Galois symmetries of knot spaces
aa r X i v : . [ m a t h . A T ] F e b GALOIS SYMMETRIES OF KNOT SPACES
PEDRO BOAVIDA DE BRITO AND GEOFFROY HOREL
Abstract.
We exploit the Galois symmetries of the little disks operads toshow that many differentials in the Goodwillie-Weiss spectral sequences ap-proximating the homology and homotopy of knot spaces vanish at a prime p . Combined with recent results on the relationship between embedding cal-culus and finite-type theory, we deduce that the ( n + 1)-st Goodwillie-Weissapproximation is a p -local universal Vassiliev invariant of degree ≤ n for every n ≤ p + 1. A knot is a smooth embedding R → R which coincides with the standardinclusion t ( t, ,
0) in the complement of [0 , K . In this paper, we show that the Goodwillie-Weissapproximation to the space of knots gives a p -local analog of Kontsevich’s integral,in a range which improves as the prime p increases. Theorem A.
Let p be a prime and n ≤ p + 1 . The ( n + 1) -st Goodwillie-Weissapproximation of the space of knots K → T n +1 ( K ) is, on path components, a universal finite type invariant of degree n over Z ( p ) , thelocalization of Z at the prime p .Furthermore, under the same assumption on n , there is a non-canonical p -localisomorphism (i.e. an isomorphism after tensoring with Z ( p ) ) π ( T n +1 ( K )) ∼ = ⊕ s ≤ n A Is where A Is is the algebra of indecomposable Feynman diagrams, that is, the quotientof the free abelian group on unitrivalent trees of degree s by the antisymmetry, IHX and
ST U relations. We now describe in more detail the main characters of this story: the
Vassilievtower and the
Goodwillie-Weiss tower .A finite-type invariant of degree n is a homomorphism from π ( K ) (viewed asa monoid with respect to concatenation) to an abelian group, whose extension tosingular knots having n + 1 double points vanishes. It is not known whether finitetype invariants – also called Vassiliev invariants – distinguish all knots, but it isknown that they abound. Examples include all so-called quantum invariants, and We gratefully acknowledge the support through: grant SFRH/BPD/99841/2014 and projectMAT-PUR/31089/2017, funded by Funda¸c˜ao para a Ciˆencia e Tecnologia; projects ANR-14-CE25-0008 SAT, ANR-16-CE40-0003 ChroK, ANR-18-CE40-0017 PerGAMo, funded by AgenceNationale pour la Recherche. Following Vassiliev, Goussarov and Birman-Lin, the extension µ of a knot invariant µ tosingular knots K having at most k double points is defined inductively by the formula: µ ( K ) = µ ( K + ) − µ ( K − ) where K + and K − denote the knots obtained from K by resolving a given doublepoint in the two possible ways. Some standard references are [27], [2]. so all invariants coming from perturbative Chern-Simons theory, as well as manyclassically studied knot invariants. Two knots are declared to be n -equivalent if allfinite type invariants of degree ≤ n agree on these knots; this defines an equivalencerelation ∼ n on π ( K ). The equivalence relation ∼ n is finer than ∼ n − , and thereforewe have a tower of surjections(0.1) π ( K ) → . . . → π ( K ) / ∼ n → π ( K ) / ∼ n − → . . . → π ( K ) / ∼ that we call the Vassiliev tower . It turns out [17, Theorem 3.1] that the concate-nation of knots endows each of the sets π ( K ) / ∼ n with the structure of a finitelygenerated abelian group. There is moreover a surjective map A In → ker[ π ( K ) / ∼ n → π ( K ) / ∼ n − ] . A deep and fundamental result in the development of the subject was Kontsevich’sconstruction of the so-called Kontsevich integral which produces a rational inverseof this map and so a (non-canonical) isomorphism( π ( K ) / ∼ n ) ⊗ Q ∼ = ⊕ s ≤ n A Is ⊗ Q . The
Goodwillie-Weiss tower , on the other hand, is a tower of topological spaces K → · · · → T n ( K ) → T n − ( K ) → · · · → T ( K )The map K → T n ( K ) records the value of an embedding R → R on subsets of R given by the complement of at most n + 1 points of (0 ,
1) and is, in a homotopicalsense, universal with respect to this data. One can hit the Goodwillie-Weiss towerwith π and obtain a tower(0.2) π ( K ) → · · · → π ( T n ( K )) → π ( T n − ( K )) → · · · → π ( T ( K )) . It is known that all the sets π ( T n ( K )) are finitely generated abelian groups andthat all the maps in the tower are compatible with this structure [8], [4]. There is aspectral sequence that computes π ( T n ( K )) (and also its higher homotopy groups): E s,t = ⇒ π t − s T n ( K ) . The E -page consists of the homotopy groups of the homotopy fibers of the maps T s ( K ) → T s − ( K ) with s ≤ n and can be expressed in terms of homotopy groups ofspheres. The d -differential can be described explicitly and we have an isomorphism E s,s ∼ = A Is − (see [10] or [24] for an alternative account).A long-standing conjecture is that the towers (0.1) and (0.2) are isomorphic; seethe last paragraph of [15] and [9, Conjecture 1.1]. Some important steps have beenmade in this direction. Voli´c thesis [28] provided the first indication, though in thehomological context. In [8], Budney-Conant-Koytcheff-Sinha showed that the map π ( K ) → π ( T n ( K )) is a degree ( n −
1) invariant. A new proof of this result, usinggrope cobordisms, will appear in [21]. The upshot is that the tower (0.1) maps tothe tower (0.2) (with a shift of degree), that is, we have a commutative ladder ofabelian groups . . . / / π ( K ) / ∼ n / / (cid:15) (cid:15) π ( K ) / ∼ n − / / (cid:15) (cid:15) . . . / / π ( K ) / ∼ (cid:15) (cid:15) . . . / / π ( T n +1 ( K )) / / π ( T n ( K )) / / . . . / / π ( T ( K )) . ALOIS SYMMETRIES OF KNOT SPACES 3
By work of Conant, Kosanovi´c, Teichner and Shi [11, 20, 21], a necessary condi-tion for the vertical maps to be isomorphisms is that the Goodwillie-Weiss spectralsequence collapses along the anti-diagonal (Theorem 6.1). Theorem A is then aconsequence of such collapse results. To put it differently, Theorem A says that thekernel of the vertical map π ( K ) / ∼ n → π ( T n +1 ( K )) can only have p -power torsionfor p < n −
1. This result is modest in comparison to the conjecture that the twotowers are isomorphic over Z . Nevertheless, it seems to be the first non-rationalcomputation about the Goodwillie-Weiss tower beyond low degrees ( n ≤ Q , our result gives a construction of anisomorphism ( π ( K ) / ∼ n ) ⊗ Q ∼ = ⊕ s ≤ n A Is ⊗ Q that is purely homotopical and does not rely on transcendantal techniques likeKontsevich’s integral. However, our splitting is not explicit, whereas Kontsevich’sintegral gives a preferred choice of splitting over the complex numbers.Theorem A is a corollary of more general results about the Goodwillie-Weisstower for knots in R d – namely Theorem B below – as we now explain. This towerconverges to the space of knots when d ≥
4, in the sense that the map fromemb c ( R , R d ) to the homotopy limit of the tower is a weak equivalence. For d = 3,the case of classical knots, the tower is known not to converge to the knot space. (Forexample, π emb c ( R , R ) is countable whereas π T ∞ emb c ( R , R ) is uncountable.)The spectral sequence associated to the Goodwillie-Weiss tower has E -pageconsisting of the homotopy groups of the layers of the tower, just like in the case d = 3. Rationally , this spectral sequence is well understood: from the work ofArone, Lambrechts, Voli´c and Turchin [1] we know that there is rational collapseat the E -page when d ≥
