote the filtered G -spectrum representing the cofiber of ι m . On the level of associated graded, we have a cofibration induced by ι m _ ( ˜ J, ˜ K ) ∈I tm +1 / I t − m +1 BS h ( i,j,m +1) ( w ( ˜ J, ˜ K ) ) −→ _ ( J,K ) ∈I tm / I t − m π ∗ m BS h ( i,j,m +1) ( w ( J,K ) ) −→ Gr t ( W m ) . Recall that π m ◦ ι m is the identity functor of J ( i,j ) m . Therefore, the map from BS h ( i,j,m +1) ( w ( ˜ J, ˜ K ) ) to π ∗ m BS h ( i,j,m +1) ( w ι m ( ˜ J, ˜ K ) ) is an equivalence for any object ( ˜ J, ˜ K ) ∈ I tm +1 / I t − m +1 . In otherwords, the first map admits a retraction, and we are left with the identification of Gr t ( W m ) with the complementary summand in the middle term. We express this summand as _ { ( J,K ) ∈ A } BS h ( i,j,m +1) ( w ( J,K ) , where the set A ⊆ I tm / I t − m denotes the collection of pairs ( J, K ) for which i l + m ij − m ∈ J and K can be augmented in O ( i,j,m ) red by adding terms on the left. Notice that the collectionof objects in A , come in two types determined by the sequence K . The first type are theones where the first term of K begins with i l + m ij − m +1 (or K is the empty sequence), andthe second type being the ones where the first term is i l + m ij − m . The boundary ∂ pairs upthe terms of the first type with those of the second, and is an equivalence between theseterms. In particular, we obtain the chain homotopy as before given by the inverse of ∂ on these terms. It follows that the cofiber of ι m is acyclic. This completes the proof oftheorem 5.1. . P ROPERTIES OF STRICT BROKEN SYMMETRIES : I
NVERSE RELATION AND REFLEXIVITY
In this section, we address the inverse relation and the property of reflection. In the for-mer, one contracts successive terms in an indexing sequence I of the form { . . . , − i, i, . . . } or { . . . , i, − i, . . . } without changing the equivariant homotopy type (upto quasi-equivalence,suspension and shift). In the latter, one reflects the indexing sequence I = { ǫ i , . . . , ǫ k i k } to the form { ǫ k i k , ǫ k − i k − , . . . , ǫ i } without changing the equivariant homotopy type.We begin with the following theorem. Theorem 6.1.
Let I ± and I ∓ denote indexing sequences of the form I ± = { ǫ i , . . . , ǫ l i l , i, − i, ǫ l +3 i l +3 , . . . ǫ k i k } , I ∓ = { ǫ i , . . . , ǫ l i l , − i, i, ǫ l +3 i l +3 , . . . ǫ k i k } , then there exists an elementary quasi-equivalence between the filtered G -spectra s B ( w I ± ) or s B ( w I ∓ ) and the shifted spectrum Σ s B ( w I red )[ − , (see 3.1 for the definition of shift) where I red = {{ ǫ i , . . . , ǫ l i l , ǫ l +3 i l +3 , . . . ǫ k i k } . The proof of the above theorem will rest on the following two claims
Claim 6.2.
The inclusion map ι i : T + −→ G i + induces a T × T equivariantly split injection ι i : S − ζ i ∧ G i + = T + ∧ T ( S − ζ i ∧ G i + ) −→ G i + ∧ T ( S − ζ i ∧ G i + ) . Proof.
We simply need to furnish an equivariant retraction. Recall that S − ζ i was the re-striction to T of a G i representation. In particular, the T -action on S − ζ i extends to a G i -action. The retraction we seek is given by the left G i -action on S − ζ i ∧ G i + µ : G i + ∧ T ( S − ζ i ∧ G i + ) −→ S − ζ i ∧ G i + . (cid:3) Claim 6.3.
The pinch map π i : S − ζ i ∧ G i + −→ T + of claim 2.5 induces a T × T equivariantlysplit surjection π i : G i + ∧ T ( S − ζ i ∧ G i + ) −→ G i + ∧ T T + = G i + . Proof.
It is easy to see that the fiber of π i is given by the spectrum G i + ∧ ( S − ζ i ∧ T T σ i + ) ,where σ i ∈ G i is any lift of the Weyl generator by the same name. It remains to constructan equivariant retraction from G i + ∧ T ( S − ζ i ∧ G i + ) to G i + ∧ T ( S − ζ i ∧ T σ i + ) . This retraction r i may be defined as follows: r i : G i + ∧ T ( S − ζ i ∧ G i + ) −→ G i + ∧ T ( S − ζ i ∧ T σ i + ) , ( g, λ, h ) ( ghσ − i , ( σ i h − ) ∗ λ, σ i ) . (cid:3) Remark 6.4.
