Introduction↩︎

The study of concordance of knots and links in the \(3\)-sphere has produced a wide range of invariants over the past decades, revealing subtle geometric and algebraic structures. The situation for \(2\)–links, i.e. smooth embeddings of copies of \(S^2\) in \(S^{4}\), is quite different. Although every \(2\)-knot is slice [1], it is still unknown whether there exist any non-slice \(2\)-links (see [2]).

For classical links, one of the most powerful families of concordance invariants is given by Milnor invariants. Introduced by Milnor in the 1950s [3], [4], these invariants generalize the linking number and capture higher-order linking phenomena through the nilpotent quotients of the fundamental group of the link complement. They have proved very effective in detecting nontrivial concordance classes.

A natural and longstanding question is whether Milnor invariants admit meaningful extensions to higher-dimensional links. Nontrivial generalizations do exist in the related context of link maps, where the embedding condition is relaxed [5], but these constructions do not directly address the embedded case, and rather concern the weaker relation of link-homotopy. In this paper, we propose a different approach. Rather than restricting attention to spherical links, we study general surface-links in 4–space, allowing components of arbitrary genus and possibly with boundary. Within this broader setting, we define Milnor-type invariants that extend the classical construction and provide new concordance invariants.

In order to define our generalization of Milnor invariants, we develop a theory of cut-diagrams. A cut-diagram consists of an oriented diagram on a surface, endowed with some labeling. This provides a combinatorial framework for studying surface-links, which can be seen as a \(2\)–dimensional version of Gauss diagrams. In particular, any diagram of a surface-link naturally gives rise to a cut-diagram, but the definition is flexible enough to allow a purely abstract treatment. To each cut-diagram, we associate a group generated by its ‘regions’  and defined by Wirtinger-type relations. In the case of a cut-diagram arising from a surface-link, this group coincides with the fundamental group of the complement. A key step in our construction is the derivation of a Chen–Milnor type presentation for the nilpotent quotients of this group (Theorem 31). This presentation, obtained by combinatorial methods, generalizes the classical result of Milnor for links [4] and provides an effective tool for computation, see Remark 60.

Building on this, we define Milnor-type invariants by considering the Magnus expansion of suitable elements associated with paths on the underlying surface. These elements generalize the notion of preferred longitudes in classical link theory. Although the resulting coefficients depend a priori on several choices, we show how to extract well-defined numerical invariants. More precisely, for each component of the surface-link and each sequence of indices in \(\{1,\cdots,n\}\), where \(n\) is the number of components of the underlying surface, we define two families of invariants: loop-invariants, associated with closed curves on the surface, and arc-invariants, associated with paths connecting boundary components. We prove the following (see Corollaries 53 and 72).

Since our invariants are extracted from essential loops and arcs on the surface, they all vanish for spherical \(2\)–links (Remark 54). However, as soon as at least one component is not simply connected, the invariants become nontrivial and, in many cases, quite effective. This is illustrated in Section 5, where a number of applications are provided:

• We give a general realization result, showing the existence of infinitely many surface-links consisting of a single torus and a union of spheres that are slice, but neither null-concordant nor link-homotopically trivial (Proposition 74). In contrast, it is known that all \(2\)-links are link-homotopically trivial [6].

• In Section 5.3, we consider a family \(\left(W_m\right)_{m\in \mathbf{N}}\) of links made of a sphere and a torus. We show that \(W_{m'}\) and \(W_{m}\) are concordant if and only if \(m=m'\) by using our Milnor invariants while, in contrast, none of the concordance invariants defined by Sato-Levine [7], Cochran [8] and Saito [9] can distinguish \(W_m\) and \(W_{m'}\) for \(m\neq m'\).

• In Section 5.4.2 we state a general link-homotopy classification for proper embeddings of disjoint punctured spheres in the \(4\)-ball, generalizing results of [10]–[12].

• In Section 5.4.1, we classify Spun links of \(3\)-components up to link-homotopy (Proposition 80), and we characterize link-homotopically trivial Spun links of any number of components (Proposition 81).

• Section 5.5 gives an obstruction for a surface-link to be concordant to a ribbon one, which in particular implies that the \(2\)-dimensional analogue of the braid closure map is not surjective (Corollary 87).

• In Section 5.6, we characterize classical links with vanishing Milnor invariants of length \(\leq 2k\) in terms of Milnor invariants of surfaces bounded by the link (Proposition 88), in connection with the \(k\)-slice conjecture proved by Igusa and Orr in [13].

We note that our construction recovers previously known extensions of Milnor invariants [10]–[12], see Section 3.4. The relation with Orr’s invariants of surface-links [14], [15] is discussed in Section 3.4.2.
Beyond the construction of invariants, another purpose of this paper is to develop of a general theory of cut-diagrams. We introduce in Sections 1.3 and 4 natural equivalence relations on cut-diagrams, corresponding to isotopy and link-homotopy of surface-links, generated by explicit local moves. This provides a combinatorial framework for studying surfaces in \(4\)–space. The authors believe that this is a promising direction for further research (see e.g. [16] for a concrete example).

The paper is organized as follows. In Section 1, we introduce cut-diagrams and discuss their connection to surface-links. Section 2 is devoted to the group and the peripheral system associated to a cut-diagram; in particular we give in Section 2.4 the main invariance results, and we state in Theorem 31 a Chen–Milnor type presentation for the nilpotent quotients of this group. Section 3 introduces Milnor invariants of cut-diagrams, hence in particular of surface-links, and discusses their properties and relation to existing constructions. In Section 4 we explain how to adapt the construction to a singular context, and derive link-homotopy invariants. Section 5 presents a range of topological applications illustrating the strength of our invariants. Finally, Section 6 contains the proofs of several technical results, including the key Theorem 31.

Definitions and Conventions↩︎

In this paper, everything is stated in the smooth category. All surfaces are compact, ordered and oriented.

A surface-link is a proper smooth embedding of a surface \(\Sigma\) in the \(4\)–ball \(B^4\). Two surface-links \(f\) and \(g\) of \(\Sigma\) are equivalent if Im\(f\) and Im\(g\) are ambient isotopic relative to boundary or, equivalently, if \(g\) is smoothly isotopic to \(f\circ \varphi\) for some diffeomorphism \(\varphi\) of \(\Sigma\). By diffeomorphism of \(\Sigma\), we always mean an orientation, component and boundary component preserving diffeomorphism from \(\Sigma\) to itself.

Two surface-links \(f\) and \(g\) of \(\Sigma\) are concordant if there exists an embedding \(W: \Sigma\times [0,1]\hookrightarrow B^4\times [0,1]\) such that, for each \(i\), the image of the \(i\)-th component of \(\Sigma\times [0,1]\) intersects \((B^3\times [0,1])\times \{0\}\), resp. \((B^3\times [0,1])\times \{1\}\), along the \(i\)-th component of Im\(f\), resp. Im\(g\), respecting the orientation. If \(\Sigma\) is not closed, we further require that \(W\left( \partial\Sigma\times [0,1]\right) = f(\partial \Sigma) \times [0,1]\).
A closed surface-link is slice if it bounds a disjoint union of \(3\)–dimensional handlebodies in the \(5\)–ball. We stress that a null-concordant (closed) surface-link is always slice, but the converse is not necessarily true (see Proposition 74); both notions however coincide in the spherical case of \(2\)–links.

