HOMOLOGY COBORDISM AND

GENERALIZATIONS OF MILNOR’S INVARIANTS

Tim D. Cochran* and Kent E. Orr**

§0. Introduction.

Two ordered links are said to be concordant if there are disjointly embedded, locally flat cylinders in S^{3} × I, transversely intersecting the boundary in the respective links. Concordance is a classical equivalence relation on links dating to work of Fox and Milnor, and playing a fundamental role in manifold theory, in singularity theory and the study of characteristic classes.

Concordant links have homology cobordant complements, implying the groups of these link complements have isomorphic nilpotent quotients. Nilpotent link group quotients were exploited in Milnor’s seminal work [M1], [M2]. This was formalized and related to concordance questions in important work of Stalling’s [St], and Stalling’s ideas were vastly extended in Bousfield [B].

Milnor, using ideas of Magnus, defined his μ̄-invariants [M2]. These are a numerical measure of the failure of distinct classical link groups to have isomorphic lower central series quotients, and in particular, of the link complements to be homology cobordant.

During the past decade, a fairly complete understanding of Milnor’s invariants has emerged. (See, for instance, [C1], [O], [HM]) As Milnor knew, his invariants obstructed the link longitudes lying in the intersection of the lower central series of the fundamental group of the link exterior. Recent work suggests slice links may have longitudes lying in the intersection of the transfinite lower central series of the group closure (See [L2], [CO2]). Unfortunately, it is still unknown if links with vanishing Milnor invariants can have a nontrivial transfinite lower central series.

Potentially, this doorway opens to previously unknown slicing obstructions. This paper attempts to lay a groundwork for tapping this resource.

For example, it is unknown if a link with vanishing Milnor invariants is concordant to a fusion of a boundary link. A fusion of a boundary link is a geometrically constructed link obtained by appropriately piecing together the components of a boundary link, and is a class of links more general than boundary links but with vanishing Milnor invariants [C1], [C2]. J. P. Levine defines an invariant which vanishes if and only if a given link is concordant to a sublink of an homology boundary link [L2]. The existence or non-existence of transfinite Milnor invariants would either distinguish Levine’s invariants from Milnor’s invariants, or give an important geometric interpretation of the vanishing of Milnor’s invariants.

The idea of this paper is simple. We attempt to reach beyond the usual lower central series into the transfinite lower central series of a link group by investigating the lower central series of finite index subgroups of link groups. This we achieve by defining Milnor invariants for links obtained as finite, often iterated, covers of the original link. We call these covering links, following [CO1]. (See that paper for examples of how concordance invariants of covering links detect links not concordant to boundary links.) To define these new invariants, we need a Milnor-Stallings type theory describing lower central series quotients of the groups of these covering links. This is already available via Stallings theorem and the observation that homology cobordisms become Z_{(p)} homology cobordisms after passing to p-fold covers.

We give mod p and Z_{(p)} version of these localized Milnor invariants, but sadly, the paper concludes with a result limiting the value of these mod p obstructions. The Z_{(p)} slicing obstructions merit further investigation.

§1.

Theorem 1. Suppose f : G → π is a homomorphism of groups which induces an isomorphism on H_{1}( ; Z_{p}) and an epimorphism on H_{2}( ; Z_{p}) for a prime p. Suppose ϕ : π → Z → Z_{p} is an epimorphism, π̃ = kernel ϕ, G̃ = kernel(ϕ ∘ f). Then the induced homomorphism f̃: G̃ → π̃ is 2-connected with Z_{p} coefficients. If G, π and H_{2}(π̃; Z) are finitely generated (for example if G is finitely generated and π is finitely presented), then f̃ is 2-connected with Z_{(p)} (and thus with Q) coefficients as well.