4. Our main technical result gives a vanishing range ofdifferentials, at a prime p : Theorem B.
Let p be a prime number. The differential d rs,t : E rs,t → E rs − r,t + r − vanishes p -locally if r − is not a multiple of ( p − d − and if t < p − s − d − . Together with a result about triviality of extensions, this gives a computation ofsome homotopy groups of T n emb c ( R , R d ). The range improves as the prime or thecodimension increases. Theorem C.
Let n be a positive integer. For any prime number p satisfying n ≤ ( p − d −
2) + 3 , there is a (non-canonical) isomorphism π i T n emb c ( R , R d ) ⊗ Z ( p ) ∼ = ⊕ t − s = i E − s,t ( T n ) ⊗ Z ( p ) for i ≤ p − d − . The proof of Theorem B stands on the relation between the Goodwillie-Weisstower and the little disks operad [25, 26, 13, 4], and the existence of Galois symme-tries on the little disks operad that we established in [3]. Our main message in thispaper is that by combining these two observations one obtains an interesting Galoisaction on the Goodwillie-Weiss tower. The numerology in Theorem B may seemesoteric at first, but it is a consequence of the action: the inequality has something
PEDRO BOAVIDA AND GEOFFROY HOREL to do with the range over which the Galois action factors through the cyclotomiccharacter.We also prove a homological statement in Section 7 which, perhaps unsurpris-ingly, is stronger. It is also easier to prove, even though the strategy is the same. In[25], Sinha constructs a cosimplicial space K • which is a model for the Goodwillie-Weiss tower. Associated to the said cosimplicial space, and given a ring R of co-efficients, there is a homological Bousfield-Kan spectral sequence E ∗ ( R ). Moduloconvergence issues, this spectral sequence is related to H ∗ (emb c ( R , R d ) , R ) whereemb c ( R , R d ) denotes the homotopy fiber of the mapemb c ( R , R d ) → imm c ( R , R d ) . Our main theorem about the homology spectral sequence is the following.
Theorem D.
Let p be a prime number. The only possibly non-trivial differentialsin the spectral sequence E ∗ ( Z p ) are d n ( d − p − for n ≥ . This theorem was announced in [18]. Asymptotically, we also recover the collapseresults for R = Q , first established by Lambrechts-Turchin-Voli´c in [22] using theformality of the little disks operad. Acknowledgments.
We wish to thank Peter Teichner, Yuqing Shi and DanicaKosanovi´c for valuable discussions.1.
Bousfield localization
In this section, we recall some facts about localization of spaces. We denote the p -adic integers by Z p . Let E be a homology theory on spaces. Definition 1.1.
A space X is E -local if for any map f : U → V such that E ( f ) isan isomorphism, the map induced by precompositionMap( V, X ) → Map(
U, X )is an isomorphism.
Theorem 1.2 (Bousfield [6]) . There exists an endofunctor L E of the category ofspaces equipped with a natural transformation id → L E satisfying the following twoconditions. (1) For each X the space L E ( X ) is E -local. (2) The map X → L E ( X ) induces an isomorphism in E -homology.Moreover, the functor L E is uniquely defined up to weak equivalence by these twoproperties. In the case where E is the homology theory H ∗ ( − , F p ), the functor L E will bedenoted L p and called p -completion, following common practice. This is justifiedby the next proposition. Proposition 1.3. If X is a simply connected space whose homotopy groups arefinitely generated, the map X → L p X induces the canonical map π ∗ ( X ) → π ∗ ( X ) ⊗ Z p on homotopy groups.Proof. See [6, Proposition 4.3 (ii)]. (cid:3)
We will need two facts about this construction.
ALOIS SYMMETRIES OF KNOT SPACES 5
Proposition 1.4.
The p -completion functor L p commutes with products up to ho-motopy. More precisely, the canonical map L p ( X × Y ) → L p ( X ) × L p ( Y ) is an equivalence.Proof. The cannonical map X × Y → L p ( X ) × L p ( Y ) is a mod p homology equiv-alence by the K¨unneth isomorphism. Moreover, the target is easily seen to be p -complete. Therefore, by uniqueness of the p -completion, L p ( X ) × L p ( Y ) must bethe p -completion of X × Y . (cid:3) The following proposition shows that under some mild assumptions on X , thehomology of X with coefficients in the p -adic integers Z p can be recovered from the p -completion. We denote by C ∗ the standard singular chains functor from spacesto chain complexes. Proposition 1.5.
Let X be a space whose homology with coefficients in Z p isdegreewise finitely generated. Then the composite C ∗ ( X, Z p ) → lim n C ∗ ( X, Z /p n ) → lim n C ∗ ( L p X, Z /p n ) is a quasi-isomorphism.Proof. We show that each of the two maps is a quasi-isomorphism. First, weobserve that each of the two towers is a tower of epimorphisms. It follows that thelimit agrees with the homotopy limit. In order to prove that the second map isa quasi-isomorphism, it is enough to prove that the map f : X → L p X is a mod p n homology equivalence for all n . This follows by an easy induction using theBockstein long exact sequence . . . → H i ( − , Z /p n ) → H i ( − , Z /p n +1 ) → H i ( − , Z /p ) → H i − ( − , Z /p n ) → . . . We now show that the map C ∗ ( X, Z p ) → lim n C ∗ ( X, Z /p n )is a quasi-isomorphism. The homology groups of the source are, by definition, thehomology groups of X with Z p coefficients and the homology groups of the targetare H i (lim n C ∗ ( X, Z /p n )) ∼ = lim n H i ( X, Z /p n ) . This isomorphism follows from a Milnor short exact sequence argument (the as-sumption made on X implies that the mod p n homology is degreewise finite andthus the lim -term vanishes by the Mittag-Leffler criterion). Thus we have to provethat the canonical map H i ( X, Z p ) → lim n H i ( X, Z /p n )is an isomorphism. This map can be factored as H i ( X, Z p ) → lim n H i ( X, Z p ) ⊗ Z /p n → lim n H i ( X, Z /p n ) . The first of these two maps is an isomorphism under the finite type assumption.The second map is also an isomorphism. This can be seen as follows: using theuniversal coefficients theorem and the left exactness of the limit functor, we havean exact sequence0 → lim n H i ( X, Z p ) ⊗ Z /p n → lim n H i ( X, Z /p n ) → lim n Tor Z p ( Z /p n , H i − ( X, Z p )) . PEDRO BOAVIDA AND GEOFFROY HOREL
But, for any finitely generated Z p -module A , the group lim n Tor( Z /p n , A ) is zero. (cid:3) Remark 1.6.
