Goodwillie's cosimplicial model for the space of long knots and its applications
GGOODWILLIE’S COSIMPLICIAL MODEL FOR THE SPACE OF LONGKNOTS AND ITS APPLICATIONS
YUQING SHI
Abstract.
We work out the details of a correspondence observed by Goodwillie betweencosimplicial spaces and good functors from a category of open subsets of the intervalto the category of compactly generated weak Hausdorff spaces. Using this, we computethe first page of the integral Bousfield–Kan homotopy spectral sequence of the tower offibrations given by the Taylor tower of the embedding functor associated to the spaceof long knots. Based on the methods in [Con08], we give a combinatorial interpretationof the differentials d mapping into the diagonal terms, by introducing the notion of ( i, n ) -marked unitrivalent graphs. Contents
Introduction 11. Background and motivation 52. A cosimplicial model constructed by Goodwillie 103. An integral homotopy spectral sequence for E mb • Introduction
Manifold calculus is a theory introduced by Goodwillie and Weiss, cf. [Wei99] and[GKW01], that produces a sequence of functors approximating a given good functor(Definition 1.1.3) from the category of open subsets of a manifold to the category
CGH ofcompactly generated weak Hausdorff spaces. Let M and N be smooth manifolds. We areinterested in studying the space Emb ∂ ( M, N, f ) of smooth embeddings that are germ equiv-alent on the boundary of M to a fixed smooth boundary-preserving embedding f : M → N .We topologise this space with the strong Whitney C ∞ -topology (Remark 1.1.5). Now we canapply manifold calculus to the embedding functor Emb ∂ ( − , N, f ) : Open ( M ) op ∂ → CGH ,sending an open subset V ⊆ M with ∂M ⊆ V to the embedding space Emb ∂ ( V, N, f ) .In this way we obtain information about the space Emb ∂ ( M, N, f ) by studying the em-bedding functor and its sequence of approximations.In this paper, we focus on analysing the following embedding functor Emb( − ) : Open ∂ (I) op → CGH V (cid:55)→ Emb ∂ ( V, R × D , c ) associated to the space K = Emb ∂ (I , R × D , c ) of long knots where c is an embeddingrepresenting the trivial long knot. Our motivation for studying this embedding functoris its close relation to knot theory, in particular, the theory of Vassiliev invariants. InSection 1 we recall some background on manifold calculus and give a precise definition of Date : December 9, [email protected]. a r X i v : . [ m a t h . A T ] D ec YUQING SHI the space K . At the end of this section, we summarise briefly the original constructionof the Vassiliev invariants and present how the theory of these knot invariants and thetheory manifold calculus relate.As mentioned above, manifold calculus associates to Emb( − ) a sequence of polynomialfunctors T n Emb( − ) (Definition 1.1.8) approximating the functor Emb( − ) , Emb( − ) . . . T n Emb( − ) T n − Emb( − ) . . . T Emb( − ) . η n η n − η r n +1 r n r n − r The induced map π ( η n (I)) : π (Emb(I)) → π (T n Emb(I)) is an additive Vassiliev invari-ant of degree at most n − and conjecturally it is the universal one, cf. [BCKS17]. It isknown that the map π ( η n (I)) is surjective, cf. [Kos20].Our approach to understanding the maps π ( η n (I)) is to compute the Bousfield–Kanhomotopy spectral sequence associated to the tower of fibrations · · · r n +1 (I) −−−−→ T n Emb(I) r n (I) −−→ T n − Emb(I) r n − (I) −−−−→ · · · r (I) −−→ T Emb(I) . (1)As a first step, we study in Section 2 the construction [GKW01, Contruction 5.1.1] thatrelates cosimplicial spaces with good functors from Open ∂ (I) to the category of spaces.In Theorem 2.11 we give a more precise formulation of this construction, summarised asthe theorem below. Theorem 2.
The category of good functors F : Open ∂ (I) op → CGH is equivalent, as asimplicial functor category, to the category of cosimplicial spaces. We were not able to find a proof of the above theorem or of [GKW01, Contruction 5.1.1]in the literature. Thus, we give a proof in Section 2, using some results of [AF15]. Therefore,we can associate a cosimplicial space E mb • to the embedding functor Emb( − ) via the aboveequivalence, which provides us with a cosimplicial model for the tower of fibrations (1),namely Tot n E mb • (cid:39) T n Emb(I)Tot E mb • (cid:39) holim n ≥ T n Emb(I) . We find the construction of this cosimplicial model very natural, yet it is the least usedone in the study of the space of long knots. There is another cosimplicial model C • of thetower of fibration (1) which is defined using the Fulton–MacPherson compactification, cf.[Sin06] and [BCKS17]. In Remark 2.24 we briefly recall the construction and propertiesof C • .The n -th level E mb n of the cosimplicial space E mb • is weakly homotopy equivalent to thecartesian product of the configuration space Conf n ( R × D ) and n copies of the sphere S .In Section 3.3, we compute the Bousfield–Kan homotopy spectral sequence { E rp,q } p,q ≥ withintegral coefficients associated to E mb • , using a presentation of the homotopy groups ofthe configuration spaces via iterated Whitehead products. More specifically, we are able togive an explicit description of the abelian groups E p,q (Proposition 3.3.5 and Remark 3.3.8).Thanks to this, we obtain a simplification of the calculation of the differentials mappinginto the diagonal terms on the E -page. OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 3
Theorem 3 (Theorem 3.3.13) . For p ≥ , the differential d : E p − ,p → E p,p can beexpressed as an explicit sum of four iterated Whitehead products. In Section 4, we give a combinatorial interpretation of the groups E p,p and E p − ,p , as wellas the differentials d : E p,p → E p − ,p , based on the methods of [Con08]. The group E p,p isisomorphic to the abelian group T p − of labelled unitrivalent trees of degree p − with atotal ordering on its leaves, modulo AS- and IHX-relations (Proposition 4.5). In Figure 1we draw an example of a labelled unitrivalent tree of degree . Figure 1.
A labelled unitrivalent tree of degree . The arrow is not partof the tree.In order to describe the groups E p − ,p , we introduce the notion of ( i, p ) -marked unitriva-lent graphs (Definition 4.10). In Figure 2 we draw an example of a marked unitrivalentgraph of degree . Figure 2. A (3 , -marked unitrivalent graph. The black dots indicate themarked nodes. The arrow is not part of the graph. Proposition 4 (Proposition 4.15) . Let D p − be the abelian group generated by ( i, p − -marked unitrivalent graphs with ≤ i ≤ p , modulo AS- and IHX sep -relations. Then wecan identify D p − with the torsion-free part of E p − ,p . The differentials d have the following interpretation using unitrivalent graphs. Theorem 5 (Theorem 4.20) . Let p ≥ . Under the identifications above, the differen-tial d : E p − ,p → E p,p maps a ( k, p − -marked unitrivalent graph Γ k,p − to the linearcombination Γ p − Γ p − ( Γ k − Γ k ) of unitrivalent trees of degree p − , where i) the linear combination Γ p − Γ p is obtained by performing the STU-relation (Def-inition 4.17) on Γ k,p − at the edge connecting the leave labelled by p − and themarked node v p − , and ii) the linear combination Γ k − Γ k is obtained by performing the STU-relation on Γ k,p − at the edge connecting the leave labelled by k and the marked node v p − . See Figure 3 for a visualisation of the graphs occurring in the theorem. We call theequivalence relation on the abelian group T p − (labelled unitrivalent trees of degree p − )generated by the image of d the STU equivalence relation (Definition 4.17). As a corollarywe have the following proposition. For p = 1 , the differential is null because the domain and the codomain are null. For p = 2 , thedifferential is an isomorphism by concrete calculation. YUQING SHI
Figure 3.
An example of d applied to a ( k, p − -marked unitrivalentgraph. The triangles are placeholders for subgraphs, which stay unmodified. Proposition 6 (Conant) . A tree τ ∈ T p − is STU -equivalent to 0 if and only if τ ∈ im( d ) under the isomorphism E p,p ∼ = T p − of Proposition 4.5. We also obtain a combinatorial interpretation of E p,p for p ≥ . Corollary 7 (Corollary 4.24) . i) E p,p is isomorphic to the abelian group generated by unitrivalent tree of degree p − ,modulo AS-, IHX-, and STU -relations, for p ≥ . ii) E , ∼ = E , ∼ = T ∼ = Z , for p = 3 . iii) E p,p = 0 , for p = 0 , , . We conclude in Section 5 by illustrating the connection between the manifold calculustower of
Emb( − ) and some geometrical and combinatorial aspects of Vassiliev knotinvariants, and mention some future work. Acknowledgements.
This text is an updated version of part of my master thesis writtenat University of Bonn. I want to thank Lukas Brantner, Danica Kosanovic, Achim Krausefor the helpful conversations during the thesis project. Also, I am thankful to TobiasDyckerhoff, the second reader of my thesis, for his feedback on this part of the thesis. Iwould like to thank Jack Davies, Gijs Heuts, Geoffroy Horel, Pablo Magni and MingcongZeng for their encouragement and feedback on the updated version. Most of all, I want toexpress my thanks to my thesis supervisor Peter Teichner for his suggestions, guidanceand support during the thesis project.
Notation.
Throughout the text, we denote by I , D n and S n the unit interval, the unit n -disk and the unit n -sphere, respectively. Situation.
We work with simplicial categories, i.e. categories enriched over the categoryof simplicial sets equipped with the Kan model structure . Most of the categories weconsider in this text are by nature enriched over the category of compactly generatedweakly Hausdorff spaces. We regard them as simplicial categories by applying the singularcomplex functor to the mapping spaces. For simplicity, we call the mapping simplicialsets obtained as above again mapping spaces . In particular, every ordinary category isa topologically enriched category with discrete morphism spaces. Thus we also considerthem as simplicial categories. For an introduction on (simplicial) enriched categories andenriched functors see [Lur09, A.1.4] and [Hir03, Chapter 9].We denote by holim the homotpy limits in simplicial categories. We refer the reader to[Hir03, Chapter 18] for a detailed explanation these homotopy limits. For the calculationof homotopy limits in the category of compactly generated weak Hausdorff spaces see[Hir03, Chapter 18.2], [Rie14, Chapter 6] and [MV15, Section 8]. See [Lur09, Appendix A.2.7] for the Kan model structure on the category of simplicial sets.
OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 5 Background and motivation
Manifold Calculus.
In this section we are going to briefly introduce the basicbuilding blocks of manifold calculus. The main references for this section are [BW13],[Wei99] and [GW99], which also contain further motivation for this subject.
Definition 1.1.1. i) Denote by
CGH the simplical category of compactly generated weak Hausdorffspaces. We will call
CGH the category of spaces for simplicity.ii) For a manifold M , denote by Open ∂ ( M ) the category of open subsets of M whichcontain ∂M with morphisms the inclusions of these open subsets. For a manifold M (cid:48) without boundary, we will simplify the notation as Open ( M (cid:48) ) . Definition 1.1.2.
A smooth codimension zero embedding i v : ( V, ∂V ) → ( W, ∂W ) betweensmooth manifolds V and W is an isotopy equivalence if there exists a smooth embedding i w : ( W, ∂W ) → ( V, ∂V ) such that i v ◦ i w and i w ◦ i v are isotopic to id ( W,∂W ) and id ( V,∂V ) respectively. Definition 1.1.3.
Let M be a smooth manifold of dimension m . A good functor on Open ∂ ( M ) is a simplicial functor F : Open ∂ ( M ) op → CGH of simplicial categories,which satisfies the following conditions:i) (isotopy invariant) If i ∈ Mor
Open ( M ) ( V, W ) is an isotopy equivalence, then F ( i ) isa weak homotopy equivalence;ii) For any filtration . . . V i ⊆ V i +1 . . . of open subsets of M , the canonical map F ( ∪ i ∈ N V i ) → holim i ∈ N F ( V i ) is a weak homotopy equivalence. Definition 1.1.4.
Let M and N be smooth manifolds.i) We define Emb(
M, N ) to be the space of smooth embeddings of M into N .ii) When M and N are smooth manifolds with boundary, we define Emb ∂ ( M, N ) tobe the space of smooth embeddings F : M → N which preserve the boundary, i.e. F ( ∂M ) ⊆ ∂N .iii) We define Emb ∂ ( M, N, f ) to be the space of smooth embeddings that coincidewith a given smooth embedding f : M (cid:44) → N near the boundary that are transverseto ∂N , i.e. Emb ∂ ( M, N, f ) := { F ∈ Emb ∂ ( M, N ) | F (cid:116) ∂N, F and f are germ equivalent at ∂M } . We topologise
Emb(
M, N ) and Emb ∂ ( M, N, f ) with the strong Whitney C ∞ -topology. Remark . Let M and N be smooth manifolds. The strong Whitney C ∞ -topology is aconvenient topology on the set C ∞ ( M, N ) of smooth maps from M to N . For a smoothmap f ∈ C ∞ ( M, N ) , a small neighbourhood of f under this topology consists of smoothmaps g such that g and all its derivatives of order k are close to f , for all k ≥ . Thecloseness here is specified by the metric of the k -jet J k ( M, N ) of mappings from M to N .For an introduction of Whitney C ∞ -topology and its properties, see [GG73, Chapter II.3]and [Hir76, Chapter 2.1] . Theorem 1.1.6 (Weiss) . Let M and N be smooth manifolds with dim M ≤ dim N , andlet f : M (cid:44) → N be a fixed smooth embedding. Then the embedding functor Emb ∂ ( − , N, f ) : Open ∂ ( M ) op → CGH V (cid:55)→ Emb ∂ ( V, N, f ) is a good functor. YUQING SHI
Proof.
