Homotopy Brunnian links and the $\kappa$-invariant

We provide an alternative proof that Koschorke's $\kappa$-invariant is injective on the set of link homotopy classes of $n$-component homotopy Brunnian links $BLM(n)$. The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techniques based on Habegger and Lin's string links. We frame the result in the language of Fox's torus homotopy groups and the rational homotopy Lie algebra of the configuration space $\text{Conf}(n)$ of $n$ points in $\mathbb{R}^3$. It allows us to express the relevant Milnor's $\mu$--invariants as homotopy periods of $\text{Conf}(n)$.


Introduction
Consider the space of n-component link maps, i.e. smooth pointed maps (1.1) Two link maps L and L are link homotopic if and only if there exists a smooth homotopy H : n i=1 S 1 × I → R 3 connecting L and L through link maps. Following [22], we denote the set of equivalence classes of n-component link maps by LM (n). Link homotopy, originally introduced by Milnor [29], can be thought of as a crude equivalence relation on links, which in addition to standard isotopies allows for any given component to pass through itself. Link homotopies can be realized on diagrams via Reidemeister moves and finitely many crossing changes within each component. In particular, observe that link homotopy theory is completely trivial for knots. Equivalently, we may consider the space of free link maps, as there is a bijective correspondence between the pointed and basepoint free theory (see Appendix A).
representing untwisted longitudes of the components of L; then G q has the presentation (c.f. [30, p. 289 The homomorphism M : F (m 1 , . . . , m n ) → Z X 1 , . . . , X n into the ring Z X 1 , . . . , X n of formal power series in the non-commuting variables {X 1 , . . . , X n } -called the Magnus expansionis defined on generators in the following way: . . . Consider the expansion of the longitude w j ∈ G q treated as a word in F (m 1 , . . . , m n ): M (w j ) = 1 + I µ(I; j)X I , (1.2) where the summation ranges over the multiindices I = (i 1 , . . . , i m ) where 1 ≤ i r ≤ n, and X I = X i 1 · · · X im for m > 0. Thus, the above expansion defines for each multiindex I the integer µ(I; j), which does not depend on q if q ≥ m. Let ∆(I) = ∆(i 1 , . . . , i m ) denote the greatest common divisor of µ(J; j) where J is a proper subset of I (up to cyclic permutations). Then Milnor's invariant µ(I; j) of L is the residue of µ(I; j) modulo ∆(I). It is well known that Milnor's higher linking numbersμ(I; j) are invariants of L up to concordance and up to link homotopy if the indices in I are all distinct. Habegger and Lin [11] provide an effective procedure for distinguishing classes in LM (n). In place of links they consider the group of string links H(n) and prove a Markov-type theorem, giving a set of moves by which a string link can vary without changing the link homotopy class of its closure. There are two types of moves: (1) ordinary conjugation in H(n); and (2) a partial conjugation, which amounts to conjugation in the normal factor of any semidirect product decomposition of H(n) induced by forgetting one strand. However, it still remains an open problem whether a complete set of "numerical" link homotopy invariants for LM (n) can be defined. For instance, the authors of [28,15,24] address this question using, among other things, the perspective of Vassiliev finite-type invariants. An alternative view, which is rarely cited in this context but which we aim to advocate here, appears in the work on higher dimensional link maps by Koschorke [21,22], Haefliger [12], Massey and Rolfsen [26], and others (e.g. [31]).
Following [21,22], recall that in the case of classical links (i.e. one-dimensional links in R 3 ), the κ-invariant is defined as where [T n , Conf(n)] is the set of pointed homotopy classes of maps from the n-torus T n to the configuration space Conf(n) of n distinct points in R 3 : Again, we consider pointed maps in (1.3) purely for convenience (see Appendix A). Note that because any link homotopy between two links L 1 and L 2 induces a homotopy of the associated maps F L 1 and F L 2 , the map κ is well-defined. Koschorke [21,22,20] introduced the following central question which is directly related to the problems mentioned above.  [22] whenever all of its (n − 1)-component sublinks are link-homotopically trivial. Denote the set of link homotopy classes of homotopy Brunnian n-links by BLM (n) ⊂ LM (n). One of the clear advantages of working with BLM (n) is that the indeterminacy of Milnor's higher linking numbers vanishes for such links, and in fact they constitute a complete set of invariants for BLM (n) (this was first proved in [29]). In order to frame the main theorem in the language of the rational Lie algebra of the based loops on the configuration space ΩConf(n) we first introduce the necessary background.