This proposition may be false if the homology of X is not finitelygenerated. As a counter-example, one can take X to be the rationalization of S .On the one hand, its p -completion is contractible. On the other hand, H ( X, Z p )is isomorphic to Q p .2. A Grothendieck-Teichm¨uller action on the little disks operads
The Grothendieck-Teichm¨uller group GT p introduced in [12] is a profinite groupequipped with a surjective map χ : GT p → Z × p called the cyclotomic character. Given a Z p -module M , we can give it a GT p -actionusing the formula γ.m := χ ( γ ) n m . We call this action the weight n cyclotomic action. Proposition 2.1.
Let M be a Z p -module equipped with a GT p -action. Supposethat M fits in a short exact sequence of GT p -modules → M ′ → M → M ′′ → . If the GT p -action on M is cyclotomic of weight n , then that is also the case for the GT p -action on M ′ and M ′′ .Proof. This is an immediate verification. (cid:3)
Proposition 2.2.
Let M and N be two Z p -modules equipped, respectively, with theweight m and weight n cyclotomic GT p -action. If m − n is not a multiple of p − ,any GT p -equivariant map f : M → N must be the zero map.Proof. Let us assume that m − n is not a multiple of p −
1. Let x be any elementof M . If f is a GT p -equivariant map, we must have χ ( γ ) m f ( x ) = χ ( γ ) n f ( x )for every γ . In particular, since χ is surjective, for any unit u of Z p we have that( u n − m − · f ( x ) = 0. Hence, it is enough to find u such that u n − m − v of F × p . Then v n − m = 1 if n − m is not a multiple of p − F × p is cyclic of order p − u be a lift of v . Then u n − m − v n − m − p and so it is non-zero modulo p , i.e. it is a unit. (cid:3) The next theorem plays a key role in what follows; it equips the p -completionof the little disks operad with an action of GT p . We denote by E d the little disksoperad of dimension d . It will be convenient to view it as an ∞ -operad (a den-droidal space satisfying the Segal condition). Applying L p objectwise, we obtain adendroidal space L p E d which, by Proposition 1.4, also satisfies the Segal condition.For more details, see [3]. ALOIS SYMMETRIES OF KNOT SPACES 7
Theorem 2.3.
Let d be an integer with d ≥ . There exists a GT p -action on L p E d such that the induced action on H d − ( L p E d (2) , Z p ) is the cyclotomic actionof weight . Moreover, if we give E the trivial GT p -action, the canonical map E → L p E d is GT p -equivariant.Proof. This is done in [3]. We recall briefly how this construction goes. For a space X , we denote by X ∧ its pro- p -completion. This is a pro-object in p -finite spaces(a space is p -finite if it has finitely many components and if each component istuncated with finite p -groups as homotopy groups) that receives a map from X andis initial for this property. There is a close relationship between this functor and the p -completion functor X L p ( X ): when X is a nilpotent space of finite F p -type(i.e. all its F p -homology groups are finite dimensional) the p -completion L p ( X )coincides with the homotopy limit of the pro-object X ∧ . This homotopy limit isdenoted R Mat in [3] and is the homotopy right adjoint of the pro- p -completionfunctor.By work of Drinfel’d, there exists a non-trivial action of GT p on E ∧ . Using aversion of the Dunn-Lurie additivity theorem established in [5] one can produce anaction of GT p on E ∧ d by writing it as a certain tensor product of ( E ) ∧ F p with theDrinfel’d action and E ∧ d − with the trivial action. Morever, with respect to thisaction, the map E ∧ → E ∧ d is GT p -equivariant. Applying the functor R Mat, weobtain the asserted GT p -equivariant map E ≃ L p E → L p E d using the fact that,for d ≥
3, the spaces E d ( k ) are nilpotent and of finite type.The statement that the action is cyclotomic of weight 1 on H d − ( L p E d (2) , Z p )follows from [3, Proof of Theorem 7.1] where it is shown that the composite GT p → Aut h ( L p E d ) → Aut h ( L p E d (2)) ∼ = Z × p coincides with the cyclotomic character. (cid:3) From this, we can fully understand the induced GT p -action on homology. Proposition 2.4.