It is straightforward to check that the composite map given by the inclusion ι i fol-lowed by the retraction r i is an equivalence r i ◦ ι i : S − ζ i ∧ G i + ∼ = −→ G i + ∧ T ( S − ζ i ∧ T σ i + ) . n proving theorem 6.1 we will only address the case of I := I ± , the other case being sim-ilar. First, let us consider the homotopy colimit hocolim J ∈I t B ( w J ) . We may decompose I t into subcategories so that this homotopy colimit may be expressed as the homotopycolimit over the following diagram hocolim J ∈I tred B ( w J ∪{ i } )hocolim J ∈I t − red B ( w J ) ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ (cid:15) (cid:15) hocolim J ∈I t − red B ( w J ∪{− i } ) o o / / O O t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ (cid:15) (cid:15) + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ hocolim J ∈I t − red B ( w J ∪{ i, − i } ) k k ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ (cid:15) (cid:15) B ( w I + red ) B ( w I + red ∪{− i } ) o o / / B ( w I + red ∪{ i, − i } ) Now the entire diagram fibers over B ( w I + ) = B ( w I + red ∪{ i } ) . In particular, we may express B ( w I + ) = B ( w I + red ∪{ i } ) as a colimit over a similar diagram B ( w I + red ∪{ i } ) B ( w I + red ) ♠♠♠♠♠♠♠♠♠♠♠♠ = (cid:15) (cid:15) B ( w I + red ∪{− i } ) o o / / O O v v ♠♠♠♠♠♠♠♠♠♠♠♠ = (cid:15) (cid:15) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ B ( w I + red ∪{ i, − i } ) h h ◗◗◗◗◗◗◗◗◗◗◗◗ = (cid:15) (cid:15) B ( w I + red ) B ( w I + red ∪{− i } ) o o / / B ( w I + red ∪{ i, − i } ) Now consider the fibration hocolim J ∈I t B ( w J ) π −→ B ( w I + ) −→ F t s B ( w I ) . We may express F t s B ( w I ) as a homotopy colimit of the pointwise cofibers, denoted as e B ( w J ) , of the above two diagrams: hocolim J ∈I tred e B ( w J ∪{ i } ) F t − s B ( w I red ) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ (cid:15) (cid:15) hocolim J ∈I t − red e B ( w J ∪{− i } ) o o / / O O t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ (cid:15) (cid:15) + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ hocolim J ∈I t − red e B ( w J ∪{ i, − i } ) j j ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ (cid:15) (cid:15) ∗ ∗ o o / / ∗ Now, using claim 6.3, it is easy to see that the following map in the above diagram F t − s B ( w I red ) −→ hocolim J ∈I tred e B ( w J ∪{ i } ) ifts to hocolim J ∈I t − red e B ( w J ∪{ i, − i } ) . Using this lift (namely by adding the negative of thelift), we may construct an inclusion of the following pushout that represents Σ F t − s B ( w I red ) ∗ ←− F t − s B ( w I red ) −→ ∗ into the above homotopy colimit diagram representing F t s B ( w I ) . The cokernel of thisinclusion is the homotopy colimit given by the G -spectrum Z , with F t Z defined as thehomotopy colimit of a diagram described below hocolim J ∈I tred e B ( w J ∪{ i } ) ∗ (cid:15) (cid:15) ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ hocolim J ∈I t − red e B ( w J ∪{− i } ) o o / / O O s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ (cid:15) (cid:15) + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ hocolim J ∈I t − red e B ( w J ∪{ i, − i } ) j j ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ (cid:15) (cid:15) ∗ ∗ o o / / ∗ Consider the associated graded complex of Z . All the above maps are trivial on the levelof associated graded and so we see that Gr t ( Z ) = _ J ∈I tred B ( w J ∪{ i } ) _ J ∈I t − red B ( w J ∪{ i, − i } ) _ J ∈I t − red B ( w J ∪{− i } ) . Using claims 6.2, 6.3 and remark 6.4, it is easy to see that the nontrivial horizontal map inthe above diagram admits an objectwise retractraction and the nontrivial slanted map isobjectwise split. It follows that the differential on Gr t ( Z ) pairs up these split summands,from which it follows that the filtered spectrum Z is acyclic. We therefore deduce that s B ( w I ± ) is quasi-equivalent to Σ s B ( w I red )[ − as we wanted to show. The above argu-ment completes the proof of theorem 6.1 and establishes theorem 2.11. Remark 6.5.
