Embedding calculus and grope cobordism of knots

We show that embedding calculus invariants $ev_n$ are surjective for long knots in an arbitrary $3$-manifold. This solves some remaining open cases of Goodwillie--Klein--Weiss connectivity estimates, and at the same time confirms one half of the conjecture that for classical knots $ev_n$ are universal additive Vassiliev invariants over the integers. In addition, we give a sufficient condition for this conjecture to hold over a coefficient group, which is by recent results of Boavida de Brito and Horel fulfilled for the rationals and for the $p$-adic integers in a range. Therefore, embedding calculus invariants are strictly more powerful than the Kontsevich integral. Furthermore, our work shows they are more computable as well. Namely, the main theorem computes the first possibly non-vanishing invariant $ev_n$ of a knot which is grope cobordant to the unknot to be precisely equal to the equivalence class of the underlying decorated tree of the grope in the associated graph complex. Actually, our techniques apply beyond dimension $3$, offering a description of the layers in embedding calculus for long knots in a manifold of any dimension, and suggesting that certain generalised gropes realise the corresponding graph complex classes.

In contrast to the study of spaces Map( , ) of maps between topological spaces, which gave rise to numerous techniques of homotopy theory, spaces Emb( , ) of smooth embeddings of manifolds seem less tractable from the homotopy viewpoint. Already at the level of components (isotopy classes of embeddings), often purely geometric arguments are used.
In this paper we relate two well-known attempts to reconcile these viewpoints in the case of long knots in a compact oriented 3-manifold with boundary. Namely, let : The first approach, Vassiliev's theory of finite type knot invariants [Vas90;Vas98], starts from the observation that having understood the space of smooth maps, we can try to describe its subspace of embeddings by studying homotopy types of the strata of the complement Map ( , ) \ K( ).
In its more geometric version [Gus00; Hab00; CT04b] this theory gives a sequence of equivalence relations ∼ for ≥ 1 on the set K( ) := 0 K( ), defined in terms of either claspers or gropes.
The second approach, the embedding calculus of Goodwillie and Weiss [Wei99;GW99], builds on the idea -having its roots in Hirsch-Smale immersion theory -that, since we understand embeddings of disjoint unions of disks, we could use them to approximate the space Emb( , ). The outcome is a certain tower of spaces P ( ) and maps ev : K( ) → P ( ), for ≥ 1. See Introduction 1 for a brief survey of both approaches, and Sections 2.2 and 5 for more details.
It was suggested early on [GW99;GKW01] that these theories should be closely related for = 3 , in which case K( 3 ) 0 Emb(S 1 , R 3 ) is precisely the abelian monoid of classical knots. The study of the relationship was initiated in [BCSS05], where the following conjecture was stated, and proven in the first non-trivial degree = 3 (see Example 2.25 below for another proof).
Conjecture 0.1 ( [BCSS05]). For each ≥ 1 the map 0 ev : K( 3 ) → 0 P ( 3 ) is a universal additive Vassiliev invariant of type ≤ − 1 over Z, that is, it is a monoid homomorphism which factors as and the induced map on the right is an isomorphism of groups.
In other words, the paths in the space P ( 3 ) should precisely encode the -equivalence relation. The work of [Vol06;Tou07;Con08] showed that, roughly speaking, graph complexes appearing in the two theories agree, while more recently a part of the conjecture was confirmed in [BCKS17]. They show that each 0 ev is an additive invariant of type ≤ − 1, namely, 0 P ( 3 ) has a group structure so that 0 ev is homomorphism of monoids, and the mentioned factorisation exists. Our joint work [KST] reproves the latter claim from the perspective of gropes, see also Theorem C.
One of the main results of the present paper is the proof of the 'surjectivity part' of Conjecture 0.1.
Theorem 0.2. For each ≥ 1 the homomorphism 0 ev : This is just a special case of Theorem 0.5 which says that the same holds for an arbitrary 3-manifold.
Starting from a geometric viewpoint, we use different models than the mentioned thread of work: capped gropes for finite type theory [CT04b] and Goodwillie's punctured knots model for embedding calculus (see [Sin09]). Moreover, in [Vol06] Volić asks 'Can one in general understand the geometry of finite type invariants using the evaluation map?', and we make a step forward in that direction.
Namely, there is a graph complex which computes the homotopy groups of P ( 3 ) and our main Theorem A implies that (see Corollary 2.19 for the precise statement) the evaluation map detects the underlying tree of a grope/clasper in the graph complex.
Analogous results hold for universal rational Vassiliev knot invariants of Kontsevich [Kon93] and Bott-Taubes [BT94; AF97], as well as for similar invariants of families of diffeomorphisms of S 4 , shown by Watanabe [Wat18] to detect the underlying graph of a family constructed using similar claspers. However, a crucial difference is that while all these invariants use integrals over configuration spaces -and so can provide results only in characteristic zero -embedding calculus is a purely topological technique for studying homotopy types of embedding spaces themselves.
Indeed, Theorem A can be used both to confirm Conjecture 0.1 rationally and show that Kontsevich and Bott-Taubes integrals factor through the tower (see Remark 2.20), and also more generally: Corollary 0.3. Let be a torsion-free abelian group. If the homotopy spectral sequence * − , ( 3 ) ⊗ for the Taylor tower of K( 3 ) collapses at the 2 -page along the diagonal, then 0 ev is a universal additive Vassiliev invariant of type ≤ − 1 over , meaning that Here * − , ( 3 ) is the usual spectral sequence for the homotopy groups of the tower of fibrations +1 : P +1 ( 3 ) → P ( 3 ) and can be related to graph complexes: its 2 -page along the diagonal was identified by Conant [Con08] as the group of Jacobi trees A (see Definition 2.7 and Theorem 2.17).
This has already been confirmed in some cases.
Corollary 0.4. The following holds.
(2) For any prime , the evaluation map 0 ev is a universal additive Vassiliev invariant of type ≤ − 1 over the -adic integers Z if ≤ + 2.
Namely, Boavida de Brito and Horel [BH20] show vanishing of higher differentials in a range for this spectral sequence in positive characteristic, implying (2). Their results also imply that * − , ( 3 ) ⊗Q collapses at the whole 2 -page (this could also be deduced from [FTW17]). Moreover, some existing low-degree computations show that the group A is torsion-free, giving (3). For proofs of both corollaries and further details see Section 2.3.2.
This was expected to hold by analogy to the famous Goodwillie-Klein connectivity formula (see Theorem 1.1), which predicts that for a 1-dimensional source manifold and a 3-dimensional target the map ev is 0-connected. For a connected source this is precisely Theorem 0.5, and in future work we plan to investigate if our method can be extended to links.
Remark 0.6. The corollaries of Theorem A stated above apply to some extent also for knots in a general 3-manifold , using analogous groups A ( ) as studied in [Kos20]; see also Remark 1.8.
Let us now introduce some notation, so that we can state Theorem A and deduce Theorem 0.5.
We will actually study the punctured knots model P ( ) even more generally: for any connected compact smooth manifold of dimension ≥ 3 with non-empty boundary. It is defined as a homotopy limit over a certain finite category and we will develop its properties in detail in Sections 3 and 4. We only restrict to oriented 3-manifolds when we later consider gropes.
Let us pick an arbitrary knot U ∈ K( ) as our basepoint and call it the unknot, and let ev (U) be the basepoint of P ( ). There is a natural map +1 : P +1 ( ) → P ( ) which is a surjective fibration and satisfies +1 •ev +1 = ev , so preserves basepoints. The fibres F +1 ( ) := fib ev U ( +1 ) are called the layers of the Taylor tower. Moreover, we also consider the homotopy fibre and in Section 3.2 we construct a map e +1 making the following diagram commute: Remarkably, both F +1 ( ) and H ( ) are related -for a priori different reasons -to the set Tree 1 ( ) of 1 -decorated trees, where Γ ∈ Tree 1 ( ) consists of a rooted planar binary tree Γ with enumerated leaves which are also decorated by elements ∈ 1 ( ), for ∈ := {1, . . . , }. See Section 2.1 for all definitions related to trees. Indeed, on one hand, the set 0 F +1 ( ) is isomorphic to the group of Lie trees Lie 1 ( ), defined as the quotient of the Z-span of Tree 1 ( ) by the antisymmetry ( ) and Jacobi relations ( ). On the other hand, from the data of a capped grope cobordism G in a 3-manifold we will construct a point (G) ∈ H ( ), with the underlying combinatorics of G also described by t(G) ∈ Tree 1 ( ).
Let B( ) be a Hall basis for the free Lie algebra L( : ∈ ), and NB( ) ⊆ B( ) the subset of words in which each letter for ∈ appears at least once. Let be the word length of ∈ NB( ). Further, for ≥ 1 denote by × the -fold product of with itself and by Ω × the space of based loops in it; for a space let + := { * } and Σ ( + ) its -fold reduced suspension. In other words, the two roles of trees, homotopy theoretic for F +1 ( ) and geometric for H ( ), are mutually compatible: we will define (Γ ) := [ (G)] for some grope G with t(G) = Γ and show that [e +1 (G)] ∈ 0 F +1 ( ) Lie 1 ( ) is precisely the class of t(G) modulo and .
Remark 0.7. There is an isomorphism Lie 1 ( ) ( −1)! . If is simply connected, we obtain Lie( ) Z ( −1)! , the arity of the Lie operad. Interestingly, in Goodwillie's homotopy calculus the -th Taylor layer for a functor : Top * → Top * is computed in terms of a spectrum (Id), and these turn out to form an operad * (Id) whose homology is precisely the Lie operad.
The following is an immediate corollary of Theorem A, and implies Theorem 0.5.
Proof of Theorem 0.5 assuming Theorem A. It is a standard fact (see (3.0.1) below) that P 1 ( ) is homotopy equivalent to the loop space Ω(S ) on the unit tangent bundle S of . Thus,

Introduction
Finite type theory. Vassiliev's study of the strata of the discriminant Map ( , 3 ) \ K( 3 ) gave rise to the filtration V * ( ) by type ≥ 1 of the group 0 (K( 3 ); ) of knot invariants with values in an abelian group , as formulated by [BL93]. A new, very active field emerged: it was shown that quantum invariants give rise to invariants of finite type [Lin91] (for example, for the Jones polynomial ( ) the coefficient next to ℎ in ( ℎ ) is of type ≤ ); that for = Q there is a universal such invariant -the Kontsevich integral [ Kon93;LM96]; and a comprehensive treatment of its target, the rational Hopf algebra of chord diagrams, was given in [Bar95].
A geometric approach to the field was introduced by Gusarov [Gus00] and Habiro [Hab00] independently, as a sequence of knot operations called surgeries on claspers (or variations) of degree ≥ 1. This gives a sequence of equivalence relations ∼ on the monoid K( 3 ) := 0 K( 3 ) and a decreasing filtration K ( 3 ) := { ∈ K( 3 ) : ∼ U} by submonoids. The work of Stanford [Sta98] exhibits a close connection of this filtration with the lower central series of the pure braid group.
By the work of Conant and Teichner [CT04b;CT04a] one can instead of claspers equivalently use capped grope cobordisms, and this is the approach we take. Gropes first appeared in the theory of topological 4-manifolds, and can be viewed as a tool for detecting 'embedded commutators' [Tei04]. See Section 5.1 for a background on gropes, and Remark 5.11 for an advantage of using them.
Notably, it has been shown that the map K( 3 ) → Z[K( 3 )] = 0 (K( 3 ); Z) defined by ↦ → − U, takes K ( 3 ) into V (Z), the dual of the Vassiliev filtration for = Z. Hence, this indeed gives a geometric version of the theory (or its primitive/additive part), since one works with knots instead of their linear combinations or invariants; see Section 2.2 for the comparison.
Additionally, the quotient of K( 3 ) by ∼ is actually an abelian group and the projection is a universal additive invariant of type ≤ − 1 [Hab00, Thm. 6.17] -meaning that any additive invariant : K( 3 ) → of type ≤ −1 factors through . However, the target here is a mysterious group and one would ideally have something combinatorially defined instead, perhaps the primitive part of the mentioned algebra of chord diagrams or the group of Jacobi trees (see Section 2.1.2).
Embedding calculus. The pioneering approach of Goodwillie and Weiss [Wei99;GW99] for studying embedding spaces 2 Emb ( , ) produces a tower of spaces, called the Taylor tower, · · · → T +1 Emb ( , ) → T Emb ( , ) → · · · → T 1 Emb ( , ) and the evaluation maps ev : Emb ( , ) → T Emb ( , ), starting from the space of immersions T 1 Emb ( , ) Imm ( , ). Since the definition of these objects is homotopy theoreticanalogously to the description of immersions due to Hirsch and Smale -we obtain an inductive way for studying the homotopy type of Emb ( , ), using a variety of tools. Indeed, a fundamental result in the field is the following theorem of Goodwillie and Klein (announced in [GW99]).
This result inspired a great deal of research on Taylor towers for various pairs ( , ).
2 One can take compact manifolds with a fixed boundary condition for all embeddings, or closed manifolds. 3 A map is -connected if it induces an isomorphism on homotopy groups below degree and a surjection on .
Relating the two theories. A different approach by [BCKS17] uses the model for T K( 3 ) from [Sin09] to equip 0 T K( 3 ) directly with an abelian group structure, so that the corresponding 0 is also a monoid map, as predicted by Conjecture 0.1 of [BCSS05]. It has been an open problem whether the group structures of [BW18] and [BCKS17] on 0 T K( 3 ) agree. We confirm this is indeed the case, see Corollary 2.23.
The authors of [BCKS17] also show that 0 is of Vassiliev type ≤ − 1, and we reprove this below for any 3-manifold . See Remark 1.8.