Proof of Theorem 1. The proof is very similar to the proof, given in [CG; Lemma 4.2], that a p-fold branched cyclic cover of a slice knot is a Z_{(p)}-homology sphere. Let ψ denote the epimorphism π ↠ Z, π_{∞} its kernel and G_{∞} the kernel of ψ ∘ f. Let f : X → Y be a map of Eilenberg-Maclane spaces inducing f on π_{1}, and M its mapping cylinder. Let X̃, X_{∞}, Ỹ, Y_{∞}, M̃, M_{∞} denote the induced covering spaces. The short exact sequences

of chain complexes 0 ⟶ C_{∗}(M_{∞}, X_{∞}) ⟶^{t−1} C_{∗}(M_{∞}, X_{∞}) ⟶^{q} C_{∗}(M, X) ⟶ 0 and

0 ⟶ C_{∗}(M_{∞}, X_{∞}) ⟶^{t^{p}−1} C_{∗}(M_{∞}, X_{∞}) ⟶^{q′} C_{∗}(M̃, X̃) ⟶ 0 induce long exact sequences in homology with Z_{p} coefficients. Considering the first of these,

⟶ H_{2}(M, X) ⟶^{∂_{∗}} H_{1}(M_{∞}, X_{∞}) ⟶^{t_{∗}−1} H_{1}(M_{∞}, X_{∞}) ⟶^{q_{∗}} H_{1}(M, X)

we see that the hypothesis that f is 2-connected implies the outer-most terms are zero and hence that t_{∗} − 1 is an isomorphism on H_{0}(M_{∞}, X_{∞}), H_{1}(M_{∞}, X_{∞}) and an epimorphism on H_{2}(M_{∞}, X_{∞}). It follows that (t_{∗} − 1)^{p} and hence t_{∗}^{p} − 1 share these properties. Now considering the second long exact homology sequence:

⟶ H_{2}(M̃, X̃) ⟶^{∂′_{∗}} H_{1}(M_{∞}, X_{∞}) ⟶^{t_{∗}^{p}−1} H_{1}(M_{∞}, X_{∞}) ⟶^{q′_{∗}} H_{1}(M̃, X̃) ⟶

we see, conversely, that H_{2}(M̃, X̃; Z_{p}) ≅ H_{1}(M̃, X̃; Z_{p}) ≅ 0. It follows immediately that f̃ is 2-connected with Z_{p} coefficients, as claimed.

If G and π are finitely generated then so are G̃, π̃, H_{1}(G̃; Z), H_{1}(π̃; Z) and H_{1}(π̃, G̃; Z). If H_{2}(π̃; Z) is finitely generated it then follows that H_{2}(π̃, G̃; Z) is also. Under these conditions the vanishing of H_{i}(π̃, G̃; Z_{p}) for i ≤ 2 implies the same for Z_{(p)} and Q coefficients. □

Remark: If π is finitely generated then f̃ will induce an isomorphism on H_{1} with Z_{p} coefficients without any assumption about f_{∗} on H_{2}(G) → H_{2}(π). For, following the above proof, we see that we still get that t_{∗}^{p} − 1 is onto. But a surjective endomorphism of the finitely generated vector space H_{1}(π̃, G̃; Z_{p}) is necessarily an isomorphism.

Remark: If G is finitely generated and π is finitely presented, Theorem 1 is true without the condition that the homomorphism ϕ : π → Z_{p} factors through Z.

Example: The second part of the theorem is false if G is not finitely generated. Let G be the semi-direct product of Q with Z where t ∈ Z acts by negation. Let π be Z, the abelianization of G, and let f be the abelianization and ϕ be the epimorphism to Z_{2}. Certainly f is 2-connected. But π̃ ≅ 2Z and G̃ ≅ Q × 2Z so f̃ is not a monomorphism on H_{1} with Z_{(2)} coefficients though f̃ is 2-connected with Z_{2} coefficients.

§2. New Concordance Invariants for Links in Z_{p}-Homology 3-spheres.

In this section we define families of link concordance invariants. In brief, our new invariants are Z_{(p)} versions of Milnor’s invariants applied to certain “covering links” of the original link.

Let R be the finite ring Z_{p} or a subring of the rationals. Recall the R lower central series of G is defined by

G_{1}^{R} = G
G_{n+1}^{R} = ker {G_{n}^{R} → R ⊗ (G_{n}^{R}/[G, G_{n}^{R}])}.

The above homomorphism is 1 ⊗ q where q is the quotient homomorphism.