Theorem 2.11 tells us that s B ( w ) depends only on w up to quasi-equivalence. How-ever, given a G -equivariant cohomology theory E G , it does not immediately follow that the cochaincomplex E ∗ G (Gr t ( s B ( w )) is well defined up to quasi-isomorphism. By invoking claim 3.6, thiswould indeed be the case if we could check that each map in the zig-zag used to establish the proofof theorem 2.11, is either injective or surjective on the level of E ∗ G (Gr) . Analyzing the proof oftheorems 5.1 and 6.1 (that feed into the proof of theorem 2.11), one observes that this condition ispurely formal for most maps since the associated graded complex for these map splits. The onlyones for which this condition needs to be verified in cohomology are the following elementary quasi-equivalences of filtered G -spectra in the proof of theorem 5.1, for any pair of indices ( i, j ) , and for < m < m ij π m : s BS h ( i,j,m ) ( w I ) −→ π ∗ m s BS h ( i,j,m +1) ( w I ) . Remark 6.6.
Let us observe that the proofs of invariance under braid and inversion relationsgiven in sections 5 and 6 actually hold for the underlying T × T -spectra s B T ( w I ) before weinduce up to U( r ) . The invariance under these relations consequently also holds for the (strict)“Bott-Samelson” spectra s B ( w I ) ∧ T S , where we have taken orbits under the right T -action. Thespectrum s B ( w I ) ∧ T S is a filtered equivariant spectrum that represents an equivariant filteredhomotopy type for the complexes studied by Rouquier in [13, 14] . We revisit the Bott-Samelsonspectra in ( [3] see in particular theorem 2.7). e end this section with an additional property of the invariant s B ( w I ) of interest. Definition 6.7. (The reflected poset I R )Given a sequence I = { ǫ i , ǫ i , . . . , ǫ k i k } , we define its reflection I R = { ǫ k i k , ǫ k − i k − , . . . , ǫ i } . Notice that given a subset J ∈ I , its reflection sequence J R is an element in I R . This gives riseto an isomorphism of the posets R : 2 I −→ I R which restricts to an isomorphism R : I −→ I R ,where I R is the poset of all non-terminal objects in I R . With the above definition in place, let us prove the reflexive property:
Theorem 6.8.
The functors B ( w J ) and B ( w J R ) ◦ R are equivalent. In particular, R induces alevelwise (honest) equivalence of filtered G -spectra R t : F t s B ( w I ) ≃ −→ F t s B ( w I R ) , t ≥ . Proof.
We require a natural equivalence between the G -spectra B ( w J ) and B ( w J R ) . Let J = { ǫ i j i j , . . . , ǫ i js i j s } be an element in I . Recall from definition 2.4 that B ( w J ) : G + ∧ T ( H i j ∧ T H i j ∧ T . . . ∧ T H i js ) , where H i = S − ζ i ∧ G i + , if ǫ i = − , H i = G i + else . Let us first consider the case of a positive braid w I , so that H i = G i + for all j . In that case,we define R on the underlying topological space by R [( g, g i j , . . . , g i js )] = [( g ( g i j . . . g i js ) , g − i js , . . . , g − i j )] . We may express the above map as R = µ ∧ T ( R i js ∧ T R i js − . . . ∧ T R i j ) , where µ is inducedby the multiplication map: µ : G × G i j × · · · × G i js −→ G, ( g, g i j , . . . , g i js ) gg i j , . . . g i js , and R i : G i → G i is the inversion map.We now extend the above description of R to the case of an arbitrary braid w I . To begin,let us observe that S − ζ i ∧ G i + admits a map of the form S − ζ i ∧ G i + −→ S − ζ i ∧ G i + ∧ G i + given by performing the diagonal on the last factor. Therefore, for an arbitrary braid w I ,we have a map of the formD : G + ∧ ( H i j ∧ H i j ∧ . . . ∧ H i js ) −→ G + ∧ (( H i j ∧ G i j + ) ∧ ( H i j ∧ G i j + ) ∧ . . . ∧ ( H i js ∧ G i js + )) . The map µ defined above can now be invoked to obtain. µ : G + ∧ (( H i j ∧ G i j + ) ∧ ( H i j ∧ G i j + ) ∧ . . . ∧ ( H i js ∧ G i js + )) −→ G + ∧ ( H i j ∧ H i j ∧ . . . ∧ H i js ) . Hence, we have a self-map M of G + ∧ ( H i j ∧ H i j ∧ . . . ∧ H i js ) given by M = µ ◦ D. Themap R i also extends to a spectrum of the form S − ζ i ∧ G i + as follows. Let J − K denote theinvolution on S − ζ i = Map( S g i , S r ) induced by conjugation with the antipode map on S g i and S r . We define R i on S − ζ i ∧ G i + to be the involution given by smashing Ad ( g − i ) ∗ ◦ J − K on S − ζ i , with the inversion map on G i + .We now define the natural equivalence R from B ( w J ) to B ( w J R ) as R : B ( w J ) −→ B ( w J R ) , R := ( R i js ∧ T R i js − ∧ T . . . ∧ T R i j ) ◦ M. It is straightforward to check from the construction that R is well-defined and indeed anatural equivalence of functors. (cid:3) . P ROPERTIES OF STRICT BROKEN SYMMETRIES : G = U( r ) AND THE M ARKOV PROPERTY
In this section, we specialize to the case of the compact Lie group G = U( r ) , whose braidgroup Br( r ) is the classical braid group on r -strands generated by the elementary classicalbraids σ , . . . , σ r − .The Markov 2 property studies the effect of taking a braid word in r -strands, and extend-ing it to a braid word in ( r + 1) -strands by augmenting it by the generator σ r ∈ Br( r + 1) .Since we will compare the spectra of broken symmetries for U( r ) and U( r + 1) , we see thatthe Markov 2 property introduces a stabilization in the strands. In order to be keep trackof the number of strands, let us set some notation. For i ≤ r , we will use the notation G ri ⊆ U( r + 1) to be the unitary form (of rank r + 1 ) in the reductive Levi subgroup withroots ± α i . We will continue to use the notation G i ⊆ U( r ) for the unitary form of rank r .Notice that for i < r , these subgroups of U( r ) and U( r + 1) are related via a block decom-position G ri = G i × S , where S denotes the last factor of the product decomposition ofthe standard maximal torus T r +1 ⊂ U( r + 1) .Let ∆ r ⊂ T r +1 denote the centralizer of the final simple root α r . More precisely, ∆ r is thesubgroup T r − × ∆ , where ∆ is the diagonal circle in the last two standard factors of T r +1 .We may re-express T r +1 as ∆ r × S , with S being identified with the last factor in thestandard decomposition of T r +1 . By construction, ∆ r centralizes the group G rr .The goal of this section is to establish the following two theorems. Theorem 7.1.
Let I = { ǫ i , · · · , ǫ k i k } denote a sequence that offers a presentation for a braidelement w ∈ Br( r ) , and let I ( r ) denote the sequence obtained by augmenting I by the index i k +1 = r . In other words, I ( r ) is a presentation for the braid element wσ r ∈ Br( r + 1) . Then thereis an elementary quasi-equivalence of U( r + 1) -spectra s B ( w I ( r ) ) −→ U( r + 1) + ∧ ∆ r Σ s B T r ( w I ) , where the action of ∆ r on s B T r ( w I ) is induced by the canonical isomorphism between ∆ r and T r given by dropping the last coordinate in ∆ r . Theorem 7.2.
Let I = { ǫ i , · · · , ǫ k i k } denote a sequence that offers a presentation for a braidelement w ∈ Br( r ) , and let I ( − r ) denote the sequence obtained by augmenting I by the index i k +1 = − r . In other words, I ( − r ) is a presentation for the braid element wσ − r ∈ Br( r + 1) . Thenthere is an elementary quasi-equivalence of U( r + 1) -spectra s B ( w I ( − r ) ) −→ U( r + 1) + ∧ ∆ r Σ − s B T r ( w I ) . The proof of theorem 7.1 rests on the following claim
Claim 7.3.