A careful study of the layers in the Taylor tower
In the homotopy theoretic part we study the space F ( ) for any smooth manifold with nonempty boundary and dimension dim( ) = ≥ 3. The upshot is the following theorem, which reformulates the first part of Theorem A.
Moreover, this space is ( ( − 3) − 1)-connected and there are explicit isomorphisms Let us give more details. Firstly, F +1 ( ) in Section 3.2 is described as the total homotopy fibre is obtained by removing | |+1 -dimensional balls from . The map is induced from the inclusion : 0 ↩→ 0 , which adds the material between two balls (see Figure 8).
Secondly, in Section 3.3 we show that this is an ( + 1)-fold loop space. Namely, F 1 ( ) ΩS and for ≥ 1: An -cube • consists of a space for each ⊆ and a compatible collection of maps : → for ∉ . The total homotopy fibre tofib( • , ) generalises the notion of a homotopy fibre of a 1-cube, see Section 2.4. 5 Of course, F +1 depends on , but we omit it from the notation.
These homotopy equivalences arise as follows.
− For and its inverse see Theorem 3.11: the map is defined using the left homotopy inverse : 0 → 0 for which adds back the -th ball and then rescales (see Figure 9).
− In Theorem 3.19 taking unit derivatives is shown to give a homotopy equivalence of contravariant cubes D• : where ⊇ 0 are obtained by gluing in a ball. In the total fibre of the latter cube the unit tangent data can be omitted, giving D as above. Thirdly, in Section 4 we describe the homotopy type of tofib Ω( ∨ S•), Ωcol , using a generalisation of the Hilton-Milnor theorem due to Gray [Gra71] and Spencer [Spe71]. Namely, we find a weak equivalence (see Theorem 4.4): This formula differs slightly from the one we gave in [Kos20], but that one can be recovered using , and once again applying the Hilton-Milnor theorem. This proves the first statement of Theorem B.

Remark 1.2. If
Σ is homotopy equivalent to a suspension, the homotopy type of F +1 ( ) was calculated in [GW99]; we can recover their result using the James splitting ΣΩΣ ∞ =1 Σ ∧ . See also [Wei99] for other description of the layers, and [BCKS17] for = 3 .
However, in neither of those approaches could we understand the comparison map, which is crucial for the proof of Theorem D; we hope that our equivalence might be of independent interest.
However, to now find maps which generate ( −3) F +1 ( ) Lie 1 ( ) we would need to invert the isomorphism (retrD ) * . For retr there is an obvious map satisfying retr • , which simply 'swings a lasso' around the missing -ball, see Figure 12. In Section 4.2 we discuss the following corollary and why it will be enough for our purposes.
Corollary 1.3. The generators of ( −2) tofib(Ω • , Ω ) are given by the canonical extensions to the total homotopy fibre of the Samelson products Γ( About the convergence. Let us point out that we obtain our results from scratch, starting with the definition of the punctured knot model and assuming only the Hilton-Milnor-Gray-Spencer theorem (whose proof is briefly recalled in Appendix B). In particular, independently of the rest of the literature we have reproved the following.
Actually, by Goodwillie-Klein Theorem 1.1 the tower converges precisely to K( ) for ≥ 4: the homotopy groups of H ( ) in degrees below ( + 1)( − 3) agree with those of F +1 ( ). We believe that our map will provide a geometric inverse to the isomorphism ( −3) ev , see Remark 1.10.
For a tower of (surjective) pointed fibrations there is an associated (non-'fringed') spectral sequence built out of long exact sequences for the homotopy groups of a fibration [BK72]. The first page is given by 1 − , := − F ( ) and the differential is the composite Corollary 1.5. The group 1 −( +1), vanishes for ≤ ( − 2), and for each = , + 1, . . . all entries in the strip 1 + ( − 2) ≤ ≤ ( + 1)( − 2) are generated by Samelson products using words of length at most in which all letters appear. In particular, the first non-vanishing slope is About configuration spaces. Taylor towers for embedding spaces are closely related to configuration spaces of manifolds Conf ( ) := Emb( , ) for a finite set . See [Sin09; BW18; FTW17] to mention just a few. They are behind the scenes in our approach as well, and we record related corollaries. As usual, let be a -dimensional compact manifold with non-empty boundary.
Proof. Each map s ∪ for ⊆ − 1 is a fibre bundle whose fibre is homeomorphic to \ ∨S . By taking fibres first in the direction of s -maps, the total fibre of the -cube ΩConf•( ), Ωs is equivalent to that of the ( − 1)-cube tofib Ω( ∨ S•), Ωcol , and this was computed in (1.1).
Sinha [Sin09] uses certain compactifications of configuration spaces to construct the mentioned model ( ) for T K( ), then employed in [BCKS17]. See Remark 3.21 for a comparison to our approach. Configuration spaces were also used by Koschorke [Kos97] to construct invariants of link maps in arbitrary dimensions. His results are very similar in spirit to ours, showing that certain invariants related to Samelson products agree with Milnor invariants for classical links.

Gropes give points in the layers of the Taylor tower
In this geometric part we specialise to = 3 (but this restriction is not essential, see Remark 1.10). We build on our joint work [KST], where we constructed points in H ( 3 ) from (simple capped genus one) grope cobordisms of degree of [CT04b]; here we extend this to any oriented 3-manifold .
Gropes. In Section 5.1 we discuss grope cobordisms, which are certain 2-complexes in modelled on trees, that 'witness' -equivalence of the two knots on 'the boundary of a cobordism'.
More precisely, one first defines (see Definition 5.1) the abstract (capped) grope Γ modelled on an undecorated tree Γ ∈ Tree( ) as a 2-complex with circle boundary built by inductively attaching surface stages according to Γ as on the left 7 of Figure 1: each leaf contributes a disk (called a cap), and each trivalent vertex a torus with one boundary component; we also fix an oriented subarc 0 ⊆ Γ = S 1 . There is a canonical embedding Γ ↩→ Γ of the tree into this 2-complex.
A (capped) grope cobordism (of genus one) on a knot ∈ K( ) modelled on Γ is a map G : Γ → which embeds all stages mutually disjointly and disjointly from except that G( 0 ) ⊆ and for ∈ the -th cap intersects transversely in a point , so that G( 0 ) < 1 < · · · < in . The degree of G is , the number of its caps. See Definition 5.2 for details and orientation conventions.
A simple example of a grope cobordism of degree 2 is shown on the right of Figure 1. Note how the two caps or 'arms' could instead be twisted and tied into knots, producing non-isotopic grope cobordisms on which are all modelled on the same tree Γ. is the union of the yellow torus and the two disks. Right: A grope cobordism G : Γ → 3 on = U, the horizontal line. The knot ⊥ G is the union of U \ G( 0 ) and the long black arc G( ⊥ 0 ), and is isotopic to the trefoil.
Moreover, we denote ⊥ 0 := Γ \ 0 and define the output knot of G by . Thus, a grope describes a modification of the knot by replacing its arc G( 0 ) ⊆ by G( ⊥ 0 ). We say that ∼ ⊥ G are -equivalent. More generally, two knots are -equivalent if there is a finite sequence of grope cobordisms of degree from one knot to the other. This gives the variant due to [CT04b] of the Gusarov-Habiro filtration K ( ; U) := { ∈ K( ) : ∼ U} mentioned above.
Another important notion related to a grope cobordism G modelled on Γ ∈ Tree( ) is its signed decoration ( , ) ∈ , where ∈ {±1} is the sign of the intersection point ∈ of and the -th cap of G, and ∈ Ω is the loop from (0) to 0 first following the unique path in the tree G(Γ), then going back along (see Definition 5.4).
7 Although our pictures sometimes seem not smooth, the corners are present only for convenience.
By appropriately framing the edges of t(G) one obtains the clasper corresponding to G, see [CT04b].
Thick gropes. For the relation to the Taylor tower it is convenient to consider thickenings (tubular neighbourhoods) of grope cobordisms which we call thick gropes. These are embeddings See Definition 5.5 for details. Finally, in Definition 5.7 we define the space Grop 1 ( ; ) of thick gropes of genus 1 and degree in on , so that taking the output knot gives a continuous map ⊥ : Grop 1 ( ; ) → K( ). This is extended below to any genus.
Theorem C. 8 If G is a thick grope of degree ≥ 1 in on a knot , then there is a path Ψ G : → P ( ) from ev ( ⊥ G) to ev ( ). Moreover, for = U this gives a continuous map We give the proof in Section 5.2, using the crucial isotopy between the two surgeries on a capped torus (Lemma 5.10). Namely, combining these isotopies for each stage of Γ gives an ( − 1)parameter family of disks D contained in the model ball B Γ and with D = Γ . Moreover, the interior of each disk G(D ) intersects the knot only inside of certain subarcs of , and so that the homotopy of G( ⊥ 0 ) back to G( 0 ) across G(D ) precisely defines a path in P ( ). The theorem immediately implies that there is a factorisation there is a sequence of thick gropes witnessing it, so concatenation of the corresponding paths in P ( ) is a path from ev to ev . In particular, as mentioned in the discussion after Conjecture 0.1, for = 3 this is equivalent to the claim that 0 ev is a Vassiliev invariant of type ≤ − 1 (this was first shown by [BCKS17]).
Remark 1.8. This equivalence to Vassiliev's theory is discussed in Section 2.2. In contrast, it is an open problem (see Remark 2.16) if such an equivalence holds for any oriented 3-manifold : the factorisation (1.2) just says that 0 ev is an invariant of -equivalence of knots in .
Remark 1.9. The remaining part of Conjecture 0.1 says that if ev is in the path component of ev U, then there exists a path between them induced from a grope forest.
In [KST] we reformulate this in terms of a map K ( 3 ) → P ( 3 ) extending ev , where we construct the simplicial space K ( 3 ) using K( 3 ) and Grop ( 3 ) for spaces of 0-and 1-simplices, respectively. We hope to prove it is a homotopy equivalence, giving a very geometric description of P ( 3 ).
Grope forests. We will also need to realise arbitrary linear combinations of decorated trees. The corresponding geometric notion which is standardly used is a grope cobordism of 'higher genus', but we instead define a slightly different notion, called a grope forest. This is an embedding F : are mutually disjoint thick gropes on whose arcs F| B Γ ( 0 ) ⊆ appear in the order of their label (see Definition 5.6). 8 The first statement for = 3 is part of the joint work [KST], and appeared first in [Shi19].
Let Grop ( ; U) := ≥1 Grop ( ; U) denote the space of grope forests of any cardinality (the component = 1 is exactly the space of thick gropes). Taking underlying trees of all the thick gropes in a grope forest gives the underlying decorated tree map This is a surjection of sets: any linear combination of 1 -decorated trees is realised by a grope forest on U (see Proposition 5.8). Furthermore, in Proposition 5.13 we extend the map from Theorem C to grope forests : Grop ( ; U) → H ( ).
Remark 1.10. One can generalise gropes to any ≥ 3 by simply replacing the model 3-ball B Γ by a -dimensional ball obtained as a ( − 2)-thickening of the 2-complex Γ . Then a thick grope in is again an embedding B Γ ↩→ so that for each 1 ≤ ≤ the neighbourhood of the -th cap ( D 2 × D −2 ) intersects in a neighbourhood of a single point ∈ .
One can similarly construct maps (G) : S ( −3) → H ( ), using that ( − 1)-dimensional normal disks to at 's give an ( − 2)-family of arcs. This gives points e +1 (G) ∈ Ω ( −3) F +1 ( ) and we believe the proofs of our main theorems below readily extend to show that e +1 is a surjection onto the first non-trivial group ( −3) F +1 ( ) for any ≥ 3. We will explore this in future work.