A surface-link map is a continuous map from \(\Sigma\) to \(\mathbb{R}^4\) or the \(4\)–ball \(B^4\) with pairwise disjoint images; if \(\partial \Sigma\neq \emptyset\), then \(\partial \Sigma\) embeds in \(S^3=\partial B^4\). Link maps correspond to the case where \(\Sigma\) is a union of spheres. Note in particular that a surface-link is a surface-link map.

Two surface-link maps \(f\) and \(g\) are link-homotopic if \(f\) is homotopic through surface-link maps to some \(f'\) such that Im\(f'\)=Im\(g\) or, equivalently, if \(g\) is homotopic through surface-link maps to \(f\circ\varphi\), for some diffeomorphism \(\varphi\) of \(\Sigma\).¹ Recall that concordance implies link-homotopy for surface-links, see [6], [19].

We will make use of the following notation and conventions:

• given two elements \(a,b\) of some group, their commutator is defined as \([a,b]:=a^{-1}b^{-1}ab\), and the conjugate of \(a\) by \(b\) is given by \(a^b:=b^{-1}ab\);

• given two normal subgroups \(G_1\) and \(G_2\) of a group \(G\), we denote by \(G_1\cdot G_2\) the normal subgroup of \(G\) made of products \(g_1g_2\) with \(g_1\in G_1\) and \(g_2\in G_2\);

• homotopies of paths should always be understood as fixing the endpoints;

• homology groups are with coefficients in \(\mathbb{Z}\), and this will be omitted in our notation.

1 Cut-diagrams↩︎

In this section we introduce cut-diagrams, which are combinatorial objects that can be thought of as generalized surface-link diagrams. Before providing a general definition, we introduce cut-diagrams when they arise from diagrams of surface-links.

1.1 An intuitive approach↩︎

Consider a surface-link diagram, which is a generic projection of a surface-link on \({\mathbb{R}}^3\times \{0\}\). This is a map \(\Sigma\looparrowright {\mathbb{R}}^3\) for a surface \(\Sigma\), with lines of transverse double points which may meet at triple points and/or end at branch points. As in knot diagrams, the over/under information on the two preimages at each line of double points is encoded by cutting off a neighborhood of the lowest preimage (see the left-hand side of Figure 1, 2 and 3 for some examples).

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Each line of double points also comes with an orientation, such that the local frame formed by a positive normal vector to the overpassing region, a positive normal vector to the underpassing region, and a positive tangent vector to the line of double points, agrees with the ambient orientation of \({\mathbb{R}}^3\).

The lowest preimages of double points form a union \(P\) of immersed circles and/or intervals in \(\Sigma\), which splits \(\Sigma\) into regions. Note that the set \(P\) is known as the lower deck in the literature (see [20]). Each triple point of the surface diagram provides an over/under information at the corresponding crossing of \(P\), which is encoded as in usual knot diagrams by splitting \(P\) into arcs, that we call cut-arcs. Each cut-arc inherits an orientation, which is the orientation of the corresponding line of double points, and we label it by the region containing the preimage with highest coordinate of the corresponding line of double points. The data of the oriented and labeled cut-arcs in \(\Sigma\) forms what we will call a cut-diagram over \(\Sigma\) for the given surface-link. An example is given in Figure 1 in the case \(\Sigma=S^2\).

We stress that the labeling respects some natural conditions which are inherited from the topological nature of the surface diagram. We investigate these conditions below, by analyzing the local cut-diagrams arising from triple and branch points.

• An example of local cut-diagram associated with a triple point is illustrated in Figure 2.

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  A crossing among cut-arcs occurs in the leftmost sheet of \(\Sigma\). There, the overpassing arc is labeled by \(H\), whereas the left and right underpassing cut-arcs, with respect to the orientation of the overpassing arc, are labeled by \(E\) and \(F\), respectively; these regions \(E\) and \(F\) are contained in the middle sheet, and are separated by an \(H\)–labeled cut-arc, with \(E\) on the left and \(F\) on the right side of this arc.

• Figure 3 shows a local cut-diagram associated to a branch point.

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  This yields a local cut-diagram with a boundary point in the interior of the surface \(\Sigma\), and the incident cut-arc is labeled by the region containing this boundary point.

We stress that Figures 2 and 3 are only examples of local models, since different cases exist depending on the local plane orientations and over/under informations.

1.2 Abstract cut-diagrams↩︎

The above discussion motivates the following definitions, leading to the abstract notion of cut-diagram. Let \(\Sigma\) be some surface, possibly with boundary.

Inspired by the discussion of the previous section, we set the following:

These two conditions are locally represented in Figure 4.

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The left-hand side of Figure 5 provides an example where the first labeling condition is violated. There, the two regions \(A\) and \(B\) are \(B\)–adjacent but are not \(C\)–adjacent; therefore the crossing of cut-arcs at the center of the figure, where an \(A\)–labeled and a \(B\)–labeled cut-arcs are \(C\)–adjacent, does not respect the first rule.

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The right-hand side of Figure 5 shows a situation where the second labeling condition cannot be satisfied. Indeed, the left \(\bullet\) endpoint sits in region \(A\), which imposes that the horizontal cut-arc must be labeled by \(A\); but this violates the second labeling condition because of the right \(\bullet\) endpoint, which lies in region \(B\).

We can finally set the main definition of this section.

As discussed in the preceding section, there is a canonical way to associate a cut-diagram to a given surface-link diagram, and the labeling conditions are automatically satisfied in this setting. We say that a cut-diagram is topological if it arises from a surface-link in this way. A number of examples can be found in Sections 3.3 and 5. We stress that not all cut-diagrams are topological.

1.3 Local moves for cut-diagrams↩︎

Recall that two surface-link diagrams represent isotopic surface-links if and only if they differ by a sequence of the seven Roseman moves given in [21]. These Roseman moves are straightforwardly translated into moves on cut-diagrams, and it is not hard to check that they are generated by the seven local cut-moves C\(i\) (\(i=0,\cdots,6\)) given in Figure 6. An alternative, equivalent set of moves is given in [16].

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Note that two embeddings of a surface having the same image in \(4\)-space may have different topological cut-diagrams, since two such cut-diagrams might differ by some diffeomorphism of \(\Sigma\). This motivates taking the following equivalence relation on cut-diagrams.

With this definition, we have the following as a consequence of Roseman’s theorem [21].

1.4 The \(1\)–dimensional case↩︎

We conclude this section by outlining an analogous notion in the context of classical knot theory.

A knot diagram is a generic immersion \(S^1\looparrowright {\mathbb{R}}^2\) with a finite number of transverse double points, endowed with the usual over/under information on the two preimages. Applying the same routine as in the surface-link case, we can consider for each crossing the preimage with lowest coordinate: this forms a collection of points on \(S^1\), that we call cut-points, which split \(S^1\) into a number of arcs, that we call regions. To each cut-point is associated a sign—the sign of the corresponding crossing, and a region—the overpassing arc at the crossing. This construction generalizes to diagrams of links and tangles; see Figure 7 for examples.

More generally, one can define abstractly a notion of \(1\)–dimensional cut-diagram, as an oriented \(1\)–manifold together with a collection of cut-points endowed with a sign and labeled by some region. (Note that the natural adaptation of Definition 4 to this \(1\)-dimensional setting yields vacuous labeling conditions.)