Let J ⊆ I denote a subsequence J = { ǫ j i j , . . . , ǫ j s i j s } . Regarding J as a subse-quence of I ( r ) , let J ( r ) ⊆ I ( r ) denote the sequence J ∪ { i k +1 } . Then there is a cofibration of U( r + 1) -equivariant spectra induced by the inclusion J ⊂ J ( r ) B ( w J ) −→ B ( w J ( r ) ) −→ U( r + 1) + ∧ ∆ r Σ B T r ( w J ) , The action of ∆ r on B T r ( w J ) is induced by the canonical isomorphism between ∆ r and T r givenby dropping the last coordinate in ∆ r . roof. Let T r +1 ⊂ G rr ⊂ U( r + 1) denote the inclusion of the standard maximal torus ( S ) × r +1 . Notice that we have a cofibration of T r +1 × T r +1 -equivariant spectra(1) T r +1+ −→ G rr + −→ S ζ r ∧ ( T r +1 σ r ) + , where σ r is the permutation matrix in U( r + 1) that permutes the last two standard coor-dinates of T r +1 , so that T r +1 σ r is a T r +1 × T r +1 -space abstractly isomorphic to T r +1 , withthe right T r +1 -action being twisted by σ r . As before, S ζ r is the compactification of the rootspace representation of the root α r , and is given a trivial right T r +1 action.Smashing equation 1, T r +1 -equivariantly with the spectra B T r +1 ( w J ) , we get a cofibration(2) B ( w J ) −→ B ( w J ( r ) ) −→ U( r + 1) + ∧ T r +1 ( H ri j ∧ T r +1 · · · ∧ T r +1 H ri js ∧ S ζ r ∧ σ r + ) . Decomposing T r +1 as ∆ r × S , and observing that the right action of ∆ r fixes the spectrum S ζ r and commutes with σ r , we may express the above cofiber as (U( r + 1) + ∧ ∆ r ∧ ( H ri j ∧ T r +1 · · · ∧ T r +1 H ri js ) ∧ S ζ r ∧ σ r + ) ∧ S S . Recall that the S -action on ( H ri j ∧ T r +1 · · · ∧ T r +1 H ri js ) ∧ σ r + is by endpoint conjugation.Incorporating the twisting by σ r on the right allows us to identify the above S -spectrumwith ( H ri j ∧ T r +1 · · · ∧ T r +1 H ri js ) , with the S -action given by twisting the conjugation actionby σ r on the right hand side. This twisted conjugation action can be identified with thestandard right multiplication action of the conjugate diagonal subgroup S ∼ = ∆ ⊆ T r +1 consisting of elements of the form ( x − , x ) in the last two factors. Recall that for i ∈ I ,we have a block decomposition G rj = G r × S . It follows that all the spectra H ri thatoccur above are free S -spectra of the form H i ∧ S , where H i denotes the correspondingspectra when J is seen as a subset of I . We may therefore express ( H ri j ∧ T r +1 · · · ∧ T r +1 H ri js ) as ( H i j ∧ T r · · · ∧ T r H i js ∧ S ) . In particular, the cofiber of equation 2 can be identified with (U( r + 1) + ∧ ∆ r ( H i j ∧ T r · · · ∧ T r H i js ) ∧ S ζ r ) ∧ ∆ S ) . Since the ∆ -action is free on the S -factor, we may drop the free S -factor and identify theabove spectrum with(3) U( r + 1) + ∧ ∆ r Σ ( H i j ∧ T r · · · ∧ T r H i js ) . Putting equation 2 and the identification 3 together, gives rise to the cofibations of U( r +1) -spectra that we seek B ( w J ) −→ B ( w J ( r ) ) −→ U( r + 1) + ∧ ∆ r Σ B T r ( w J ) . (cid:3) Let us use the above claim to prove theorem 7.1. Let us first recall the categories used indefining the spectra s B ( w I ( r ) ) . Definition 7.4. (The poset I ( r ) and the functor B r ( w J ) )Let I ( r ) ⊂ I ( r ) denote the poset subcategory of subsets in I ( r ) that do not contain the terminalobject. Consider the functor B r from I ( r ) to U( r + 1) -spectra that sends J ∈ I ( r ) to B ( w J ∩ I ) .It is clear that B ( w J ) = B r ( w J ) if J ⊆ I . In particular, the above functor is a natural extensionof the functor B on I . Let us also observe that one has a canonical natural transformation T : B r −→ B induced by the inclusions J ∩ I ⊂ J ( r ) . ow consider the following commutative diagram hocolim J ∈I ( r ) t B r ( w J ) hocolim T (cid:15) (cid:15) / / B r ( w I + ) (cid:15) (cid:15) hocolim J ∈I ( r ) t B ( w J ) (cid:15) (cid:15) / / B ( w I ( r ) + ) (cid:15) (cid:15) hocolim J ∈I t U( r + 1) + ∧ ∆ r Σ B T r ( w J ) / / U( r + 1) + ∧ ∆ r Σ B T r ( w I + ) . It is clear from claim 7.3 that the right vertical sequence is a