For any k , the GT p -action on H n ( d − ( E d ( k ) , Z p ) is cyclotomicof weight n .Proof. The homology of E d is the operad of d -Poisson algebras; it is generatedby a commutative product of degree 0, a Lie bracket of degree ( d − H ∗ ( E d ( k ) , Z p ) is aquotient of tensor powers of H ∗ ( E d (2) , Z p ). The statement of the proposition holdsfor k = 2 by Theorem 2.3 and the general case follows from Proposition 2.1. (cid:3) The Goodwillie-Weiss tower and p -completion Generalities about p -completion of towers. For a tower of based spacesand basepoint preserving maps, · · · → T n → T n − → · · · → T there is an associated spectral sequence [7, IX.4] whose E page is E − s,t ( T ) = π t − s hofiber( T s → T s − )and the first differential d : E − s,t ( T ) → E − ( s +1) ,t ( T ) is the composite π t − s hofiber( T s → T s − ) → π t − s T s → π t − s − hofiber( T s → T s − ) PEDRO BOAVIDA AND GEOFFROY HOREL where the second map is the boundary homomorphism.We assume that each T i has abelian homotopy groups (including π ) and thatthe induced maps on homotopy groups are group homomorphisms (e.g. a towerof 2-fold loop spaces). This is the case in our examples below, and having thishypothesis avoids subtleties which arise when defining this spectral sequence ingeneral, when π and π are not abelian groups. The differential d r has degree( − r, r − F s π ∗ (holim T ) bethe image of the map π ∗ (holim T ) → π ∗ T s . Using the Milnor short exact sequence,we see that, up to a lim -term, the group π ∗ (holim T ) is the limit of the tower ofepimorphisms(3.1) · · · → F s π ∗ (holim T ) → F s − π ∗ (holim T ) → · · · → F π ∗ (holim T )(indeed the limit of this tower can easily be identified with lim s π ∗ T s ) and there areinclusions ker[ F s π ∗ (holim T ) → F s − π ∗ (holim T )] ⊂ E ∞− s,s + ∗ ( T ) . For these inclusions to be isomorphisms, a sufficient condition is thatlim r E r − s,t vanishes [7, IX.5.4]. Under this hypothesis, the term lim s π ∗ T s also vanishes, andso the spectral sequence converges in the sense that the limit of the tower (3.1) is π ∗ (holim T ) and the associated graded of this tower is given by the E ∞ -page. Thevanishing of lim r E r − s,t happens, for example, if the connectivity of the layers of thetower T goes to ∞ as n → ∞ .Before we get to our main example, we need to discuss the effect of applying p -completion L p to a certain tower of spaces. Suppose { X n } n ∈ N is a tower of simplyconnected based spaces with finitely generated homotopy groups. Assume furtherthat the layers are also simply connected and consider the corresponding tower T = { Ω X n } of two-fold loop spaces and maps. We wish to compare the tower T with a tower T ( Z p ) · · · → Ω L p X n → Ω L p X n − → · · · → Ω L p X and an analogous spectral sequence whose E -page is E − s,t ( T ( Z p )) = π t − s Ω hofiber( L p X s → L p X s − ) . A warning must be issued at this point: taking homotopy fibers does not commutewith L p in general. However, under the simple connectivity assumption, we havethat hofiber( L p X s → L p X s − ) ≃ L p hofiber( X s → X s − ) . Therefore,
Proposition 3.1.
For each positive integer r , the map E r − s,t ( T ) → E r − s,t ( T ( Z p )) is given by algebraic p -completion, i.e., tensoring with Z p . ALOIS SYMMETRIES OF KNOT SPACES 9
Proof.
For r = 1, this follows from what was just said and the fact that if a space Y has homotopy groups that are finitely generated abelian, then the map π ∗ Y → π ∗ L p Y is a p -completion, in the sense that it is isomorphic to the canonical map π ∗ Y → π ∗ Y ⊗ Z p . For a finite r , the result follows by the exactness the functor A A ⊗ Z p . (cid:3) Now, suppose that we are in a situation for which there is convergence of thespectral sequence of the tower T , e.g. if the connectivity of the layers of T growsto ∞ . Then we have a tower approximating π ∗ (holim T ( Z p ))(3.2) · · · → F sp → F s − p → · · · → F p where F sp is the image of π ∗ (holim T ( Z p )) → π ∗ T ( Z p ) s , and satisfying E ∞− s,s + ∗ ( T ( Z p )) ∼ = ker( F sp → F s − p ) . This is related to the tower { F s } which approximates π ∗ (holim T ). Namely, thetower (3 .
2) is a p -completion of the tower (3 . Remark 3.2.
Without convergence hypothesis, the tower (3 .
2) may not be a p -completion of the tower (3 . p -completion can fail to commutewith infinite limits. For the same reason, the map π ∗ (holim T ) → π ∗ (holim T ( Z p ))may not be a p -completion.3.2. The Goodwillie-Weiss tower for long knots.
Let d ≥
3. The Goodwillie-Weiss tower approximating the space of long knots emb c ( R , R d ) · · · → T k emb c ( R , R d ) → · · · → T emb c ( R , R d )is a tower of 2-fold loop spaces. We briefly explain this here, following [4]. The k -th term, k ≥
2, can be described via a homotopy fiber sequence T k emb c ( R , R d ) → ΩInjLin( R , R d ) Ω f −−→ Ω Map h ( E , E d ) ≤ k where InjLin denotes the space of injective linear maps, and Map h ( E , E d ) ≤ k de-notes the space of derived maps between k -truncated operads. (That is, we restrictthe operads to operations with at most k inputs.) The map f is essentially anevaluation map (see [4] for a precise description using configuration categories).The space InjLin( R , R d ) is weakly equivalent to Map h ( E , E d ) ≤ and with thatin mind it is not hard to see that f is a section of the truncation map g k : Map h ( E , E d ) ≤ k → Map h ( E , E d ) ≤ . Therefore, we have a weak equivalence T k emb c ( R , R d ) ≃ Ω hofiber( g k ) . Moreover, the map T k +1 emb c ( R , R d ) → T k emb c ( R , R d )can be identified with the double loop map of the obvious maphofiber( g k +1 ) → hofiber( g k ) . In order to simplify notation, we denote the tower k Ω hofiber( g k ) by T . For a positive integer n , we write n for the finite set { , . . . , n } . The first pageof the spectral sequence associated to T has the following description: whenever s >
2, we have E − s,t ( T ) = π t thofiber (cid:0) S ⊂ s emb( S, R d ) (cid:1) where thofiber means the total homotopy fiber of the cube ; when s ≤
2, we have E − s,t ( T ) = 0. This identification follows from: Proposition 3.3 (G¨oppl [16]) . Let P be an operad having weakly contractible P (0) and P (1) . Let us fix a map E → P that we use as a base point for each of thespaces Map h ( E , P ) ≤ n . For n ≥ , there is a homotopy fiber sequence Ω n − thofiber ( S ⊂ n P ( S )) → Map h ( E , P ) ≤ n → Map h ( E , P ) ≤ n − where the n -cube S P ( S ) is given by compositions with -arity operations. The homotopy groups of this total homotopy fiber are well-understood. Werecord this in the following two elementary and well-known statements.
Lemma 3.4.
Let X and Y be two ( n − dismensional cubes. Let p : X → Y and s : Y → X be two maps of cubes with s a homotopy section of p (i.e. p ◦ s is homotopic to the identity). Then, for any k ≥ , if we denote by C the n -dimensional cube given by the map p , we have π k (thofiber( C )) ∼ = ker ( π k (thofiber( X )) → π k (thofiber( Y ))) where the map on the right hand side is the map induced by p .Proof. In general, there is a fiber sequencethofiber( C ) → thofiber( X ) p −→ thofiber( Y )and the existence of the map s implies that the long exact sequence of homotopygroups splits into short exact sequences:0 → π k (thofiber( C )) → π k (thofiber( X )) → π k (thofiber( Y )) → (cid:3) Proposition 3.5.