See [Wei99, Proposition 1.4]. (cid:3)
Now we are going to introduce the approximation sequence for good functors producedby manifold calculus.
Definition 1.1.7.
Denote by [ n ] the set { , , . . . , n } .i) Define the category Pow ([ n ]) as the category whose objects are the subsets of [ n ] and the morphisms are inclusions of subsets.ii) Define the full subcategory Pow ([ n ]) (cid:54) = ∅ of Pow ([ n ]) , whose objects are the non-empty subsets of [ n ] . Definition 1.1.8.
For a manifold M without boundary, a good functor F is a polynomialfunctor of degree at most n if for every open subset U ∈ Open ( M ) and A , A , . . . , A n pairwise disjoint closed subsets of M which lie in U , the ( n + 1) -cube χ : Pow ([ n ]) → CGH S (cid:55)→ F ( U \ ∪ i ∈ S A i ) is homotopy cartesian, i.e. the natural map χ ( ∅ ) → holim S (cid:54) = ∅ χ ( S ) is a weak homotopyequivalence. In other words, F ( U ) → holim S (cid:54) = ∅ F ( U \ ∪ i ∈ S A i ) is a weak homotopyequivalence. Remark . One obtains the definition of polynomial functors of degree at most n formanifolds M with boundary by replacing Open ( M ) with Open ∂ ( M ) and requiring thateach A i has empty intersection with ∂U so that F ( U \ ∪ i ∈ S A i ) is well-defined.The name ‘polynomial functor’ may come from the following criterium for polynomialfunctions. Lemma 1.1.10.
A smooth function p : R → R satisfies (cid:88) S ⊆ [ n ] ( − S p (cid:32)(cid:88) i ∈ S x i (cid:33) = 0 (1.1.11) for any collection of real numbers x , x , . . . , x n if and only if p is a polynomial of degreeat most n . (cid:3) Definition 1.1.12.
Let M be a smooth manifold and let n ∈ N . Define Open n∂ ( M ) to bethe full subcategory of Open ∂ ( M ) whose objects are the open subsets W of M that arediffeomorphic to N( ∂M ) (cid:116) ( (cid:70) k R m ) with ≤ k ≤ n . Here N( ∂M ) denotes a (non-fixed)tubular neighbourhood of ∂M in M . Definition 1.1.13.
Let M be a manifold and let F be a good functor. The n -th Taylorapproximation T n F of F is the homotopy right Kan extension Open n∂ ( M ) op Open ∂ ( M ) op CGH i n F T n F of F | Open n∂ ( M ) op along the inclusion i n : Open n∂ ( M ) op (cid:44) → Open ∂ ( M ) op , together with thenatural transformation η n : F → T n F coming from the universal property of the homotopyright Kan extension. Written as a homotopy limit, the functor T n F is T n F ( V ) := holim W ⊆ VW ∈ Open n∂ ( M ) F ( W ) . OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 7
Example 1.1.14 ([Wei99, Section 0]) . Let
M, N be smooth manifolds (without bound-ary). The first Taylor approximation T Emb( V ) of the embedding functor Emb( − , N ) isweakly homotopy equivalent to the immersion functor Imm( − , N ) , which associates toan open subset V ⊆ M the space of immersions Imm(
V, N ) . In other words, the “linear”approximation of Emb(
M, N ) is the space Imm(
M, N ) of “local” embeddings. Proposition 1.1.15 (Weiss) . Using the notation from Definition 1.1.13, the functor T n F toghether with the natural transformation η n has the following properties. i) The functor T n F is a polynomial functor of degree at most n , ii) For any V ∈ Open n∂ ( M ) , the map η n ( V ) is a weak homotopy equivalence. iii) If F is a polynomial functor of degree at most n , then η n is a weak equivalence, i.e. η n ( V ) is a weak homotopy equivalence for every V ∈ Open ∂ ( M ) . iv) If µ : F → G is a natural transformation where G is a polynomial functor of degreeat most n , then (up to weak equivalence) the natural transformation µ factorsthrough T n F .Proof. See [Wei99, Theorem 3.9, Theorem 6.1]. (cid:3)
Remark . In other words, the natural transformation η n : F → T n F is the bestapproximation of F by a polynomial functors of degree at most n and T n F is unique upto weak homotopy equivalence. Definition 1.1.17.
Let F be a good functor. The Taylor tower of F is the tower ofnatural transformations r i : T i F → T i − F with i ≥ , that are induced by the inclusion Open i − ∂ ( M ) (cid:44) → Open i∂ ( M ) . F. . . T n +1 F T n F T n − F . . . T F. η n η n − η n +1 η r n +2 r n +1 r n r n − r Definition 1.1.18.
Let M be a smooth manifold and let F : Open ∂ ( M ) → CGH be agood functor. The Taylor tower of F converges if for every V ∈ Open ∂ ( M ) , the canonicalmap η ( V ) : F ( V ) → holim n (cid:54) =0 T n F ( V ) is a weak homotopy equivalence.The Taylor tower of a good functor does not converge in general. However, for someembedding functors, we have the following convergence criterium. Theorem 1.1.19 (Goodwillie–Weiss) . Let M and N be two smooth manifolds. Then theTaylor tower of the embedding functor Emb( − , N ) converges, if dim N − dim M ≥ .Proof. See [GW99, Corollary 2.5]. (cid:3)
Thus, in the codimension case, where interesting knot theory is developed, thisconvergence theorem is not applicable. Nevertheless, manifold calculus allows us to studythe space of knots from a homotopy theoretic viewpoint, as shown in the later sections.For this purpose, let us first recall the basic notions of knot theory.1.2. Long knots and Vassiliev’s knot invariants.
Classically, knot theory studiessmooth embeddings of S into S up to isotopy. A long knot is an embedding from I to R × D coinciding with a fixed linear embedding near the boundary. We consider longknots instead of knots in this paper because of technical convenience. For example, thespace K of long knots has an E -algebra structure induced by concatenation, cf. [BCKS17,Section 4] and [Bud08]. The one point compactification of each long knot induces an We denote by E the little 1-cubes operad, which is a concrete model for A ∞ -operad in CGH . YUQING SHI isomorphism between π ( K ) and π (Emb(S , S )) . Thus for the study of knot invariantswith values in an abelian group A , i.e. elements of H (Emb(S , S ); A ) , it does no harm touse long knots instead of knots. However, note that Emb(S , S ) and the space of longknots K have different higher homotopy groups, cf. [Bud08, Theorem 2.1]. Definition 1.2.1.
Fix the embedding c : I → R × D , t (cid:55)→ (0 , , − t ) and define thespace K of long knots as Emb ∂ (I , R × D , c ) . Elements of K are called long knots .For the joy of the reader, see Figure 4 for an example of a long knot. Figure 4.
An example of a long knot.
Definition 1.2.2.
Two long knots K , K ∈ K are called isotopic if there is a smoothmap F : I × I → R × D such that F | I ×{ } = K and F | I ×{ } = K , and F | I ×{ t } ∈ K forevery t ∈ I . We call F an isotopy between K and K , and write K ∼ K Definition 1.2.3. A knot invariant with values in a set R is a map f : π ( K ) → R .Classifying knots up to isotopy has always been a central problem in knot theory, sofinding computable knot invariants plays an important role. The knot invariants that weare interested in are the so called finite type invariants, which are a collection of knotinvariants discovered by Vassiliev, see [Vas90] for Vassiliev’s original approach and [Bar95]for an alternative explanation. There is a precise definition of Vassiliev invariants usingcombinatorics, which links the study of Vassiliev invariants to Chern–Simon theory andalgebraic structures of Feynman diagrams, cf. [Bar95] and [Kon94]. For motivation, wesketch Vassiliev’s original approach here:Instead of focusing on one specific knot invariant, Vassiliev considered the whole set H ( K ; A ) of all knot invariants with values in a given abelian group A . The main steps ofhis computation are the following:i) Embed K in the space C ∞ ∂ (I , R × D , c ) of all smooth maps from I into R × D which are germ equivalent with c on the boundary ( Definition 1.2.1).ii) Compute the homology of the complement of K in C ∞ ∂ (I , R × D , c ) .iii) Use Alexander duality to obtain H • ( K ; A ) , and in particular H ( K ; A ) .In order to perform step ii) and iii), Vassiliev finds a filtration by finite dimensionalvector spaces { Γ i } i ∈ N , which approximate the space C ∞ ∂ (I , R × D , c ) . Intersecting thissequence with C ∞ ∂ (I , R × D , c ) \ K yields a filtration σ ⊆ σ ⊆ · · · ⊆ σ n ⊆ σ n +1 ⊆ · · · ⊆ C ∞ ∂ (I , R × D , c ) \ K . Now, Vassiliev computes H • (C ∞ ∂ (I , R × D , c ) \ K ; A ) via a homological spectral sequenceassociated to this filtration. Furthermore, this filtration gives a filtration of the homology In the smooth settings, isotopy and ambient isotopy are equivalent. Vassiliev used another mapping space instead of K , which is homotopy equivalent to K . For simplicity,we just write K instead. OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 9 groups H • (C ∞ ∂ (I , R × D , c ) \ K ; A ) . In each of the finite dimensional vector spaces wecan apply Alexander duality to obtain a filtration V A ⊆ V A ⊆ · · · ⊆ V An ⊆ V An +1 ⊆ · · · ⊆ H ( K ; A ) . Finally, a
Vassiliev invariant of degree at most n with values in A is defined to be anelement of V An . Remark . Let K ∈ C ∞ ∂ (I , R × D , c ) be a smooth map. We call a point p ∈ im( K ) a singularity of K if K − ( p ) contains more than one element. The filtration ( σ i ) i ≥ of C ∞ ∂ (I , R × D , c ) \ K arises by distinguishing K by the type and the number of itssingularities. Thus it appears natural to conjecture that the system of Vassiliev invariantsclassifies knots. On the other hand, it is still open whether Vassiliev invariants detect theunknot. Notation 1.2.5.
We abbreviate the embedding functor
Emb ∂ ( − , R × D , c ) associatedto the space K as Emb( − ) , since this is the only embedding space we are going to considerin the rest of the text.The convergence for the Taylor tower of the embedding functor Emb( − ) correspondingto the space K indeed fails, because the set π ( K ) is countable, but the homotopy limit ofthe corresponding Taylor tower can be shown (formally) to be uncountable. However, thenatural transformations η n in the Taylor tower of Emb( − )Emb( − ) . . . T n Emb( − ) T n − Emb( − ) . . . T Emb( − ) η n η n − η r n +1 r n r n − r (1.2.6)do provide us with a sequence of knot invariants, i.e. π ( η n (I)) : π ( K ) → π (T n Emb(I)) for every n ∈ N . These are actually finite type invariants: Theorem 1.2.7 (Budney–Connant–Koytcheff–Sinha) . Let n ∈ N . The map π ( η (I)) isan additive finite type knot invariant of degree at most n − .Proof. See [BCKS17, Theorem 6.5]. (cid:3)
Remark . Let A be an abelian group. An additive knot invariant is a abelian monoidhomomorphism π ( K ) → A , where the abelian monoid structure of π ( K ) is inducedby connected sum of knots. Thus, one non-trivial point in Theorem 1.2.7 is to give π (T n Emb(I)) an abelian monoid structure such that it is compatible with the connectedsum of knots. The authors of [BCKS17] solve this by defining compatible E -algebrastructures on the spaces K and T n Emb(I) , cf. [BCKS17, Section 4]. For more survey onthe operadic structures on the space of long knots (not necessarily in codimension ), cf.[Bud08], [Sin06] and [DH12]. Conjecture 1.2.9 (Budney–Connant–Koycheff–Sinha) . Let n ∈ N . The map π ( η n (I)) isthe universal additive finite type knot invariant of degree at most n − . We refer the readers to [Hab00], [CT04a], [CT04b] and [Sta00] for various geometricand combinatorial descriptions of universal additive Vassiliev invariants. If the conjectureis true, then we expect π (T n Emb(I)) to be isomorphic to the abelian groups generatedby certain combinatorial diagrams. The rest of the text aims at explaining one method of The connected sum of knots is up to isotopy abelian because one can shrink one knot in the connectedsum and slide it through the other knot in the connected sum. understanding the homotopy groups (at least π ) of T n Emb(I) , namely, the computationof a Bousfield–Kan homotopy spectral sequence of the following tower of fibration: · · · r n +1 (I) −−−−→ T n Emb(I) r n (I) −−→ T n − Emb(I) r n − (I) −−−−→ · · · r (I) −−→ T Emb(I) . In order to do this, we introduce in the next section a cosimplicial space associated to theembedding functor
Emb( − ) .2. A cosimplicial model constructed by Goodwillie
Goodwillie observed that a good functor on
Open ∂ (I) corresponds to a cosimplicialspace, and vice versa, cf. [GKW01, Section 5]. The cosimplicial spaces arise naturally inthis way enjoy nice properties and can be considered as cosimplicial models for the Taylortower of the corresponding good functors (Definition 2.14). Because certain details areleft out in the paper [GKW01] for the construction of the cosimplicial spaces and theirproperties, we reformulate this correspondence in terms of equivalence of simplicial functorcategories and give proofs in full detail, using some results from [AF15]. This equivalencefurther facilitates the computations in Section 3.3. Definition 2.1.