Let L̃ = {K_{1}, …, K_{m}} denote an ordered, oriented link of circles embedded in Σ, an oriented Z_{p}-homology sphere (p prime). Milnor’s \overline{μ}-invariants can be extended, in a straightforward fashion, to links in Z_{p}-homology spheres using the R-lower central series of G = π_{1}(Σ\L) where R = Z_{p}, Z or Q. A sketch of these extensions follows. Choose a meridional map ϕ : F ⟨x_{1}, …, x_{m}⟩ ⟶ G which, by Alexander Duality, induces an isomorphism on H_{1}( ; R). Let t_{n}(G) = G/G_{n}^{R}. R is implicit and determined by context.

Lemma 2.1 (see [M3] for the case R = Z): The meridional homomorphism

ϕ_{n} : t_{n}(F) ⟶ t_{n}(G)

is surjective and its kernel is the normal closure of {[x_{i}, w_{i,n}] | 1 ≤ i ≤ m} where ϕ_{n}(w_{i,n}) represents an i^{th} longitude of L̃. Thus t_{n}(G) has a presentation

⟨t_{n}(F) | [x_{i}, w_{i,n}] 1 ≤ i ≤ m⟩ .

Proof of 2.1. Choose an embedding g of a wedge W of m circles into Σ whose image is L̃ together with a set of base paths compatible with ϕ. The map F ⟶ P ≡ π_{1}(Σ\W) is 2-connected with R coefficients and thus induces isomorphisms t_{n}(F) ⟶ t_{n}(P) ([St], see also [BK; Lemma 3.7]). Moreover, Σ\L is obtained from Σ\W by adding m 2-cells (and one 3-cell) along the commutators [x_{i}, ℓ_{i}] where ℓ_{i} is a parallel of K_{i} on ∂(Σ\W). Hence t_{n}(G) is the desired quotient of t_{n}(P) (if R ⊆ Q use the fact that ⊗R is exact). □

There are also modp, Z_{(p)}, and Q versions of the Magnus embedding. Let P be the group of units in ring of power series in the non-commuting variables {t_{i} | 1 ≤ i ≤ m} with coefficients in R. Let h be the Magnus homomorphism F ⟶^{M} P given by h(x_{i}) = 1 + t_{i}. This is an embedding [S, §6]. This induces a map F/F_{n} ⟶^{M} P/ ∼ modulo terms of total degree at least n + 1. Therefore M induces an embedding M : t_{n}(F) → P/ ∼. Hence, given a multi-index I = i_{1}i_{2} … i_{n−1}j of length n, we can define μ(I) for L̃ to be the coefficient of t_{i_{1}}t_{i_{2}} … t_{i_{n−1}} in M(w_{j,n}) where ϕ_{n}(w_{j,n}) = ℓ_{j} in t_{n}(G). If the longitudes of L̃ each lie in G_{n−1} then by 2.1, w_{j,n} is unique in t_{n}(F) and hence \overline{μ}(I) is a unique element of R.

Corollary 2.2. The longitudes of L̃ lie in G_{n−1}^{R} if and only if ϕ_{n} : t_{n}(F) → t_{n}(G) is an isomorphism.

Proof of 2.2. Suppose ℓ_{i} ∈ G_{n−1}^{R} so ℓ_{i} = 0 in t_{n−1}(G). By induction we may assume ϕ_{n−1} is an isomorphism, so w_{i,n−1} = 0 in t_{n−1}(F). If R = Z_{p} it follows that w_{i,n} ∈ F_{n−1}^{Z_{p}} so the relations [x_{i}, w_{i,n}] are redundant. If R ⊆ Q it follows that w_{i,n} = Σa_{j} ⊗ r_{j} where a_{j}^{m_{j}} ∈ F_{n−1}^{Z} for some m_{j} ≠ 0 and consequently [x_{i}, w_{i,n}] = 0 in t_{n}(F).