The underlying tree is detected in the Taylor tower
The first step on the journey relating the homotopy theory of punctured knots and the geometry of gropes was to connect them both to the language of decorated trees: they generate the group of components of the layers and also underlie gropes. It remains to show their compatibility via More explicitly, for a thick grope G : B Γ → on U with the underlying tree εΓ , we claim e +1 (G) = εΓ ∈ 0 F +1 ( ) Lie 1 ( ).
We prove this in Section 6 using the above Corollary 1.3: it is enough to show that the Samelson product Γ( is homotopic to the map (the initial coordinate of D e (G) ): The maps D and were constructed in Theorem B. The idea of the proof is to use inductive descriptions of both Samelson products (see Lemma B.5) and thick gropes, to reduce to checking that D e (G) is homotopic to a certain pointwise commutator map.
A crucial step for this reduction is our description of the map in Appendix A as This is a map on S obtained by gluing together along faces 2 different maps ( ) ℎ , each defined on as a certain 'ℎ -reflection' of the original map across a face of .
Hence, every generator of 0 F +1 ( ) is in the image of 0 e +1 . However, to prove that this map of sets is surjective we must also realise linear combinations. Recall that they arise as underlying trees of grope forests, so the following is all we need (for a proof see the end of Section 6).
Proof of Theorem A. The statements about F +1 are contained in Theorem B.
It remains to construct the map making the lower triangle in the following diagram commute: Observe that Theorem E is equivalent to the commutativity of the outer square. Thus, it is enough to pick any set-theoretic section of t, or in other words, let ( ) := [ (F)] for any grope forest F ∈ Grop ( ; U) with the underlying tree t(F) = .

The outline
In the preliminary Section 2 we define several variants of trees and survey finite type theory, and in Section 2.3 prove the corollaries of Theorem A announced at the beginning.
In Section 3 we study the punctured knots model P ( ) for any -manifold with boundary, ≥ 3. In Section 3.3 we show the layers are iterated loop spaces, while in Section 4 we determine their homotopy type, and describe geometrically the generators of the first non-trivial homotopy group, proving Theorem B. In Section 4.2 we outline the proof strategy for Theorem D.
In Section 5 we are concerned with 3-manifolds. Firstly, Section 5.1 presents a self-contained account of gropes. Then in Section 5.2 we prove Theorem C and its extension for grope forests, and in Section 5.3 describe points e +1 (G) and e +1 (F) explicitly. Finally, main Theorems D and E are proven in Section 6; the proof of Theorem D is by induction, using two auxiliary lemmas.
The construction of an explicit homotopy equivalences (from total fibres of certain cubes to iterated loop spaces) is deferred to Appendix A, while Appendix B provides background on Samelson products and contains two important lemmas about their inductive behaviour.
Throughout the paper we aim to make both homotopy theory and geometry accessible without assuming much background. The mutually related Examples 3.1, 4.7, 4.8, 5.15, and Figure 27 all describe the lowest degree computation for a 3-manifold , which is also the induction base in the proof of Theorem D. The induction step is outlined in Example 4.9.

Trees
Fix a finite nonempty set and an integer ≥ 2.
Definition 2.1. A (vertex-oriented uni-trivalent) tree is a connected simply connected graph with vertices of valence three or one and with cyclic order of the edges incident to each trivalent vertex, called the vertex orientation. In the pictures this is specified by the positive orientation of the plane.
A rooted tree Γ ∈ Tree( ) is a tree with one distinguished univalent vertex (the root) and all other univalent vertices (the leaves) labelled in bijective manner by the set . The grafting of two rooted trees Γ j ∈ Tree( j ) for j = 1, 2 is the rooted tree , as the quotient of the free abelian group on the set of rooted trees by the following relations (dots represent the remaining unchanged part of an arbitrary tree): : . . .
We now relate Lie trees to words (Lie monomials) in the free Lie algebra.
Definition 2.2. Let L ( ) = L( : ∈ ) be the free N 0 -graded Lie algebra over Z, with each having degree | | = − 2. Thus, the degree of a word ∈ L ( ) is | | = ( − 2) where is the length of , that is, the total number of letters in .
The normalised Lie algebra NL ( ) is the Lie subalgebra of L ( ) generated by the words in which every letter appears at least once. 9 Let Lie ( ) ⊆ NL ( ) be its subgroup generated by the words in which each letter appears exactly once. This is precisely the part of degree | |( − 2) of NL ( ).
If = 2 one can assign to a tree Γ ∈ Lie( ) a Lie word 2 (Γ) ∈ Lie 2 ( ) using vertex orientations, so that the grafting of trees precisely corresponds to the Lie bracket. This gives an isomorphism Lie( ) Lie 2 ( ) since relations (2.1) correspond to the antisymmetry and Jacobi relations in Lie 2 ( ).
Lemma 2.3. If is ordered, there is an isomorphism of abelian groups : Lie( ) − → Lie ( ) defined inductively on | | by ↦ → and for = 1 2 and Γ j ∈ Tree( j ) by Hence, one can think of graded Lie words also as Lie trees, but keeping in mind that for odd the isomorphism introduces a sign. For the proof of the lemma see the end of Appendix B.
For := {1, . . . , } we write Tree( ) := Tree( ) and Lie( ) := Lie( ). Their elements can alternatively be drawn in the plane as in Figure 2: the root and leaves are attached to a fixed horizontal line according to their increasing label, with the root labelled by 0 (the edges might intersecting, but this is not part of the data). The vertex orientation is still induced from the plane.

Decorated trees
For manifolds with non-trivial fundamental group we need more general trees. Let , be as above, and be a set. We write := Map( , ) (so = is the cartesian product), and if = 1 , then we assume := dim by convention. If we let Tree ( ) := Tree( ) × , then Lie ( ) is the quotient of Z[Tree ( )] by the relations analogous to , from (2.1), which respect decorations in the natural way. We denote elements of Tree ( ) by Γ , where Γ ∈ Tree( ) and := ( ) ∈ ∈ , and call them -decorated trees. Namely, these are rooted trees whose leaves are labelled bijectively by , and additionally for each ∈ the edge incident to the leaf is assigned an element ∈ , called a decoration.
When = 1 we view a 1 -decorated tree Γ as a homotopy class of a map : Γ → which takes the root and leaves to some fixed arc in (a knot in practice). The decoration ∈ 1 for the -th leaf is the homotopy class of the loop , which goes from (0) (the basepoint) to the leaf ( ) along the unique path in the tree connecting the -th leaf and the root, then back from ( ) to (0) along . More precisely, the basepoint of is fixed in some 0 ∈ and the described loop should be conjugated by the piece of between 0 and ( ).
Remark 2.5. The group Lie ( ) is equal to Λ −1 ( , + 1) from the work of Schneiderman and Teichner [ST14], where the more general groups Λ ( , ) were used as targets for obstruction invariants for pulling apart surfaces in ; see [ST14, Lem. 2.1] for this identification.

Jacobi diagrams and Jacobi trees
In the theory of finite type invariants for = 3 one considers more general uni-trivalent graphs, with univalent vertices 0, . . . , | | − 1 (the leaves) and vertex-oriented trivalent vertices (as in Definition 2.1). The degree of such a graph is half of the total number of vertices. Here we use the convention from Figure 2 to draw graphs in the plane (now without roots). However, in geometric finite type theory one can reduce to considering trees only, see Section 2.2. where the 2 relation is given by applying in two different ways: Here a Jacobi diagram has degree and exactly one loop (i.e. the first Betti number 1 ( ) = 1) and 0 and lie on the loop of and are neighbours of the leaf 0 or respectively (see Figure 4).  But this follows from (2.4) via ( , ) = ( , 0 ) = ( , ), since we can assume that the distance of the root to the unique loop of is one. Indeed, in the relation (2.1) let 0 be the vertex joining Γ 2 and Γ 3 , with the root 0 in Γ 1 . Then two of the terms have distance from 0 to the loop smaller than the third, so we can proceed by induction.
The natural inclusion A ↩→ A sends a Jacobi tree to itself viewed as a Jacobi diagram modulo , and lands in the primitives of the Hopf algebra A := ⊕ ≥0 A , see [Con08; Kos20].

Geometric finite type theory
Let us briefly review classical and geometric approaches to finite type theory; for book treatments see [Oht02;CDM12]. We restrict to the case of classical (long) knots K( 3 ) := 0 K( 3 ) (monoid under the connected sum #, with unit the unknot U); for general 3-manifolds, see [Kos20].
A singular knot is an immersion : 3 with finitely many transverse double points, which agrees with U near boundary. Each double point can be resolved by pushing the two strands off each other in two different ways, and all possible resolutions of give The associated graded of this filtration is related to the Hopf algebra of chord diagrams: those Jacobi diagrams from Definition 2.6 in Section 2.1.2 which have no trivalent vertices. Namely, for a singular knot with double points one has pairs of points on the source interval which are identified by ; we record each pair by a chord to get a chord diagram on of deg( ) = .
This assignment is surjective, but far from being injective. However, there is a well-defined map R which takes a chord diagram to the class [ ( ) ] ∈ V V +1 , where ( ) is any singular knot with the chord diagram ( ) = . One can check that this is well-defined and vanishes on diagrams which have an 'isolated chord', as 1 from (2.3) (since our knots are not framed) and on the 4 relations, certain linear combinations of four chord diagrams (coming from triple points).
Actually, 4 is a consequence of the relation from Definition 2.6, so there is a linear map Moreover, by Bar-Natan this is an isomorphism [Bar95] (the proof was given over Q, but it actually applies integrally). Combining this with the previous paragraph gives a surjection of finitely generated abelian groups (2.6) called the realisation map.
It is an open problem if its kernel is non-trivial, and a potential inverse is classically called a universal Vassiliev knot invariant of type ≤ over Z.
Namely, a knot invariant : Here is an abelian group and is just a map of sets.
Definition 2.9. Let be a ring, a graded -module and := ≥1 its completion. A map filtered -linear map inducing an isomorphism of the associated graded -modules. Equivalently, and for each ≥ 1 the map is an invariant of type ≤ , It is an open problem if they agree, but some progress was made in [Les02] (note that there may be several universal invariants over the same coefficient ring since only the 'bottom part' | V is determined). As a consequence, Therefore, the kernel of R consists of torsion elements, but it is unknown if A has any.
The Bar-Natan's isomorphism (2.5) gives more power to the theory as it is relatively easy to construct interesting invariants of Jacobi diagrams, called weight systems. For example, invariants are obtained by interpreting each trivalent vertex as the Lie bracket in a fixed semisimple Lie algebra and the horizontal line as its representation. Actually, symbols of quantum invariants of knots are precisely these weight systems, but by [Vog11] this is a strict subset of all of them.
However, introducing trivalent vertices raises the question of their geometric interpretation, as we had for chords. Several different answers are summarised in the following theorem.
Theorem 2.11. For , ∈ K( 3 ) and ≥ 1 the following are equivalent: (1) − ∈ V or, equivalently, and are not distinguished by any invariant of type ≤ −1; (2) can be obtained from by a finite sequence of infections by pure braids lying in the -th lower central series subgroup; (3) can be obtained from by a surgery on a simple strict forest clasper of degree ; (4) can be related to by a finite sequence of simple capped genus one grope cobordisms of degree . In this case we say that and are -equivalent and write ∼ .
The equivalence (1) ⇔ (2) is due to [Sta98], The idea behind all these descriptions is to view a crossing change as the simplest move, of degree one, in a whole family of moves. Namely, a chord guides a crossing change (a homotopy passing through the corresponding singular knot), while moves of higher degrees are certain iterations of the 'trivalent' move: grab three strands of the knot and tie them into the Borromean rings. We make this precise in Section 5.1 using the last approach of the theorem (see Remark 5.3).
Let us define the Gusarov-Habiro filtration by sumbonoids Then the theorem implies that it maps to the Vassiliev-Gusarov filtration: This is what we call the geometric approach, as we are back to working with knots, instead of their linear combinations -or dually, their invariants 0 (K( 3 ); ). In terms of invariants of finite type, the next lemma shows that we are restricting to the study of those which are additive, that is, monoid maps from K( 3 ) to abelian groups.
Lemma 2.12. An additive invariant has type ≤ if and only if it vanishes on K +1 ( 3 ). That is, : K( 3 ) → is a monoid map vanishing on K +1 ( 3 ) if and only if the linear extension In this setting we pass from Jacobi diagrams A to its subgroup A ⊆ A of Jacobi trees, and from the realisation map R to its 'tree part' R , defined as the unique map completing the diagram

is a monoid map if and only if its linear extension vanishes on
Recall from Section 1.1.2 that Grop ( 3 ; U) is the space of degree grope forests, ⊥ is the output knot map and t the underlying tree map. The following describes the exact relation between the two realisation maps.
Theorem 2.13 ([Hab00; CT04a; Oht02], see also [Kos20]). There is a structure of an abelian group on the set K ( 3 ) ∼ +1 so that R is map of abelian groups.
Moreover, R vanishes on relations , and 2 , and fits into the commutative diagram of abelian groups There is also a similar notion of a universal additive invariant.
Definition 2.14. Let = 0 ⊇ 1 ⊇ · · · be a filtered -module and := lim . A universal additive Vassiliev knot invariant over is a map of filtered monoids : K( 3 ) → which induces an isomorphism on the associated graded -modules, that is ).
Remark 2.15. Here we of course consider K( 3 ) is a filtered monoid with the Gusarov-Habiro We can define the completion of K over with respect to the Gusarov-Habiro filtration as the inverse limit Then for a universal additive Vassiliev knot invariant over , the induced map : K( 3 ) → is an isomorphism of complete filtered -modules.
Similarly as before, a universal additive invariant indeed satisfies a universality property: any Note however that this definition is more flexible than Definition 2.9, since the completion on is with respect to a filtration instead of a grading. For example, we could take = Z and = K( 3 ), so the obvious map K( 3 ) → lim K( 3 ) ∼ satisfies the conditions. This is precisely universal additive invariant of Habiro [Hab00, Thm. 6.17].
A universal additive Vassiliev invariant over Q can be constructed as the logarithm either of the Kontsevich or the Bott-Taubes integral from (2.7), which are both grouplike [Kon93; AF97]: Recall that Conjecture 0.1 asserts that the evaluation map 0 ev factors through an isomorphism This is equivalent to the claim that is a universal additive Vassiliev invariant over Z in the sense of Definition 2.14, where the filtration on the target is by abelian groups . See also (2.12).
Remark 2.16. The previous discussion largely generalises to long knots in any oriented 3-manifold .