The inspiration for the proof of the Main Theorem comes from the techniques introduced by the first author in [3] and from the algebraic proof of [7,8]. This seems like a promising approach for answering Question 1.1, not least because a generalization of the n = 3 case suffices to completely answer the question for 3-component links (see in particular [7,Section 5]). Moreover, Corollary 1.2 implies that the µ-invariants of homotopy Brunnian links can be computed by Chen's iterated integrals [13], which in this light appear as generalized Gauss integrals and hence as a possible source of invariants for fluid flows [1, p. 176] (see also [18,19]). This paper is structured as follows. In Section 2, we gather basic facts about string links and use them to provide an algebraic characterization of BLM (n). Then, in Section 3 we review algebraic properties of the torus homotopy groups [T n−1 , ΩConf(n)] and, in anticipation of the proof of the Main Theorem, recognize an analogue of BLM (n) in this context. Finally, we prove the theorem in Section 4. By the usual abuse of notation we often do not make a distinction between equivalence classes and their representatives. Theoretical Physics, and the organizers of the program Knotted Fields, 2012, where a part of this work has been completed. This research was supported in part by the National Science Foundation under Grant No. PHY11-25915. The second author acknowledges the support of DARPA YFA N66001-11-1-4132.

String links
Following Habegger and Lin [11], let H(n) be the group of link homotopy classes of ordered, oriented, parametrized string links with n components. There is a split short exact sequence of groups where H(n; i) is the copy of H(n − 1) given by the map δ i which deletes the ith strand. The normal subgroup C(n; i) is isomorphic to RF (n − 1), the reduced free group on the n − 1 generators shown in Figure 1. The exact sequence (2.1) is split by the map s i : H(n; i) −→ H(n) which just adds one trivial strand to any element of H(n; i) as the ith strand. As a result we have the semidirect product decomposition (for each i) be the factorization of ρ with respect to the above decomposition. A partial conjugation of ρ by The Markov-type theorem for homotopy string links is then Some known facts about H(n) are: (1) H(n) is torsion-free and nilpotent of class n − 1, H(n) k−1 /H(n) k is a free abelian group of rank (k − 2)! n k . In the following we will focus on the copy of C(n; n) ∼ = RF (n − 1) in H(n) generated by τ 1 , . . . , τ n−1 where τ k := τ n,k . Recall that the reduced free group RF (n − 1) is the quotient of F (n − 1) given by adding the relations [τ i , gτ i g −1 ] = 1 for all i and all g ∈ F (n − 1). We list known and useful facts (see [3]) about RF (n − 1) below.
(4) RF (n − 1) admits the presentation where σ is a permutation of {2, . . . , k}. Then Consider the homomorphism where δ i is as defined in (2.1). Clearly, Observe that elements of ker δ have a natural geometric meaning: they are precisely the string links which become trivial after removing any of their components, which we naturally call Brunnian string links and denote by BH(n); i.e., BH(n) := ker δ. In the ensuing lemma we choose to treat BH(n) as a subgroup of C(n; n) ∼ = RF (τ 1 , . . . , τ n−1 ). Moreover, Proof. The fact that H(n) n−1 ⊂ Z(H(n)) follows immediately from nilpotency (i.e. length n commutators are all trivial in H(n)). Thus (i) implies (ii). Clearly BH(n) ⊂ ker δ n = C(n; n), so any z ∈ BH(n) can be written in the normal form (2.7). We claim that z = λ n−1 . Indeed, for each k < n − 1 and any I = (i 1 , . . . , i k ), consider τ (I, σ) given in (2.6). Pick j ∈ {1, . . . , n − 1} such that j = i r for all i r ∈ I, which is possible since k < n − 1. The map δ j : C(n; n) −→ C(n; n, j) which deletes the jth strand is given on generators as so we have δ j (τ (I, σ)) = τ (I, σ). Therefore, δ j (λ k ) = 1 for all j = 1, . . . , n − 1, contradicting the fact that z ∈ ker δ j for all j. Hence, the normal form (2.7) implies z = λ n−1 and therefore (2.10) follows as well as the first part of (i). The second identity in (i) is immediate: (2) implies that H(n) n−1 = H(n; i) n−1 C(n; i) n−1 , but this is just C(n; i) n−1 since H(n; i) n−1 is trivial by (1). Remark 2.3. The proof of the fact that {τ (n, σ)} generates BH(n) is fully analogous to that of claim (a) in Appendix B.