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This notion is reminiscent of Gauss diagrams [22]. Indeed, cut-points can be thought of as heads of arrows in a Gauss diagram, where the label represents the region where the tail is attached. Note however that the relative position of tails attached to a same region is not specified in this language. So \(1\)–dimensional cut-diagrams should rather be thought of as Gauss diagrams modulo the local move which exchanges two adjacent tails. In fact, by translating Reidemeister moves into this langage of \(1\)–dimensional cut-diagrams, one recovers the theory of welded knotted objects. Hence cut-diagrams can be seen as a \(2\)–dimensional generalization of Gauss diagrams and/or welded theory. This will however not be pursued in this paper, whose main focus is surface-links. Consequently, in what follows, the term cut-diagram will be exclusively used for the \(2\)–dimensional setting of Section 1.2.

2 Group and peripheral systems of cut-diagrams↩︎

One important property of cut-diagrams is that they contain all the relevant information to define ‘fundamental’ groups and their distinguished elements, meridians and longitudes.

Fix a surface \(\Sigma\) with connected components \(\Sigma_1, \cdots, \Sigma_n\), and let \(\mathcal{C}\) be a cut-diagram over \(\Sigma\).

2.1 The group of a cut-diagram↩︎

We will see in Proposition 25 that equivalent cut-diagrams have isomorphic groups. Moreover, this notion is sensible from a topological point of view, as the following illustrates.

In the sequel, we will be mainly interested in the following quotients.

For any \(q\in\mathbb{N}\), the \(q\)-th nilpotent group of \(\mathcal{C}\) is the nilpotent quotient \(N_qG(\mathcal{C})\) of \(G(\mathcal{C})\), and we have \[N_qG(\mathcal{C})\cong \leavevmode \kern.1em \raise .25ex \the\scriptfont 0 \bar{F}\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 \bar{F}_q\!\cdot\! W }.\]

2.2 Generic path and associated group elements↩︎

A path \(\gamma:[0,1]\to \Sigma\) is generic if \(\gamma(0)\) and \(\gamma(1)\) are disjoint from \(\mathcal{C}\), and \(\gamma\big(]0,1[\big)\) meets \(\mathcal{C}\) transversally in a finite number of regular points of \(\mathcal{C}\), that is, points that are neither double points nor endpoints.

Given a generic path \(\gamma\) on \(\Sigma\), we associate unique words \(\widetilde{w}_\gamma\) and \(w_\gamma\) of the regions of \(\mathcal{C}\) as follows. For the \(k\)-th intersection point between \(\gamma\) and \(\mathcal{C}\) met when running along \(\gamma\) according to its orientation, denote by \(A_k\) the label of the cut-arc met at this point, and by \(\varepsilon_k=\pm 1\) the local sign of this intersection point. This sign is \(1\) if and only if the local orientation given by the orientation of \(\mathcal{C}\) and \(\gamma\) agrees with the ambient orientation of \(\Sigma\). Then the word \(\widetilde{w}_\gamma\) associated to \(\gamma\) is given by \[\widetilde{w}_{\gamma}:=A_1^{\varepsilon_1}\cdots A_{|\gamma\cap\mathcal{C}|}^{\varepsilon_{|\gamma\cap\mathcal{C}|}},\] whereas the normalized word \(w_\gamma\) is given by \[w_\gamma:=A^{-|\gamma|} \widetilde{w}_{\gamma},\] where \(A\) is the region where \(\gamma\) starts, and \(|\gamma|\) is the sum of the exponents \(\varepsilon_i\) in \(\widetilde{w}_{\gamma}\) such that \(A_i\) is in the same connected component as \(\gamma\); see Figure 9 for an example.

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Any generic path on \(\Sigma\) gives rise in this way to elements \(\widetilde{w}_{\gamma}\) and \(w_{\gamma}\) in the group \(G(\mathcal{C})\).

2.3 Longitudes↩︎

We will now define a notion of longitudes, associated to homotopy classes of paths on \(\Sigma\).

For each \(i\in\{1,\ldots,n\}\), pick a basepoint \(p_i\in\Sigma_i\) in the interior of some region \(R_i\). This provides in particular a preferred set of normal generators for \(G(\mathcal{C})\) (Remark 15). Then, thanks to Lemma 16, we can define the \(i\)-th loop-longitude map \[\lambda^l_i \colon \begin{array}{ccc} \pi_1(\Sigma_i,p_i) & \longrightarrow & G(\mathcal{C})\\[.2cm] [\gamma] & \longmapsto & w_{\gamma} \end{array},\] where \([\gamma]\) is an element of \(\pi_1(\Sigma_i,p_i)\) represented by a generic loop \(\gamma\) based at \(p_i\). Observe that \(\lambda^l_i\) is actually a group homomorphism. Indeed, for any two loops \(\gamma_1,\gamma_2\) based at \(p_i\), it follows from Remark 15 (2) that \(w_{\gamma_1.\gamma_2}=R_i^{-|\gamma_1|-|\gamma_2|}\widetilde{w}_{\gamma_1}\widetilde{w}_{\gamma_2}=R_i^{-|\gamma_1|} \widetilde{w}_{\gamma_1}R_i^{-|\gamma_2|} \widetilde{w}_{\gamma_2}=w_{\gamma_1}w_{\gamma_2}\).

In general, \(\mathcal{C}\) has an infinite number of loop-longitudes, and it will sometimes be useful to consider only a finite generating set of these loops.

Observe that any system of loop-longitudes determines the whole loop-longitude maps \(\lambda^l_i\), since these maps are homomorphisms.

If \(\partial\Sigma\neq\emptyset\), we shall also consider ‘boundary-to-boundary’ longitudes. For each \(i\in\{1,\ldots,n\}\) such that \(\partial\Sigma_i\neq\emptyset\), we fix (boundary) marked points \(b_{ij}\), \(j\in\{0,\ldots,\vert \partial\Sigma_i\vert-1\}\) on each boundary component of \(\Sigma_i\), and disjoint from \(\mathcal{C}\). This provides an ordering of the components of \(\partial\Sigma_i\), and the one containing \(b_{i0}\) is regarded as a ‘preferred’ boundary component of \(\Sigma_i\).

We stress that the well-definedness of arc-longitudes requires the choice of the marked points \(\{b_{ij}\}\), and that these are kept fixed at all times.

In the sequel, we shall simply use the terminology longitude when referring to either a preferred loop- or arc-longitude. We denote by \[\lambda_i:\, \pi_1(\Sigma_i,p_i)\sqcup {\mathcal{A}}\big(\Sigma_i,\{b_{ij}\}_j\big)\rightarrow G(\mathcal{C})\] the \(i\)-th longitude map defined by combining \(\lambda^l_i\) and \(\lambda^\partial_i\).

2.4 Peripheral system↩︎

The notions introduced so far can now be gathered into the following.

It follows in particular from Proposition 8 that this peripheral system is an invariant of surface-links. This is in agreement with Remark 22, as it corresponds to the topological peripheral system. In fact, when restricted to the surface-link case, we have a stronger invariance property, Theorem 27 below, which uses the following notion.

It is well-known that equivalence classes of nilpotent peripheral systems of links in \(S^3\) are concordance invariants [25]. This remains true for surface-links.