cofibration. Let us notice thatthe map hocolim T is also a cofibration. To see this, recall that the functor B ( w J ) agreeswith the functor B r on the full sub-category generated by J ∈ I ( r ) t that do not contain i r +1 . This sub-category has a terminal object I . In particular, the cofiber of hocolim T isdetected on the full sub-category of objects J containing i r +1 . This category is equivalentto I t , and one may identify the cofiber with hocolim J ∈I t U( r + 1) + ∧ ∆ r Σ B T r ( w J ) usingclaim 7.3. This shows that the left vertical sequence is a cofibration. Taking horizontalcofibers of the above diagram gives rise to a cofibration of filtered U( r + 1) -spectra s B r ( w I ( r ) ) s T −→ s B ( w I ( r ) ) −→ U( r + 1) + ∧ ∆ r Σ s B T r ( w I ) , where the filtered U( r + 1) -spectrum s B r ( w I ( r ) ) is defined to have filtrates F t s B r ( w I ( r ) ) given by the cofiber of the top horizontal map. It remains to show that s B r ( w I ( r ) ) isacyclic. From the definition of s B r ( w I ( r ) ) , the associated graded of the filtered U( r + 1) -spectrum s B r ( w I ( r ) ) is easily computed to be Gr t ( s B r ( w I ( r ) )) = Gr t − ( s B ( w I )) ∨ Gr t ( s B ( w I )) , with the differential identifying the obvious summands. The null homotopy is straight-forward to construct, completing the proof of theorem 7.1.The proof of theorem 7.2 is similar to the above and rests on the following claim similarto claim 7.3. We sketch the argument below, leaving the details to the reader Claim 7.5.
Let J ⊆ I denote a subsequence J = { ǫ j i j , . . . , ǫ j s i j s } . Regarding J as a subse-quence of I ( − r ) , let J ( − r ) ⊆ I ( − r ) denote the sequence J ∪ {− i k +1 } . Then there is a cofibrationof U( r + 1) -equivariant spectra induced by the map J ( − r ) → J B ( w J ( − r ) ) −→ B ( w J ) −→ U( r + 1) + ∧ ∆ r Σ − B T r ( w J ) . The proof of this claim is formally the same as that of claim 7.3 and starts with the cofi-bration sequence of T r +1 × T r +1 spectra induced by the inclusion T r +1 σ r ⊆ G rr S − ζ r ∧ ( T r +1 σ r ) + −→ S − ζ r ∧ G rr + −→ T r +1+ −→ Σ S − ζ r ∧ ( T r +1 σ r ) + . We leave it to the reader to complete the argument along the lines of claim 7.3.The proof of theorem 7.2 is now very similar to that of theorem . . One begins by defin-ing a functor B − r from I ( − r ) to U( r + 1) -spectra that sends J ∈ I ( − r ) to B ( w J ∪{− i k +1 } ) .It is clear that B ( w J ) = B − r ( w J ) if − i k +1 ∈ J . The rest of the proof follows from chasing adiagram similar to the one described in the proof of theorem . . We leave it to the readerto complete the proof. . T HE INVARIANT s B ( L ) OF LINKS
By now it it clear that the invariant s B ( w I ) that has been studied in the previous few sec-tions enjoys several important properties. Theorem 4.2 shows that s B ( w I ) is equivalentto its cyclic permutation s B ( w I ) , which is known as the Markov 1 property. In theorem2.11 we showed that s B ( w I ) did not depend on the indexing sequence I used to presentthe braid word w . And finally, in the case G = U( r ) , theorems 7.1 and 7.2 demonstratedthat s B ( w I ) satisfied an interesting property on adding an index to append I , which is avariant of the Markov 2 property.Of particular importance is the Markov 2 property, which we turn our attention to for themoment. Recall that by theorems 7.1 and 7.2, we have elementary quasi-equivalences s B ( w I ( r ) ) −→ U( r + 1) + ∧ ∆ r Σ s B T r ( w I ) ,s B ( w I ( − r ) ) −→ U( r + 1) + ∧ ∆ r Σ − s B T r ( w I ) . The difference in the number of suspensions in these two equivalences is easily correctedwhen we normalize and consider the invariant s B ( w ) of definition 2.10. A more subtleissue is that the invariant s B ( w I ( ± r ) ) is equivalent to the spectrum obtained by inducingthe T r -spectrum s B T r ( w I ) to a U( r + 1) spectrum, along a non-standard copy of the torus T r ⊂ U( r + 1) given by ∆ r . The following discussion describes how one may resolve this. Claim 8.1.