Conversely now suppose ϕ_{n} is an isomorphism implying that [x_{i}, w_{i,n}] = 0 in t_{n}(F). Recall that, by definition, [ℓ_{i}] ∈ G/G_{2}^{Z_{p}} contains x_{i} with zero exponent sum, and hence the same is true of [w_{i,n}] ∈ F/F_{2}^{Z_{p}}. Now we require a result that, under this condition, [x_{i}, w_{i,n}] ∈ F_{n}^{Z_{p}} implies w_{i,n} ∈ F_{n−1}^{Z_{p}}. The proof of Lemma 5 of [Levine; Link Concordance and Algebraic Closure, p. 242] for the case R = Z works also for the cases R = Z_{p} and R ⊆ Q using the fact that n^{th} order commutators may be detected in all cases by the “Magnus” embedding. It follows then that ℓ_{i} ∈ G_{n−1}^{Z_{p}} if R = Z_{p} and if R ⊆ Q, that ℓ_{i} raised to a power lies in G_{n−1}. In each case one concludes ℓ_{i} ∈ G_{n−1}^{R}. □

Therefore R-valued (R = Z_{p}, Z_{(p)}, Q) Milnor’s \overline{μ}-invariants can be defined, generalizing Milnor, for any link in a Z_{p}-homology 3-sphere. (The first non-vanishing Z_{(p)} and Q invariants are equivalent invariants.) Moreover these can be seen to be unchanged by a “Z_{p} concordance”. By a p-concordance between L̃_{0} and L̃_{1} in Σ_{0} and Σ_{1} respectively, we mean that there is an oriented Z_{p} homology cobordism W between Σ_{0} and Σ_{1} and a proper flat embedding C of m annuli such that (∂W, C ∩ ∂W) = (Σ_{0}, L̃_{0}) ⨿ (−Σ_{1}, L̃_{1}) (as in [CO1, p. 522]). Then it is well known that the exterior W \C of the p-concordance is an R-homology cobordism between the link exteriors and that such an homology cobordism induces isomorphisms on the quotients t_{n}(G) [Stallings]. Therefore the \overline{μ}-invariants of L̃_{0} and L̃_{1} coincide when the links are p-concordant.

Now, given any link L = {J_{0}, J_{1}, …, J_{k}} in S^{3} or indeed in any Z_{p}-homology 3-sphere, which has all linking numbers zero mod p, we shall describe how to associate various “covering links L̃” to L in such a way that varying L in its p-concordance class only varies L̃ in its p-concordance class. Our proposed “new” μ̃-invariants then are simply the “R-valued \overline{μ}-invariants” of the “covering link” L̃.

Definition 2.3: Let μ̃_{R}(L) denote one of the R-valued Milnor invariants of a covering link L̃ associated to L as described in this chapter.

Following [CO; §1] exactly, given L, we construct the p-fold cyclic cover of S^{3} branched over J_{0} to yield Σ and let L̃ be the inverse images of the components. In [CO; §1] we usually excluded the branch set but here we usually include it. The covering link is canonically oriented but not canonically ordered. To do so entails specifying the “first” component for each set of p-lifts of a component of L. The first such choice will not actually affect invariants since Z_{p} acts as an equivalence of links. Therefore there are actually k − 1 arbitrary choices. Hence if k > 2, the μ̃(L) suffer from greater ambiguity than Milnor’s invariants and perhaps one should view μ̃(L) as a set. It does not seem fruitful to invent a notation for this at this time. Note that the property that all μ̃-invariants vanish is unambiguous with respect to ordering of components. One extremely important feature

is that this process can be iterated to yield more complicated covering links ˜ L̃. Note that a composite of p-fold branched covers may not even result in a regular covering. Thus we have defined a vast tree of potentially new and useful link invariants.

Proposition 2.4. If L is a link in a Z_{p}-homology sphere then the set of μ̃-invariants for L vanishes if and only if the longitudes of the appropriate covering links of L lie in each term of the corresponding R-lower central series.

Proof of 2.4. Use 2.2 and the fact that the Magnus maps are embeddings. □

Proposition 2.5. If L and L′ are p-concordant then their μ̃-invariants (corresponding to iterated p-fold covers) agree.

Proof of 2.5. See the discussion preceeding 2.3.

§3. What the μ̃-invariants may detect.

In this section we to assess what these may be measuring above and beyond Milnor’s \overline{μ}-invariants. Although there are philosophical arguments that these invariants should detect more than Milnor’s invariants alone, we show that all currently-known links with vanishing \overline{μ}-invariants also have zero value for these generalized invariants.