We have π ∗ thofiber ( S ⊂ n P ( S )) ∼ = n − \ i =0 (cid:0) ker π ∗ ( s i ) : π ∗ P ( n ) → π ∗ P ( n − (cid:1) where the maps s i : P ( n ) → P ( n − are the n edges of the cube originating from P ( n ) .Proof. Let us call C the cube under consideration. Let z be any point of the space P (0) (the choice does not matter since this space is contractible). The map s i is then defined as operadic composition with z in the ( i + 1)-st input. The cube C can be viewed as a map of ( n − X → Y with X ( S ) = P ( S ∪ { n } )and Y ( S ) = P ( S ) for S ⊂ n −
1. The map from X to Y is induced by operadiccomposition with z in the input labelled n .Recall that we have chosen a map E → P . Let x in P (2) be the image of anypoint in E (2) (again the choice is irrelevant). Using operadic composition with x we obtain a homotopy section Y → X as in the previous lemma. By the previouslemma, we deduce that π ∗ thofiber( C ) is the kernel of the map induced by s n − on ALOIS SYMMETRIES OF KNOT SPACES 11 π ∗ thofiber( X ). Iterating this reasoning ( n −
1) more times, we obtain the desiredresult. (cid:3)
We now fix a prime p . As in the discussion in the beginning of the section, wewant to investigate the tower · · · → Ω L p hofiber( g k ) → Ω L p hofiber( g k − ) → . . . In line with the notation of the previous subsection, we denote this tower by T ( Z p ).We also introduce another tower – denoted T ′ ( Z p ) – of the form · · · → Ω hofiber( g ′ k ) → Ω hofiber( g ′ k − ) → . . . where g ′ k is the truncation map for the operad L p E d : g ′ k : Map h ( E , L p E d ) ≤ k → Map h ( E , L p E d ) ≤ . Lemma 3.6.
The canonical map T ( Z p ) → T ′ ( Z p ) is a weak equivalence of towers.As such, it induces an isomorphism of the associated spectral sequences.Proof. We compare the layers. According to Propositions 3.3 and 3.5, the n th layerof T ′ ( Z p ) has homotopy groups n − \ i =0 (cid:0) ker π ∗ ( s i ) : ( π ∗ E d ( n )) ⊗ Z p → ( π ∗ E d ( n − ⊗ Z p (cid:1) . By Proposition 3.1, the homotopy groups of the layers of T ( Z p ) are the p -completionof the homotopy groups of the layers of T . The homotopy groups of the layers of T are also identified using Propositions 3.3 and 3.5. Taking p -completion amountsto tensoring the homotopy groups with Z p , which is an exact functor since Z p istorsion-free, and so the result follows. (cid:3) The important corollary of the above discussion is the following.
Corollary 3.7.
The tower T ′ ( Z p ) has an action of GT p . Therefore, the spectralsequence E r − s,t ( T ( Z p )) ∼ = E r − s,t ( T ′ ( Z p )) has an action of GT p .Proof. This is a straightforward consequence of Theorem 2.3. (cid:3)
Remark 3.8.
In [19, Appendix D], Kassel and Turaev construct a Galois actionon the Vassiliev tower over Q p . As we show later, the Vassiliev tower over Q p isisomorphic to the Goodwillie-Weiss tower π T ( Q p ) (defined in the next section).But we do not know whether the action from Corollary 3.7 agrees with the one ofKassel and Turaev.It is also not absurd to wonder whether there is a relation between the tower T ′ ( Z p ) and the profinite knots of Furusho [14], who also come with an action of theGrothendieck-Teichm¨uller group. Remark 3.9.
There is a closely related tower · · · → T k → · · · → T approximating the spaceemb c ( R , R n ) = hofiber(emb c ( R , R n ) → ΩInjLin( R , R d ))and whose k -term is T k = hofiber( T k → ΩInjLin( R , R d )) ≃ Ω Map h ( E , E d ) ≤ k . This tower is identified (Sinha [25]) with the tower associated to a pointed cosim-plicial space K • · · · → holim ∆ ≤ k K • → holim ∆ ≤ k − K • → · · · → holim ∆ ≤ K • = K The first page of the spectral sequence associated to this tower T agrees with thatof the tower T , except on the column s = 2. All the discussion above applies to thetower T without any additional difficulty. In section 7 we study the Bousfield-Kanhomology spectral sequence of this cosimplicial space.4. Main theorem
In this section, we prove the following Theorem from which Theorem B of theintroduction is an easy consequence.
Theorem 4.1.
Let p be a prime number. The differential d r − s,t : E r − s,t ( T ( Z p )) → E r − s − r,t + r − ( T ( Z p )) vanishes if r − is not a multiple of ( p − d − and if t < p − s − d − . We begin by recalling the theorem of Hilton-Milnor in a form that we will need.Given a finite set R , we denote by W ( R ) a set of Lie words in the finite set R thatforms a basis for the free Lie algebra on R . For w ∈ W ( R ) we write | w | for itslength. Theorem 4.2 (Hilton-Milnor) . There is a weak equivalence Y w ∈ W ( R ) ′ Ω S | w | ( d − ≃ −→ Ω _ R S d − ! where the symbol Q ′ stands for the weak product (union of finite products). Let T ( Q p ) be the tower analogous to T ( Z p ) obtained by replacing all instancesof L p Z , Z a space, by their rationalizations ( L p Z ) Q . As in the previous section,we have a GT p -action on T ( Q p ) and so a GT p -action on the associated spectralsequence E ( T ( Q p )) (c.f. Corollary 3.7). Note also that by exactness of the functor − ⊗ Z p Q p , we have the following isomorphisms for any r , s and tE r − s,t ( T ( Q p )) ∼ = E r − s,t ( T ( Z p )) ⊗ Q ∼ = E r − s,t ( T ) ⊗ Q p the first of which is GT p -equivariant. Theorem 4.3.