Let T denote the circle group. Consider the injection e r : T r −→ T × U( r ) , ( t , . . . , t r ) ( t r , ∆[ t t − r , t t − r , . . . , t r − t − r , , where ∆[ t t − r , t t − r , . . . , t r − t − r , denotes the diagonal subgroup of U( r ) with the correspondingentries. Given an indexing set of the form I = { i , . . . , i k } , define the e r -lift of broken symmetries B ( w I , e r ) to be the T × U( r ) -space B ( w I , e r ) := ( T × U( r )) × e r B T ( w I ) . Then one has a canonicalequivalence of T × U( r + 1) -spaces ( T × U( r + 1)) × e r +1 ◦ ∆ r B T ( w I ) = ( T × U( r + 1)) × T × U( r ) B ( w I , e r ) . Identifying T with the center of U( r ) , consider the homomorphism induced by group multiplica-tion in U( r ) , m : T × U( r ) −→ U( r ) . Let K ( m ) denote the kernel of m . Then B ( w I , e r ) is a freeK ( m ) -space, and the map m extends to an equivalence of stacks B ( w I , e r ) = ( T × U( r )) × e r B T r ( w I ) −→ U( r )) × T r B T r ( w I ) = B ( w I ) . Proof.
The first part of the claim, including freeness of B ( w I , e r ) as a K ( m ) -space, is straight-forward to verify. To see the equivalence of stacks, one simply needs to observe that thecomposite map, m ◦ e r : T r −→ U( r ) is the standard inclusion. (cid:3) As an immediate corollary of the above claim, we see
Corollary 8.2.
The e r -lift of (strict) broken symmetries s B ( w I , e r ) := ( T × U( r )) + ∧ e r s B T ( w I ) is invariant under the second Markov move. Let K ( m ) be the kernel of the multiplication map m .Then s B ( w I , e r ) is a filtered free K ( m ) -spectrum, and we have an equivalence of spectra inducedalong m U( r ) + ∧ T × U( r ) s B ( w I , e r ) ∼ = s B ( w I ) . onvention 8.3. We will continue to work with the model of strict broken symmetries given by s B ( w I ) = U( r ) + ∧ T r s B T r ( w I ) , with the understanding that one must replace it with the T × U( r ) -spectrum s B ( w I , e r ) as incorollary 8.2 in order for the invariance under the second Markov move to be manifest. Definition 8.4. ( s B as an invariant of links)Given a link L described by the closure of a braid word on r strands, define the normalized, filtered U( r ) -equivariant spectrum as described in 2.10 s B ( L ) := s B ( w ) = Σ l ( w I ) s B ( w I )[ ̺ I ] , where w ∈ Br( r ) is any braid with presentation w I , that represents the link L . This normalizationcorrects for the filtration shifts and suspensions that one encounters in proving invariance underthe various properties. This topological normalization may differ from other algebraic normaliza-tions, see remark 8.8. Having verified all the required properties: Braid invariance, invariance under the twoMarkov moves as well as inversion, we conclude
Theorem 8.5.
As a function of a link L that is described by the closure of a braid word on r -strands, the filtered U( r ) -spectrum s B ( L ) is well-defined up to quasi-equivalence. In particular,the limiting equivariant stable homotopy type s B ∞ ( L ) is a well-defined link invariant. Now let E G denote a family of equivariant cohomology theories indexed by the collection G = U( r ) , with r ≥ , and compatible under restriction E U( r ) ∼ = ι ∗ E U( r +1) , where ι : U( r ) −→ U( r + 1) . We do not assume that E is multiplicative for now. In the case of multiplicative theories,one will require some more structure (see [3] section 4). Definition 8.6. (ISN-Type equivariant cohomology theories)Given a family of equivariant cohomology theories { E U( r ) , r ≥ } as above, we call them ISN-type theories if the elementary quasi-equivalences flagged in remark 6.5 (called B-type maps), andthose of 7.1 and 7.2 (called M2a and M2b-type maps resp.) induce injective, surjective or nullmaps on the associated graded complex. For such theories, claims 3.6 and 3.7 show that the quasi-isomorphism type of the bi-complex E s U( r ) (Gr t ( s B ( L ))) is well defined up to a shift in bi-degree.By claim 3.7 and remark 3.2, we see that the shift will depend only on the number of elemantaryquasi-equivalences between two braid presentations of the link L that induce null maps on theassociated graded complex. Below we list how the cohomology theories we consider in [3, 4] behaveunder the B, M2a and M2b-type elementary quasi-equivalences above. Equivariant cohomology Theory B-type maps M2a-type maps M2b-type mapsSingular Cohomology (untwisted) [3]( §
2) Injective Null NullSingular Cohomology (twisted) [3]( §
4) Injective Null InjectiveDominant K-theory [4] Injective Null Injective iven an ISN-type equivariant cohomology theory, an obvious strategy to construct groupvalued link invariants from s B ( L ) is to study the spectral sequence that computes the co-homology E ∗ U( r ) ( s B ∞ ( L )) by virtue of the underlying filtration. The E -term of this spec-tral sequence is the cohomology of the complex E ∗ U( r ) (Gr t ( s B ( L )) , described in example3.3. Theorem 8.7.