The relation between Brunnian string links and Brunnian links is revealed in the following result, which is a consequence of Lemma 2.2 and Theorem 2.1.
Proposition 2.4. The restriction of the Markov closure operation defined in (2.4) to BH(n) is injective, and the image BH(n) equals BLM (n).
Proof. In order to see that · is injective on BH(n), let ρ ∈ BH(n). Since BH(n) = C(n; i) n−1 , the factorization Hence, all partial conjugations of ρ are just ordinary conjugations and, since BH(n) ⊂ Z(H(n)), they act trivially on ρ. Thus Markov closure is injective on BH(n).
In order to see BH(n) = BLM (n), observe that the inverse image of [1] ∈ LM (m) under · : H(m) −→ LM (m) contains only 1 ∈ H(m). Indeed, Theorem 2.1(b) tells us that any element of the inverse image of [1] under · has to be related to 1 by conjugations or partial conjugations. Obviously, both of these operations act trivially on 1, which proves the claim.
Let us fix a representative link L ∈ BLM (n) and by Theorem 2.1(a) let ρ ∈ H(n) be such that ρ = L. Since any (n − 1)-component sublink δ i (ρ) of L is trivial, the fact proven above implies that δ i (ρ) = 1 for any 1 ≤ i ≤ n, and therefore ρ ∈ ker δ = BH(n). to the based loop space ΩConf(n) of the configuration space Conf(n). The product in T (n) comes from the loop multiplication or equivalently the coproduct of suspensions. In the notation of Fox, who introduced torus homotopy groups [10], T (n) is denoted by τ n (Conf(n)) and equivalently defined as π 1 (Maps(T n−1 , Conf(n)), * ), where the basepoint * in the case of T (n) is defined to be the constant map. In the following, we will freely alternate between both ways of representing T (n) given in (3.1). Letting P k be the k-skeleton of T n−1 , we have the filtration

Torus homotopy groups
For each ordered multiindex I, observe that P k = I,|I|=k S I , where We have a decreasing filtration of groups  π k (ΩConf(n)), (3.3) where S I := S I /(S I ∩ P |I|−1 ).
Another way to see (3.3) is via the homotopy equivalence Then, each monomorphism j # I of (3.4) can be regarded as induced from the restriction in the above bouquet to the Ith factor ΣS |I| of ΣT n−1 . We have the following lemma which is a special case of a lemma due to Fox [10]: and j ∈ J. Then for any α ∈ π I (ΩConf(n)), β ∈ π J (ΩConf(n)), we have Now, we aim to understand the free part of T (n), which we will be denoted by T F (n). Consider the following elements of T (n) obtained from the generators {B j,i } of L(Conf(n)), given as adjoints where ( ) is the length 1 multiindex. In view of the fact that {B k,i } are free generators of π 1 (ΩConf(n)), {t k,i ( )} are infinite cyclic and generate a subgroup of T (n) which we call the free part of T (n) and denote by T F (n).  Proof. By Theorem 5.4 of [25], the kth stage T F (n) k of the lower central series of T F (n) is generated by the following simple r-fold commutators for r ≥ k: with no additional assumptions on the order or distinctiveness of (j 1 , . . . , j r ), (i 1 , . . . , i r ), and ( 1 , . . . , r ). For (10), observe that if k = n, any multiindex ( 1 , . . . , n ) has to have repeating entries, since 1 ≤ s ≤ n − 1 for 1 ≤ s ≤ n. Then Lemma 3.1(ii) tells us that any n-fold commutator as in (3.8) is trivial, proving (10). Moreover, from Lemma 3.1(i) and (3.6), . . , t jr,ir ( r )] ∈ π ( 1 ,..., r ) (ΩConf(n)) free ⊂ π ( 1 ,..., r ) (ΩConf(n)).