Proofs of corollaries
As mentioned in the introduction, an important result is the identification due to Conant of the diagonal of the second page of spectral sequence for the tower P ( ).
We saw in Corollary 1.5 that the first page has Lie( ) on the diagonal, so Conant identifies the image of the first differential as 2 ⊆ Lie( ). See also [Shi19].

The diagram
We now summarise the preceding discussion in the diagram All objects here are abelian groups except that 0 Grop using that 0 ev factors though the quotient by ∼ , which is a corollary of Theorem C, see (1.2).
and Conjecture 0.1 precisely claims that this inclusion is an equality. Thus, on the left side of (2.9) there is a priori no vertical map.
Proof. The subdiagram comprised of solid arrows commutes by Theorem 2.13. The upper rectangle commutes by Theorem E, see (1.3). It remains to check that the triangle on the right commutes.
To this end, let and [ ] ∈ A its class. By the surjectivity of t we can find

The spectral sequence and the universality
From the bottom triangle in the diagram (2.9) we deduce Corollary 0.3: if for some ≥ 1 the map (mod * >1 ) is an isomorphism, then the other two maps are isomorphisms as well. More generally, if is a torsion-free abelian group and the map (mod * >1 ) : A ⊗ → ker( 0 +1 ) ⊗ is an isomorphism, then both R ⊗ and +1 ⊗ are isomorphisms.
If for some this is the case for all degrees below a fixed degree , then 0 ev is a universal additive Vassiliev invariant of type ≤ − 1 over , meaning that there is an isomorphism (2.11) Indeed, this follows by induction by tensoring the sequences in (2.10) by and using its flatness.
From this we concluded in Corollary 0.4 that the isomorphism (2.11) holds for = Q for all ≥ 1, and the -adics = Z in the range ≤ + 2, using the results of [BH20] (see also Remark 2.21).
Furthermore, they also show that for ≤ + 2 there is a splitting To deduce the integral result, we use that the kernels of both R and +1 must consist of torsion elements, and by [Gus94] and [Man16,Sec. 3.5] the group A is torsion-free for ≤ 6.
Note that conversely, if there is a non-trivial higher differential hitting the diagonal, then not both R ⊗ and +1 ⊗ can be injective.
Remark 2.20. There exists an inverse to R ⊗ Q obtained as the logarithm of either the Kontsevich integral [Kon93] or the Bott-Taubes integrals [BT94; AF97] (see the end of Section 2.2). Hence, ⊗ Q agrees with these invariants, implying that the configuration space integrals factor through the embedding calculus tower: where the map on the right is given by some splittings over Q, making the diagram commute.
Remark 2.21. Let us remark that the collapse of the spectral sequence * − , ( )⊗Q which converges to the rational homotopy groups of K( ) was shown earlier by [ALTV08] but only for ≥ 4.
The collapse of the corresponding homology spectral sequences for the Taylor tower of K( ) for any ≥ 3 was shown by [LTV10; Son13; Mor15]. However, it is not clear if those arguments can be extended to show that the homotopy collapse for = 3. This follows from the results of [FTW17], and more directly from [BH20].

Further consequences and examples
Non-convergence. The rational collapse along the diagonal is also used in the following argument of Goodwillie, which we include for completeness.
Proposition 2.22 (Goodwillie). The set 0 P ∞ ( 3 ) is uncountable, so the Taylor tower does not converge to K( 3 ), i.e. the map ev ∞ : is surjective for every ≥ 1, but not injective since we saw that ker( 0 +1 ) ⊗ Q A ⊗ Q, and these groups are non-trivial for all ≥ 2.
Nevertheless, recalling that we denote the Gusarov-Habiro completion by K( we have an exact sequence This is obtained by taking limits in (2.10), using Milnor's lim 1 -sequence and lim 1 K( 3 ) ∼ = 0, as the maps in that tower are surjective. Thus, if Conjecture 0.1 is true then ev ∞ is an isomorphism.

Group structures.
Corollary 2.23. Two group structures on the set of components of Taylor stages constructed by [BW18] and [BCKS17] are equivalent, i.e. 0 T 0 as abelian groups. More generally, any two group structures on 0 T K( 3 ) respecting the connected sum of knots must agree.
Proof. The models are weakly equivalent, so there is a bijection of sets : Using that 0 ev is also a monoid homomorphism we have Examples.
Example 2.25. Therefore, the first non-trivial knot invariant from embedding calculus is Using the linking of certain 'colinearity submanifolds' of configuration spaces, [BCSS05] show that 0 ev 3 agrees with the unique Vassiliev invariant 2 of type ≤ 2 taking value 1 on . Classically, 2 is given as the second coefficient of the Conway polynomial (lifting the Arf invariant) and induces Our approach not only recovers 0 ev 3 = 2 but also lifts this computation to the fibres via the map e 3 : H 2 ( 3 ) → F 3 ( 3 ). Namely, by the previous example for any ∈ K( 3 ) there exists a grope forest F of degree 2 from to U. By the extension of Theorem C for grope forests we get a point (F) ∈ H 2 ( 3 ) and by definition 0 ev 3 ( ) = [e 3 (F)] ∈ Lie(2). Now, our main Theorem E says that this element is the class of the underlying trees of F. Actually, in Example 4.8 we do this computation for the case = , as a warm-up problem for the proof of that theorem.
We get precisely 0 ev 3 ( by the uniqueness of 2 . If we then for computing the coefficient 2 ( ) use the Hopf invariant Lie(2) ⊆ 3 (S 2 ∨ S 2 ) → Z given as the linking number lk( −1 ( 1 ), −1 ( 2 )) for a suitable representative : S 3 → S 2 ∨ S 2 of the desired homotopy class, then we are in the colinearity story of [BCSS05].

Homotopy limits and the notation
A diagram over a small category C is a functor • : C → Top to the category of topological spaces. Let Top C denote the category of diagrams over C. The categories we will be using are the cube We will need the notion of a homotopy limit of a diagram • ∈ Top C ; for an introduction see [BK72] or [MV15]. This is the space holim( •) ∈ Top, also written holim ∈C , defined as the mapping space Firstly, |C ↓ • | ∈ Top C is the diagram which sends ∈ C to the classifying space of the category (C ↓ ), called the overcategory, whose objects are morphisms → in C, and arrows are triangles over in C. Recall that the classifying space |D| of a category D is the geometric realisation of the nerve of D -the simplicial set whose -simplices are sets of -composable arrows in D.
Finally, the mapping space between two objects in Top C is defined as the set of natural transformations between the two diagrams and is seen as a subspace of ∈C Map |C ↓ |, from which it inherits the topology. In other words, a point ∈ holim( •) consists of a collection of maps : |C ↓ | → which are compatible with respect to the morphisms in C.
The crucial property of a homotopy limit is its homotopy invariance: if • → • is a map of diagrams such that each → is a weak equivalence, then the induced map holim • → holim • is a weak equivalence as well. ⊆ of the face whose barycentric coordinates in \ are zero. Therefore, in this case we have and a point ∈ holim( •) consists of maps : Δ → such that for all ∌ ⊆ [ ] the following diagram commutes Similarly, there is a levelwise homeomorphism of cubes |P ↓ • | • , since |P ↓ | is a cube whose coordinates are indexed by , and the map \ : ↩→ for ⊆ ⊆ is the inclusion of the face whose coordinates in \ are zero. But now for • ∈ Top P we have holim( •) ∅ , since ∅ ∈ P is the initial object.
However, we can instead take the homotopy limit of the punctured -cube with ∅ omitted, and compare it to ∅ . More precisely, for • ∈ Top P define the total homotopy fibre by the homotopy fibre of the natural map (see Definition 3.5). We will show in Lemma 3.6 that ∈ tofib( •) can also be given as a suitable collection : → , using the following lemma.
Lemma 2.26. There is a levelwise homeomorphism of cubes where (C Δ) is the cone on the barycentric subdivision of Δ .
Sketch of the proof. Each h should map the initial vertex of the cube (i.e. the one with coordinates 0 ) to the cone point of Δ , and the 1-faces to the barycentrically subdivided simplex. This will clearly respect maps in the diagrams. We omit writing out explicit formulae.
Notation 1 (Spaces). Let := [0, 1] denote the unit interval, P the space of free paths [0, 1] → in a space , and P * the subspace of paths that start at the basepoint * ∈ . Further, Σ is the reduced suspension of and Ω is its based loop space.
For ∈ P we write : (0) (1) ⊆ . The use −1 or 1− to denote the inverse path. The concatenation of loops will be denoted by · , and the commutator by [ , ] := · · −1 · −1 . All manifolds have non-empty boundary and S ⊆ is the unit tangent bundle.
Notation 2 (Main objects). See also Notation 3 for the notation related to the punctured knots model, Notation 4 for manifolds 0 , and the beginning of Section 4 for ∨ S .

The punctured knots model
Throughout this section is a connected compact smooth manifold of dimension ≥ 3 with non-empty boundary. Recall that we fix : [0, ) (1 − , 1] ↩→ and consider the space whose elements we simply call knots. We equip it with the Whitney ∞ topology, and choose an arbitrary knot U ∈ K( ) for the basepoint. The -th Taylor approximation of K( ), for ≥ 0, is defined as the homotopy limit Example 3.1. In degree = 2 the space P 2 ( ) is the homotopy limit of the punctured 3-cube Thus, ∈ P 2 ( ) consists of three once-punctured knots, for each two of them an isotopy between their restrictions to twice-punctured knots, and three two-parameter isotopies of thrice-punctured knots connecting restrictions of respective isotopies of twice-punctured knots.
Remark 3.2. The condition ≡ near for ∈ Emb ( \ , ) (also for K( )) can be replaced by the requirement that is 'flat' outside of . This clearly gives equivalent spaces.
Notation 3. To save space we shall denote Emb ( \ , ) by E and write where the ambient manifold will be clear from the context. We also simply write E := E ∪{ } and equip each E with the basepoint U , so : E → E with ∉ ⊆ [ ] is a based map.
The upper row forms an ( + 1)-cube also denoted by E• ∪ +1 , but with the index now in P[ ].
For ∈ K( ) and ∈ P ∅ [ ] we denote by For two indices , ∈ [ ] we define Here by abuse of notation is both in the source and in the image U ⊆ . By Remark 3.2 we assume ( ±∞ ) = ±∞ for any ∈ E . Lastly, denote by the midpoint of the interval , +1 . Given ∈ K( ) various restrictions are mutually compatible, so assemble to give a map K( ) → lim E•. Composing it with the canonical map from the limit to the homotopy limit gives the evaluation map More explicitly, for ∈ P ∅ [ ] this is given as the constant family Actually, for ≥ 2 there is a homeomorphism K( ) lim E • . This follows since having at least three different punctures ensures that all ∈ E are pairwise disjoint, apart from agreeing on intersections. However, for = 1 this does not hold, since {0} | 1 and {1} | 0 potentially intersect. Instead, we shall see below that lim E 1 Remark 3.3. By a family version of the isotopy extension theorem, is a locally trivial fibre bundle [Pal60]. In particular, it is a Serre fibration, whose fibre is the space of embeddings of into which miss the punctured unknot U , namely fib( ) = Emb ( , \ U ).