Assume that E U( r ) is a family of ISN-type U( r ) -equivariant cohomology theories.Then, given a link L described as a closure of a braid w on r -strands as above, one has a spectralsequence converging to E ∗ U( r ) ( s B ∞ ( L )) and with E -term given by E t,s = M J ∈I t / I t − E s U( r ) ( B ( w J )) ⇒ E s + t + l ( w I )U( r ) ( s B ∞ ( L )) . The differential d is the canonical simplicial differential induced by the functor described in defi-nition 2.6. In addition, the terms E q ( L ) are invariants of the link L for all q ≥ . Our goal in the followup [3, 4] will be to give two examples of ISN-type equivariantcohomology theories. The first will be the usual (Borel-equivariant) singular cohomologyH U( r ) , and the second will be a version of equivariant K-theory known as Dominant K-theory [2], denoted by n K U( r ) , and built from level n representations of the loop groupof U( r ) . We also incorporate twistings on our cohomology theories. We will describe theframework to incorporate twisting in [3]. Remark 8.8.
We make note here that the normalization of the spectrum s B ( L ) we have defined indefinition 8.4 is purely topological in nature, and may differ from the standard normalization forlink invariants once we identify the cohomology of s B ( L ) with such an invariant. The table givenin definition 8.6 indicates how the topological normalization is sensitive to the various cohomologytheories we will consider in the sequel.
9. T HE p - COMPLETION , E
TAL ´ E HOMOTOPY TYPE AND THE F ROBENIUS
In this very brief section, we point out a piece of algebraic structure that appears on p -completing our constructions. We have kept this section brief since it deviates from thegeneral geometric flavor of the arguments we have been describing in this document. Weplan to return to this structure in the future.Let us revert back to a general compact connected Lie group G , and assume that it is theunitary form of the complex points of a Chevalley group scheme G Z . For instance, wemay take G Z to be GL( r ) Z in the case G = U( r ) . The groups G i in definition 2.1 admit Z -forms given by the corresponding split reductive Levi factors. It follows that for positivesequences I , the spaces B ( w I ) also admit Z -forms B Z ( w I ) .Now Etal´e homotopy theory [1] allows us to compare the Etal´e homotopy type of schemesover the algebraic closure F q , with the analytic space of complex points after p -completionat any prime p = q . It follows from these ideas that the p -completion of their respectivesuspension spectra are also equivalent. In particular, we recover the action of the Frobe-nius automorphism ψ q on the p -complete spectrum L H Z /p B ( w I ) , where L H Z /p denotesBousfield localization with respect to mod- p homology. ontinuing to work with positive sequences I , the reader can confirm that maps used toshow the independence of B ( w I ) on the presentation w I (upto quasi-equivalence) are allalgebraic. In particular, the action of the Frobenius ψ q on p -complete spectra of brokensymmetries extends to an action on the p -complete spectra of strict broken symmetriesL H Z /p s B ( w I ) . Since one can think of the diagram that defines s B ( w I ) as the diagram ofcomplex points of a simplicial scheme defined over Z , we conclude Theorem 9.1.
Let I be a positive sequence, and let p = q be distinct primes, then the stable p -completion of the Etal´e homotopy type of the simplicial scheme s B F q ( w I ) is independent of thepresentation w I upto quasi-equivalence in the category of p -complete spectra endowed with anaction of a (Frobenius) automorphism ψ q . R EFERENCES
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EPARTMENT OF M ATHEMATICS , J
OHNS H OPKINS U NIVERSITY , B
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