Let d ≥ . The Q p -vector space E − s,t ( T ( Q p )) is zero unless t = n ( d − for some n ≥ s − . If t = n ( d − , the GT p -actionis cyclotomic of weight n .Proof. We assume s >
2, otherwise E − s,t ( T ( Q p )) is zero. Then E − s,t ( T ( Q p )) = π t thofiber (cid:0) S ⊂ s L p emb( S, R d ) (cid:1) ⊗ Q . Regard the s -cube S L p emb( S, R d ) as an ( s − R ⊂ s − χ R ALOIS SYMMETRIES OF KNOT SPACES 13 where χ R is the map L p emb( R ∪ { s } , R d ) → L p emb( R, R d ) forgetting the pointlabelled s . Then the total homotopy fiber of the original cube is identified withthe total homotopy fiber of the ( s − R hofiber χ R . This ( s − R φ R := L p ( ∨ R S d − ) . where the maps collapse wedge summands. This follows from the Fadell-Neuwirthfiber sequence ∨ R S d − → emb( R ∪ { s } , R d ) → emb( R, R d )and the fact that L p commutes with taking homotopy pullbacks of simply connectedspaces. By Proposition 3.5, the canonical projection(4.1) thofiber( φ ) → L p (cid:0) ∨ s − S d − (cid:1) is injective on homotopy groups. Now, the GT p action on E − s,t ( Q p ) comes from a(basepoint preserving) action on the cube S L p emb( S, R d ), or equivalently, froman action on the cube φ . As such, the projection map (4.1) is GT p -equivariant.This means that to understand the action on E − s,t ( Q p ) it is enough to understandthe action on π t L p (cid:0) ∨ s − S d − (cid:1) ⊗ Q ∼ = π t (cid:0) ∨ s − S d − (cid:1) ⊗ Q p . Using Hilton-Milnor theorem (Theorem 4.2) or, alternatively, Milnor-Moore, therational homotopy groups of a wedge of ( d − n ( d −
2) + 1, for n ≥
1. The integer n records the size of the word inthe Hilton-Milnor decomposition. In the total homotopy fiber of the ( s − R
7→ ∨ R S d − , n must be at least s − ι ∈ π d − ( S d − ) ofeach wedge summand must occur at least once. This establishes the first part ofthe theorem. The second part is a consequence of the fact that the GT p -action on π n ( d − ( ∨ S d − ) ⊗ Q p is cyclotomic of weight n . This is the content of Proposition4.4 below when n = 1. For higher n ’s, these homotopy groups are generated byWhitehead products of elements in π d − , and since the GT p action exists at thelevel of spaces it must be compatible with Whitehead products. (cid:3) Proposition 4.4.
The GT p action on π d − L p ( ∨ k S d − ) ∼ = ⊕ k Z p is cyclotomic of weight .Proof. For the duration of this proof, we will drop L p from the notation and so, fora space X , we keep X as notation for L p X .By Proposition 2.4, the statement holds for k = 1. For k >
1, set ℓ = k + 1 andconsider the commutative squareemb( ℓ, R d ) emb( k, R d )emb(2 , R d ) emb(1 , R d ) forget k + 1forget 2 where the left map is induced by the inclusion f : 2 → ℓ given by f (1) = j , f (2) = k + 1; and the right map is induced by the map 1 → k selecting j . Theinduced map on horizontal homotopy fibers has the form p j : ∨ k S d − → S d −
14 PEDRO BOAVIDA AND GEOFFROY HOREL and it corresponds to collapsing all summands except the j -th one to the basepoint.All the (based) maps in the square are GT p -equivariant since they correspond tocomposition maps in the operad E d involving nullary operations, and so p j is a GT p -equivariant map.Let α be an element of GT p . To describe the action of α on the wedge ∨ k S d − we must describe, for each pair 1 ≤ i, j ≤ k , the composition S d − i −−−→ ∨ k S d − α −→ ∨ k S d − p j −→ S d − where incl i means the inclusion of the i -th summand. Since p j ◦ α = α ◦ p j , thiscomposite equals α ◦ p j ◦ incl i , which is α if i = j and is trivial otherwise. (cid:3) Proof of Theorem 4.1.
Let W denote the total homotopy fiber of the ( s − R
7→ ∨ R S d − , for s >
2. We look at the p -torsion in the homotopy groups of W .By the naturality in the Hilton-Milnor theorem (Theorem 4.2), it follows thatΩ W ≃ Y w ′ Ω S | w | ( d − where w runs over the words in the letters 1 , . . . , s − atleast once . As such, the smallest word in the product has length s −
1, and thesphere of smallest dimension is S ( s − d − .By a famous result of Serre, the homotopy groups π ∗ S ℓ are p -torsion free for ∗ ≤ ℓ + 2 p − ℓ ≥
3. Therefore, the homotopy groups π ∗ W are p -torsion freewhenever ∗ ≤ N where N = ( s − d −
2) + 2 p − , (remember d ≥ s ≥ s − d −
2) + 1 ≥ t ≤ N , we have an inclusion E − s,t ( T ( Z p )) ֒ → E − s,t ( T ( Z p )) ⊗ Q ∼ = E − s,t ( T ( Q p )) . Therefore, by Theorem 4.3, in the range t ≤ N , the group E − s,t ( T ( Z p )) is zerounless t = n ( d −
2) + 1, in which case GT p acts with weight n . The same is truefor the successive pages E r − s,t by Proposition 2.1. Proposition 2.2 completes theproof. (cid:3) We can now prove Theorem B of the introduction.
Proof of Theorem B.
Recall that Z ( p ) denotes the ring of p -local integers. Propo-sition 3.1 gives an isomorphism of spectral sequences E ∗∗ , ∗ ( T ( Z p )) ∼ = E ∗∗ , ∗ ( T ) ⊗ Z Z p ∼ = ( E ∗∗ , ∗ ( T ) ⊗ Z Z ( p ) ) ⊗ Z ( p ) Z p Then the result follows from the fact that a map f of Z ( p ) -modules vanishes if andonly if f ⊗ Z ( p ) Z p vanishes. (cid:3) Some consequences
Corollary 5.1.
The spectral sequence E ∗− s,t ( T ) ⊗ Z Q collapses at the E -page.Proof. Let d r − s,t be a differential in that spectral sequence with r >
2. Up tochoosing p a large enough prime, we may assume that t < p − s − d − r − p − d − E r − s,t ( T ) ⊗ Z Z ( p ) by Theorem B so it is also zero after inverting p . (cid:3) ALOIS SYMMETRIES OF KNOT SPACES 15
Remark 5.2.
For d ≥
4, this result is due to Arone, Lambrechts, Turchin andVoli´c (see [1]). The case d = 3 does not appear in loc. cit. and, to the best of ourknowledge, does not appear elsewhere in the literature. The reason seems to bethat the relative formality of the map of operads E → E was not known when [1]was written. Corollary 5.3.
Let p be a prime. For n ≤ ( p − d −
2) + 3 , the spectral sequenceassociated to T n ( Z p ) , computing π ∗ T n emb c ( R , R d ) ⊗ Z p , collapses at the E -page for ( − s, t ) satisfying t < p − s − d − .Proof. Let T ( s ) := 2 p − s − d − A be the region of the second-quadrant consisting of those ( − s, t ) such that t < T ( s ). By theorem 4.1, the firstpossibly non-trivial differential in the region A is d R : E R − s,t → E R − s − R,t + R − . with R = ( p − d −
2) + 1. Since the first non-zero column is s = 3, the firstpossibly non-trivial differentials land in E R − − R,t + R − . But the groups E − s, ∗ arezero whenever s > n . So E R − − R,t + R − is zero, and so are the target groups ofhigher differentials from the region A . Therefore, the spectral sequence collapses inthe region A whenever n < R + 3. (cid:3) Remark 5.4.