Proof of Main Theorem
For convenience let us restate our main result: Main Theorem. The restriction of κ to BLM (n) is injective. Moreover, (i) the image κ(BLM (n)) of BLM (n) is contained in a copy of π n (Conf(n)) inside [T n , Conf(n)] and it is a free, rank (n − 2)! module generated by the iterated Samelson products Let s = (0, . . . , 0) be the basepoint of T n and let each factor be parametrized by the unit interval. Distinguish the following subsets of T n : and define the maps This result follows from Satz 12, Satz 20 in [32] (see [22, p. 305]), but for completeness we provide an independent argument in Appendix C. Next, we turn to the proof of the Main Theorem. We will work with C(n; n), which is a copy of RF (n − 1) inside H(n), and begin by constructing a homomorphism φ : C(n; n) −→ T F (n) = [ΣT n−1 , Conf(n)] free via the canonical homomorphism F (τ 1 , . . . , τ n−1 ) −→ T F (n) defined on generators by Observe, by Lemma 3.1(ii), that any commutator in {t n,i (i)} with repeats is trivial. Therefore, as a direct consequence of the presentation in (4), we can pass to the quotient and obtain φ : C(n; n) −→ T F (n), as required. We will need the following relation between the κ-invariant of (1.3) and φ.
Hence, Lemma 3.3 implies that φ • ι is a monomorphism with image equal to BT F (n). Further, the Fact stated above yields the commutative diagram Further, by Proposition 2.4 the Markov closure map · is a bijection on BH(n). Therefore, the injectivity of κ follows, proving (i) of the theorem modulo the above Fact. We also have the set identity κ(BLM (n)) = p # • j # N (BT F (n)), which implies that κ(BLM (n)) has the structure of a Lie Z-module with basis given by the iterated Samelson products B(n, σ) from (4.1).
Proof of the Fact. As a first step, we show commutativity of (4.5) on the generators {τ i } of C(n; n). Given a multiindex I, we adapt the notation S I and S I from (3.2) and (9), respectively, to T n . Observe that, because all strands of τ i except the nth and the ith can be collapsed to the basepoint (see Figure 1), κ( τ i ) factors, up to homotopy, through Since the ith and nth strands of τ i link once, the restriction κ( τ i )| S i ×Sn has degree 1 after composing with the projection Π i,n : Conf(n) −→ Conf(2) ∼ = S 2 onto the ith and nth coordinates. Further, κ( τ i )| S i ×Sn factors through where the second equality is obtained by passing to adjoints. This shows that Diagram 4.5 commutes on the generators. Now, let τ ∈ C(n; n) be a word of length k in {τ i }; specifically Working with a combed representative of τ (as in Figure 2), we let be such that the restriction of the t parameter to [t j−1 , t j ] parametrizes τ i j in (4.8). Recall from (4.2) the subsets A j := A t j and S n , which are the (n − 1)-torus with fixed last coordinate t j and the nth coordinate circle, respectively. We have the following cofibration diagram together with the induced exact sequence of sets and groups: Since κ( τ ) restricted to any A j is null and Conf(n) is simply-connected, there exists some z ∈ [ k ΣT n−1 , Conf(n)] such that h # (z) = κ( τ ). Let r j : k ΣT n−1 −→ ΣT n−1 be the projection onto the jth factor and let z j = r # j (z). Clearly z = z 1 · . . . · z k where · is the coproduct in [ΣT n−1 , Conf(n)]. Since we can choose the representative of τ i j to be 1 · . . . · 1 · τ i j · 1 · . . . · 1, repeating the above reasoning yields where the last identity follows from the definitions of h, r j , and p. But then we know from (4.7) that κ( τ i j ) = p # (t n,i j (i j )), so it follows that the adjoint of z j is t n,i j (i j ). Since the adjoint of the coproduct is the loop product, we conclude that the adjoint of z equals φ(τ ), and κ( τ ) = h # (z) = p # (φ(τ )). Figure 3 shows the intuition -which originated in Section 5 of [7] -behind the above argument.
Therefore, (4.11) follows from (1.2) and the definition of the µ-invariants from Section 1.
This ends the proof of the Main Theorem. Corollary 1.2 is a direct consequence of the fact that the homotopy periods of the Samelson products {B(n, σ)} in π n−1 (ΩConf(n)) can be obtained from the general methodology of Sullivan's minimal model theory [34], or Chen's iterated integrals theory [13]. Consult [19] for a basic derivation of such an integral in the case of three component links.