2.5 A Chen–Milnor Theorem for cut-diagrams↩︎

In this subsection we give an analogue, for cut-diagrams, of the so-called Chen–Milnor presentation for the nilpotent quotients of fundamental groups of link complements [4].

A central tool in establishing the Chen–Milnor presentation for links is an inductively defined sequence of maps \(\eta_q\). These are not only a central theoretical tool, but they also provide an effective algorithm to compute the image of preferred longitudes in nilpotent quotients, by expressing recursively each \(i\)-th meridian as a conjugate of a chosen meridian \(R_{i}\). The major difference with the link case is that, in the context of cut-diagrams, there is no canonical way to express these conjugations; as a matter of fact, we shall fix paths joining any region to the basepoint of its connected component.

Fix a basepoint \(p_i\) on each connected component \(\Sigma_i\), away from \(\mathcal{C}\). Regions of \(\mathcal{C}\) living in \(\Sigma_i\) will be denoted by \(R_{ij}\), with \(R_i:=R_{i0}\) the region containing \(p_i\). We also denote by \(F=\langle R_i \rangle\) and \(\bar{F}=\langle R_{ij} \rangle\) the free group generated by the \(R_i\)’s and the \(R_{ij}\)’s, respectively.

Following Section 2.2, a word \(v_{ij}:=\widetilde{w}_{\alpha_{ij}}\in \bar{F}\) can be associated to each road \(\alpha_{ij}\). Notice that the relation \(R_{ij}=R_i^{v_{ij}}\) holds in \(G(\mathcal{C})\) (see Remark 15).

As the notation suggests, this sequence of homomorphisms a priori depends on the choice of road network \(\alpha\). However, for simplicity we will often drop the \(\alpha\) from the notation.

Now for each component \(\Sigma_i\) of \(\Sigma\), pick a system of loop-longitudes \(\mathcal{L}_i(\Sigma):=\{w_{ij}\}_j\) associated with a collection of loops \(\{\gamma_{ij}\}_j\) based at \(p_{i}\) (see Definition 18). The proof of the following key result is postponed to Section 6.3.

3 Milnor invariants for cut-diagrams and surface-links↩︎

The purpose of this section is to define Milnor invariants for any cut-diagram over a surface \(\Sigma\). In the classical setting of links in the \(3\)–sphere, Milnor invariants are extracted from the coefficients in the Magnus expansion of the preferred longitudes, seen in the nilpotent quotients of the fundamental group of the link exterior. Now, as discussed in Section 2, the notion of longitudes for cut-diagrams comes in two flavors: loop-longitudes and arc-longitudes. As a matter of fact, although coming from a same global construction that associates a collection of ‘Milnor numbers’ to any generic path on \(\Sigma\) (Section 3.1), Milnor invariants defined from loop and arc-longitudes have rather different behaviors, and we shall discuss them separately (Sections 3.2.1 and 3.2.2). Although we are going to define here invariants for the cut-diagram \(\mathcal{C}\), we emphasize that they are determined by its nilpotent peripheral systems, as defined in Section 2.4.

Let us recall some notation: \(\Sigma=\sqcup_{i=1}^n\Sigma_i\) is a surface, and \(\mathcal{C}\) is a cut-diagram over \(\Sigma\). For all \(i\), we pick a basepoint \(p_i\) in the interior of \(\Sigma_i\), which specifies an \(i\)-th meridian \(R_i\); we denote by \(F:=\langle R_i\rangle\) the free group generated by these meridians. We also fix, for each \(i\in\{1,\ldots,n\}\) such that \(\partial\Sigma_i\neq\emptyset\), a markedpoint \(b_{ik}\) on each boundary component of \(\Sigma_i\) (\(k\in\{0,\ldots,\vert \partial\Sigma_i\vert-1\}\)). Finally, we fix a system of loop-longitudes \(\mathcal{L}_i(\Sigma)\) for each \(i\) (Definition 18).

3.1 Milnor numbers associated with a longitude↩︎

Fix an integer \(q\ge 1\). By Theorem 31, the meridians \(R_k\) provide a generating set for \(N_q G(\mathcal{C})\). Given a generic path \(\gamma\) in \(\pi_1(\Sigma_i,p_i)\sqcup {\mathcal{A}}\big(\Sigma_i,\{b_{ij}\}_j\big)\) for some \(i\in\{1,\ldots,n\}\), we can thus consider a representative word \(\omega_\gamma\) in \(\{R_k^{\pm 1}\}\) for the associated \(i\)-th longitude in \(N_qG(\mathcal{C})\).³ In what follows we shall often blur the distinction between the path \(\gamma\) and the associated \(i\)-th longitude, seen either in \(G(\mathcal{C})\) or in \(N_qG(\mathcal{C})\).

In particular, we obtain in this way a collection of integers \(\mu_\mathcal{C}(I;\omega_\gamma)\), one for each sequence \(I\) of integers in \(\{1,\ldots,n\}\), from the representative word \(\omega_\gamma\). Of course, the coefficients \(\mu_\mathcal{C}(I;\omega_\gamma)\) are not determined by the \(i\)-th longitude \(\gamma\), as they depend on the choice of the representative word \(\omega_\gamma\). But Theorem 31 tells us precisely how two different representatives may differ. Indeed we have a presentation \[\label{eq:Presentation} N_q G(\mathcal{C}) \cong \Big\langle R_i \;\big|\; F_q \,; \, \big[R_i,\omega_{ij}\big] \textrm{ for all i,j} \Big\rangle,\tag{1}\] where, for each \(i,j\), the \(\omega_{ij}\)’s are given word representatives in \(\{R_k^{\pm 1}\}\) for the elements of the system of \(i\)-th loop-longitudes \(\mathcal{L}_i(\Sigma)\). (We do not need here the explicit formula for \(\omega_{ij}\), given in Theorem 31 in terms of the Chen homomorphisms.)

In order to extract invariants from the numbers \(\mu_\mathcal{C}(I;\omega_\gamma)\), we introduce the following indeterminacies.

We stress that the indeterminacies of Definition 34 only involve loop-longitudes on \(\Sigma\), and more precisely the chosen systems of loop-longitudes. We also stress that, while the definition of \(m_\mathcal{C}(Ii)\) only uses the system of \(i\)-th loop-longitudes \(\mathcal{L}_i(\Sigma)\), the indeterminacy \(\Delta_\mathcal{C}(Ii)\) involves all systems \(\mathcal{L}_j(\Sigma)\) such that index \(j\) appears in the sequence \(Ii\), since the definition involves cyclic permutation.

This allows for the following.

We stress that a choice of one meridian per component has been made for defining \(\overline{\mu}_\mathcal{C}(I;\gamma)\); this will be addressed in the next subsection 3.2. The definition of \(\Delta_\mathcal{C}(Ii)\) also uses this choice, as well as the choice of a system of loop-longitudes for each component: in the rest of this subsection, we show that \(\Delta_\mathcal{C}(Ii)\) is actually independent of these choices. We will need the next technical lemma.

As a first application of Lemma 40 we have the following, which shows the independence of the indeterminacy \(\Delta_\mathcal{C}(Ii)\) on the choice of systems of loop-longitudes; we also state this result for \(\gcd\big\{\Delta_\mathcal{C}(Ii),m_\mathcal{C}(Ii)\big\}\) in preparation for the next subsection (see Definition 45).