The zeroth and first Taylor stages
Given ∈ E for some ≥ 0 we can start 'shortening' its both ends until only the flat parts −∞ and ∞ remain. In other words, there is a deformation retraction of E = Emb ( \ , ) onto the point U ∈ E . Hence, P 0 ( ) = E 0 is contractible as well.
Next, as we mentioned above, the limit of the diagram is not homeomorphic to the space of knots. Instead, it is given as are embeddings} -the space of those immersions which are embeddings when restricted to \ 0 or \ 1 . Actually, since both maps in the diagram E 1 • are fibrations (see Remark 3.3), the limit is equivalent to the homotopy limit. 11 Hence we have the upper row in the commutative diagram Let us now explain the rest of the diagram. As mentioned in the preliminaries, the homotopy limit can be computed from any levelwise homotopy equivalent diagram: The first equivalence is induced from the weak equivalences Emb( , ) → Imm( , ) for a disjoint union of disks, see for example [Cer61]. The second equivalence is induced from the unit derivative maps, giving paths in the unit tangent bundle S , with PS S (both endpoints free) and P * S * (one endpoint fixed). One can check that the homotopy limit of the rightmost diagram is Ω(S ), so one has the triangle in (3.4).
On the other hand, the strict limit in the middle diagram is clearly lim I 1 • Imm ( , ). The fact that this is also its homotopy limit is non-trivial: by a theorem of Smale [Sma58] the restriction maps for immersions are also fibrations. This also implies that immersions form a polynomial functor of degree at most 1, that is, T Imm ( , ) Imm ( , ) for all ≥ 1. Here we similarly define T Imm ( , ) := holim I • , using I := Imm ( \ , ). See [Wei05; GW99].
However, to obtain P 1 ( ) Ω(S ) we did not need Smale's result. Finally, observe that as a consequence of this discussion the inclusion : lim E 1 • ↩→ lim I 1 • of 'special' immersions into all immersions is -maybe surprisingly -a weak equivalence.
Before proving this, we recall the notion of a total homotopy fibre and its properties.
Here U ∈ are the basepoints and is the natural map sending ∈ ∅ to the collection of constant maps ( ) , each equal to the image of under ∅ : ∅ → . This factors as where is the restriction map to the homotopy limit of a subdiagram and const is the canonical map from the limit to the homotopy limit. Since has the initial object, const is a weak equivalence. Hence, hofib( ) and hofib( ) are weakly equivalent.
Even something stronger is true.
Proof. See [Goo92] for several descriptions of total homotopy fibres and inspiration for this proof. Consider the mapping path space 12 of . The natural projection : is a fibration and fib (U ∅ ) ( ) = hofib (U ∅ ) ( ), and the latter is our definition of tofib( •).
We will construct a homeomorphism : → holim and a commutative diagram: holim It will immediately follow that is a fibration as well (as the composition of a fibration and a homeomorphism) with the fibre homeomorphic to tofib( •).
Let ( , ) ∈ , so ∈ ∅ and : → holim | P Proof of Proposition 3.4. Using the decomposition of E +1 • into subcubes from (3.2) and the fact that the homotopy limits can be computed 'iteratively', we obtain 14 a homeomorphism: Thus, P +1 ( ) is the homotopy limit of the diagram on the right which has only two mapsso a homotopy pullback. The map is an analogue of ev but for +1 -punctured knots E +1 := Emb ( \ +1 , ), while +1 * : P ( ) → holim P ∅ [ ] (E• ∪ +1 ) is the induced map on the homotopy limits from the maps +1 , so it punctures at +1 every punctured knot in the family.
Remark 3.7. In Section 3.3 we will see that BF +1 ( 3 ) is an -fold loop space, so one can try to show that P +1 ( 3 ) = hofib(P ( 3 ) → BF +1 ( 3 )) is a double loop space by induction on ≥ 1 and showing that +1 * is a map of double loop spaces. Such deloopings were shown in other models by [Tur14] and [BW18], but in this approach one could check if they also exist for some other . 14 To prove this one uses homeomorphisms Δ ∪ +1 CΔ which assemble into a map (CΔ) • → Δ • (see Footnote 13); for details see [Goo92] or [MV15, Lemma 5.3.6].

The layers and homotopy fibres of evaluation maps
Since in Proposition 3.4 we found F +1 ( ) tofib P[ ] (E• ∪ +1 ), let us define e +1 : H ( ) → F +1 ( ) as the map induced on total homotopy fibres from the map +1 • : E • → E• ∪ +1 of ( + 1)-cubes. This again 'punctures at +1 every punctured knot in the family'. Using different descriptions of total homotopy fibres from the proof of Lemma 3.6 we immediately have the following. This completes the diagram (0.2) from the introduction.
The total homotopy fibre of an ( + 1)-cube can also be computed 'iteratively', by first taking homotopy fibres in one arbitrary direction and then finding the total fibre of the resulting -cube. This is similar to Footnote 14, and uses +1 = × ; see [Goo92] or [MV15] for a proof. For the first direction we choose the one which 'punctures at zero', that is, we take homotopy fibres of 0 . Since by Remark 3.3 these maps are fibrations, we can instead take the actual fibres. (3.7) The basepoint of E is U := U \ , so writing the first fibre out, we get Thus, F is the space of embeddings of the arc 0 into the complement in of the punctured unknot U 0 with condition that they agree near the boundary 0 with U 0 (see Figure 7). The maps in the -cube (F•, ) are restriction maps as before : F → F , with ∉ ⊆ . Similarly, the -cube (F +1 • , ) can be given by F +1 Emb ( 0 , \ U 0 +1 ) with the restriction maps +1 : F +1 → F +1 . However, note that one of the vertices of the cube computing H ( ) is the space of knots itself. This is in contrast to the cube for F +1 ( ), in which the piece +1 is always absent -precisely this will allow us to compute its homotopy type in the next two sections.
Namely, the space F +1 can equivalently be described as the space of embeddings of the arc 0 in the complement in of the -dimensional balls B +1 ⊆ obtained as neighbourhoods of the pieces +1 of the punctured unknot U 0 +1 (recalling Notation 3). For example, the lower picture in Figure 8 corresponds to the point in F +1 from the bottom of Figure 7 for + 1 = 6 and = {2, 3, 5}. Let us introduce notation for these complements. Thus, for ⊆ there is a homeomorphism F +1 Emb ( 0 , 0 ) and is the composition with Corollary 3.10. The -cube (F +1 , ) is levelwise homeomorphic to the one obtained from the -cube ( 0 , ) by applying the functor Emb ( 0 , −).

Delooping the layers
In this section we make a crucial step for the description of the homotopy type of F +1 ( ) by constructing a homotopy equivalence from F +1 ( ) to an -fold loop space (Section 3.3.1), and then deloop once more (Section 3.3.2).

The initial delooping
Theorem 3.11. For the ( + 1)-st layer F +1 ( ) tofib(F +1 • , ) of the Taylor tower for K( ), ≥ 0, there is a contravariant -cube (F +1 • , ) and an explicit homotopy equivalence : tofib We prove this using Proposition 3.16, which says that such a homotopy equivalence exists for any cube which has an -fold left homotopy inverse. After we define this notion and state that proposition, we proceed to construct maps giving such an inverse (F +1 • , ) for our cube (F +1 • , ). All proofs about left homotopy inverses are deferred to Section A.
A left homotopy inverse (a retraction up to homotopy) for a map : → is a map : → such that • Id . For our purposes it is crucial to specify a homotopy ℎ from Id to • : Moreover, we have * −1 hofib( ) * hofib( ) ker( * ) and split short exact sequences since * is a section in the long exact sequence of homotopy groups for . Actually, more is true.
Lemma 3.12. Given the data of (3.8) there are explicit inverse homotopy equivalences One can generalise this from 1-cubes (maps), to diagrams over P for ≥ 1 as follows.
Definition 3.13. Let • = •∌ − − → • be an -cube with ≥ 1, seen as a 1-cube of ( − 1)cubes. A left homotopy inverse for • is the data of a diagram:  To repeat this procedure and get a homotopy equivalence from the total fibre of an -cube to an -fold loop space, we need a left homotopy inverse for each . We also need suitable conditions for homotopies ℎ , in order to avoid obtaining cubes which are commutative only up to homotopy. (1) For each ⊆ and ∈ \ a map : → is given such that (3.12) These equations ensure that for 0 ≤ ≤ there is a well-defined -cube obtained from by replacing the arrows by for + 1 ≤ ≤ . 15 In particular, 0 = ( • , ).
(2) For each 0 ≤ ≤ and ∈ [0, 1] a map of diagrams is given, such that ℎ • (0) = Id and ℎ • (1) = • • • . 15 We see as an ( − )-cube of -cubes; maps in -cubes are -maps, while maps between them are -maps. Hence is a diagram over P( − ) × P ( \ − ); in particular, 0 is a contravariant cube. However, all these categories are isomorphic to P( ) if we appropriately rename the vertices. Thus, an -fold left homotopy inverse for is given by the following data for each 0 ≤ ≤ : (3.14) Proposition 3.16. Given the data of (3.14) there is a sequence of homotopy equivalences
Let us now turn to applying them in our situation: Theorem 3.11 will follow from Proposition 3.16 once we construct an -fold left homotopy inverse (F +1 , ) for (F +1 , ). Recall that by Corollary 3.10 the latter is obtained by applying Emb ( 0 , −) to the -cube ( 0 , ).
From this we get the desired (F +1 , ) by applying Emb ( 0 , −) to ( 0 , ), i.e. letting = • − and ℎ • • − gives a cube satisfying conditions of Definition 3.15, and proving Theorem 3.11. Now to prove Theorem 3.17 we first define a left homotopy inverse for each in the sense of (3.8), and then revisit the construction to ensure that all mentioned conditions are satisfied. Proof. Let > := { ∈ : > } and let +1 := min{( > ) ∪ { + 1}} (this is the smallest index in which is bigger than , or + 1 if that set is empty). Consider the inclusion map +1 : 0 ↩→ 0 ∪ B +1 which adds back the ball B +1 . We visualise this by erasing B +1 as in Figure 9.
Observe that 0 ∪ B +1 and 0 are isotopic as submanifolds of by an ambient isotopy drag > ( ) : which, loosely speaking, elongates B by gradually dragging the right hemisphere of S −1 to the right, until it equals the right hemisphere of S +1 −1 . We will define a specific parametrisation in the proof of Theorem 3.17 below (it will indeed depend on > and not only on +1 ). It remains to provide a homotopy ℎ between Id 0 and the composite The composition of the first two maps adds to 0 the material B +1 \B , which is diffeomorphic to a ball (note that B −1, +1 \ (B −1 ∪ B ) = B +1 \ (B ∪ B +1 )). Now adding this material gradually gives an isotopy add : 0 ↩→ such that im(add 0 ) = 0 and im(add 1 ) = 0 ∪B +1 . We can parametrise this so that im(add ) = im( ( )) for each ∈ [0, 1], so the two isotopies can be composed into the desired homotopy Proof of Theorem 3.17. We now ensure that the maps and ℎ constructed in the previous proof satisfy conditions of Definition 3.15. We are still free to specify a particular parametrisation of the ambient isotopy drag > ( ) : → , which we roughly described as a 'dragging move', acting non-trivially only in a tubular neighbourhood of +1 . Firstly, for ⊆ \ { , } the conditions (3.12) and (3.11) are respectively equivalent to having that the following left diagram commute for > and the right diagram for, say, < : Lastly, the condition (3.13) is equivalent to having that for each ∈ [0, 1] the following left square commute if > and the right square if < : Again, the case on the left is clear since drag > ( ) = drag > ( )), and for the right one we should ensure that For the second equality we used that drag > (1) • drag > (1) = drag > (1) • drag > (1) again by the induction hypothesis (3.17). Now observe that { > } = { > }, so the last expression equals drag > (1) −1 • drag > ( ) • drag > (1), finishing the induction step.

The final delooping
Our approach is motivated by the following observation about the diagram as in Section 3.0.1: We can use the weak equivalences of fibrations given there to find the homotopy type of the layer F 1 ( ) := fib( ) = fib( 0 1 ) Emb ( 0 , \ U 01 ) =: F 1 ∅ , namely: Note how the disjointness condition with U 01 is lost in fib( 0 1 ) in the case of immersions. The equivalence F 1 ( ) Imm ( 0 , ) ΩS takes the unit derivative D : → S of : 0 ↩→ and makes it into a closed loop based at ( 0 , ì ), by concatenating it with the unit derivative of any arc U 0 , which agrees with U 0 except near endpoints, at which its derivative is −ì instead. Namely, the endpoints of D are ( 0 , ì ) and ( 0 , ì ) (see Figure 10). We now analogously determine the homotopy type of each F +1 := fib( 0 ) = Emb ( 0 , 0 ) for possibly empty = { 1 , . . . , } ⊆ . Namely, recall that 0 := from Notation 4, and consider the space (3.20) We analogously have the composite of the inclusions which forget embeddedness and the disjointness between 0 and U 0 +1 , and a similar derivative map ↦ → (D U 0 ) · (D ) 1− . To see that D is a weak equivalence we can simply replace by in the diagram (3.19). This is actually a homotopy equivalence since the spaces are of the homotopy type of complexes; for an alternative proof see [Kos20], where D is shown to arise also as a homotopy equivalence for a certain left homotopy inverse of 0 . Now observe that we also have maps : → for ∉ as in the proof of Lemma 3.18, since the presence of the index 0 was irrelevant there (note that ∅ : Remark 3.20. With tangent data now gone, define D : F +1 → Ω using simply U 0 := U 0 .
Remark 3.21. As mentioned in the introduction, the authors of [BCKS17] use Sinha's cosimplicial model where Conf 3 is a compactified configuration space of points in 3 together with tangent vectors [Sin09]. To compute F ( 3 ) they use the cosimplicial identity s • d = Id, which precisely says that the codegeneracy s (forgetting the -th point in the configuration) is a strict left inverse for d (doubling the -th point).
Remark 3.22. Actually, the maps D factor through spaces Ω 0 , giving the following diagram in which if either all upward or all downward arrows are omitted, the resulting diagram commutes. By Theorem 3.19 the first and last dashed maps form cubes with homotopy equivalent total fibres. By Theorem 3.17 there are the cubes (Ω 0• , Ω ) and its -fold left homotopy inverse (Ω 0• , Ω ).
However, it is important to point out that there are no -maps between spaces Ω which would form a cube equivalent to our original cube (F +1 , ). The reason lies in the fact that the information about the disjointness of 0 with B 0 1 is lost when we pass from 0 to . More precisely, there are no maps → when is smaller than all indices in , but otherwise we do have (and the homotopies ℎ ( ) do restrict to and also commute with D by construction).