We define the following two categories.i) The simplex category ∆∆∆ consists of the objects [ n ] = { , , . . . , n } ⊆ N with n ≥ ,and the morphisms are the order-preserving maps.ii) Denote by ∆∆∆ + the category of finite, totally ordered sets. The morphisms areorder-preserving maps. Remark . The simplex category ∆∆∆ is equivalent to the category of non-empty finitetotally ordered sets, which we also denote, by abuse of notation, by ∆∆∆ . Definition 2.3. i) A cosimplicial space is a functor X • : ∆∆∆ → CGH .ii) An augmented cosimplicial space Y • + is a functor Y • + : ∆∆∆ + → CGH . Notation 2.4.
Let X • be a cosimplicial space. Denote by X k the value X • ([ k ]) , for k ≥ . Notation 2.5.
By restricting an augmented cosimplicial space Y • + to the subcategory ∆∆∆ ,we obtain the associated cosimplicial space , which we denote as Y • . Convention 2.6.
By the totalisation
Tot X • of a cosimplicial space X • we always meanthe totalisation Tot (cid:102) X • of a fibrant replacement (cid:102) X • of X • , with respect to the modelstructure introduced in [BK72, Section X.4.6]. Similarly, by the n -th partial totalisation Tot n X • of X • we always mean Tot n (cid:102) X • . Remark . We have natural weak homotopy equivalences
Tot X • (cid:39) holim (cid:102) X • Tot n X • (cid:39) holim k ≤ n (cid:102) X k Definition 2.8.
Let
Open ∂ (I) fin be the full subcategory of Open ∂ (I) whose objectsare the open subsets of I that contain ∂ I and have only finitely many path connectedcomponents, i.e. Ob (
Open ∂ (I) fin ) := { I } ∪ (cid:91) n ≥ Ob (
Open n∂ (I)) Proposition 2.9.
A good functor on
Open ∂ (I) op is determined up to weak homotopyequivalence by its restriction on Open ∂ (I) opfin .Proof. This follows by Definition 1.1.3.ii) of good functor. (cid:3)
OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 11
Remark . The restriction of a good functor on
Open ∂ (I) op to Open ∂ (I) opfin is isotopyinvariant in the sense of Definition 1.1.3.i). An isotopy invariant functor on Open ∂ (I) opfin fullfills Definition 1.1.3.ii) automatically.In the following theorem, we reformulate the correspondence described in [GKW01,Construction 5.1.1] between cosimplicial spaces and good functors from Open ∂ (I) opfin tothe category of spaces. Theorem 2.11.
Let κ : Open ∂ (I) opfin → ∆∆∆ + be the functor κ ( V ) := π (I \ V ) . Denoteby Fun g ( Open ∂ (I) opfin , CGH ) the category of good functors from Open ∂ (I) opfin to CGH .Precomposing a cosimplicial space with the functor κ induces an equivalence Fun (∆∆∆ + , CGH ) κ ◦− −−→ Fun g ( Open ∂ (I) opfin , CGH ) of simplicial enriched functor categories.Remark . This theorem says that the category of good functors from the
Open ∂ (I) opfin to the category of spaces is equivalent to the category of cosimplicial spaces. Corollary 2.13 ([GKW01, Remark 5.1.3]) . Let F : Open ∂ (I) opfin → CGH be a good functor.Denote by F • an augmented cosimplicial space such that κ ◦ F • (cid:39) F . Then F • has thefollowing properties: i) For V ∈ Open ∂ (I) opfin , we have that F • ( π (I \ V )) (cid:39) F ( V ) . ii) We have
Tot n F • (cid:39) T n F (I) and Tot F • (cid:39) holim n ≥ T n F (I) . Definition 2.14.
Using the notations of the above corollary, we call a cosimplicialspace F • satisfying the i) and ii) of Corollary 2.13 a cosimplicial space associated to thegood functor F .Because of F • enjoys the property of Corollary 2.13.ii), we call F • a cosimplicial modelfor the tower of fibrations · · · r n +1 (I) −−−−→ T n F (I) r n (I) −−→ T n − F (I) r n − (I) −−−−→ · · · r (I) −−→ T F (I) , obtained by evaluation of the Taylor tower of F on I . Remark . Let n = π (I \ V )) − . Denote by E mb • a cosimplicial space associatedto Emb( − ) . Corollary 2.13.i) implies that E mb n is weakly homotopy equivalent to theCartesian products of this configuration space (points of embeddings) and copies of S (tangent vectors at the embedded points). We will use this relation to configuration spacesin Section 3.Now we work towards proofs of Theorem 2.11 and Corollary 2.13, which is, to the bestof our knowledge, not available in the literature. Let us begin by introducing severalcategories. Definition 2.16. i) Define the category
Man m of smooth oriented m -dimensional manifolds. Objectsof Man m are smooth oriented manifolds of dimension m , and the morphisms arethe orientation-preserving smooth embeddings.ii) Define the simplicial category M an m of smooth oriented m -dimensional manifolds.This category has the same objects as Man m , and the morphisms are spaces oforientation-preserving smooth embeddings, equipped with Whitney C ∞ -topology.iii) Define the full subcategory Disc m of Man m whose objects are finite disjoint unionsof R m and R ≥ × R m − .iv) Define the full simplicial subcategory D isc m of M an m that has the same objectsas Disc m . v) Let M be a smooth oriented manifold of dimension m . Define the category Disc m / M := Disc m × Man m Man m / M , where Man m / M is the slice-category over M . Objects of Disc m / M are embeddingsof finite disjoint unions of R m and R ≥ × R m − into M .vi) Let M be a smooth oriented manifold of dimension m . Define the simplicialcategory D isc m / M := D isc m × M an m M an m / M , where M an m / M is the over category over M .vii) Let M be a smooth oriented manifold of dimension m . Define the subcategory Isot m / M of Disc m / M which has the same objects as Disc m / M , but only the mor-phisms that are isotopy equivalences. Proposition 2.17 (Ayala–Francis) . The canonical functor
Disc m / M (cid:44) → D isc m / M induces an equivalence of simplicial categories Disc m / M [ Isot − / M ] (cid:39) D isc m / M , where Disc m / M [ Isot − / M ] is the (Dwyer-Kan) localisation of Disc m / M at Isot m / M .Proof. See [AF15, Proposition 2.19]. (cid:3)
Remark . In [AF15], the authors uses the language of ∞ -categories (quasi-categories).For a translation between simplicial categories and ∞ -categories, see [Lur09, Section 1.1.3,1.1.4 and 1.1.5].For our application, we consider m = 1 and M = I . Definition 2.19.
Define the subcategory
Disc ∂ / I of Disc / I whose objects are the embed-dings such that the boundary ∂ I of I is in the image.In the same way, let Isot ∂ / I be the subcategory of Isot / I whose objects are theembeddings such that ∂ I is in their images. Explicitly, objects of Disc ∂ / I are embeddingsof the form [0 , (cid:15) ) (cid:116) n (cid:71) k =1 R (cid:116) (1 − (cid:15) (cid:48) , (cid:44) → I , n ∈ N . Using the same proof strategy as Proposition 2.17, we have
Corollary 2.20.
The canonical functor
Disc ∂ / I → D isc ∂ / I induces an equivalence ofsimplicial categories Disc ∂ / I (cid:104)(cid:0) Isot ∂ / I (cid:1) − (cid:105) (cid:39) D isc ∂ / I . (cid:3) Proposition 2.21. i) (Ayala–Francis) The functor S : (cid:0) D isc ∂ / I (cid:1) op → ∆∆∆ + ( i : V (cid:44) → I) (cid:55)→ π (I \ i ( V )) is an equivalence of simplicial categories. See [DK80] for more details on Dwyer–Kan localisation.
OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 13 ii)
The functor
Im :
Disc ∂ / I → Open ∂ (I) fin , ( i : V (cid:44) → I) (cid:55)→ im( i ) is an equivalence ofordinary categories. Let Isot (I) be the subcategory of
Open ∂ (I) fin which has thesame objects as Open ∂ (I) fin with only the morphisms that are isotopy equivalences.Then Im | Isot ∂ / I : Iso ∂ / I → Isot (I) is an equivalence of categories.Proof. i) We can prove that the functor is essentially surjective and fully faithful. Thisis true since the every connected component of each space of morphism of D isc ∂ / I iscontractible. See also [AF15, Lemma 3.11].ii) First Im is essentially surjective, because the boundary of I is in the image of i for any i ∈ Isot ∂ / I . For any two objects i : V (cid:44) → I and i : V (cid:44) → I in Disc ∂ / I , themorphism set Mor
Disc ∂ / I ( i , i ) is either empty or a one element set. Also the morphismset Mor
Open ∂ (I) (im( i ) , im( i )) is either empty or has one element. Thus Im is fully faithful.Similarly it follows that Im | Isot ∂ / I is an equivalence. (cid:3) Corollary 2.22.
We have the following equivalences of simplicial categories
Open ∂ (I) fin [ Iso (I) − ] (cid:39) Disc ∂ / I (cid:104)(cid:0) Isot ∂ / I (cid:1) − (cid:105) (cid:39) D isc ∂ / I S (cid:39) ∆∆∆ op+ . (cid:3) Proof of Theorem 2.11.
Recall the functor κ : Open ∂ (I) opfin → ∆∆∆ + with κ ( V ) = π (I \ V ) from Theorem 2.11. This functor fits into the equivalences of Corollary 2.22, i.e. we havethe following commutative diagram ∆∆∆ op+ D isc ∂ / I Open ∂ (I) fin Open ∂ (I) fin [ Iso (I) − ] Disc ∂ / I (cid:104)(cid:0) Isot ∂ / I (cid:1) − (cid:105) S (cid:39) L κ κ loc (cid:39) (cid:39) (2.23)where κ loc is the induced map of κ by the universal property of the localisation functor L ,since κ sends an isotopy equivalence morphisms in Open ∂ (I) fin to isomorphisms in ∆∆∆ + .By the universal property of localisation and isotopy invariance of good functors,precomposing with the localisation functor L induces an equivalence Fun g (cid:0) Open ∂ (I) fin [ Iso (I) − ] op , CGH (cid:1) L ◦− −−→ Fun (
Open ∂ (I) opfin , CGH ) of simplicial categories. By the equivalences of Corollary 2.22, the map κ loc in the diagram(2.23) also induces an equivalence Fun (∆∆∆ + , CGH ) κ loc ◦− −−−−→ Fun (cid:0)
Open ∂ (I) fin [ Iso (I) − ] op , CGH (cid:1) of simplicial categories.Therefore, the map κ induces an equivalence Fun (∆∆∆ + , CGH ) κ ◦− −−→ Fun g ( Open ∂ (I) opfin , CGH ) of simplicial functor categories. (cid:3) Proof of Corollary 2.13.
Let F • be a cosimplicial space such that κ ◦ F • (cid:39) F . ThenCorollary 2.13.i) follows by composition of functors.As for the proof of Corollary 2.13.ii), We have Tot n F • = holim F • | ∆ ≤ n F • (cid:39) holim L ( Open n∂ (I)) F loc (cid:39) holim Open n∂ (I) F where L is the localisation functor in diagram (2.23) and F loc denotes the induced functorfrom Open ∂ (I) fin [ Iso (I) − ] op to CGH by F . The first equality is by definition. The secondweak homotopy equivalence comes from the equivalences of categories from Corollary 2.22. The third weak homotopy equivalence follows by observing that under the fully faithfulfunctor
Fun (cid:0)
Open ∂ (I) fin [ Iso (I) − ] op , CGH (cid:1) L ◦− −−→ Fun (
Open ∂ (I) opfin , CGH ) a limit cone over a functor Open ∂ (I) fin [ Iso (I) − ] op → CGH restricts to a limit cone overa functor Open ∂ (I) opfin → CGH . (cid:3) Remark . There is another cosimplicial model C • for the above tower of fibrations,constructed via a compactification of Conf n ( R × D ) , cf. [BCKS17, Section 3] and [Sin09,Section 6]. Compared with E mb • , the cosimplicial space C • has the advantage that it isgeometric and various versions of C • have already been used in context concerning finitetype invariants of -manifold, cf. [AS94] and [BT94]. Using this cosimplicial space C • , theauthors of [BCKS17] give the space Tot n C • (cid:39) T n Emb(I) an E -algebra structure, whichis used to define the abelian group structure on π (T n Emb(I)) mentioned in Remark 1.2.8,cf. [BCKS17, Corollary 4.13, Section 5.7]. We are exploring whether we can define similarmultiplicative structure on the cosimplicial space E mb • . Remark . One way to generalise Theorem 2.11 in higher dimension, i.e. good functorsfrom
Open ∂ ( M ) op with dim M > , is to consider configuration categories con( M ) associated to a smooth manifold M . See [BW18] for a detailed explanation on this subject.3. An integral homotopy spectral sequence for E mb • To a cosimplicial space E mb • (Definition 2.14) associated to the functor Emb( − ) , wecan associate the Bousfield–Kan homotopy spectral sequence { E p,q } q ≥ p ≥ with integralcoefficients, cf. [BK72, Chapter X]. In this section, we will first briefly recall someproperties of this spectral sequence (Section 3.1) and then give a concrete computation ofthe d -differentials that map into the diagonal terms (Section 3.3). For the computation,we make use of the calculation of the homotopy groups for Conf n ( R × D ) , which we willrecall in Section 3.2.3.1. A spectral sequence for cosimplicial spaces.