3.2 Milnor invariants for cut-diagrams↩︎

At this point, each generic path, and in particular each longitude of a cut-diagram, yields Milnor numbers. We now need to analyse how these numbers provide actual invariants of the cut-diagram. On one hand, the choice of meridians \(\{ R_k\}\) must be discussed. On the other hand, comparing longitudes for two different cut-diagrams is in general not possible, as there is no natural identification between the elements of the fundamental group of the underlying surfaces.

3.2.1 Milnor invariants for loop-longitudes↩︎

We first focus on the case where \(\gamma\) is an element of \(\pi_1(\Sigma_i;p_i)\) representing an \(i\)-th loop-longitude. By Lemma 40, the residue class \(\overline{\mu}_\mathcal{C}(I;\gamma)\) remains unchanged when replacing \(\gamma\) by a conjugate. Actually, Lemma 40 implies that, for fixed meridians \(\{p_i\}\), Milnor numbers for loop-longitudes define group homomorphisms from \(\pi_1(\Sigma_i,p_i)\) to \(\leavevmode \kern.1em \raise .25ex \the\scriptfont 0 {\mathbb{Z}}\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 \Delta(Ii){\mathbb{Z}} }\). Since the latter is abelian, these homomorphisms factor through the abelianization \(H_1(\Sigma_i)\). We have the following.

Proposition 42 leads to the following.

Proposition 42 and Lemma 40 readily give:

Note that Milnor maps \(M_\mathcal{C}^{Ii}\) are not invariants of the equivalence class of the cut-diagram \(\mathcal{C}\), as they are not in general invariant under pre-composition by an automorphism of \(H_1(\Sigma_i)\), see Remark 48 below.  However, we can define numerical invariants of the cut-diagram \(\mathcal{C}\) from loop-longitudes, which essentially record the images of Milnor map.

We next prove that Milnor loop-invariants are determined by the equivalence class of the nilpotent peripheral system:

3.2.2 Milnor invariants from arc-longitudes↩︎

We now turn to the case where \(\gamma\) is an \(i\)-th arc-longitude, meaning in particular that \(\Sigma_i\) is not closed. We actually assume here that each component of \(\Sigma\) has nonempty boundary. Note that this is the case for \(2\)-string links [11] and concordances in \(4\)–space [12], which we aim at generalizing here, see Section 3.4.

Recall that for \(i\in \{1,\cdots,n\}\), we have marked points \(\{b_{ij}\}\) on each boundary component of \(\Sigma_i\) (\(j\in \{0,\ldots,\vert\partial \Sigma_i\vert-1\}\)), among which the point \(b_{i0}\) indicates the preferred boundary component. This provides a canonical choice of \(i\)-th meridian, materialized by the basepoint \(p_i=b_{i0}\), and hence well-defined Milnor numbers associated to each arc-longitude. In other words, as a corollary of Proposition 36, Milnor numbers associated to arc-longitudes are well-defined invariants of cut-diagrams over \(\Sigma\).

In general, there are however several, possibly infinitely many different arc-longitudes running from \(b_{i0}\) to some other marked point \(b_{ij}\). But using the action of loop-longitudes noted in Definition 21, all arc-longitudes can be recovered from a single one. More precisely, for any two arcs \(\gamma, \gamma'\) running from \(b_{i0}\) to \(b_{ij}\), there exists some \(\tau\in \pi_1(\Sigma_i,b_{i0})\) such that \(\gamma'= \tau\cdot\gamma\) (see Definition 21). By Lemma 40, we then have for any sequence \(I\) of integers in \(\{1,\ldots,n\}\), \[\mu_\mathcal{C}(I;\gamma') \equiv \mu_\mathcal{C}(I;\gamma) + \mu_\mathcal{C}(I;\tau) \mod\Delta_\mathcal{C}(Ii).\] Since \(\tau\) decomposes as a product of elements in a system of loop-longitudes (and their inverses), Lemma 40 further implies that \[\mu_\mathcal{C}(I;\gamma') \equiv \mu_\mathcal{C}(I;\gamma) \mod \nu_\mathcal{C}(Ii),\] where \(\nu_\mathcal{C}(Ii)=\gcd\big\{\Delta_\mathcal{C}(Ii),m_\mathcal{C}(Ii)\big\}\) was introduced in Definition 45.

We can thus define intrinsic Milnor invariants from arc-longitudes, as follows.

Since the notion of equivalence of peripheral systems fixes the boundary (Definition 23), the proof of Theorem 47 also provides the following.

3.3 Milnor invariants of surface-links↩︎

We call Milnor invariants of a cut-diagram \(\mathcal{C}\) the collection of invariants defined in Sections 3.2.1 and 3.2.2, namely Milnor loop-invariants \(\nu_\mathcal{C}(Ii)\) and Milnor arc-invariants \(\nu^\partial_\mathcal{C}(I;ij)\), for all indices \(i,j\) and all sequences of indices \(I\).

By Theorems 47 and 51, Milnor invariants only depend on the nilpotent peripheral systems. We thus have the following, as a consequence of Theorem 27.

Let us give a couple of basic examples; more can be found in Section 5. We address Milnor loop-invariants in Examples 55 and 57, then Milnor arc-invariants in Example 59 .

3.4 Relation to previous works↩︎

We now discuss the relationship between our generalized Minor invariants with several related works.

3.4.1 Milnor invariants of \(2\)-string links and concordances↩︎

Milnor arc-invariants generalize the invariants defined in [10]–[12]. These earlier works indeed correspond to the case where \(\Sigma\) is a union of annuli properly embedded in \(B^4\). Note that in this setting, loop-longitudes are given by parallel copies of the boundary components, so that both \(\Delta_\mathcal{C}(I)\) and \(\nu_{\mathcal{C}}(I)\) involve (classical) Milnor invariants of the link at the boundary. See [12] for a complete description (note that in [12], the indeterminacy \(\nu_{\mathcal{C}}(Ii)\) of Definition 49 is denoted by \(\overline{\Delta}(Ii)\)).

3.4.2 Orr invariants↩︎

Orr introduced in [14], [15] (based) concordance invariants \(\theta_q^l\) for (based) codimension \(2\) embeddings (\(q\in \mathbb{N}\)). The construction in the surface-link case can be roughly summarized as follows.

Let \(L\) be an embedding of an \(n\)–component surface \(\Sigma\) into \(S^{4}\), and let \(G(L)\) be the fundamental group of its complement. Suppose that the nilpotent quotients satisfy \(\leavevmode \kern.1em \raise .25ex \the\scriptfont 0 G(L)\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 G(L)_q }\cong \leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F_q }\), where \(F\) is the free group of rank \(n\).