A discussion is in order concerning why one might hope that these invariants carry more information than the ordinary Milnor’s \overline{μ}-invariants of L. Recall that K. Murasugi has shown that the information contained in Milnor’s \overline{μ}-invariants of L is captured by ordinary linking numbers between certain components of certain covering links L̃ associated to L by certain nilpotent covers of the exterior of L. Applying this philosophy to μ̃(L) we might expect that μ̃(L) is captured by ordinary linking numbers between components of covering links obtained from L by doing a (sequence of) p-fold branched covers, then a nilpotent cover. Since the composition of these covering spaces will not, in general, be a nilpotent cover, nor even a regular cover, there is no reason to expect that our μ̃(L) are zero if Milnor’s \overline{μ}(L) are zero. Nevertheless, the only known class of links in S^{3} which have vanishing Milnor’s \overline{μ}-invariants are fusions of boundary links (perhaps slightly more general is the class of sublinks of homology boundary links, but no examples of such are known except fusions of boundary links [Cochran-Levine]) and these also have vanishing μ̃(L) as we shall prove below. The discussion will also serve to focus on precisely what the μ̃(L) may be detecting over and above the set of \overline{μ}(L).

Proposition 3.1. Suppose L is a link and G its group. Suppose there exists a homomorphism ψ : G → P such that

i) ψ induces an isomorphism on H_{1}( ; Z_{p})

ii) H_{2}(P; Z_{p}) = 0 Then the Z_{p}-valued μ̃-invariants vanish for L. If, in addition,

iii) P is finitely presented then the Z_{(p)} and Q-valued μ̃-invariants vanish for L.

Proof of 3.1. Let ϕ : F ⟨x_{1}, …, x_{m}⟩ ⟶ G be a meridional map. Then ψ and ψ ∘ ϕ are 2-connected with Z_{p} coefficients so, by Theorem 1.1, ψ̃ : G̃ → P̃ and { \atop ψ ∘ ϕ} : F̃ → P̃ are 2-connected with Z_{p} coefficients. Hence ϕ̃ : F̃ → G̃ induces isomorphisms ϕ̃_{n} : t_{n}(F̃) → t_{n}(G̃) for all n. Note that ϕ̃ is a meridional map for the covering link L̃ so ϕ̃_{n} may be used to compute μ̃(L). If ℓ̃_{j} is a longitude of a component of L̃ then ℓ̃_{j} ∈ G̃_{n−1}^{R}, by 2.2. Hence by 2.4, all “first-stage” μ̃ vanish for L. This argument may then be iterated since the 2-connectivity of { \atop ψ ∘ ϕ} implies H_{2}(P̃; Z_{p}) = 0. □.

Corollary 3.2. If any of the following holds for the link L in S^{3} then all μ̃ invariants vanish for L

i) L is concordant to a sublink of an homology boundary link; (in particular if L is a fusion of a boundary link).

ii) L is concordant to an E-link in the sense of [Cochran LCIHT; p. 643].

iii) L is an F̂-link in the sense of [Levine; LCACII, p. 580].

iv) The longitudes of L lie in a subgroup N of G which is the normal closure in G of

a finite number of elements, and such that N = [N, G].

v) The link group G has the same algebraic closure as the free group on a set of meridians (i.e., F̂ ≅ Ĝ in the sense of [Levine; LCACII]).

If any of the following holds for the link L in S^{3} then all of the Z_{p}-valued μ̃-invariants vanish for L.

vi) the longitudes of L lie in every term of the transfinite lower central series of the

link group; vii) the link group has the same Bousfield-Kan Z_{p}-completion as the free group with

the isomorphism induced by a meridian map [B]. (In fact conditions i and ii of 3.1 are equivalent to this condition vii).

Proof of 3.2. There are implications i⇒ii⇒iii⇒iv⇒v by (respectively) [Cochran, Theorem 2.9], [Levine, II, proof of Proposition 10], [Levine, II, definition on page 580]. By Proposition 6 of [Levine II], F̂ is a direct limit of finitely-presented groups P_{n} with H_{2}(P_{n}; Z) = 0 and H_{1}(P_{n}; Z) ≅ H_{1}(F̂; Z) ≅ Z^{m} so the map G ⟶ Ĝ ≅ F̂ factors through a map G ⟶ P satisfying the hypotheses of 3.1 iii. Hence a link satisfying v) has vanishing μ̃-invariants.