We begin by introducing techniquesthat we need for the computation of Bousfield–Kan spectral sequences. The main referencefor this section is [BK72, Chapter X].
Notation 3.1.1.
Let X • : ∆∆∆ → CGH be a cosimplicial space. For ≤ i ≤ n , denote itscoface map by δ i : X n − → X n and its codegeneracy maps by s i : X n +1 → X n .Given a cosimplicial space X • : ∆∆∆ → CGH , there is a tower of fibrations (cf. [BK72,Chapter 6, Section 6.1]) · · · → Tot n +1 X • → Tot n X • → · · · → Tot X • → Tot X • . (3.1.2)Denote by L n +1 X • the homotopy fibre of Tot n +1 X • → Tot n X • and L X • := Tot X • .Applying Bousfield–Kan homotopy spectral sequence, cf. [BK72, Section X.6], to the towerof fibrations (3.1.2), we obtain a spectral sequence calculating the homotopy groups of Tot X • , whose first page is given by E p,q = π q − p ( L p X • ) , where q ≥ p ≥ , and E p,q = 0 otherwise. With the help of the cosimplicial structure, we cancalculate the homotopy groups of the spaces L p X • and the differentials d : E p − ,p → E p,p in terms of the homotopy groups of X p . OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 15
Proposition 3.1.3 (Bousfield–Kan) . Given a cosimplicial space X • : ∆∆∆ → CGH , we have π q − p ( L p X • ) ∼ = π q (cid:32) X p ∩ p − (cid:92) i =0 ker( s i ) (cid:33) ∼ = π q ( X p ) ∩ p − (cid:92) i =0 ker( s i ∗ ) , where the push-forward s i ∗ : π q ( X p ) → π q ( X p − ) is induced by the codegeneracy maps s i .Proof. See [BK72, Section X.6.2]. (cid:3)
Proposition 3.1.4 (Bousfield–Kan) . Given a cosimplicial space X • , the first page of theBousfield–Kan homotopy spectral sequence of X • is given by E p,q ∼ = π q ( X p ) ∩ p − (cid:92) i =0 ker( s i ∗ ) , where q ≥ p ≥ , and the push-forward s i ∗ : π q ( X p ) → π q ( X p − ) is induced by the codegen-eracy maps s i . The differential d : E p,q → E p +1 ,q on the first page is given by x (cid:55)→ p +1 (cid:88) i =0 ( − i δ i ∗ ( x ) , where the push-forward δ i ∗ : π q ( X p ) → π q ( X p +1 ) is induced by the coface maps δ i on X p .Proof. See [BK72, Chapter X.7]. (cid:3)
Homotopy groups of
Conf n ( R × D ) . From Section 3.1 we see that we need tocompute the homotopy groups of E mb n for n ≥ , in order to compute the Bousfield-Kanspectral sequence associated to the cosimplicial model E mb • . By Remark 2.15 we knowthat E mb n relates closely to the configuration spaces of n points in R × D . Therefore,let us gather some information about the homotopy groups of configurations spaces in thissection. The main reference for this section is [FN62] and [FH01]. Definition 3.2.1.
Let M be a smooth manifold (possibly with boundary). Define the configuration space Conf n ( M ) of n ≥ points on M as Conf n ( M ) := { ( x , . . . , x n ) ∈ ( M \ ∂M ) n | x i (cid:54) = x j for i (cid:54) = j } . Now we focus on
Conf n ( R × D ) for n ≥ . Convention 3.2.2.
We define Conf ( R × D ) := { (0 , , − , (0 , , } ⊆ ∂ ( R × D ) . Situation 3.2.3.
Let us fix the following points of R × D . Define e := (1 , , ∈ R × D ,and q := (0 , , , and q k = q + 4( k − e for k ≥ Also define the set of points Q := ∅ and Q k := { q , q , . . . , q k } . Theorem 3.2.4 (Fadell–Neuwirth) . For n ≥ and n ≥ k ≥ , the map pr k,n : Conf n ( R × D \ Q k ) → R × D \ Q k ( x , x , · · · , x n ) (cid:55)→ x is a fibre bundle whose fibre is homeomorphic to Conf n − ( R × D \ Q k +1 ) . For k ≥ , themap pr k,n admits a cross section .Proof. See [FN62, Theorem 2]. (cid:3) We make this conventions because we will see in the next section that E mb n (cid:39) Conf n ( R × D ) × (cid:0) S (cid:1) n for n ≥ . The case k = 0 works since we are looking at Euclidean spaces. Thus we can compute π ∗ (Conf n ( R × D )) inductively via the splitting long exactsequences for the fibre bundles pr k,n for ≤ k ≤ n and n ≥ . And we can conclude thefollowing corollary. Corollary 3.2.5 (Fadell–Neuwirth) . For n ≥ and i ≥ , we have π i (Conf n ( R × D )) ∼ = n − (cid:77) k =0 π i ( R × D \ Q k ) ∼ = n − (cid:77) k =1 π i ( ∨ k S ) . In particular,
Conf n ( R × D ) is simply connected.Proof. See [FN62, Corollary 2.1]. (cid:3)
Now we are going to introduce a set of generators for π (Conf n ( R × D )) , which wewill use in the computations of Section 3.3. Definition 3.2.6.
For ≤ i (cid:54) = j ≤ n , define the map x ij as the composition of thefollowing two maps S → Conf n − j +1 ( R × D \ Q j − ) x (cid:55)→ ( q i + x, q j +1 , . . . , q n ) , and Conf n − j +1 ( R × D \ Q j − ) (cid:44) → Conf n ( R × D )( x , . . . , x n − j +1 ) (cid:55)→ ( q , . . . , q j − , x , . . . , x n − j +1 ) . Proposition 3.2.7.
The maps x ij : S → Conf n ( R × D ) for ≤ i < j ≤ n generate thegroup π (Conf n ( R × D )) .Proof. The image S ij := im( x ij ) of x ij is homeomorphic to a 2-sphere. For a fixed j with ≤ j ≤ n , the space S j ∨ S j ∨ · · · ∨ S j − ,j ⊆ R × D is homotopy equivalentto the space R × D \ Q j − . Note that for every i with ≤ i < j , the map x ij is thepositive generator of π ( S ij ) . Thus, by Hurewicz isomorphism theorem, the maps x ij with i = 1 , . . . , j − generates the group π ( S j ∨ S j ∨ · · · ∨ S j − ,j ) ∼ = π ( R × D \ Q j − ) .Now let j varies and apply Corollary 3.2.5, we have that the maps x ij for ≤ i < j ≤ n generate the group π (Conf n ( R × D )) . (cid:3) Remark . The proof provides a decomposition of π ∗ (Conf n ( R r × D )) as π ∗ (Conf n ( R r × D )) ∼ = n (cid:77) j =2 π ∗ ( S j ∨ · · · ∨ S ij ∨ · · · ∨ S j − ,j ) , where for ≤ i < j ≤ n the positive generator of S ij is x ij . Thus by the following theoremof Hilton about the homotopy groups of wedges of spheres, we reduce the computation ofhomotopy groups of Conf n ( R × D ) to homotopy groups of spheres. Definition 3.2.9 ([Hil55],[Whi78, Page 511–512]) . Let T := S r +1 ∨ S r +2 ∨ · · · ∨ S r k +1 and denote by ι i the positive generator of π r i +1 (S r i +1 ) . Note that ι i can be considered asan element of π r i +1 ( T ) via the canonical embedding S r i +1 (cid:44) → T . We consider basic products eventually as homotopy classes, but to get a well-defined definition, onehas to first define basic products as ‘formal’ products, cf. [Whi78, Page 511–512]. The index set P inTheorem 3.2.10 below is then the set of formal basic products, and this ensures a posteriori that we donot have to distinguish formal products and Whitehead products of homotopy classes. OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 17 i) The basic products of weight 1 are the elements ι , ι , · · · , ι k . We order the setof basic products of weight by ι < ι < · · · < ι k . We define basic productsof weight bigger than recursively. A basic product of weight ω is a Whiteheadproduct [ a, b ] , where a and b are both basic products of weights α < ω and β < ω respectively such thata) α + β = ω and a < b , andb) if b is defined as the Whitehead product [ c, d ] of basic products c and d , thenwe have c ≤ a .We declare every basic product of weight ω to be greater than any basic product ofsmaller weight. We order the set of basic products of weight ω lexicographically, i.e.for two basic products [ a, b ] and [ a (cid:48) , b (cid:48) ] of weight ω , we set [ a, b ] < [ a (cid:48) , b (cid:48) ] if a < a (cid:48) ,or a = a (cid:48) and b < b (cid:48) .ii) Thus a basic product p of weight ω is a suitably bracketed word in the symbols ι i for i = 1 , . . . , k . Assume ι i appears w i times in p . We define the height h ( p ) of p as (cid:80) ki =1 r i w i . Theorem 3.2.10 (Hilton) . Using the notation of Definition 3.2.9, let P be the set of(formal) basic products of ι , . . . , ι k . We have π ∗ ( T ) ∼ = (cid:77) p ∈ P π ∗ (S h ( p )+1 ) . where the direct summand π ∗ (S h ( p )+1 ) is embedded in π ∗ ( T ) by composition with the basicproduct p ∈ π h ( p )+1 ( T ) .Proof. See [Whi78, Theorem 8.1]. (cid:3)
The Whitehead products of x ij for ≤ i, j ≤ n and i (cid:54) = j of π (Conf n ( R × D )) satisfysome relations, which we will use in the computation in the Section 3.3. Proposition 3.2.11 (Hilton, Nakaoka–Toda, Massey–Uehara) . Let X be a topologicalspace. Then the Whitehead product [ − , − ] on π ∗ ( X ) is bilinear, antisymmetric and satisfiesthe Jacobi identity, i.e. for α ∈ π a +1 ( X ) , β ∈ π b +1 ( X ) and γ ∈ π c +1 ( X ) we have i) [ α, β + γ ] = [ α, β ] + [ α, γ ] and [ α + β, γ ] = [ α, γ ] + [ β, γ ] , ii) [ α, β ] = ( − ( a +1)( b +1) [ β, α ] , and iii) ( − c ( a +1) [ α, [ β, γ ]] + ( − a ( b +1) [ β, [ γ, α ]] + ( − b ( c +1) [ γ, [ α, β ]] = 0 .Proof. See [Hil61], [UM57] and [NT54]. (cid:3)
Proposition 3.2.12.
The element x ij ∈ π (Conf n ( R × D )) for ≤ i, j ≤ n and i (cid:54) = j satisfy the following relations: i) x ij = − x ji , ii) [ x ij , x jk ] = [ x ji , x ik ] = [ x ik , x kj ] , if n ≥ ; iii) [ x ij , x kl ] = 0 , if { i, j } ∩ { l, m } = ∅ and n ≥ .Proof. See the proof of [FH01, Theorem 3.1]. (cid:3)
A homotopy spectral sequence for the Taylor tower of
Emb( − ) . In this sectionwe perform some computation of the integral homotopy Bousfield–Kan spectral sequenceof cosimplicial space E mb • (Definition 2.14) associated to the embedding functor Emb( − ) (Notation 1.2.5). Recall that this spectral sequence aims at analysing the homotopy limitof the tower of fibrations · · · → T n Emb(I) → T n − Emb(I) → · · · → T Emb(I) , according Corollary 2.13. By Remark 2.15 we have the weakly homotopy equivalence E mb n (cid:39) Emb( V ) (cid:39) Conf n ( R × D ) × (cid:0) S (cid:1) n , (3.3.1)for any V ∈ Open ∂ (I) opfin such that π (I \ V ) ∼ = [ n ] . For the computation of the homotopyspectral sequence associated to E mb • we need to compute the induced maps on homotopygroups of the coface and codegeneracy maps. From (3.3.1) we have π ∗ ( E mb n ) ∼ = π ∗ (Conf n ( R × D )) × ( π ∗ (S )) n , and by abuse of notation, we consider x ij , for ≤ i < j ≤ n , as elements of π ∗ ( E mb n ) under the natural inclusion.Let l ∈ N and ≤ l ≤ n . Recall the notation from Theorem 2.11. Let V n +1 be anopen subset of I with ∂ I ⊆ V n +1 such that κ ( V n +1 ) = [ n + 1] . We obtain an open subset V n ⊆ V n +1 by removing the ( l + 2) -th subinterval of V n +1 \ ∂ I . Then the codegeneracymap s l for E mb • is the induced restriction map Emb( V n +1 ) → Emb( V n ) , i.e. forgetting theembedding of the ( l + 2) -th interval. With respect to the homotopy equivalence 3.3.1, wecan write s l with ≤ l ≤ n concretely as s l : Conf n +1 ( R × D ) × (cid:0) S (cid:1) n +1 → Conf n ( R × D ) × (cid:0) S (cid:1) n ( x , . . . , x n +1 ) × ( v , . . . , v n +1 ) (cid:55)→ ( x , . . . , (cid:100) x l +1 , . . . , x n +1 ) × ( v , . . . , (cid:100) v l +1 , . . . , v n +1 ) , Precomposing with the map from S representing the generators x ij , we obtain thefollowing Proposition 3.3.2.