The natural homomorphisms \(F\longrightarrow \leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F_q }\) and \(G(L)\longrightarrow \leavevmode \kern.1em \raise .25ex \the\scriptfont 0 G(L)\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 G(L)_q }\cong \leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F_q }\) induce continuous maps \[f_q^n:K(F,1)\longrightarrow K\left(\leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F_q },1\right)\text{~~and~~} \phi_q^n:S^{4}\setminus L\longrightarrow K\left(\leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F_q },1\right)\] respectively, where \(K(G,1)\) denotes an Eilenberg-MacLane space for a group \(G\). The map \(\phi_q^n\) extends to a map \(\overline{\phi_q^n}\) from \(S^{4}\) to \(\mathrm{Cone}(f_q^n)\), the mapping cone of the map \(f_q^n\). The fundamental class \(\theta_q^n\in\pi_{4}\left(\mathrm{Cone}(f_q)\right)\) of \(\overline{\phi_q^l}\) is an invariant of \(L\) for each \(q\ge 1\), and Orr shows that these \(\theta_q^l\) actually are (based) concordance invariants.

The definition of Orr’s invariants thus requires that \(\leavevmode \kern.1em \raise .25ex \the\scriptfont 0 G(L)\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 G(L)_q }\cong \leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F_q }\). Observe that this condition is fulfilled if \(H_1(\Sigma)=0\), which is the (stronger) assumption taken in [14] for the definition. We note that \(\leavevmode \kern.1em \raise .25ex \the\scriptfont 0 G(L)\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 G(L)_q }\cong \leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F_q }\) if and only if all Milnor loop-invariants of \(L\) with length \(<q\) vanish. Hence Orr invariants actually are invariants for surface-links with vanishing Milnor loop-invariants.

Moreover, as discussed in Remark 73, we expect that Orr’s construction can be adapted to create new link-homotopy invariants of surface-links.

4 Non-repeated Milnor invariants and link-homotopy↩︎

Cut-diagrams were, for a large part, motivated by the study of surface-links and their Milnor concordance invariants. But Milnor’s work was first introduced in the context of link-homotopy [3], and it is natural to extend the theory of cut-diagrams to this setting.

4.1 Self-singular cut-diagrams↩︎

Recall that a surface-link map is generically an immersion, with only a finite number of singularities which are transverse double points. In diagrams, a singularity corresponds to the local model given on the left-hand side of Figure 12.

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The procedure explained in Section 1.1, that associates a cut-diagram to a surface-link diagram, extends naturally to surface-link maps. As Figure 12 illustrates, each singularity gives rise to a pair of internal endpoints, represented by a white dot \(\circ\). Note also from the figure that a terminal cut-arc incident to a \(\circ\) endpoint, is labeled by the region containing the paired \(\circ\) endpoint, which belongs to the same connected component of the surface. This leads to the following general definition.

In figures, we encode the local orientation at a \(\circ\) endpoint by a sign, which is positive if and only if the incident cut-arc is locally oriented outwards.⁷ We stress that the pairing of \(\circ\) endpoints will not be used in this paper, but is crucial in [16], where self-singular cut-diagrams are used to define link-homotopy invariants of link maps.

Now, Roseman moves have been extended to the case of surface-link maps in [11], where three self-singular Roseman moves were introduced. It is an easy exercise to translate the three self-singular Roseman moves of [11] into local moves on self-singular cut-diagrams (see [16]), and to show that they are generated by the four local moves SC\(i\) (\(i=1,\cdots,4\)) of Figure 13.

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As a consequence of [11], we have:

4.2 Reduced peripheral system↩︎

Let \(\mathcal{C}\) be a self-singular cut-diagram over \(\Sigma\). Since internal endpoints play no role in its definition, the group \(G(\mathcal{C})\) can be defined for \(\mathcal{C}\) in the same way as in Definition 9. However, in this case, longitudes are no longer well-defined elements in \(G(\mathcal{C})\), and we consider the following quotient, first introduced by Milnor [3].

Now, for each component \(\Sigma_i\) of \(\Sigma\), let \(R_i\) be a choice of meridian, specified by a basepoint \(p_i\). For each \(i\in\{1,\ldots,n\}\), we have by Lemma 65 a well-defined map \[\lambda_{i,R}:\, \pi_1(\Sigma_i,p_i)\sqcup {\mathcal{A}}\big(\Sigma_i,\{b_{ij}\}_j\big)\rightarrow \leavevmode \kern.1em \raise .25ex \the\scriptfont 0 \textrm RG(\mathcal{C})\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 N_{(i)} }\] defined by composing the \(i\)-th longitude map \(\lambda_i\) of Section 2.4 with the projection map from \(G(\mathcal{C})\) to \(\leavevmode \kern.1em \raise .25ex \the\scriptfont 0 \textrm RG(\mathcal{C})\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 N_{(i)} }\). Elements in the image of \(\lambda_{i,R}\) are called reduced \(i\)-th longitudes. These are cosets of the form \(\lambda\cdot N_{(i)}\), where \(\lambda\) is a word representing an \(i\)-th longitude.

Definition 23 naturally induces a notion of equivalence for reduced peripheral systems.

As noted in Remark 15, \(G(\mathcal{C})\) is normally generated by the meridians \(\{R_i\}\). It follows, by a result of Habegger and Lin [28], that its reduced quotient is nilpotent of order at most \(n\), so that \(\textrm RG(\mathcal{C})\cong N_q\textrm RG(\mathcal{C})\) for any integer \(q\ge n\).

As in Section 2.5, pick for each component \(\Sigma_i\) of \(\Sigma\), a system of loop-longitudes \(\mathcal{L}_i(\Sigma)=\{w_{ij}\}_j\) in \(G(\mathcal{C})\), and recall that \(F:=\langle R_i\rangle\). We have the following Chen–Milnor type presentation for \(\textrm RG(\mathcal{C})\), similar to Theorem 31.

The proof is an adaptation of that of Theorem 31, and is given in Section 6.4.

We next prove the following ‘reduced version’  of Proposition 25.

4.3 Non-repeated Milnor invariants and link-homotopy invariance↩︎

We have the following ‘reduced version’  of Proposition 36.

This allows for the following, which in particular defines non-repeated Milnor invariants of surface-links and surface-link maps.

By arguments similar to those proving Theorems 47 and 51, we deduce that two self-singular cut-diagrams with equivalent reduced peripheral systems, have same non-repeated Milnor invariants. More precisely:

Combining Theorem 71 with Proposition 68, we obtain that non-repeated Milnor invariants of self-singular cut-diagrams are invariant under the moves SC\(i\), \(i=1,\cdots,4\). In particular, we have the following.

5 Examples and applications↩︎

In this section, we gather several properties and topological applications of Milnor invariants of surface-links.

5.1 Realization result, and sliceness vs. null-concordance↩︎

We first give a general realization result for Milnor loop-invariants, which also highlights the subtle difference between sliceness and null-concordance for surface-links.

Pick two positive integers \(m\) and \(n\) with \(n \geq 2\). Let \(O_{n-1}\) be a copy of the \((n-1)\)-component trivial \(2\)-link in the \(4\)-sphere \(S^4\). The fundamental group of the complement of \(O_{n-1}\) is a free group generated by \(x_1,\ldots ,x_{n-1}\), where each \(x_i\) is represented by a meridian of the \(i\)-th component. Consider a loop \(\gamma_n^m\) in the complement of \(O_{n-1}\) representing the element \[[x_1,[x_2,[x_3,\cdots,[x_{n-2},x_{n-1}]\cdots]]]^m\] in the free group. Here, we may assume that \(\gamma_n^m\) lies in a \(3\)-sphere in \(S^4\setminus O_{n-1}\). Let \(T_n^m\) be the boundary of a tubular neighborhood of \(\gamma_n^m\) in this \(3\)-sphere. We consider the surface-link \(B_n^m\) defined by \[B_n^m:=O_{n-1}\cup T_n^m,\] consisting of a single torus and \(n-1\) spheres. Note that the surface-links \(H\) (Figure 8) and \(B\) (Figure 10) coincide with \(B_2^1\) and \(B_3^1\), respectively. The following actually generalizes the first computations given in Examples 55 and 57.