Homotopy type of the layers
We now express the homotopy type of F +1 ( ) in terms of suspensions of the space of loops on the iterated product of with itself. Such computations go back to [GW99, Section 5] for the case when has the homotopy type of a suspension, and also [BCKS17] for = 3 . However, as mentioned in the introduction, even then, those results do not suffice for our purposes as we need a geometric interpretation of the homotopy classes (see Section 4.2 below): for the discussion in Section 6 it will be crucial to determine the class in 0 F +1 ( ) of a geometrically described point ∈ F +1 ( ), for = 3.  Finally, it is not hard to see that contractions can be chosen so that commutes with col .
The assumption ≠ ∅ is essential as the lemma does not hold for closed manifolds. For another proof see [GW99,Cor. 5.3]. Together with Theorems 3.11 and 3.19 we obtain the following.
Corollary 4.2. There are homotopy equivalences F 1 ( ) Ω(S ) and for ≥ 1 x Assume ≥ 1 and let us determine the homotopy type of tofib( ∨ S•, col) using some classical results which we now recall, referring the reader to Appendix B for more details.   → ΩΣ → ΩΣ similarly to the notation (4.1).
where NB( ) ⊆ B( ) consists of those words in which every for ∈ appears at least once. Therefore, By the naturality of equivalences Ω( ∨ S ) (ΩΣ ) × Ω from (4.2) and ℎ from (4.4) there is an equivalence of contravariant -cubes We will show that the total homotopy fibre of the first cube is the desired product over NB( ). The map proj for ∈ is a projection onto the factors corresponding to those words ∈ B( ) which also belong to B( \ ). These are precisely the words in which does not appear.
For ⊆ let be the product of factors ∈ B( ) in which for each ∈ the letters appears at least once. Now, one clearly has hofib(proj 1 01 : 0 × 01 → 0 ) 01 and more generally: This follows by induction using the iterative description of total fibres (see [MV15,Ex. 5.5.5]).
In Section 4.1 we compute the first non-trivial homotopy group of F +1 ( ) and describe its generators as maps to tofib Ω( ∨ S•). Then in Section 4.2 we discuss how to transform this into direct maps to F +1 ( ), along with a strategy for the main proofs in Section 6, and some examples.

The first non-trivial homotopy group
We will now restate Theorem B as two propositions. Recall from Section 2.1 that the group Lie 1 ( ) ( ) Z[( 1 ) ] ( −1)! is generated by decorated trees Γ ∈ Tree 1 ( ) consisting of a tree Γ ∈ Tree( ) together with decorations ∈ 1 for ∈ .
Observe that the generator of Σ Ω × + corresponding to ( 1 , . . . , ) ∈ ( 1 ) is represented by the -fold suspension Σ of the based map It remains to determine explicit generators of ( −2) tofib Ω( ∨ S•). Recalling from (3.15) that the map forgetting homotopies induces an injection (4.7) where we use the adjoint of Σ 1+ ( −2) ( ) which is for ∈ S 1 and ì ∈ S −2 given by Let us now simplify these generating maps.     Finally, as the target is a loop space, we have their Samelson products (see Figure 14 for = 2): Now, since forg * : ( −2) tofib Ω • → ( −2) Ω is injective in this setting as well, the generators of the source group are represented by the extensions of the maps (4.9) to the total fibre using the canonical null-homotopies of -by 'pulling up' through the ball B . Similarly, we would need to define Samelson products Γ( ) using some embedded analogue of commutators. However, this is not immediate: F +1 is not an -space in an obvious way, as concatenation of arcs 0 ↩→ might result in a non-embedded arc. If this has been done, the generators of ( −2) tofib(F +1 • , ) would be canonical extensions of Γ( ), again by (3.15).
The delooping map. If such embedded commutators had been constructed, then the map ) would be very easy: it is given as the composition of the map forg which forgets all null-homotopies, and the inclusion Ω F +1 ↩→ tofib(F +1 • , ).
We do not, however, pursue defining such embedded commutators directly, as we will not need them. Namely, for = 3 we will in Section 5 directly construct points (G) ∈ H ( ) using gropes -which can indeed be seen as embedded commutators, see Remark 5.3 -and then prove that e +1 ( (G)) : S ( −2) → tofib(F +1 • , ) are precisely the generators, using the following strategy. To prove Theorem D in Section 6 we will show that for = 3 and G = e +1 ( (G)) ∈ F +1 ( ), the point coming from a thick grope G : B Γ → on U, we have where ( , ) is the signed decoration of G.
The proof will be based on the fact that both (G) and Samelson products are constructed inductively. For the former see Section 5 and for the latter see Lemma B.5.
Furthermore, for the proof of Theorem E we will use that =1 ε Γ is represented by the following pointwise product -again by the Eckmann-Hilton argument: We note that the same strategy should work for appropriate notion of gropes for any ≥ 4, as mentioned in Remark 1.10. Let us now illustrate our discussion so far on several examples.
Example 4.7 ( = 2). The punctured cube E 2 • computing P 2 ( ) was displayed in Example 3.1. On one hand, F 2 ( ) tofib(F 2 ) is the total homotopy fibre of the top square: On the other hand, if we complete E 2 • with the initial vertex K( ) := Emb ( , ), then the space H 1 tofib(F ) is the total fibre of the bottom square: and e 2 : H 1 ( ) → F 2 ( ) is the obvious upward map (see Section 3.2). Now by Theorem 3.11, we have The map 1 ∅ corresponds to 1 ∅ : 1 = \B 12 → ∅ which adds back the ball B 12 and rescales using the map drag, see the proof of Lemma 3.18. Next, by Theorem 3.19 the derivative and by Lemma 4.1 the retraction induce equivalences Finally, hofib(Ωcol 1 ∅ ) Ω(S 1 ∨ (S 1 ∧ Ω )) by the Grey-Spencer Lemma 4.3, so we have The generators for hofib(Ωcol 1 ∅ ) are the canonical extensions of 1 and 1 , while for hofib(Ω 1 ∅ ) they are the extensions of 1 and 1 . See Figure 12. The generators for hofib( 1 ∅ ) are the extensions of 1 and 1 (Figure 15). For example, the extension of 1 : S −2 → F 2 1 to hofib( 1 ∅ ) uses the family of obvious null-homotopies of 1 ∅ ( 1 ) across the -ball B 1 .
We apply Theorems 3.11 and 3.19 to get homotopy equivalences Hence, the first non-trivial homotopy group is Lie 1 (2) in degree 2( − 2), and for the last total fibre in the first row the generating maps are the canonical extensions of the maps forg(retr) −1 * ( 1 1 , 2 2 ) = 1 1 , 2 2 ∈ 2 Ω 12 . (4.11) Example 4.9 (Sketch of the proof of Theorem D for = 3). Assume t 2 (G) = 2 2 1 1 for a thick grope G in a 3-manifold , using the underling forest map from Proposition 5.8. The construction in Section 5.3 produces the point and so we obtain a class forg(D e 3 (G)) ∈ 2 Ω 12 . Then Theorem D asserts that this class agrees with (4.11). One can visualise this by comparing Figure 14 with Figure 21.
A close look at the definition of implies that ( e 3 (G)) {12} is obtained from the square-family of loops Ψ G (−) {12} 0 by 'reflections relative to the balls B 1 and B 2 ', see Proposition A.6.
In Lemma 6.2 we will show that D 12 Ψ G (−) {12} 0 : 2 → F 3 12 → Ω 12 is homotopic to the commutator of certain loops corresponding to degree 1 gropes out of which G is built (the two caps of G). On the other hand, the Samelson product [ 1 1 , 2 Throughout this section is an oriented 3-manifold with non-empty boundary. In 5.1 we define grope cobordisms and their modifications, and in 5.2 we prove Theorem C.

Abstract gropes
For a finite non-empty set we defined the set of (rooted vertex-oriented uni-trivalent) trees Tree( ) in Definition 2.1. Now we define certain 2-complexes modelled on such trees.
Definition 5.1. A punctured torus is a genus one compact oriented surface with one boundary component , see Figure 16. We fix an oriented subarc 0 ⊆ and view as the plumbing of two ordered bands 1 and 2 . The core curve j ⊆ j is oriented for j = 1(2) in the same (opposite) manner as the component of j which contains a part of 0 (so built inductively on | | as follows.
− For = { } the only tree is Γ = and we let Γ simply be an oriented disk, the -th cap, with a chosen oriented subarc 0 of the boundary.
− For | | ≥ 2 any tree Γ ∈ Tree( ) is obtained by grafting two trees of lower degrees Thus, abstract gropes Γ 1 and Γ 2 are defined by the induction hypothesis. Let Γ be the result of attaching both of them to a single punctured torus , called the bottom stage of Γ , via orientation-preserving homeomorphisms Note that Γ has precisely | | caps (this is its degree), labelled bijectively by . By a stage of a grope we mean any of the punctured tori or caps it contains; each stage of a grope is an oriented surface. Observe that a thickening of the 2-complex Γ , that is, the union of products of all stages with an interval, is homeomorphic to B 3 (see also Figures 22 and 24).
Abstract gropes are just combinatorial objects: there is a 1-1 correspondence between them and rooted trees. In fact, the tree Γ on which Γ was modelled can be seen as its subset Γ ⊆ Γ , called the underlying tree of Γ , using the following construction (equivalent [CST07, Def. 16 & Sec. 3.4]).
The root of Γ is the initial point * of 0 ⊆ Γ , and each trivalent vertex of Γ is the intersection point 1 ∩ 2 in the corresponding grope stage. The leaves of Γ are the centres of caps. Finally, * the edges are obtained at each stage as in Figure 17, and are cyclically ordered using the order ( 1 , 2 ), which agrees with the corresponding vertex orientation in Γ.

Grope cobordisms
Grope cobordisms are particular embeddings of abstract gropes into a 3-manifold .
Definition 5.2. Let ∈ K( ) and Γ ∈ Tree( ) for a finite nonempty set ⊆ N. A (simple capped genus one) grope cobordism 16 on modelled on Γ is is an embedding G : Γ → into the complement of except that: − G( 0 ) ⊆ ( 0 ) and the orientations of these arcs agree; − for each ∈ , the -th cap intersects 0 := ( \ 0 ) transversely in exactly one point ∈ 0 , which is the centre of the cap and which belongs to ( ) ⊆ 0 .
We see G as a cobordism between and the output knot ⊥ G : Remark 5.3. Note that the arc G( ⊥ 0 ) is oriented oppositely in ⊥ G than as a subset of G, as usual for oriented cobordisms. The crucial observation is that G( ⊥ 0 ) is an 'embedded commutator' of the curves G( 1 ) and G( 2 ), as for the Borromean link, see Figure 19 and [Tei02; Tei04].

The underlying decorated tree
We now extend the underlying tree Γ ∈ Tree( ) of a grope cobordism G : Γ → on a knot to a 1 ( )-decorated tree (such trees were defined in Definition 2.4).
Definition 5.4. Let ⊥ : → be the path from ( 0 ) to obtained as the image under G of the unique path in the tree Γ ⊆ Γ from the root to its -th leaf. Let [ , ( 0 )] be the image of between ∈ ( ) ∩ G( Γ ) and ( 0 ) (see Figures 18, 19, 21).
Then we have a loop in given by Let := sgn( ) ∈ {±} be the sign of the intersection of ( ) and the -th cap of G. The tuple ( , ) ∈ is called the signed decoration of G. In other words, is obtained by gluing two different paths from ( 0 ) to : the obvious one along , and the one that goes 'through the grope', following G(Γ) ⊆ G( Γ ) ⊆ . 16 Non-simple, non-capped and higher genus gropes are also considered in the literature, but will not be needed in our discussion. However, grope forests defined below in Definition 5.6 are related to higher genus grope cobordisms. Figure 18. Two grope cobordisms of degree 1 on : ↩→ (the horizontal line). In both cases ⊥ G is the union of black and red arcs; the left one is contained in 3 ⊆ and is isotopic to the trefoil. The signed decorations are respectively ε(G) = 1 = −1 and +1 1 = [ 1 ] ∈ 1 . 1 2 0 Figure 19. A grope cobordism G : Γ → 3 on = U with the underlying tree G(   Figure 19. Right: 'Swinging' the bands of ⊥ G is an isotopy which introduces twists into the bands giving a projection of the right-handed trefoil, cf. Example 2.25.  Figure 21. A grope cobordism G in a manifold whose underlying decorated tree is 2 2 1 1 , since 1 = 2 = +1. If = 3 and is the unknot, then ⊥ G is the figure eight knot.