Let i, j, l, n ∈ N and ≤ i < j ≤ n + 1 and ≤ l ≤ n and n ≥ . i) We have s l ∗ ( x ij ) = x i − ,j − if l < i − x i,j − if i − < l < j − x i,j if l > j − otherwise ii) Denote by s l ∗ ( c ) : π r (Conf n +1 ( R × D )) → π r (Conf n ( R × D )) the restriction ofthe map s l ∗ to the π r (Conf n +1 ( R × D )) component.Denote by Z the set of basic products of the elements x i,j containing x u,l +1 or x l +1 ,v for ≤ u ≤ l and l + 2 ≤ v ≤ n + 1 . Under the isomorphism inTheorem 3.2.10, the kernel of the map s l ∗ ( c ) is isomorphic to (cid:76) p ∈ Z π r (S h ( p ) + 1) ,for r ≥ . iii) Denote by s l ∗ ( t ) : ( π r (S )) n +1 → ( π r (S )) n the restriction of the map s l ∗ on the ( π r (S )) n +1 component.For r ≥ , the map s l ∗ ( t ) is the canonical projection where forgetting the l -thcomponent. Thus the kernel of s l ∗ ( t ) is isomorphic to (0) l − × π r (S ) × (0) n − l .Proof. i) and iii) follows from the description of s l right above the proposition.For the proof of ii), let us abbreviate s l ∗ ( c ) by s l ∗ in this part of the proof. Note thatfor n ≥ , we have s l ∗ ( x uv ) = 0 if and only if u = l + 1 or v = l + 1 . Therefore, for n ≥ and z ∈ Z , we have s l ∗ ( z ) = 0 by the naturality of the Whitehead product. Thus s l ∗ factorsthrough π r (Conf n ( R × D )) / (cid:76) p ∈ Z π r (S h ( p ) + 1) π r (Conf n +1 ( R × D )) π r (Conf n ( R × D )) π r (Conf n ( R × D )) / ⊕ p ∈ Z π r (S h ( p ) + 1) s l ∗ p ¯ s l ∗ OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 19
By inspecting the value of s l ∗ on x ij , we conclude that for two basic products w ≤ w with s l ∗ ( w k ) (cid:54) = 0 for k ∈ { , } , we have s l ∗ ( w ) ≤ s l ∗ ( w ) . Also the heights of w k and s l ∗ ( w k ) are the same.So ¯ s l ∗ sends a basis of π r (Conf n ( R × D )) / (cid:80) p ∈ Z π r (S h ( p )+1 ) (i.e. basic products thatare not in Z ) injectively to a basis of π r (Conf n ( R × D )) . Thus ¯ s l ∗ is injective, whichimplies that the kernel of s l ∗ is isomorphic to (cid:76) p ∈ Z π r (S h ( p )+1 ) , via the isomorphism fromTheorem 3.2.10. (cid:3) Similar analysis of the definition of the coface maps tells us that these maps δ l of E mb • corresponds to “breaking” the embeddings of the ( l + 1) -th interval into the embedding oftwo subintervals. Therefore, one representative for the map δ l with < l < n + 1 is thefollowing: Conf n ( R × D ) × (cid:0) S (cid:1) n → Conf n +1 ( R × D ) × (cid:0) S (cid:1) n +1 ( x , . . . , x n ) × ( v , . . . , v n ) (cid:55)→ ( x , . . . , x l , x l + (cid:15)v l , x l +1 , . . . , x n ) × ( v , . . . , v l , v l , v l +1 , . . . , v n ) where the scalar (cid:15) ∈ R is so chosen that ( x , . . . , x l , x l + (cid:15)v l , x l +1 , . . . , x n ) is a well-definedpoint in Conf n +1 ( R × D ) . For l = 0 and l = n + 1 , we have δ (( x , . . . , x n ) × ( v , . . . , v n )) = ( x − + (cid:15)e, x , . . . , x n ) × ( e, v , . . . , v n ) δ n +1 (( x , . . . , x n ) × ( v , . . . , v n )) = ( x , . . . , x n , x +1 + (cid:15) (cid:48) e, ) × ( v , . . . , v n , e ) , where x − = (0 , , − and x +1 = (0 , , and e = (0 , , .By explicit calculation we obtain the following Proposition 3.3.3.
Let i, j, l, n ∈ N such that n ≥ and ≤ i < j ≤ n and ≤ l ≤ n +1 . i) For n ∈ N and n ≥ , we have δ l ∗ ( x ij ) = x i +1 ,j +1 if l < ix i,j +1 + x i +1 ,j +1 if l = ix i,j +1 if i < l < jx i,j + x i,j +1 if l = jx ij otherwise . ii) Denote by y k a generator for the k -th component π (S ) of ( π (S )) n . We have that δ l ∗ ( y k ) = y k +1 if l < kx k,k +1 + y k + y k +1 if l = ky k otherwise . (cid:3) Now we can compute E p − ,p and E p,p in the homotopy spectral sequence associated tothe cosimplicial space E mb • . Corollary 3.3.4.
Let l, n, r ∈ N and n ≥ and ≤ l ≤ n and r ≥ , and recall thenotations from Proposition 3.3.2. For the degeneracy map E mb n +1 s l −→ E mb n , we have ker s l ∗ = ker s l ∗ ( c ) × ker s l ∗ ( t ) and n − (cid:92) l =0 ker s l ∗ ∼ = n − (cid:92) l =0 ker s l ∗ ( c ) × (0) n Proof.
We have that s l ∗ = s l ∗ ( c ) × s l ∗ ( t ) . (cid:3) Proposition 3.3.5. i) For p ≥ and ≤ i < p − , let T be the set of basic products of the elements x i,p − of height p − , such that each x i,p − appears exactly once. Let F be the set of basicproducts of elements x i,p − of height p − , such that one x k,p − appears exactlytwice and all other x i,p − appear exactly once. Then we have E p − ,p ∼ = (cid:77) T π p (S p − ) ⊕ (cid:77) F π p (S p ) (3.3.6) where π p (S p − ) and π p (S p ) are embedded in π p (Conf p − ( R × D )) by compositionwith the basic products in T and F respectively. ii) For p ≥ , let H be the set of basic products of height p − of the elements in x i,p for ≤ i ≤ p − such that each x i,p appears exactly once. Then E p,p ∼ = (cid:77) H π p (S p ) , (3.3.7) where the direct summands π p (S p ) are embedded in π p (Conf p ( R × D )) by compo-sition with the basic products in H .Proof. i) Recall from Proposition 3.1.4 that E p − ,p ∼ = π p ( E mb p − ) ∩ (cid:84) p − l =0 ker( s l ∗ ) . ByCorollary 3.3.4 we only need to consider the π p (Conf p − ( R × D )) component of π p ( E mb p − ) .In other words, E p − ,p ∼ = π p (Conf p − ( R × D )) ∩ p − (cid:92) l =0 ker( s l ∗ ( c )) . Recall from Corollary 3.2.5 that π p (Conf p − ( R × D )) ∼ = p − (cid:77) j =2 π p ( S j ∨ S j ∨ · · · ∨ S j − ,j ) , and x ij is the positive generator of S ij , ≤ i < j ≤ p − . For a fixed j , let { b ( j ) k } k ∈ N bethe set of basic products of the elements x ij for i = 1 , . . . , j − . Using Theorem 3.2.10,we have π p (Conf p − ( R × D )) ∼ = (cid:77) We denote the torsion-free part of E p − ,p by E p − ,p / tors , i.e. thesummand (cid:76) F π p (S p ) in Equation 3.3.6. Proposition 3.3.12. For p ≥ , denote by D sep p the set of iterated Whitehead products ofthe elements x i,p − for i = 1 , . . . , p − with the following properties: i) For every w ∈ D sepp , there exists one x k ( w ) ,p − that appears exactly twice and allother x i,p − with ≤ i ≤ p − and i (cid:54) = k ( w ) appear exactly once. ii) Every w ∈ D sepp is of the form w = [ c , c ] where c is an iterated Whiteheadproduct of elements x i,p − with i ∈ I and c is an iterated Whitehead product ofelements x i,p − with j ∈ J such that I, J ⊆ { , . . . , p − } , I ∩ J = { k ( w ) } and I ∪ J = { , , . . . , p − } .Then, E p − ,p / tors is generated by D sep p .Proof. Denote by D p the set of iterated Whitehead products of the elements x i,p − with i = 1 , . . . , p − satisfying only condition i). Using the same argument as in the proof ofProposition 3.3.5, we see that D sep p ⊆ E p − ,p and D p ⊆ E p − ,p . In particular, the basicproducts in F , which are a basis of E p − ,p / tors , are contained in D p . We have reduced thedesired statement to the following claim which we prove by induction. Claim. For p ≥ , any element of D p can be written as a linear combination of elementsof D sep p using only the Jacobi identity and antisymmetry relations (Proposition 3.2.11). For p = 4 , the claim follows by listing all the elements of D and using the Jacobiidentity of the Whitehead product.Assume that the claim is true for all p ≤ n with n ≥ . Let p = n + 1 and con-sider (cid:101) w = [ a , a ] ∈ D n +1 . Without loss of generality, we can assume that x ,n is therepeated element in (cid:101) w . If the two copies of x ,n appear in a and a separately, then (cid:101) w isalready an element of D sep p . Otherwise, both copies of x ,n appear in either a or a , saythey appear in a . By assumption a is a Whitehead product of elements x m,n . Definethe set M := { m ∈ N | x m,n appears in a } . So we know M ≤ n − and ∈ M . Theelement x m,n appears exactly once in a , for (cid:54) = m ∈ M There is a bijection r : { x m,n | m ∈ M } → { x i, M +1 | i = 1 , . . . , M } such that x ,n ismapped to x , M +1 . Define a (cid:48) ∈ D M +2 by replacing each occurrence of x m,n a by r ( x m,n ) . OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 23 By an inductive assumption, we can write a (cid:48) as a finite sum a (cid:48) = (cid:80) i ∈ I [ c (cid:48) i , c (cid:48) i ] such that [ c (cid:48) i , c (cid:48) i ] ∈ D sep M +2 and x , M +1 appears exactly twice in [ c i , c i ] . Thus, by replacing each x i, M +1 by r − ( x i, M +1 ) we obtain a = (cid:80) i ∈ I [ c i , c i ] such that [ c i , c i ] is a Whiteheadproduct of the elements x m,n with m ∈ M , where x ,n appears exactly twice and x m,n appears exactly once for m (cid:54) = 1 .Therefore (cid:101) w can be written as (cid:101) w = (cid:88) i ∈ I (cid:2) [ c i , c i ] , a (cid:3) = (cid:88) i ∈ I ( − (cid:15) (cid:2) [ c i , a ] , c i (cid:3) + ( − (cid:15) (cid:2) [ a , c i ] , c i (cid:3) , where (cid:15) and (cid:15) denote the signs which come from the Jacobi identity for the Whiteheadproduct. For every i ∈ I , we have that [[ c i , a ] , c i ] , [[ a , c i ] , c i ] ∈ D sep n +1 . Thus (cid:101) w is alinear combination of elements of D sep p . (cid:3) The upshot is that it is sufficient, for the computation d : E p − ,p → E p,p , to computeits evaluation on w for every w ∈ D sep p . Theorem 3.3.13. Let w = [ c , c ] ∈ D sep p , say with repeated occurrence of x k,p − . Wewrite c and c as c = [ . . . x i,p − . . . x k,p − . . . ] and c = [ . . . x k,p − . . . x j,p − . . . ] . Theindex k will be fixed through out the proposition. Then we have d ( w ) = ∂ k ( w ) + ∂ p − ( w ) , where ∂ k ( w ) = ( − k (cid:2) [ . . . x i (cid:48) ,p . . . x k,p . . . ] , [ . . . x k +1 ,p . . . x j (cid:48) ,p . . . ] (cid:3) + ( − k (cid:2) [ . . . x i (cid:48) ,p . . . x k +1 ,p . . . ] , [ . . . x k,p . . . x j (cid:48) ,p . . . ] (cid:3) and ∂ p − ( w ) = ( − p − (cid:2) [ . . . x i,p − . . . x k,p − . . . ] , [ . . . x k,p . . . x j,p . . . ] (cid:3) + ( − p − (cid:2) [ . . . x i,p . . . x k,p . . . ] , [ . . . x k,p − . . . x j,p − . . . ] (cid:3) , where i (cid:48) = i if i < k and i (cid:48) = i + 1 if i > k and j (cid:48) = j if j < k and j (cid:48) = j + 1 if j > k . Before we prove the theorem, let us take a look at an example of computation of d . Example 3.3.14. Let k = 2 and p = 8 and w ∈ D sep8 of the form w = (cid:2) [[ x , x ] , x ] , [[ x , x ] , [ x , x ]] (cid:3) . Now let us calculate d ( w ) using the formulas in Proposition 3.3.13. We have ∂ ( w ) = (cid:2) [[ x , x ] , x ] , [[ x , x ] , [ x , x ]] (cid:3) + (cid:2) [[ x , x ] , x ] , [[ x , x ] , [ x , x ]] (cid:3) , and ∂ ( w ) = − (cid:2) [[ x , x ] , x ] , [[ x , x ] , [ x , x ]] (cid:3) − (cid:2) [[ x , x ] , x ] , [[ x , x ] , [ x , x ]] (cid:3) . Proof of Theorem 3.3.13. First we proof the proposition for k ≤ p − . Claim 1. For l (cid:54) = k , p − , p , each of the elements ( − l − δ l − ∗ ( w ) , ( − l δ l ∗ ( w ) and ( − l +1 δ l +1 ∗ ( w ) can be written canonically as a sum of two iterated Whitehead products of x i,p with ≤ i < p such that every summand of ( − l δ l ∗ ( w ) appears in ( − l − δ l − ∗ ( w ) or ( − l +1 