5.2 Milnor invariants of Spun links↩︎

Spun links refer to a classical construction due to Artin [29], which produces surface-links from classical tangles as follows. Consider in \({\mathbb{R}}^4\) the upper \(3\)–dimensional space \({\mathbb{R}}^3_+=\{(x,y,z,0) \, \vert\, x,y\in{\mathbb{R}},\, z\ge 0\}\); the \((x,y)\)–plane \(P_{xy}=\{(x,y,0,0)\, \vert\, x,y\in{\mathbb{R}}\}\) sits as the boundary of \({\mathbb{R}}^3_+\). Given a compact \(1\)–manifold \(T\) properly embedded in \({\mathbb{R}}^3_+\), the Spun of \(T\) is the surface-link \(\mathop{\mathrm{Spun}}(T)\) obtained by spinning \(T\) around \(P_{xy}\) inside \({\mathbb{R}}^4\supset{\mathbb{R}}^3_+\): \[\label{eq:spun} \mathop{\mathrm{Spun}}(T) = \left\{\big(x,y,z\cos \theta,z\sin \theta \big)\, \vert \, (x,y,z,0)\in T,\, \theta\in [0,2\pi] \right\}.\tag{2}\] Observe that a closed component of \(T\) yields a toroidal component of \(\mathop{\mathrm{Spun}}(T)\), while a knotted arc in \(T\) produces a spherical component.

Using a similar spinning process in \({\mathbb{R}}^3\) around the line \(\{(x,0,0)\, \vert\, x\in{\mathbb{R}}\}\), a diagram \(D\) of \(T\) provides a natural surface-link diagram of \(\mathop{\mathrm{Spun}}(T)\), where each crossing yields a circle of double points without triple or branch point. And in turn, spinning the \(1\)–dimensional cut-diagram \({\mathcal{C}}_D\) associated to \(D\) (in the sense of Section 1.4) provides a \(2\)–dimensional cut-diagram \({\mathcal{C}}_S\) for \(\mathop{\mathrm{Spun}}(T)\), where each (\(1\)–dimensional) region of \({\mathcal{C}}_D\) yields a (\(2\)–dimensional) region of \({\mathcal{C}}_S\), and each cut-point of \({\mathcal{C}}_D\) yields a closed cut-arc of \({\mathcal{C}}_S\) with same label and oriented according to the sign of the cut-point. In particular, the toroidal components of \({\mathcal{C}}_S\) are nothing but the circle components of \({\mathcal{C}}_D\) times \(S^1\), while the spherical components of \({\mathcal{C}}_S\) are given by the interval components of \({\mathcal{C}}_D\) times \(S^1\) (see Figure 14 for an example.)

Our Milnor invariants are well-behaved under the Spun construction, in the sense that they relate naturally to the (classical) Milnor invariants of \(T\):

5.3 A classification result up to concordance↩︎

We now compare the relative strength of our Milnor invariants with other concordance invariants of surface-links in the literature.

For every \(m\in{\mathbb{N}}\), we define \(W_m\) as the spun surface-link obtained by spinning the tangle \(X_m\) described in Figure 14.

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Many of the known concordance invariants of surface-links cannot detect this family of Spun links. Indeed, for all \(m\in {\mathbb{N}}\), one can check that

• the Sato–Levine invariant [7] vanishes on \(W_m\),

• Cochran’s derivation invariants [8] all vanish on \(W_m\),

• Saito’s invariants [9] of \(W_m\) are equal for all values of \(m\in{\mathbb{N}}\),

• since \(W_m\) is link-homotopic to a union of trivially embedded torus and sphere for all \(m\in{\mathbb{N}}\), all link-homotopy invariants vanish on \(W_m\).

In contrast, using Milnor loop-invariants we show the following.

5.4 Link-homotopy classification results↩︎

The next two applications provide link-homotopy classification results for Spun links and for knotted punctured spheres.

5.4.1 Spun links up to link-homotopy↩︎

Milnor showed in [3] that links of \(3\) components are classified up to link-homotopy by non-repeated Milnor invariants of length \(\le 3\). Using the good behavior of our Milnor invariants under the Spun construction, we show the following.

Of course the link-homotopy classification for \(2\)–component Spun links is also implied by Proposition 80, for example by adding a third trivial and unlinked component.

For Spun links with an arbitrary number of components, we also have the following consequence of Lemma 76.

5.4.2 Link-homotopy classification of knotted punctured spheres↩︎

We next mention another link-homotopy classification result derived from our construction, whose proof is given in a separate paper [31].

Let \(n\ge 2\) be some integer, and \(p_1,\ldots,p_n\) some positive integers. A knotted punctured spheres is any surface-link which is a proper embedding of \(\sqcup_{i=1}^n S^2_{p_i}\) into the \(4\)–ball \(B^4\), where \(S^2_{p_i}\) is the \(2\)–sphere with \(p_i\) holes, and such that the boundary is mapped to a fixed trivial link with \(\sum_i p_i\) components in \(S^3=\partial B^4\). Notice that taking \(p_i=1\) for all \(i\) yields the notion of linked disks introduced by Le Dimet in [32], while taking \(p_i=2\) for all \(i\) produces \(2\)–string links, as studied in [11]. We have the following classification result, which generalizes [11].

Note that knotted punctured spheres do not contain any nontrivial loop longitude, since the boundary is assumed to be a trivial link. Hence the classifying invariants of the statement are actually Milnor arc-invariants.

5.5 An obstruction for ribbonness↩︎

Recall that a ribbon immersed \(3\)–manifold in \(4\)–space is an immersion such that the only singularities are \(2\)–disks with two preimages, one embedded in the interior and the other properly embedded. A surface-link is ribbon if its image is the boundary of some ribbon immersed \(3\)–dimensional handlebodies in \(4\)–space (note that the latter is in general not unique). Ribbon surface-links are the natural analogues of the notion of ribbon knots introduced by Fox in the classical dimension. Spun links are examples of ribbon surface-links.

We now show how Milnor maps can be used to obstruct ribbonness. Let \(S\) be the image of an embedding of some surface \(\Sigma=\sqcup_{i=1}^n \Sigma_i\) in \(4\)–space, and let \(Ii\) be some sequence of indices.

Since the Milnor map \(M^{Ii}\) up to automorphisms of \(H_1(\Sigma_i)\) is an invariant for nilpotent peripheral systems up to equivalence (Remark 48), it is a concordance invariant. In particular, the rank of the free kernel is a concordance invariant of surface-links

Now, suppose that the surface-link \(S\) is ribbon. If the \(i\)-th component of \(S\) has genus \(g\), then any ribbon immersed handlebodies \(H\) bounded by this component induces a rank \(g\) submodule \(\mathop{\mathrm{Ker}}\big(H_1(\Sigma_i)\hookrightarrow H_1(H)\big)\), which is clearly contained in the free kernel of any Milnor map on \(H_1(\Sigma_i)\). This implies the following obstruction result for ribbonness.