Note that the corollary has nothing to offer at the prime 2.
Remark 5.5.
Theorem 4.1 has something to say about the homotopy groups of π ∗ ( T n ) even outside of the range of the previous corollary. For instance, let usconsider π ( T n ) ⊗ Z p in the case of knots in R . This abelian group has filtrationwhose associated graded is ⊕ s ≤ n E ∞− s,s ( T n ( Z p )). The corollary tells us that we haveisomorphisms E − s,s ∼ = E ∞− s,s in the range s ≤ p + 2. For p + 3 ≤ s ≤ p + 1, the only differential that can hit E − s,s is d p , therefore, we have an isomorphism E ∞− s,s ∼ = E p +1 − s,s ∼ = E − s,s / Im( d p )For 2 p + 2 ≤ s ≤ p , the group E ∞− s,s will be the quotient of E − s,s by the image of d p and d p − . This pattern continues.In the remainder of this section, we prove some results about extensions leadingto the proof of Theorem C from the introduction. Let us fix a unit u ∈ Z p suchthat the residue of u modulo p is a generator of F × p . For M a Z p -module and i aninteger, we denote by M ( i ) the Z p [ t, t − ]-module M where t acts as multiplicationby u i . When i = 0, we simply write M for M (0). Proposition 5.6.
Let M be a Z p -module of finite type. Let i and j be two integerssuch that ( p − does not divide i − j . Then the group Ext Z p [ t,t − ] ( M ( i ) , N ( j )) iszero.Proof. Since M can be written as finite direct sums of copies of Z p and Z /p n ,wthout loss of generality, we may assume that M is either Z p or Z /p n . The shortexact sequence 0 → Z /p n → Z /p n − → Z /p → induces an exact sequenceExt Z p [ t,t − ] ( Z /p ( i ) , N ( j )) → Ext Z p [ t,t − ] ( Z /p n +1 ( i ) , N ( j )) → Ext Z p [ t,t − ] ( Z /p n ( i ) , N ( j ))Therefore, by induction on n , we can reduce the case M = Z /p n to the case M = Z /p . The short exact sequence0 → Z p .p −→ Z p → Z /p → Z p [ t,t − ] ( Z p ( i ) , N ( j )) → Ext Z p [ t,t − ] ( Z /p ( i ) , N ( j )) → Ext Z p [ t,t − ] ( Z p ( i ) , N ( j ))therefore, the case M = Z /p follows from the vanishing of Hom Z p [ t,t − ] ( Z p ( i ) , N ( j ))and of Ext Z p [ t,t − ] ( Z p ( i ) , N ( j )). The vanishing of the Hom-group is similar to theproof of Proposition 2.2. In order to compute the Ext-group, we use the followingprojective resolution of Z p ( i )0 → Z p [ t, t − ] . ( t − u i ) −−−−→ Z p [ t, t − ] → Z p → . We obtain an exact sequence N → N → Ext Z p [ t,t − ] ( Z p ( i ) , N ( j )) → u j − u i . Since j − i is not a multiple of( p − u j − u i is a unit of Z p and the result follows. (cid:3) Proposition 5.7.
Let M be a Z p -module of finite type equipped with a GT p -action.Let n and m be two integers with ≤ n − m < p − . Assume that M sits at thetop of a GT p -equivariant tower M = M n p n −→ M n − p n − −−−→ . . . → M m p m −−→ where each map p k is surjective. Assume further that, for each k , the GT p -actionon the kernel F k of p k is cyclotomic of weight k . Then there is a non-canonicalisomorphism of Z p -modules M ∼ = F n ⊕ . . . ⊕ F m . Proof.
It suffices to prove that, for each k , the short exact sequence0 → F k → M k → M k − → Z p ( M k − , F k ). Since the shortexact sequence is GT p -equivariant, the obstruction lies in the image of the obviousmap Ext Z p [ GT ] ( M k − , F k ) → Ext Z p ( M k − , F k ) . Pick an element u of Z × p such that the residue of u modulo p is a generator of F × p .Pick a lift t of u in GT p . This choice induces a homomorphism of Z p -algebras Z p [ t, t − ] → Z p [ GT ]and we deduce that the obstruction lies in the image of the obvious mapExt Z p [ t,t − ] ( M k − , F k ) → Ext Z p ( M k − , F k ) ALOIS SYMMETRIES OF KNOT SPACES 17
Hence it is enough to show that the group Ext Z p [ t,t − ] ( M k − , F k ) is zero. We willin fact prove that the group Ext Z p [ t,t − ] ( M i , F k ) = 0 for any m ≤ i < k . We provethis by induction on i . For i = m , we haveExt Z p [ t,t − ] ( M m , F k ) = 0by the previous proposition. If the statement is true for i , then, we use the longexact sequence induced by the short exact sequence0 → F i +1 → M i +1 → M i → (cid:3) Using this proposition, we can prove Theorem C from the introduction.
Proof of Theorem C.
The analogous statement with Z p replaced by Z ( p ) followsfrom Corollary 5.3 and the previous proposition. If M and N are finitely generated Z ( p ) -modules, the canonical mapExt( M, N ) ⊗ Z p → Ext Z p ( M ⊗ Z p , N ⊗ Z p )is an isomorphism. This can be reduced to the case where M and N are either Z ( p ) or Z /p k in which case this is an easy computation. It follows that if an extensionof Z ( p ) -modules splits over Z p , it also splits over Z ( p ) . (cid:3) Universality
We now come to the proof of theorem A from the introduction. To lightennotations, we denote the space of knots emb c ( R , R ) by K in this section, aswe did in the introduction. Given two knots f, g , we write f ∼ n − g if f and g share the same type n − π K ;the set of equivalences classes is denoted π ( K ) / ∼ n − . It is not hard to see thatthe operation of concatenation makes this set into a commutative monoid. Thiscommutative monoid is in fact a finitely generated abelian group as shown in [17]. Theorem 6.1.