δ l +1 ∗ ( w ) with opposite sign. By abuse of notation we hide the inner brackets of iterated Whitehead products when we write anelements as done here. We note first that δ l − ∗ ( x in ) = δ l ∗ ( x in ) = δ l +1 ∗ ( x in ) for i (cid:54) = l − , l and l + 1 . Moreover, δ r ∗ ( x in ) = (cid:40) x i +1 ,p , if i > l + 1 x i,p , if i < l − for r = l − , l and l + 1 .Without loss of generality, we write w = [ . . . x l,p − . . . x l − ,p − . . . x l +1 ,p − . . . ] , where weshow only the elements that are interesting for us. Note that in this presentation of w ,the order in which the elements x l,p − , x l − ,p − and x l +1 ,p − appear does not play a role.We calculate δ r ∗ ( w ) for r = l − , l and l + 1 : δ l − ∗ ( w ) = [ . . . x l +1 ,p . . . x l − ,p + x l,p . . . x l +2 ,p . . . ] δ l ∗ ( w ) = [ . . . x l,p + x l +1 ,p . . . x l − ,p . . . x l +2 ,p . . . ] δ l +1 ∗ ( w ) = [ . . . x l,p . . . x l − ,p . . . x l +1 ,p + x l +2 ,p . . . ] Thus we see that [ . . . x l +1 ,p . . . x l − ,p . . . x l +2 ,p . . . ] of δ l ∗ ( w ) appears in δ l − ∗ ( w ) and the term [ . . . x l,p . . . x l − ,p . . . x l +2 ,p . . . ] of δ l ∗ ( w ) appears in δ l +1 ∗ ( w ) . Since the signs in front of δ r ∗ with r = l − , l and l + 1 in d alternate, we see that in d ( w ) the terms of ( − l δ l ∗ ( c ) arecancelled by terms of ( − l − δ l − ∗ ( w ) and ( − l +1 δ l +1 ∗ ( w ) as desired. Claim 2. For l = k , after cancelling terms of ( − k δ k ∗ ( w ) by terms of ( − k − δ k − ∗ ( w ) and ( − k +1 δ k +1 ∗ ( w ) as in Claim 1. , the remaining terms of ( − k δ k ∗ ( w ) is equal to ∂ k ( w ) . For convenience of the proof, we write without loss of generality w = [ . . . x k,p − . . . x k − ,p − . . . x k,p − . . . x k +1 ,p − . . . ] . Again note that for i (cid:54) = k − , k , k + 1 , we have δ k − ∗ ( x i,p − ) = δ k ∗ ( x i,p − ) = δ k +1 ∗ ( x i,p − ) ,and note δ k − ∗ ( w ) = [ . . . x k +1 ,p . . . x k − ,p + x k,p . . . x k +1 ,p . . . x k +2 ,p . . . ] δ k ∗ ( c ) = [ . . . x k,p + x k +1 ,p . . . x k − ,p . . . x k,p + x k +1 ,p . . . x k +2 ,p . . . ] δ k +1 ∗ ( w ) = [ . . . x k,p . . . x k − ,p . . . x k,p . . . x k +1 ,p + x k +2 ,p . . . ] . Thus after cancelling with terms of δ k − ∗ ( w ) and δ k +1 ∗ ( w ) , the remaining term of δ k ∗ ( w ) is ( − k [ . . . x k,n +1 . . . x k − ,p . . . x k +1 ,p . . . x k +2 ,p . . . ]+ ( − k [ . . . x k +1 ,p . . . x k − ,p . . . x k,p . . . x k +2 ,p . . . ] . Writing w as w = [ c , c ] = [[ . . . x i (cid:48) n . . . x kn . . . ] , [ . . . x kn . . . x j (cid:48) n . . . ]] , the remaining termsof δ k ∗ ( w ) look like ( − k (cid:2) [ . . . x i (cid:48) ,p . . . x k,p . . . ] , [ . . . x k +1 ,p . . . x j (cid:48) ,p . . . ] (cid:3) + ( − k (cid:2) [ . . . x i (cid:48) ,p . . . x k +1 ,p . . . ] , [ . . . x k,p . . . x j (cid:48) ,p . . . ] (cid:3) , which we recognise as ∂ k ( w ) as desired. Claim 3. After cancelling with terms of ( − p − δ p − ∗ ( w ) and ( − p δ p ∗ ( w ) , the remainingterm of ( − p − δ p − ∗ ( w ) is ∂ p − ( w ) .In order to prove this claim, we write w = [ c , c ] as in Proposition 3.3.13. Recallthat δ p − ∗ ( x i,p − ) = x i,p − + x i,p , so we have δ p − ∗ ( w ) = [ δ p − ∗ ( c ) , δ p − ∗ ( c )]= (cid:2) [ . . . x i,p − + x i,p . . . x k,p − + x k,p . . . ] , [ . . . x k,p − + x k,p . . . x j,p − + x j,p . . . ] (cid:3) . Recall that by assumption c and c are iterated Whitehead products of the ele-ments x i,p − with i ∈ I and x j,p − with j ∈ J where I, J ⊆ { , . . . , p − } and I ∩ J = { k } OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 25 and I ∪ J = { , . . . , p − } . Furthermore, each x i,p − with i ∈ I appears exactly once in c and similarly each x j,p − with j ∈ J appears exactly once in c .Again recall that E p − ,p / tors is a subgroup of π p ( E mb p − ) ∼ = π p (Conf p − ( R × D )) . Foran element w ∈ π p (Conf p − ( R × D )) of the form [ . . . [ x i,p − + x i,p , x j,p − + x j,p ] . . . ] with i (cid:54) = j , we have by Proposition 3.2.12 the following equality (cid:2) . . . [ x i,p − + x i,p , x j,p − + x j,p ] . . . (cid:3) = (cid:2) . . . [ x i,p − , x j,p − ] . . . (cid:3) + (cid:2) . . . [ x i,p , x j,p ] . . . (cid:3) . Thus by induction on the number of brackets the brackets, we have δ p − ∗ ( c ) = [ . . . x i,p − . . . x k,p − . . . ] + [ . . . x i,p . . . x k,p . . . ] δ p − ∗ ( c ) = [ . . . x k,p − . . . x j,p − . . . ] + [ . . . x k,p . . . x j,p . . . ] . and thus δ p − ∗ ( w ) = (cid:2) [ . . . x i,p − . . . x k,p − . . . ] , [ . . . x k,p − . . . x j,p − . . . ] (cid:3) + (cid:2) [ . . . x i,p . . . x k,p . . . ] , [ . . . x k,p . . . x j,p . . . ] (cid:3) + (cid:2) [ . . . x i,p − . . . x k,p − . . . ] , [ . . . x k,p . . . x j,p . . . ] (cid:3) + (cid:2) [ . . . x i,p . . . x k,p . . . ] , [ . . . x k,p − . . . x j,p − . . . ] (cid:3) . Let us now look at δ p − ∗ ( w ) and δ p ∗ ( w ) . We have δ p ∗ ( w ) = (cid:2) [ . . . x ik . . . x k,p − . . . ] , [ . . . x k,p − . . . x j,p − . . . ] (cid:3) . Assume without loss of generality that x p − ,p − appears in c , and write c of theform c = [ . . . x i,p − . . . x k,p − . . . x p − ,p − . . . ] . We get δ p − ∗ ( w ) = (cid:2) [ . . . x i,p . . . x k,p . . . x p − ,p + x p − ,p . . . ] , [ . . . x k,p . . . x j,p . . . ] (cid:3) = (cid:2) [ . . . x i,p . . . x k,p . . . x p − ,p . . . ] , [ . . . x k,p . . . x j,p . . . ] (cid:3) + (cid:2) [ . . . x i,p . . . x k,p . . . x p − ,p . . . ] , [ . . . x k,p . . . x j,p . . . ] (cid:3) . Thus after cancelling with the terms of ( − p − δ p − ∗ ( w ) and ( − p δ p ∗ ( w ) , the remainingterm of ( − p − δ p − ∗ ( w ) is ( − p − (cid:2) [ . . . x i,p − . . . x k,p − . . . ] , [ . . . x k,p . . . x j,p . . . ] (cid:3) + ( − p − (cid:2) [ . . . x i,p . . . x k,p . . . ] , [ . . . x k,p − . . . x j,p − . . . ] (cid:3) , which is exactly ∂ p − ( w ) as desired.Finally one can prove Claim 1 for k = p − analogously. Then we can explicitlywrite down δ r ∗ ( w ) for r = p − , p − , p − and p and obtain the desired formula in theproposition. (cid:3) Remark . Note that in the calculations of the proof we only changed the indicesof x i,p − for i = 1 , . . . , p − , whereas the bracketing of c and c was not changed at all. Moreexplicitly, the expressions [[ . . . x i (cid:48) ,p . . . x k,p . . . ] , [ . . . x k +1 ,p . . . x j (cid:48) ,p . . . ]] in the formula of ∂ k ( w ) and [[ . . . x i,p − . . . x k,p − . . . ] , [ . . . x k,p . . . x j,p . . . ]] in the formula of ∂ p − ( w ) have thesame bracketing as the one of c . The term [[ . . . x i (cid:48) ,p . . . x k +1 ,p . . . ] , [ . . . x k,p . . . x j (cid:48) ,p . . . ]] in ∂ k ( w ) and [[ . . . x i,p . . . x k,p . . . ] , [ . . . x k,p − . . . x j,p − . . . ]] in ∂ p − ( w ) have the same bracketingas c . We note in advance here that the bracketing determines the shape of the unitrivalentgraphs in the combinatorial interpretation in the next section. Combinatorial interpretation Since we computed the abelian groups E p,p , E p − ,p and the differential d : E p,p → E p − ,p of the spectral sequence associated to the cosimplicial model E mb • in the previous section,we would also know E p,p . In this section we give a combinatorial interpretation of E p,p and E p − ,p and the differentials d , based on the calculation of Proposition 3.3.5 andTheorem 3.3.13. As a corollary, we obtain a graphic interpretation of the groups E p,p .From these interpretations we will see that this spectral sequence relates closely to thetheory of Vassiliev invariants. Definition 4.1. A unitrivalent graph Γ is a graph whose nodes have only degree or ,together with a cyclic order on the edges at each node. We call the nodes of degree leaves and nodes of degree trivalent nodes . When Γ has n leaves, we define a labelling (ortotal ordering) on Γ to be a bijection of the set { , , . . . , n } to the set of leaves. Denoteby UTG the collection of labelled unitrivalent graphs. We define the degree of Γ as thenumber of nodes divided by .When we draw a labelled unitrivalent graph, we place the leaves on an oriented line,ordered according to the labelling. Unless explicitly mentioned, the cyclic orders ofthe trivalent nodes are given counterclockwise. See Figure 5 for an example of labelledunitrivalent graph. Figure 5. A unitrivalent graph of degree 8. The arrow is not part of thegraph. Definition 4.2. We define the following relations on Z [UTG] :i) Two labelled unitrivalent graphs Γ and − Γ are AS-related if Γ and Γ are thesame up to the cyclic order at one node. This is depicted in Figure 6.ii) Let Γ be a labelled unitrivalent graph. Let e be an edge in Γ between two trivalentnodes v and w . Then Γ is IHX-related to the difference Γ (cid:48) − Γ (cid:48)(cid:48) of the followingtwo labelled unitrivalent graphs Γ (cid:48) and Γ (cid:48)(cid:48) : Let { e, e (cid:48) v e (cid:48)(cid:48) v } be the ordered set ofedges at the node v , i.e. we have e < e (cid:48) v < e (cid:48)(cid:48) v (cyclic order). In the same way, let { e, e (cid:48) w , e (cid:48)(cid:48) w } be the ordered set of edges at the node w . The graph Γ (cid:48) arises from Γ by deleting the edge e and the nodes v and w of Γ , and adding an edge e (cid:48) and twotrivalent nodes v (cid:48) and w (cid:48) such that the ordered set of edges at v (cid:48) are { e (cid:48) , e (cid:48)(cid:48) v , e (cid:48) w } and the ordered set of edges at w (cid:48) are { e (cid:48) , e (cid:48)(cid:48) w , e (cid:48) v } . The unitrivalent graph Γ (cid:48)(cid:48) isconstructed in a similar fashion. For Γ (cid:48)(cid:48) the ordered set of edges at the nodes v (cid:48) is { e (cid:48) < e (cid:48)(cid:48) v < e (cid:48)(cid:48) w } and at the nodes w (cid:48) is { e (cid:48) < e (cid:48) w < e (cid:48) v } . This is depicted in Figure 7. Figure 6. A visualisation of the AS-relation. OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 27 Figure 7. A visualisation of the IHX-relation. Definition 4.3. Denote by UTT p the set of labelled unitrivalent trees of degree p . Wedenote by T p the abelian group generated by UTT p modulo AS- and IHX-relations, i.e. T p := Z [UTT p ] / ∼ AS , ∼ IHX . Construction 4.4. For p ≥ , denote by T p the set of iterated Whitehead products ofthe elements x ip with ≤ i < p such that x ip appears at most once in a given iteratedWhitehead product. We are going to construct a one-to-one correspondence T p Ψ T (cid:29) Φ T labelled unitrivalent tree of degree at most p − together with a monotone bijection of their labellingwith a subset of { , . . . , p } containing p . Define the length of τ ∈ T p to be the total number of occurrences of x i,p with i = 1 , . . . , p in τ . We will define Ψ T inductively on the length n of τ . Define Ψ T ( x ip ) to be the degree labelled unitrivalent tree consisting of two nodes labelled by i and p and an edge connectingthem. Assume that for all τ k ∈ T p of length k with k ≤ n − < p , we have that Ψ T ( τ k ) is adegree k − unitrivalent tree with labelling L τ k := { i ∈ N | x ip appears in τ k } ∪ { p } . For atree τ n = [ τ (cid:48) , τ (cid:48)(cid:48) ] ∈ T p of length n , we know that τ (cid:48) and τ (cid:48)(cid:48) are elements of T p and of lengthsmaller than n . By induction hypothesis, both Γ (cid:48) := Ψ T ( τ (cid:48) ) and Γ (cid:48)(cid:48) := Ψ T ( τ (cid:48)(cid:48) ) have a leafwith label p . We define the labelled unitrivalent tree Ψ T ( τ n ) to be the tree that arises byjoining the tree Γ (cid:48) and Γ (cid:48)(cid:48) at the respective leaves labelled by p , and joining to this jointpoint a new leaf labelled by p . The set of labels of Ψ T ( τ n ) is L τ (cid:48) ∪ L τ (cid:48)(cid:48) . This constructionis depicted in Figure 8, where also the cyclic order at the joint node is indicated. Figure 8. Unitrivalent trees Γ (cid:48) , Γ (cid:48)(cid:48) , and the unitrivalent tree obtainedfrom Γ (cid:48) and Γ (cid:48)(cid:48) by joining their leaves labelled by p .Now we proceed to define the inverse map Φ T . For a labelled unitrivalent tree Γ ofdegree with the set of labellings { i, p } , set Φ T ( Γ ) = x ip . Assume that we have alreadydefined Φ T for unitrivalent trees of degree smaller than n with n < p − . Every unitrivalenttree Γ n of degree n can be depicted as in Figure 8. Then define Φ T ( Γ n ) := [ Φ T ( Γ (cid:48) ) , Φ T ( Γ (cid:48)(cid:48) )] ,where Γ (cid:48) and Γ (cid:48)(cid:48) are the trees depicted in Figure 8. By construction, the two maps Ψ T and Φ T are inverse to each other. Proposition 4.5. Recall the group E p,p ∼ = (cid:76) H Z from Proposition 3.3.5. For p ≥ , theconstruction above induces an isomorphism E p,p ∼ = T p − of groups. Lemma 4.6. Let J p ⊆ T p be the set of iterated Whitehead products of the elements x ip with i = 1 , . . . , p − such that each x ip appears exactly once in an iterated Whiteheadproduct. Then we have E p,p ∼ = Z [ J p ] / ∼ , where ∼ denotes the antisymmetry and Jacobi identity relations from Proposition 3.2.11.Proof. Recall H from Proposition 3.3.5, and note that H ⊆ J p ⊆ E p,p . Thus any elementof J p \ H can be written as linear combination of elements of H . Furthermore, thislinear combination is produced by applying the Jacobi identity and antisymmetry relationto the element, cf. [Hal50, Theorem 3.1]. As a result, we obtain the desired a groupisomorphism E p,p ∼ = Z [ J p ] / ∼ . (cid:3) Proof of Proposition 4.5. We are going to define group homomorphisms Z [ J p ] ψ T (cid:29) φ T Z [ UTT p − ] , such that the induced map ψ T and φ T on the quotients E p,p and T p − are inverse to eachother. We will define our morphisms on generators and extend linearly to the whole group.For p = 2 , define ψ T ( x ) = Γ and φ T ( Γ ) = x , where Γ is the labelled unitrivalenttree of degree and with labelling set { , } . There is no relation to consider when passingto the quotients E , and T . Thus ¯ ψ T and ¯ φ T are inverse to each other by definition.For p ≥ and v = [ v , v ] ∈ J p , define ψ T ([ v , v ]) = ( − L + L × > L ) Ψ T ([ v , v ]) where L i := { j ∈ N | x jp appears in v i } ∪ { p } and L × > L := { ( a, b ) ∈ L × L | a > b, and a, b (cid:54)∈ L ∩ L } . To see that the anti-symmetry of the Whitehead product corresponds to the AS-relation,we look at ψ T ([ v , v ] + ( − L × L ) [ v , v ]) which equals ( − L + L × > L ) Ψ T ([ v , v ]) + ( − L × L )+ L × > L )+ L Ψ T ([ v , v ]) . (4.7)Recall from Construction 4.4 that the only difference between the tree Ψ T ([ v , v ]) and thetree Ψ T ([ v , v ]) is the cyclic order at the trivalent node which is adjacent to the leaf withlabel p , i.e. Ψ T ([ v , v ]) ∼ AS − Ψ T ([ v , v ]) . Note that the sum of signs in Equation . is L + L × > L ) + L × L ) + L × > L ) + L = L + L × L ) + L + ( L − L − , because L × > L ) + L × > L ) = ( L − L − . Thus we obtain that theelement in Formula 4.7 AS-related to 0.As for the Jacobi identity, take v = [ v , [ v , v ]] ∈ J p with v i ∈ π l i (Conf p ( R × D )) where l i = L i . Then ψ T (cid:0) ( − ( l − l [ v , [ v , v ]] + ( − ( l − l [ v , [ v , v ]] + ( − ( l − l [ v , [ v , v ]] (cid:1) = ( − (cid:15) Ψ T ([ v , [ v , v ]]) + ( − (cid:15) Ψ T ([ v , [ v , v ]]) + ( − (cid:15) Ψ T ([ v , [ v , v ]]) , (4.8)where (cid:15) , (cid:15) and (cid:15) are the suitable signs. Again by Construction 4.4, we have that ( − (cid:15) [ Ψ T ([ v , [ v , v ]])] + ( − (cid:15) [ Ψ T ([ v , [ v , v ]])] + ( − (cid:15) [ Ψ T ([ v , [ v , v ]])] is IHX-related to , and thus the element in Equation 4.8 is IHX-related to . Therefore, ¯ ψ T is well-defined. Note that L i is the set of labels of the tree Ψ T ( v i ) , and thus L i is the number of leaves of this tree. OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 29 Let Γ p − be a labelled unitrivalent tree of degree p − , drawn as in Figure 8, define φ T ( Γ p − ) = ( − L + L × > L ) Φ T ( Γ p − ) , where L and L is the set of labels of Γ (cid:48) and Γ (cid:48)(cid:48) respectively.Similar to the discussion of ψ T , one can show that φ T is well-defined. Therefore, themaps ψ T and φ T are inverses to each other by construction. (cid:3) Remark . Based on the methods in [Con08, Section 4], Construction 4.4 and Propo-sition 4.5 are generalisations of [Con08, Definition 4.5, Proposition 4.6] and [Con08,Proposition 4.7] respectively. Our proofs also add missing details of [Con08, Proposi-tion 4.7]. Definition 4.10. Let i, j ∈ N with i (cid:54) = j . An ( i, j ) -marked unitrivalent graph Γ ij is aunitrivalent graph of degree j together with two marked nodes v i and v j that satisfy thefollowing properties:i) The underlying graph of Γ is connected and has exactly one simple cycle .ii) The two marked nodes v i and v j are adjacent to the leaf with label i and j ,respectively, and lie on the simple cycle.Denote by UTG i,j the set of ( i, j ) -marked unitrivalent graphs.In Figure 5, we can mark the nodes adjacent to the leaf with lable 3 and 8, respectively,and obtain a (3, 8)-marked unitrivalent graph, see Figure 9. Figure 9. A (3 , -marked unitrivalent graph. The blacks dots indicatethe marked nodes. The arrow is not part of the graph. Definition 4.11. For p ≥ , define D p to be the abelian group generated by the collectionof ( i, p ) -marked unitrivalent graphs with ≤ i < p , modulo AS- and IHX sep -relations, i.e. D p := Z [ ∪ p − i =1 UTG i,p ] / ∼ AS , ∼ IHX sep , where the IHX sep -relation is the usual IHX-relation, except that the edge e , which appearsin Definition 4.2.ii), is not allowed to be an edge that adjacent to the marked nodes. Construction 4.12. Recall from Proposition 3.3.12 the definition of D sep p for p ≥ . Weare going to construct a one-to-one correspondence D sep p Ψ D (cid:29) Φ D p − (cid:91) i =1 UTG i,p − . We use Ψ T and Φ T from Construction 4.4 to define Ψ D and Φ D . A simple cycle is defined as a loop in the graph with no repetitions of nodes and edges allowed, otherthan the repetition of the starting and ending nodes. Figure 10. A visualisation of the labelled unitrivalent graph Ψ D ( w ) .For w = [ c , c ] ∈ D sep p , say with repeated occurrence of x k,p − , apply Ψ T to c and to c .We obtain the following two labelled unitrivalent trees Γ := Ψ T ( c ) and Γ := Ψ T ( c ) .Define Ψ D ( w ) to be the following ( i, p − -marked unitrivalent graph.For a ( i, p − -marked unitrivalent diagram Γ i,p − , we can draw Γ i,p − as in Figure 10.Then define Φ D ( Γ i,p − ) := [ Φ T ( Γ ) , Φ T ( Γ )] . By construction the maps Ψ D and Φ D areinverse to each other. Proposition 4.13. Let w = [[ c , [ c , c ]] , c ] be an iterated Whitehead product in D sep p , saywith c i ∈ π l i (Conf p − ( R × D )) . We have in Z [ D sep p ] the separated Jacobi identity , i.e. ( − ( l − l (cid:2) [ c , [ c , c ]] , c (cid:3) + ( − ( l − l (cid:2) [ c , [ c , c ]] , c (cid:3) + ( − ( l − l (cid:2) [ c , [ c , c ]] , c (cid:3) = 0 . Proof. Apply the Jacobi identity to [ c , [ c , c ]] . (cid:3) Remark . Note that the usual Jacobi identity is not well-defined in Z [ D sep p ] . Proposition 4.15. Recall the group E p − ,p / tors from Proposition 3.3.12. For p ≥ , theconstruction above induces an isomorphism E p − ,p / tors ∼ = D p − of groups. Similar to Lemma 4.6, we obtain the following presentation of E p − ,p / tors . Lemma 4.16. We have E p − ,p / tors ∼ = Z [ D sep p ] / ∼ , where ∼ denotes relation induced by antisymmetry and the separated Jacobi Identity. (cid:3) Proof of Proposition 4.15. Similar as in the proof of Proposition 4.5, we can define grouphomomorphisms Z [ D sep p ] ψ D (cid:29) φ D Z [ p − (cid:91) i =1 UTG i,p − ] . such that the induced map ψ D and φ D are on the quotients are inverses to each other. Let w = [ c , c ] ∈ D sep p , say with repeated occurrence of x k,p − . For i = 1 , , define the set oflabels L i := { j ∈ N | x j,p − appears in c i } ∪ { p − } . Define ψ D ( w ) := ( − L × > L ) Ψ D ( w ) .Let Γ k,p − ∈ UTG k,p − , for example drawn as in Figure 10. Denote by L i the labellingof Γ i with i = 1 , . Define φ D ( Γ k,p − ) := ( − L × > L ) Φ D ( Γ k,p − ) . OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 31 Similar as in the proof of Proposition 4.5 we can check by explicit computation that theinduced maps ¯ ψ D and ¯ φ D on the quotients E p − ,p / tors and D p − are well-defined. Moreprecisely, antisymmetry corresponds to AS-relation and the separated Jacobi identitycorresponds to IHX sep -relation. By construction the maps ψ D and φ D are inverse toeach other. (cid:3) Definition 4.17. Define the following relations on Z [ UTG ] :i) Let Γ be a labelled unitrivalent graph. Let e be the edge connecting the leaflabelled by n and the adjacent trivalent node v . Then Γ is ST U -related to thedifference Γ (cid:48) − Γ (cid:48)(cid:48) of the following two labelled unitrivalent graphs Γ (cid:48) and Γ (cid:48)(cid:48) Denote by { e, e , e } the ordered set of edges at the node v , i.e. e < e < e (cyclicorder). The graph Γ (cid:48) arise from Γ the following steps: First, delete the edge e ,the node v and the leaf labelled by n . Second, add two leaves, labelled by n and n + 1 with adjacent edges e and e respectively. Third, relabel the leaves labelledwith m by m + 1 if m > n . The graph Γ (cid:48)(cid:48) is constructed similar to Γ (cid:48) . For Γ (cid:48)(cid:48) theadjacent edges of the leaves labelled by n and n + 1 are e and e respectively. TheSTU-relation is depicted in Figure 11. Figure 11. A visualisation of the STU-relation.ii) Let Γ be a labelled unitrivalent graph. Then Γ (cid:48) − Γ (cid:48)(cid:48) is STU -related to Γ (cid:48) − Γ (cid:48)(cid:48) ,where Γ (cid:48) − Γ (cid:48)(cid:48) is obtained by performing the STU-relation at the leaf of Γ labelledby n , and Γ (cid:48) − Γ (cid:48)(cid:48) is obtained by performing the STU-relation at the leaf of Γ labelled by m . The STU -relation is depicted in Figure 12. Figure 12. A