Thick gropes
Let us observe that a regular neighbourhood of a grope cobordism G is diffeomorphic to a 3-ball G : B 3 ↩→ , since inductively we are just thickening the punctured torus and attaching cancelling 3-dimensional 2-handles, see Figure 22. This 3-ball G intersects the knot in the neighbourhood G( ) ⊆ of the intersection points ∈ ( ), for some arcs ⊆ B 3 with ∩ B 3 = , 1 ≤ ≤ . It is convenient to fix a choice of such a neighbourhood G as follows; we pick some > 0. Note that G := G| Γ is a grope cobordism on in the sense of Definition 5.2. We can thus also define an underlying decorated tree ε(G)Γ (G) of a thick grope as in Definition 5.4. Moreover, we define the output of G as the knot ⊥ G := ⊥ (G) = 0 ∪ G( ⊥ 0 ), see Definition 5.1 and 5.2. Conversely, for a given grope cobordism G and a choice of its regular neighbourhood, there is a unique thick grope G whose image is precisely that neighbourhood and G| Γ = G.

Grope forests
Recall from Theorem 2.11 that two knots are -equivalent if there exist a sequence of grope cobordisms between them. An analogue in our setting is a disjoint collection of thick gropes.
Definition 5.6. A grope forest of degree and cardinality ≥ 1 on a knot is a map such that G : B Γ ↩→ are mutually disjoint thick gropes on modelled on Γ ∈ Tree( ), and whose arcs G ( 0 ) ⊆ ( 0 ) appear in ( 0 ) in the decreasing order of their indices ≥ ≥ 1.
The output knot ⊥ F is obtained from by replacing each interval G ( 0 ) by the arc G( ⊥ 0 ) (the order in which replacements are done is irrelevant by the disjointness assumption).
Note that for a fixed ∈ we allow an arbitrary order of intersections of ( ) with the -th caps of different gropes, see Figure 23 for an example with cap 1 (G 1 ) < cap 1 (G 2 ), but cap 2 (G 2 ) < cap 2 (G 1 ). Figure 23. A grope forest F = G 1 G 2 of degree 2 is a thickening of the depicted 2-complex.
Grope forests are suitable for defining 'spaces of gropes' in a straightforward manner. Proposition 5.8. There is a surjection of sets which sends a grope forest =1 G : Proof. To prove that this map is well-defined, first consider = 1. For a fixed tree Γ the only allowed isotopies of thick gropes modelled on Γ -that is, paths in the space Emb (B Γ , ) -are those isotopies of the 3-ball B Γ which preserve the property that each special arc ⊆ B Γ is mapped into ( ). Such an isotopy cannot change the homotopy classes (G ), so Γ (G ) is an invariant.
Similarly, the sign (G ) as defined in Section 5.1.3 is positive if and only if the orientation G ( ) agrees with the orientation of . This is preserved during an isotopy.
An analogous argument applies to grope forests ≥ 1, considering one thick grope at a time.
For the surjectivity, let =1 ε Γ be a linear combination of decorated trees, ε ∈ {±1}. Any ∈ ( 1 ) can be represented by a tuple of disjointly embedded loops ⊆ . Thus, there is a map Γ → which embeds the edges mutually disjointly, maps the -th leaf to a point ∈ ( ) and has the associated path (from ( 0 ) to along Γ and then back along ) isotopic to . Thicken this to a ball to get a thick grope G , introducing a twist to one cap if ε = −1.
This can be done so that G are mutually disjoint (as they are neighbourhoods of 1-complexes), and that the order G ( 0 ) is decreasing with , so this defines a desired grope forest.

Gropes give paths in the Taylor tower
Let G be a thick grope in on a knot modelled on Γ ∈ Tree( ). According to Theorem C there is a path in P ( ) between the evaluation of the output knot and of the original knot In this section we prove this based on ideas from [KST], and also show there is a continuous map We reformulate the theorem as the following proposition.
Proposition 5.9. For G as above there is a continuous map We will finish the proof by checking that Ψ G is well-defined thanks to conditions (5.4). The continuity of (5.1) follows as well, since the space Grop 1 ( ; U) of thick gropes of degree on U from Definition 5.7 was given the subspace topology Grop 1 ( ; U) ⊆ Emb(B 3 , ).

The symmetric surgery
Let us first construct a 1-parameter family of disks D ⊆ B Γ , ∈ Δ 1 , for Γ an abstract grope modelled on the unique tree of degree = 2 (Figure 24). This consists of a punctured torus (yellow) and two caps bounded by its core curves 1 (blue) and 2 (orange). There is a classical construction of ambient surgery on a punctured torus ⊆ , using an embedded disk whose interior is disjoint from and with boundary a simple closed curve on .
Namely, we take out a neighbourhood of the curve ⊆ and glue to the newly created boundary two parallel copies of , so that is turned into an embedded disk.
Hence, when our abstract grope of degree 2 is embedded as a grope cobordism we can do two different ambient surgeries on it: on the first (respectively second) cap as depicted in the leftmost (rightmost) part of Figure 25. Note that the thick grope specifies concrete push-offs of caps. In addition, one can do both surgeries at once, called the symmetric surgery (or contraction), as depicted in the middle part of Figure 25. The following lemma says that there actually exists a whole 1-parameter family of disks containing the three disks we have described. . We now specify an isotopy from D 1 ⊆ B Γ to D 2 ⊆ B Γ , which passes through the symmetric surgery, using Figure 25 as an accurate model of these disks.
First isotope the interior of the blue band of D 1 by pushing it across the interior of the ball B 2 , until we arrive at the symmetric surgery. In more detail, as increases from 0 to 1 2 we let the blue band 'stick more and more to the bottom and top orange disks', as shown in Figure 26, so that when = 1 2 the band has transformed into the union of the two orange disks and the yellow region. The two 'sticking curves' (inside of the two orange disks, copies of the cap 2) are specified by an isotopy : 0 ↩→ D 2 which we fixed at the beginning of this section (also, smoothen the corners).
Symmetrically, for increasing ∈ [ 1 2 , 1] the isotopy uses the ball B 1 to stretch the distinguished yellow region of the symmetric surgery, using the sticking curves on the blue disks as a guide, until reaching the position of the orange band for = 1.
Remark 5.11. It is precisely this isotopy that is a crucial ingredient for the connection between the geometric calculus and the Taylor tower. To construct paths in P ( ) using claspers instead, it would be necessary to fix a 1-parameter family of homotopies of Borromean rings whenever one component is erased, but for trees of higher degrees these homotopies will increase in complexity.
Instead, gropes precisely keep track of all necessary homotopies in a canonical way, missing in the clasper picture. Moreover, we will use our exact choice of the isotopy in the crucial Lemma 6.2.

Families of disks
We now generalise the Symmetric Isotopy Lemma 5.10 to trees of any degree ≥ 2. We view Δ as the simplicial set obtained by barycentric subdivision from the standard simplex with the vertex set (see Figure 5).
Proposition 5.12. Let a finite set ≠ ∅ and a tree Γ ∈ Tree( ) there is a continuous map Γ : Δ −→ Emb D 2 , B Γ describing a family D := im Γ ( ) ⊆ B Γ of neatly embedded disks in the model ball such that where ⊆ Δ denotes the top dimensional simplex to which belongs and | | its set of vertices.
Proof. We prove this by induction on | |. For | | = 1 we have Γ = and |Δ | = Δ 0 = { }, so we need to construct only one disk D ⊂ B Γ whose boundary is the boundary of the grope and such that int(D ) 1 ≠ ∅. Clearly, we can just let D := Γ , since in this case the abstract grope is itself a disk, intersecting 1 in one point.
Assume that we have defined the desired family for any tree of degree < for some ≥ 2, and consider with | | = and a tree Γ ∈ Tree( ) such that Pick ∈ Δ and let us define D ⊆ B Γ , using the identification Δ Δ 1 ★ Δ 2 , the join of two simplices. Thus, is given as a linear combination = (1 − ) 1 + 2 , ∈ [0, 1], j ∈ Δ j . The ball B Γ is by definition the plumbing of the balls B Γ j for j = 1, 2, and since 1 ≤ | j | ≤ | | − 1, by induction hypothesis we have maps Γ j satisfying (5.4). In particular, we have disks D j ⊆ B Γ j . holds. Since D is contained in B, which is a sufficiently small neighbourhood of the disks D 1 and D 2 , it will intersect an arc only if one of those disks did. Hence, by the induction hypothesis belongs either to | 1 | or | 2 |, but | | = | 1 | | 2 | by the definition of the join.

Let us pick some neat tubular neighbourhoods
In particular, for = 2 we have = (1− )+2 = 1+ and so D = D 1+ is precisely the isotopy from Lemma 5.10. Note how for an abstract grope of degree each torus stage gives one independent parameter for the family, so there are − 1 parameters in total (remember that |Δ | = Δ −1 ).

The end of the proof: isotopies across disks
Proof of Proposition 5.9. As announced in (5.3) at the beginning of the section, we use the isotopy of the previous proposition for = and the given thick grope to define for ∈ Δ and ∈ [0, 1] We clearly have P G (0) = ( ⊥ G) 0 = G( ⊥ 0 ) and P G (1) = 0 = G( 0 ) for all ∈ Δ . We claim that thanks to the condition (5.4), the map Ψ G as defined in (5.2) is well-defined, that is This is clear for ⊆ [ ] such that 0 ∈ , since we then constantly have the punctured unknot U . On the other hand, for 0 ∉ we need to check that for each ì ∈ Δ and ∈ [0, 1] we have Equivalently, if the interior of G(D ) intersects some ( ), then ∈ . Indeed, if intG(D ) ∩ ( ) ≠ ∅, then we must have int(D ) ∩ ≠ ∅, since G is an embedding. But then (5.4) implies that ∈ | |. As is obtained by inclusion from the face Δ , the maximal simplex that contains it must be contained in Δ . Hence, ∈ | | ⊆ . Proof. If F = =1 G : =1 B Γ ↩→ , then each G can be viewed as a thick grope on . Indeed, it has G ( 0 ) ⊆ 0 and the conditions for all the arcs , ∈ , are satisfied. Therefore, by Theorem C we have a path Ψ G in P ( ) from ev ( ⊥ G ) to ev , which was constructed in Proposition 5.9 using the arcs P G ( ) : 0 ↩→ \ 0 . For a fixed ∈ [0, 1] and ⊆ these arcs are pairwise disjoint for varying = 1, . . . , , because of the mutual disjointness of G . Hence, we can concatenate them to get an arc

Grope forests give points in the layers
. We then define Ψ F analogously to the definition of Ψ G in (5.2), by letting To see that this is a continuous map on the space of grope forests, note that moving within a component in that space preserves the order of roots of thick gropes, so arcs always get concatenated in the same order. Since the topology is given as the subspace topology of the space Emb( small deformations of grope forests lead to small deformations of each of the arcs, keeping them disjoint.
Remark 5.14. A perhaps more obvious choice for Ψ F would simply be the concatenation of the paths in P ( ).
This will actually give an equivalent point e ( F) ∈ F ( ), essentially because F ( ) is an iterated loop space and -while our definition was concatenation in the 0 -direction, this definition corresponds to the concatenation in the 'diagonal' Ω direction. However, our choice will make the proof of Theorem E straightforward.
We omit the proof, only indicating that the two choices D e ( F) ∈ Ω tofib( • , ) can be compared using the description of e ( F) in terms of the ℎ-reflections of Proposition A.6. Note that this discussion implies that concatenation of thick gropes into a grope forest can be seen as a partially defined -space or 1 structure on the space H ( ).
Example 5.15 (degree 1). We now demonstrate the map Ψ on an example in the lowest degree.
A grope cobordism G on modelled on Γ = 1 is simply a disk (see Figure 18 for examples) guiding a crossing change homotopy from 0 = ⊥ G to 1 = . The corresponding thick grope G is a tubular neighbourhood of this disk.
Its underlying decorated tree is 1 for some element ∈ 1 ( ). The disk family in this case consists It is the isotopy between ( ⊥ G) 1 and 1 -the crossing change homotopy, now unobstructed since 1 is gone. See also Figure 27 below for the corresponding points (G) ∈ H 1 ( ) and e 2 (G) ∈ F 2 ( ).
Since H ( ) := hofib(ev ) tofib ⊆[ ] (E ), in the latter coordinates this is given by where h • is the homeomorphism of cubes from (2.13), needed for the translation from the definition of a total fibre as a homotopy fibre to its description in terms of maps of cubes.