Let R be a commutative ring that is torsion-free (e.g. Z , Q , Z p , Z ( p ) ). The evaluation map e n : π K → π T n K ⊗ R is a universal Vassiliev invariant of degree n − over R if the canonical map E − k,k ⊗ R → E ∞− k,k ⊗ R is an isomorphism for all k ≤ n .Proof. In [8], and [21], it it was shown that e n is well-defined on equivalence classesand as such factors through e n : ( π ( K ) / ∼ n − ) → π T n K . The statement of thetheorem is equivalent to the statement that e n ⊗ R is an isomorphism.We argue inductively, using the commutative square of group homomorphisms π ( K ) / ∼ n − π T n Kπ ( K ) / ∼ n − π T n − K e n e n − and compare the map between vertical kernelsΦ n − → E ∞− n,n where Φ n − denotes the kernel of the left vertical map. In [11], Conant-Teichnerconstruct a surjective homomorphism R n : E − n,n → Φ n − and in [20], Kosanovi´cshows that the composition E − n,n → Φ n − → E ∞− n,n agrees with the canonical map. Since the composition is an isomorphism by hypoth-esis, we then have that both maps are isomorphisms. It follows by an application ofthe five lemma that e n ⊗ R is an isomorphism if and only if the the homomorphism E − n,n ⊗ R → E ∞− n,n ⊗ R is an isomorphism, as claimed. (cid:3) Proof of Theorem A.
The first part follows from Theorem 6.1 in conjunction withTheorem B. The second part is subsumed by Theorem C. (cid:3) The homology Goodwillie-Weiss spectral sequence
The Goodwillie-Weiss tower for emb c ( R , R d ), the homotopy fiber of the inclusionemb c ( R , R d ) → imm c ( R , R d ) , can also be described as follows. Consider the category O k whose objects are openproper subsets T of R containing ( −∞ , ∪ [1 , + ∞ ) and having at most k + 2connected components. A morphism T → R is an isotopy connecting T ⊂ R to asubset of R ⊂ R , and which fixes ( −∞ , ∪ [1 , + ∞ ) pointwise. Then T k emb c ( R , R d ) = holim T ∈O op k emb c ( T, R d ) . The Goodwillie-Weiss tower agrees with the tower associated to a cosimplicial space,the one coming from the usual filtration of ∆ by the subcategories ∆ ≤ k spannedby the objects [ n ] satisfying n ≤ k . This was first proved by Sinha [25]. Roughly,the relation comes from the equivalence between the topological category O k andthe opposite of ∆ ≤ k . It follows that the functor emb c ( − , R d ) defines a cosimplicialspace whose homotopy limit is the limit of the Goodwillie-Weiss tower. In thissection, we review an operadic construction of this cosimplicial space and use it tostudy the associated homology Bousfield-Kan spectral sequence.Recall that a multiplicative operad is a non-symmetric topological or simplicialoperad P together with the data of a map α : A → P where A denotes the non-symmetric associative operad. From such data, one can construct a cosimplicialspace X • as we now recall (see [25, Definition 2.17]). We denote by u the image in P (0) of the unique point in A (0) and by m the image in P (2) of the unique pointin A (2).In degree q , our cosimplicial space will be given by X q = P ( q ). The cofaces d i : P ( q ) → P ( q + 1) are as follows. The two outer cofaces d and d q +1 take x ∈ P ( q ) to m ◦ x and m ◦ x respectively. The inner coface d i with i ∈ { , . . . , q } takes x to x ◦ i m . The codegeneracy s i : P ( q ) → P ( q −
1) takes x ∈ P ( q ) to x ◦ i u .In what follows, we will start with a map ˜ A → P where ˜ A is not quite theassociative operad but is merely weakly equivalent to it. The following propositionwill be useful in that situation. We denote by Op ns the category of non-symmetricoperads in simplicial sets. ALOIS SYMMETRIES OF KNOT SPACES 19
Proposition 7.1.
Let ˜ A be a non-symmetric operad and let w : ˜ A → A be a mapof non-symmetric operads. Consider the adjunction w ! : Op ˜ A/ns ⇆ Op A/ns : w ∗ where w ∗ is precomposition by w and w ! is its left adjoint. Then, if w is a weakequivalence, the adjunction is a Quillen equivalence.Proof. First, we easily verify that the left adjoint sends ˜ A → P to the bottom mapin the following pushout square. ˜ A / / (cid:15) (cid:15) P (cid:15) (cid:15) A / / Q The result is then completely standard once we know that the model category Op ns is left proper (see [23, Corollary 1.12]). (cid:3) Fix a linear inclusion R → R d . This induces a map of non-symmetric operads E ns → E d where E ns is the non-symmetric little 1-disks operad. The unique map w : E ns → A being a weak equivalence, we can apply the derived functor of w ! (using the notation of Proposition 7.1) to it and get a map A → P for some non-symmetric operad P . Moreover, by Proposition 7.1 this map is weakly equivalentto E ns → E d in the arrow category of Op ns . The cosimplicial space associated tothis map is what we will denote by K • . It is identified with the cosimplicial spacementioned in the beginning of the section.We can do the exact same construction but starting from the map L p E → L p E d (noticing that L p E ≃ E ) and we obtain a cosimplicial space that we denoteby L p K • . Since the map L p E → L p E d is GT p -equivariant (Theorem 2.3), thecosimplicial space L p K • d gets a GT p -action.Applying the functor C ∗ ( − , A ) to the cosimplicial space K • , we get a bicomplexand hence a spectral sequence E ∗∗ , ∗ ( A ) with E p,q ( A ) = H q ( K − pd , A )and whose d is the alternating sum of the coface maps. Our main theorem aboutthis spectral sequence is the following (Theorem D in the introduction). Theorem 7.2.
Let p be a prime number. The only possibly non-trivial differentialsin the spectral sequence E ∗ ( Z ( p ) ) are d n ( d − p − for n ≥ .Proof. As in the proof of Theorem B, it suffices to prove the theorem for the spectralsequence E ∗ ( Z p ). As we said above, we have a GT p -action on the cosimplical space L p K • d . Using Proposition 1.5, we obtain an action on C ∗ ( X, Z p ) and hence on thespectral sequence E ∗ ( Z p ). Moreover, the action on E ∗∗ , ( d − k is cyclotomic of weight k . Indeed, this is true on the E page by Proposition 2.4 and therefore on any pageby Proposition 2.1. The result now follows from Proposition 2.2. (cid:3) In particular, we recover the following theorem.
Corollary 7.3 (Lambrechts-Turchin-Voli´c, [22]) . The spectral sequence E ∗ ( Q ) col-lapses at the second page.Proof. The proof is analogous to the proof of Corollary 5.1. (cid:3)
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ALOIS SYMMETRIES OF KNOT SPACES 21
Dept. of Mathematics, IST, Univ. of Lisbon, Av. Rovisco Pais, Lisboa, Portugal
E-mail address : [email protected] Universit´e Sorbonne Paris Nord, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse,France´Ecole normale sup´erieure, DMA, CNRS, UMR 8553, 45 rue d’Ulm, 75230 Paris Cedex05, France
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