visualisation of the STU -relation. Remark . Note that the STU -relation is finer than the STU-relation. Proposition 4.19. The STU -relation is well-defined on Z [UTT p ] for p ≥ .Proof. This is because the STU -relation does not change the connectivity and degree ofthe unitrivalent graphs, and it also does not add simple cycles to the graphs. (cid:3) Recall the computation of the differential d : E p − ,p → E p,p from Proposition 3.3.13.Note that in the computation only d | E p − ,p / tors is relevant, and by abuse of notation we willwrite d instead of d | E p − ,p / tors in the following. By Proposition 4.5 and Proposition 4.15we can consider d as a map between two groups of unitrivalent graphs as follows Ψ T ◦ d ◦ Φ D : D p − → T p − . By abuse of notation we will also denote this map between D p − and T p − by d . Thefollowing proposition describes what d means on the level of unitrivalent graphs. Theorem 4.20. Viewed on generators, the differential d maps a ( k, p − -markedunitrivalent graph Γ k,p − to the linear combination Γ p − Γ p − ( Γ k − Γ k ) of labelled unitrivalenttrees, where Figure 13. An example of d applied to a ( k, p − -marked unitrivalentgraph. The triangles are placeholders for subgraphs, which stay unmodified.i) Γ p − Γ p is obtained from performing the STU-relation on Γ k,p − at the edge con-necting the leaf labelled by p − and the marked node v p − , and ii) Γ k − Γ k is obtained from performing the STU-relation on Γ k,p − at the edge con-necting the leaf labelled by k and the marked node v k . For a pictorial illustration of d , see Figure 13. Proof. Denote by w = [ c , c ] ∈ D sep p , say with repeated occurrence of x k,p − , a elementin E p − ,p that corresponds to Γ k,p − under the isomorphism from Proposition 4.15. Write(without loss of generality) c = [ . . . x i,p − . . . x k,p − . . . ] and c = [ . . . x k,p − . . . x j,p − . . . ] .Recall the notations from the proofs of Proposition 4.5 and Proposition 4.15, we have ψ T ( c ) = ( − (cid:15) Γ ψ T ( c ) = ( − (cid:15) Γ ψ D ( w ) = ( − (cid:15) w Γ w , where Γ , Γ and Γ w are depicted in Figure 14. We will discuss the signs (cid:15) , (cid:15) and (cid:15) w atthe end. Figure 14. Visualisations of Γ , Γ and Γ w .Recall the formula for ∂ k ( w ) from Proposition 3.3.13. We write ∂ k ( w ) = ( − k [ c k , c k +12 ] + ( − k [ c k +11 , c k ] , where c k , c k +12 , c k +11 and c k correspond exactly to the four Whitehead brackets in theformula of ∂ k ( w ) . Thus we have ψ T ( c k ) = ( − (cid:15) Γ k ψ T ( c k +12 ) = ( − (cid:15) Γ k +12 ψ T ( c k +11 ) = ( − (cid:15) Γ k +11 ψ T ( c k ) = ( − (cid:15) Γ k , OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 33 where Γ k , Γ k +12 , Γ k +11 and Γ k are the labelled unitrivalent trees depicted belowNote that the underlying unlabelled trees of Γ ki and Γ k +1 i for i = 1 , are the same asthe ones for Γ i respectively. Furthermore, we have ψ T ([ c k , c k +12 ]) = ( − (cid:15) k Γ k ψ T ([ c k +11 , c k ]) = ( − (cid:15) k Γ k where Γ k and Γ k are the labelled unitrivalent trees depicted in Figure 15. Figure 15. Descriptions of the labelled unitrivalent trees Γ k and Γ k .We perform the same steps for ∂ p − ( w ) = ( − p − [ c , c p ] + ( − p − [ c p , c ] , thus obtain ψ T ( c p ) = ( − (cid:15) Γ p ψ T ( c p ) = ( − (cid:15) Γ p , where Γ p and Γ p are the labelled unitrivalent trees depicted in Figure 16. Figure 16. Descriptions of the labelled unitrivalent trees Γ p , Γ p ,Γ p and Γ p Furthermore, ψ T ([ c , c p ]) = ( − (cid:15) p Γ p and ψ T ([ c p , c ]) = ( − (cid:15) p Γ p , where Γ p and Γ p aredepicted in Figure 16. Reviewing the leaves of Γ ik and Γ ip as placed on the oriented line,we get the following illustrations of the treesThus, we have that ψ T ( d ( w )) = ( − k (cid:0) ( − (cid:15) k Γ k + ( − (cid:15) k Γ k (cid:1) + ( − p − (cid:0) ( − (cid:15) p Γ p + ( − (cid:15) p Γ p (cid:1) . Now let us check the signs. For i = 1 , , let L i be the set of labels of Γ i . Thus we have (cid:15) w = (cid:15) + (cid:15) + L × > L ) . where (cid:15) and (cid:15) appeared right at the beginning of the proof. Let L ji be the set of labelsof Γ ji and L pi be the set of labels of Γ pi , for j ∈ { k, k + 1 } and i ∈ { , } . Then we have (cid:15) k = (cid:15) + (cid:15) + L k + L k × > L k +12 ) ,(cid:15) k = (cid:15) + (cid:15) + L k +11 + L k +11 × > L k ) ,(cid:15) p = (cid:15) + (cid:15) + L + L × > L p ) ,(cid:15) p = (cid:15) + (cid:15) + L p + L p × > L ) . Recall Remark 3.3.15 and thus observe that L = L k = L k +11 = L p . Furthermore,we have L k +11 × > L k ) = L k × > L k +12 ) + 1 , where the +1 arises from the pair ( k + 1 , k ) ,and L p × > L ) = L × > L p ) + 1 , where the +1 arises from the pair ( p, p − . Thuswe can write ψ T ( d ( w )) as ψ T ( d ( w )) = ( − k + (cid:15) k ( Γ k − Γ k ) + ( − p − (cid:15) p ( Γ p − Γ p ) . (4.21)For the signs we use (cid:0) L k +11 × > L k (cid:1) = L \ { k } ) × > ( L \ { k } ))+ { k + 1 } × > ( L \ { k } ))+ L \ { k } ) × > { k } ) , L × > L p ) = L \ { k } ) × > ( L \ { k } )) + L \ { k, p − } ) , L \ { k, p − } ) = { k + 1 } × > ( L \ { k } )) + L \ { k } ) × > { k + 1 } ) p − k − L \ { k } ) × > { k + 1 } ) + L \ { k } ) × > { k } ) . Thus we we have k + (cid:15) k + p − (cid:15) p = 2 p − , Here we join the trees at the leaves labelled by k . Note that Construction 4.4 only considers the casewhere we join at the leaves labelled by p . The construction for k inplace p is analogous, however, one hasto check that both constructions apply to the same tree yield the same iterated Whitehead product. OODWILLIE’S COSIMPLICIAL MODEL FOR KNOTS AND ITS APPLICATIONS 35 i.e. the signs before ( Γ k − Γ k ) and ( Γ p − Γ p ) in Equation 4.21 are different.Consider the sign in front of Γ k,p − and Γ p in the equations ψ D ([ c , c ]) = ( − (cid:15) w Γ k,p − and ψ T ([ c , c p ]) = ( − (cid:15) p Γ p . First recall that (cid:15) w = (cid:15) + (cid:15) + L × > L ) ,(cid:15) p = (cid:15) + (cid:15) + L + L × > L p ) . Observe that we have L × > L p ) = L × > L ) + L \ { k, p } ) , where the term L \ { k, p } ) arises because of the set of pairs { ( p − , x ) ∈ L × L p | x (cid:54) = k, p } . Moreoverwe know that L + L \ { k, p } ) = p − . So the sign difference between ψ D ([ c , c ]) and ψ T ([ c , c p ]) is ( − p − , which cancels with the sign in front of [ c , c p ] in the expressionof ∂ p − ( w ) . Therefore, (cid:15) w and (cid:15) p have the same parity. In other words, Γ k,p − and Γ p havethe same sign.In conclusion, we are justified to write d ( Γ k,p − ) = Γ p − Γ p − (cid:0) Γ k − Γ k (cid:1) . (cid:3) Corollary 4.22 (Conant) . Let τ ∈ T p − , then τ ∈ im( d ) if and only if τ is STU -relatedto 0.Proof. “ ⇐ ” It follows from the formula for d in the previous theorem that the imageof the generators under d are STU -related to 0. Thus, any element in im( d ) is alsoSTU -related to 0.“ ⇒ ” It is sufficient to prove that any linear combination of the form Γ p − Γ p − ( Γ k − Γ k ) with ≤ k ≤ p − , lies in the image of d . Note that Γ k − Γ k is the result of performinga STU-relation at the trivalent node adjacent to the leaf with label k in Γ w = ¯ ψ D ( w ) , cf.Figure 14. Similarly, Γ p − Γ p is the result of performing a STU-relation at the trivalentnode adjacent to the leaf with label p − in Γ w . Therefore, we can obtain the linearcombination Γ p − Γ p − Γ k + Γ k via performing STU-relations on a ( k, p − -markedunitrivalent graphs at its two marked trivalent nodes. By Proposition 4.15, the domain ofthe differential d is exactly the set of ( k, p − -marked unitrivalent graphs. (cid:3) Remark . Our proofs of Theorem 4.20 and Corollary 4.22 supplement the proof of[Con08, Proposition 4.8] with more details. With the notion of marked unitrivalent graph,we are able to give the combinatorial interpretation of the map d , instead of only itsimage. Corollary 4.24. i) For p ≥ , the group E p,p is isomorphic to the abelian group generated by unitrivalenttrees of degree p − , modulo AS-, IHX-, and ST U -relations. ii) For p = 3 , we have E , ∼ = T ∼ = Z . iii) For p = 0 , , , we have E p,p = 0 .Proof. i) follows from Lemma 4.6 and Corollary 4.22. ii) follows from the fact that themap d : E , → E , is trivial (Proposition 3.3.10) and E , ∼ = T (Proposition 4.5) iii)follows from Proposition 3.3.10. (cid:3) Theorem 4.25 (Conant) . Let n ∈ N . We have E n,n ⊗ Q is isomorphic to the rationalvector space A In − ⊗ Q generated by unitrivalent trees of degree n − modulo AS-, IHX-,and SEP-relations.Proof. See [Con08, Theorem 3.3]. (cid:3) Remark . We refer the readers to [Bar95], [CT04a], [CT04b] and [Hab00], for the closerelation between Vassiliev knot invariants and unitrivalent graphs. As already points outby Bott in [Bot95], studying the groups E rn,n , and especially the passage from E n,n to E ∞ n,n may be another approach to the theory of Vassiliev invariants. Conclusion and further work The connection between Vassiliev invariant and the embedding functor Emb( − ) can beexpressed in the following diagram with n ≥ π (T n Emb(I)) E ∞ n,n E n,n E n,n π ( K ) π ( K ) / ∼ C n G n − Z [ UTT n − ] AS, IHX, STU Z [ UTT n − ] AS, IHX .(cid:63) η n (I) ¯ η n (I) ¯ η n (I) | G n − [CT04a] ∼ = ∼ = (5.1)Let us first explain and give references to some of the notations in the diagram:i) Recall that E in,n with i ∈ { , , ∞} denotes the diagonal terms of the homotopyBousfield–Kan spectral sequence associated to the tower of fibrations · · · → T n Emb(I) → T n − Emb(I) → · · · → T Emb(I) with E ∞ n,n ∼ = ker (cid:0) π (T n Emb(I)) → π (T n − Emb(I)) (cid:1) ii) The clasper surgery equivalence ∼ C n is an equivalence relation on the set π ( K ) of isotopy classes of knots defined by Habiro in [Hab00], and he proves that thecanonical map π ( K ) → π ( K ) / ∼ C n is the universal additive Vassiliev invariant ofdegree n − .iii) The group G n − has a geometric interpretation using gropes. There is an isomor-phism G n − ∼ = ker (cid:0) π ( K ) / ∼ C n → π ( K ) / ∼ C n − (cid:1) A grope is an embedded (CW)-complex in R , whose boundary components areknots. Conant and Teichner [CT04a; CT04b] explain the connection among gropes,clasper surgery and Vassiliev invariants.In this text we showed the commutativity of the right most square in the above diagram,by giving combinatorial interpretations to our computations of the spectral sequence. Thekey ingredient for the calculations is the equivalence between the category of cosimplicialspaces and the category of good functors from Open ∂ (I) op to spaces, which provides uswith a cosimplicial space E mb • associated to the embedding functor Emb( − ) .We find it interesting that the upper row of Diagram 5.1 is algebraic in nature whereasthe lower row is geometric and combinatorial. If Conjecture 1.2.9 holds, the map ¯ η n (I) would be an isomorphism of groups. 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