Appendix A. Independence of the basepoints
Consider the function space of, respectively, free and based n-component link maps where {e 1 , e 2 , . . . , e n } is a set of n distinct points in R 3 which serve as basepoints for each component. Now, the free link homotopy classes and pointed link homotopy classes can be defined as LM (n) = π 0 (LMaps(S 1 . . . S 1 , R 3 )), LM 0 (n) = π 0 (LMaps 0 (S 1 . . . S 1 , R 3 )).
Since we are only concerned with the free part of ker Ψ it suffices to work rationally and show the following about the B(n, σ)s: (a) they span BT F (n) ⊗ Q, (b) they are linearly independent in L(Conf(n)) = π * (ΩConf(n)) ⊗ Q. be any length (n − 1) commutator with i k = 1. We need to "shuffle" the elements of b I to get B n,1 to the front. If k < n − 1, then the shuffling is easy because b I = [a, B n,i k+1 , . . . , B n,i n−1 ], where a = [B n,i 1 , . . . , B n,1 ] is a commutator of length ≤ n − 2. The inductive hypothesis implies that a is a linear combination of products beginning with B n,1 and, consequently, bilinearity of the Samelson product implies that b I is a linear combination of the B(n, σ)s. By the inductive hypothesis the inner part [a n−3 , B n,1 ] of the first term is a linear combination of products beginning with B n,1 and therefore bilinearity of [ , ] implies again that this term is a combination of products beginning with B n,1 . The second term [[B n,1 , B n,i n−2 ], a n−3 ] requires repeated applications of the Jacobi identity, which we carry out up to sign. Since a n−3 = [a n−4 , B n,i n−3 ], we have Part (b) follows by considering L(Conf(n)) as a subalgebra of the universal enveloping algebra U L(Conf(n)). Each B(n, σ) is then an element of U L(Conf(n)) and can be expanded in the tensor product as B(n, σ) = [B n,1 , B n,σ(2) , . . . , B n,σ(n−1) ] = B n,1 ⊗ B n,σ(2) ⊗ . . . ⊗ B n,σ(n−1) + {other terms not starting with B n,1 }.
This completes the proof of Lemma 3.3.
Appendix C. Proof of Lemma 4.1 Note that in place of p # • j # N we may prove injectivity of the map p # • j # n : π n (Conf(n)) −→ [T n , Conf(n)], where j # n : π n (Conf(n)) −→ [ΣT n−1 , Conf(n)] is the adjoint of j # N . We simplify further by considering the projection q : T −→ T/T (n−1) ∼ = S n , where T (n−1) is the (n − 1)-skeleton of T := T n , and observing that, up to homotopy equivalence, q # : π n (Conf(n)) −→ [T n , Conf(n)] satisfies q # = p # • j # n . We wish to think about T as T = T (n−1) ∪ ϕ B n , where B n is an embedded n-ball in T and ϕ : ∂B n → T (n−1) is the attaching map. We consider the associated cofibration (where we used the fact that T/T (n−1) ∼ = S n ). The Barratt-Puppe action [5] of π n (Conf(n)) on [T, Conf(n)] arises from the map T −→ T ∨ T/T (n−1) = T ∨ S n which collapses the boundary of the embedded ball B n in T to a point. We indicate this action by a ∈ π n (Conf(n)), f ∈ [T, Conf(n)].
One property of this action, which is clear from the definition, is that for any a, c ∈ π n (Conf(n)) we have a • q # (c) = q # (c) a = q # (a + c).
Turning to the proof of the injectivity of q # , suppose a, b ∈ π n (Conf(n)) are such that q # (a) = q # (b). Thanks to the above property of the action, [1] = q # (a − a) = q # (a) −a = q # (b) −a = q # (b − a).
By exactness of the cofibration sequence it follows that b − a is in the image of the map Σϕ # : [ΣT (n−1) , Conf(n)] −→ π n (Conf(n)). But Σϕ # , as shown below, is trivial and so a = b, which implies the claim.
To see that Σϕ # is trivial, note that there exists a map s : ΣT −→ ΣT (n−1) such that the composite is homotopic to the identity. Indeed, T is a product of circles and therefore ΣT is a wedge of spheres (as in (3.5)). Hence it suffices to choose s to be the projection onto those factors of ΣT in the wedge which correspond to ΣT (n−1) . Then, is homotopic to Σϕ. But the composition of Σϕ and ⊂ is null, which implies the triviality of Σϕ # . This completes the proof of Lemma 4.1.