Preliminaries for the induction step
It will be convenient to consider trees labelled by a finite set . (setup) Let G be a thick grope on U modelled on a tree Γ ∈ Tree( ) obtained by grafting together Γ 1 ∈ Tree( 1 ) and Γ 2 ∈ Tree( 2 ) with 1 2 = . Let ∈ {±1} and ∈ Ω with ∈ be the signed decorations of G.
In order to prove (6.1) for = we first simplify the map D G : S → Ω as follows. By Proposition A.6 we have In words, along their 0-faces. These maps were defined inductively in Definition A.5 by (for = min ) Since D is applied pointwise, we obtain In the Commutator Lemma 6.2 we will show that D ( G ) is homotopic to a certain commutator map and in the Reflections Lemma 6.3 generalise this to all ℎ -reflections D G ℎ . Having these homotopies collected in Corollary 6.4, we will be able to finish the proof of Theorem D.
The Commutator Lemma. For each ⊆ we now study the map . Using the inductive nature of Definition 5.5 we can write the thick grope G as the plumbing (see Figures 16 and 22) of two thick gropes G j := G| B Γ j modelled on trees Γ j for j = 1, 2.
More precisely, the boundary of the abstract grope Γ j = j has its corresponding distinguished subarc + j ⊆ j . Thus, the map G j : B Γ j ↩→ can be seen as a thick grope modelled on Γ j ∈ Tree( j ) on the knot obtained from U as follows: replace G( 0 ) ⊆ U 0 by the arc G( + j ), together with some arcs connecting their endpoints (the dotted arcs in the model Γ ⊆ B Γ on the left of Figure 29). Observe that the newly produced knot is isotopic to U by an isotopy across the shaded region.
Thus, we can also isotope G j , so that the boundary of its bottom stage is as in the right picture, and hence it is instead a thick grope from U to ⊥ G j := (U \ G( + j )) ∪ G( j \ + j ). Actually, for G j to be a grope on U we also need to reparametrise U so that punctures indeed have labels 1 ≤ ≤ | j |. Also, G 2 should have the orientation of all stages reversed. Thanks to these modifications we have points (G j ) ∈ H | j | ( ) for j = 1, 2. However, we now immediately reparametrise back to get the analogous maps j G j : j → F | j |+1 j with j := ∩ j . In other words, although formally G j are not thick gropes on U, we can easily switch back and forth between the viewpoints. Moreover, we define recalling from Remark 3.22 that the last map is well-defined on the image of D j .
By the following result each loop D G ( ì ) is either the commutator of loops D j G j ( ì j ) or some time of a canonical null-homotopy. We use the convention [0, 1 2 ] ∅ = ∅ and [ 1 2 , 1] ∅ = ∅.
Lemma 6.2. Assume | | ≥ 2. The map D G is homotopic to the composition of the map ( 1 , 2 ) : → 1 × 2 which permutes the coordinates, and the map G : 1 × 2 → Ω given by where ( ) is the time of the canonical null-homotopy · −1 const 0 for a loop ∈ Ω . In Figure 30 these are shown as blue lines, and the subspace on which G is constant is contracted. Proof. Assume = ∅. We have ∅ G ( ) = ( ⊥ G) 0 = G( ⊥ 0 ) and D ∅ ∅ G ( ) = (U 0 ) · G( ⊥ 0 ) 1− is exactly the loop G( Γ ), the boundary of the bottom stage. Since the bottom stage is a punctured torus, it collapses onto the 1-skeleton. This homotopes the boundary onto the commutator Assume now ≠ ∅ and recall that for h ( ì ) = ( , ) ∈ C (Δ ) the arc G ( ì ) := Ψ G ( ) 0 ( ) is the time of an isotopy across the disk G(D ) (see Proposition 5.9): as ∈ [0, 1] increases, the arc ⊥ 0 is being homotoped to 0 across D using a foliation which we are still free to specify. The disk D ⊆ B Γ was in turn defined as the time ∈ [0, 1] of the symmetric isotopy (Lemma 5.10) between the two disks obtained by surgery on the bottom stage along D 1 ⊆ B Γ 1 or D 2 ⊆ B Γ 2 (see Proposition 5.12), where = (1 − ) 1 + 2 ∈ Δ = Δ 1 ★ Δ 2 , with j ∈ Δ j . Without loss of generality, assume < 1 2 , so D looks like in the left of Figure 31. The loop D G ( ì ) is obtained by closing up the arc Ψ G ( ) 0 ( ) using for all ì the same arc U 0 . Thus, we can collapse this U 0 to the basepoint 0 throughout the whole family, so that D G ( ì ) becomes, for a fixed , a basepoint preserving homotopy of the loop G( Γ ) to const 0 .
We now specify the foliation of D in such manner that this homotopy is first done across the two parallel copies of D 1 (vertical disks in Figure 31) and the two pieces of D 2 (two regions lying flat), until for = 1 2 we have completely exhausted the parts of D which come from the caps. We are then left with a band ( ) as on the right of Figure 31 and we let the rest of the homotopy be the 'vertical' contraction onto the vertical line containing 0 , followed by its collapse onto 0 .
To further simplify these homotopies we collapse throughout the family the remaining pieces of the surgered torus in D onto its skeleton. So 'parallel copies of curves' get identified similarly as for = ∅. The final result is as on the left of Figure 32: for any ∈ Δ , ≤ 1 2 our D G ( ì ) became the commutator of the loops D Here is such that = 1 2 corresponds to ( 1 , 2 ) = (1, ) and at this moment D 1 G 1 ( ì 1 ) = const 0 , while D 2 G 2 ( ì 2 ) is the curve on the right of Figure 32. Hence, for = 1 2 and a fixed we have D G ( ì ) = [const 0 , ] and our null-homotopy across ( ) for ≥ 1 2 indeed becomes the canonical null-homotopy ↦ → | [0, ] · 1− | [1− ,1] after the collapse.
Corollary 6.4. The homotopies from the last lemma glue to a homotopy The end of the proof Assume inductively that (6.1) is true for all G as in (setup) with | | < . Let | | = and let us prove that Γ( ) and D ( G ) Firstly, G is the plumbing of thick gropes G 1 and G 2 modelled respectively on Γ j ∈ Tree( j ) and with signed decorations ( , ) ∈ j . Since both | j | < the induction hypothesis implies that The first map in the formula is the Samelson product which was shown in Lemma B.5 to be obtained by canonically trivialising all the faces of the map 17 Plugging in (6.8) we get Γ( ) • ( 1 , 2 ) for the map obtained by trivialising the faces of the map 18 The map is depicted in Figure 33, with the map (6.9) given as the green square with the two coordinate axes ì j ∈ j (so it is an -cube). Trivialising this on the boundary corresponds to putting the green square into a bigger one and filling in the intermediate region by null-homotopies · −1 * along straight blue lines. Here ∈ Ω is some value of (6.9) on the boundary of the inner -cube, and so is indeed constant on the boundary of the outer -cube.
We now show that agrees with the map ⊆ G ℎ , so Corollary 6.4 will finish the proof. 17 More precisely, in that lemma we had , j := Ω j \ • if ∈ j , but we have already abused the notation when we decided to write := Ω \ • S (cf. (4.6)). 18 Here we denote D = Ω where the second equality is by (6.3), the third holds because the commutator bracket is applied pointwise and the last equality is by definition of G ℎ in Lemma 6.3.
Similarly, for ì 1 ∈ [0, 1 2 ] 1 we have a blue line null-homotopy, where ì 2 runs in [ 1 2 , 1] 2 , so In other words, for F = =1 G : =1 B Γ ↩→ with t(G ) = ε Γ , and denoting F := e +1 (F), we need to show This was reduced in Section 4.2 to proving that D ( F ) : S → Ω is homotopic to a map realising the class on the right, namely, the pointwise product =1 Γ ( ) which takes ì ∈ S to the concatenation of the loops Since each G is a thick grope on U with the underlying decorated tree t(G ) = ε Γ , the maps Recall the definition of F in (5.7). Similarly as in the proof of the Commutator Lemma 6.2, there is a homotopy between D F and the pointwise product of D ( G ) -since we can collapse U 0 for all loops in the family. This extends to all ℎ-reflections as in the Reflections Lemma 6.3 -there we had commutators of loops and here just their pointwise concatenations.
Therefore, using the same arguments as in the proof of Theorem D we can conclude

A On left homotopy inverses
In this section we prove Lemmas 3.12 and 3.14, and Propositions 3.16 and A.6.
Proof of Lemma 3.12. Consider the commutative square Its total homotopy fibre is according to Lemma 3.6 given by where · is path concatenation and the square , : 2 → can be viewed as a path of paths in two different ways, or as a path of loops -from its zero-edges to its one-edges, see the picture in (A.2).
Since clearly 1 • −1 1 = Id and 1 is a weak equivalence, −1 1 is a homotopy inverse for 1 (alternatively, there is a homotopy −1 1 • 1 Id by gradually introducing back the coordinate ).
Definition A.2. In the situation of the previous proof let us define the ℎ-reflection of by Note that this is a path from * to 0 ∈ . We now rewrite as ). When attempting to prove Proposition 3.16 by induction, one might run into needing that a similar map is a loop map, which is not the case. However, instead of delooping from 'below' (first delooping F +1 for example), we need to start delooping from 'above', using the following lemma.
Proof of Lemma 3.14. We have two 1-cubes of ( − 1)-cubes, namely the original cube • which uses maps and the cube • which uses instead, so we can write Hence, the induced map ℎ * ( ) on the total fibres is precisely a homotopy ( • ) * Id.
Proof. The left hand side is by (A.9) equal to −1 applied to the map (denoting := −1 ) The concatenation in the −1 -direction can be interchanged with the one in the -direction (this is another manifestation of the Eckmann-Hilton principle), so we obtain Finally, applying −1 gives the desired right hand side in the statement of the lemma. Therefore, we make the following definition.
where the last map is the fold and the first map can be explicitly defined, see [Whi78]. It is precisely the adjoint of the Samelson product [ 1 , 2 ] : 1 ∧ 2 → Ω(Σ 1 ∨ Σ 2 ) from (B.5). If = S −1 are spheres, this is the attaching map S 1 + 2 −1 → S 1 ∨ S 2 of the top cell in S 1 × S 2 .
Actually, for = Ω ∈ Σ the Samelson products ( ) 'generate the homotopy type of '. A more precise statement is the Hilton-Milnor theorem below, for which we need a bit more notation. Firstly, since (B.5) is a map into a loop space, there is a unique multiplicative extension Moreover, given Lie words 1 and 2 we can take the pointwise product 1 ( ) · 1 ( ) (pointwise concatenate loops) as we saw above. Therefore, if B( ) denotes a Hall basis for the free Lie algebra L( : ∈ ), we can define the map ℎ := : where the source is the weak product, defined as the filtered colimit of products over the finite subsets of B( ). Thus, points in it have all but finitely many coordinates equal to the basepoint.

Samelson products for trees
Let us now consider Samelson products for Lie words ( ) in which each letter , ∈ , appears exactly once, for a finite ordered set . This is the ungraded case for now, with | | = 0. Recall from Section 2.1 there is an isomorphism 2 : Lie( ) → Lie 2 ( ) (see also next subsection).
In the proof of Theorem D in Section 6 we will need the following observation. Let ( , ) ⊆ be a cube with maps : → and assume we are given maps : S −2 → Ω for ∈ . For ⊆ with ∈ let us denote More precisely, for each ì ∈ ( −2 ) we glue in the standard null-homotopy ( ì ) · ( ì ) −1 * of loops in Ω to extend to a bigger cube on whose boundary it is constant.
Here we defined a map on (S −2 ) ∧ ( −2 ) by giving it on the cube so that it is constant on the boundary. The proof of the lemma is clear from definitions. See Section 4.2 for how it is used.
Proof of Lemma 2.3. We define by linearly extending the definition in the lemma and check it descends to the quotient by (2.1). We write Γ := (Γ) for short.
To this end, let and be the images under of the linear combinations as in (2.1), but with roots instead of dots. It suffices to show that these are trivial, since then will also vanish on any tree in which or appears as a subtree.
and note that by the last tree is equal to − . . .
Therefore, is equal to where we have used the identities Now again plugging in (2|13) + (1|3) = | Γ 1 || Γ 2 | we get that the terms in the parenthesis are precisely those of the graded Jacobi relation (2.2), which holds in Lie ( ).
Finally, is clearly a surjection and an inverse −1 can be constructed in an analogous wayand will imply it is well-defined modulo graded antisymmetry and Jacobi relations.