There is an implication vi⇒vii derived as follows. Let G_{δ} be the intersection of the (integral) transfinite lower central series of G. By Stallings’ exact sequence [Stallings, p. 170], H_{2}(G) ⟶ H_{2}(G/G_{δ}) is surjective. Since H_{2}(G) is generated by the tori on ∂(S^{3}\L), if ℓ_{i} ∈ G_{δ} these classes become spherical and hence vanish in H_{2}(G/G_{δ}) implying that the latter group is zero. Thus the composition of a meridional map F ⟶ G with the projection G ⟶ G/G_{δ} is 2-connected with any coefficients. It follows that this composition induces an isomorphism EF ⟶ E(G/G_{δ}) between the Bousfield Z_{p}-homology localizations [B, §1]. Since G ⟶ G/G_{δ} is also 2-connected, it induces an isomorphism EG ⟶ E(G/G_{δ}) so E(F) ≊ E(G) as claimed. A similar argument shows that i) and ii) of 3.1 imply vii) of 3.2. Conversely if F ⟶^{ϕ} G induces an isomorphism ϕ̂: EF ⟶ EG then the composition of G ⟶ EG with (ϕ̂)^{−1} satisfies 3.1 i) and ii) since H_{2}(EF; Z_{p}) ≅ H_{2}(F; Z_{p}) ≅ 0. □

It is not known whether or not the vanishing of Milnor’s \overline{μ}-invariants for L ⊆ S^{3} implies any of conditions i)–vii), since the only links which have been shown to have zero Milnor’s invariants are fusions of boundary links. Since the vanishing of Milnor’s invariants is equivalent to the longitudes lying in G_{ω} = ⋂_{n=1}^{∞} G_{n}, we see that the μ̃ may obstruct the a priori stronger condition iv) above. However it is known that the vanishing of Milnor’s invariants implies that G has the same Z_{p}-completion as the free group and this leads to the following disappointing result that the Z_{p}-valued μ̃-invariants are not very useful.

Proposition 3.3. If L is a link in S^{3} for which Milnor’s \overline{μ}-invariants vanish, then the Z_{p}-valued μ̃-invariants also vanish for L. (We are not claiming these hypotheses imply vi) or vii) of 3.2).

Proof of 3.3. If the Milnor’s \overline{μ}-invariants vanish for L then it is well known that the meridian map F ⟶^{ϕ} G induces an isomorphism between the nilpotent completions (the Z-completions in the sense of [BK; §12]). Recall that the R-completion of G, which we denote \overline{G}_{R}, is \underleftarrow{lim} t_{n}^{R}(G). Since \overline{F}_{Z} ≅ \overline{G}_{Z}, \overline{F}_{Z_{p}} ≅ \overline{G}_{Z_{p}} induced by ϕ. If 1 ⟶ G̃ ⟶^{i} G ⟶ Z_{p} ⟶ 1 induces a “one stage” covering link L̃ of L then ϕ̃ : F̃ ⟶ G̃ is a meridian map for L̃. In the exact sequence 1 ⟶ F̃ ⟶^{j} F ⟶ Z_{p} ⟶ 1, the generator t of Z_{p} acts on H_{1}(F̃; Z_{p}) ≅ Z^{1+mp} nilpotently. This is true since t_{∗}^{p} = id so (t_{∗} − id)^{p} = 0. Since H_{1}(F̃; Z_{p}) ≅ H_{1}(G̃; Z_{p}) the same holds for 1 ⟶ G̃ ⟶^{i} G ⟶ Z_{p} ⟶ 1. Therefore, by [BK; p. 63–64], these induce short exact sequences between the Z_{p}-completions 1 ⟶

\overline{F̃} ⟶^{j̄} \overline{F} ⟶ Z_{p} ⟶ 1, 1 ⟶ \overline{G̃} ⟶^{ī} \overline{G} ⟶ Z_{p} ⟶ 1. Consequently a diagram chase

shows that ϕ̃ induces an isomorphism \overline{F̃} ⟶ \overline{G̃}. Then by 2.2 and 2.4, the Z_{p}-valued μ̃- invariants of L (corresponding to L̃) vanish. This argument may then be iterated to show that all Z_{p}-valued μ̃-invariants vanish. □

Perhaps the Z_{(p)}-valued obstructions will detect links with vanishing Milnor invariants.

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