Let us give a concrete application. For all \(m\in{\mathbb{Z}}\), let \(A_{\!m}\) be the surface-link shown on the left-hand side of Figure 15. It is obtained by spinning a \(3\)-component link as illustrated and, while spinning, moving the third unknot component \(m\) times around the second one.

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The surface-link \(A_{\!1}\) appears in [11], where it was proved that it is not link-homotopic to any ribbon surface. Using free kernels, we can prove the following.

Recall from [11] that \(2\)-string links are surface-links in \(B^4\) of properly embedded annuli, with boundary a fixed unlink in \(S^3\) (see Section 5.4.2). There is a natural \(2\)-dimensional analogue of the braid closure map, that closes a \(2\)-string link into knotted tori in \(4\)–space, see [10]. We have the following noteworthy consequence of Proposition 86.

5.6 \(k\)-slice links↩︎

The notion of \(k\)-slice classical links was introduced independently by Orr [15] and Cochran [33], as follows. An \(n\)-component link \(L=K_1\cup\cdots\cup K_n\) in the boundary \(\partial D^4\) of the \(4\)–ball \(D^4\) is \(k\)-slice if there is a disjoint union \(S=S_1\cup\cdots\cup S_l\) of properly embedded surfaces in \(D^4\) with \(\partial S_i=K_i~(i=1,\ldots,n)\), such that the composition of the homomorphism \(\pi_1(S_i)\longrightarrow \pi_1(D^4\setminus S)\) induced by the inclusion from \(S_i\) to \(D^4\setminus S\) obtained by pushing \(S\) in the normal direction, with the natural projection \(\pi_1(D^4\setminus S)\longrightarrow N_k\pi_1(D^4\setminus S)\), is the trivial homomorphism for each \(k\).

Igusa and Orr [13] showed that a link is \(k\)-slice if and only if all its Milnor invariants of length \(\leq 2k\) vanish, a result which is known as the \(k\)-slice conjecture.

Observe that for each \(i\) and all \(k\), the composition \[\pi_1(S_i)\longrightarrow \pi_1(D^4\setminus S)\longrightarrow N_k\pi_1(D^4\setminus S)\] is trivial if and only if Milnor loop-invariant \(\nu_S(Ii)\) vanish for any sequence \(I\) of length \(\leq\) \(k-1\). Hence we have the following result, which relates Milnor invariants in dimensions 3 and 4.

6 Chen homomorphisms and Chen-Milnor presentations for cut-diagrams↩︎

We give in this section all technical results involving the Chen homomorphisms introduced in Section 2.5. We prove in particular Theorems 31 and Theorem 67.

Let \(\mathcal{C}\) be a cut-diagram over \(\Sigma=\sqcup_{i=1}^n \Sigma_i\), with a choice of basepoint \(p_i\) on each connected component \(\Sigma_i\). Recall that we denote by \(R_{ij}\) all regions in \(\Sigma_i\), with \(R_i:=R_{i0}\) being the region containing \(p_i\). Recall also the free groups \(F=\langle R_i \rangle\) and \(\bar{F}=\langle R_{ij} \rangle\). Finally, fix a road network \(\alpha\) for \(\mathcal{C}\) based at the point \(p_i\). Recall that \(\alpha_{ij}\in \alpha\) denotes the road running from \(p_i\) to region \(R_{ij}\), and that \(v_{ij}=\widetilde{w}_{\alpha_{ij}}\) denotes the associated word in \(\bar{F}\).

6.1 Some preliminary results↩︎

We first derive some elementary properties of the Chen homomorphisms \(\eta_q\). We will make use of the following elementary fact.

Recall from Notation 12 that \(W\) is the normal subgroup of \(\bar{F}\) generated by all Wirtinger relations in \(G(\mathcal{C})\).

6.2 Nilpotent quotients↩︎

We first give here a presentation for the nilpotent quotients of \(G(\mathcal{C})\) with as many generators as components in \(\Sigma\), and only nilpotent and meridian/loop-longitude commutation relations. The Chen–Milnor presentation of Theorem 31, proved in the next subsection, will significantly reduce the number of relations.

To do so, we define walls of \(\mathcal{C}\), which are connected components of regular points of \(\mathcal{C}\). They actually correspond to regular ‘walls’ between adjacent regions. Note that each wall is included in a cut-arc of \(\mathcal{C}\), hence inherits a labeling by some region.

Consider some wall \(a\) of \(\mathcal{C}\), say on the \(i\)-th component and labeled by some region \(R\). Recall from Section 2 that the Wirtinger relation \(R_{ik}=R_{ij}^R\) holds in \(G(\mathcal{C})\), where \((R_{ij},R_{ik})\) is the pair of \(R\)–adjacent regions at \(a\). We define the word \(w_a\in \bar{F}\) by \[w_a:=v_{ij}Rv^{-1}_{ik}, \label{eq:w95a}\tag{6}\] which is the (not necessarily normalized) word associated to the \(p_i\)–based loop defined by connecting the road \(\alpha_{ij}\) to \(\alpha_{ik}^{-1}\) by a segment crossing \(a\) transversally.

We have the following seemingly weaker form of Lemma 93, with an infinite number of commutator relations. This version shall however be useful.

6.3 Chen–Milnor presentation for nilpotent quotients: proof of Theorem 31↩︎

Recall that we pick, for each component \(\Sigma_i\) of \(\Sigma\), a system of loop-longitudes \(\mathcal{L}_i(\Sigma):=\{w_{ij}\}_j\) associated with a collection of \(p_i\)-based loops \(\{\gamma_{ij}\}_j\) comprising a generating set for \(\pi_1(\Sigma_i;p_i)\).

The proof of Theorem 31 builds on the presentation given in Proposition 94. The idea is to show that the infinite family of commutation relations in Proposition 94, involving all generic loops on \(\Sigma_i\), can be realized by just the commutation relations involving the finite collection of loops in \(\{\gamma_{ij}\}_i\). In fact, we shall prove more precisely that \(F_q\cdot L_{(q)}=F_q\cdot V_{(q)}\), where

• \(L_{(q)}\) is the normal closure of \(\left\{[R_i,\eta_q(w_\gamma)]\;| \;\textrm{\gamma any p_i--based generic loop on \Sigma_i}\right\}\) in \(F\),

• \(V_{(q)}\) is the normal closure of \(\left\{[R_i,\eta_{q}(w_{ij})]\;| \;w_{ij}\in \mathcal{L}_i(\Sigma) \right\}\) in \(F\).

The inclusion \(F_q\!\cdot\!V_{(q)}\subset F_q\!\cdot\!L_{(q)}\) is clear. To prove the converse inclusion, it suffices to show that the map \(\leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F_q\!\cdot\!V_{(q)} } \longrightarrow \leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F_q\!\cdot\!L_{(q)} }\), induced by the identity, is injective. This is a consequence of Proposition 96 below. Indeed, since any loop on \(\Sigma_i\) is homotopic to a product of elements in \(\{\gamma^{\pm 1}_{ij}\}_j\), this proposition in particular implies that \([R_i,\eta_{q}(w_\gamma)]\equiv 1\) mod \(F_{q}\!\cdot\!V_{(q)}\) for any \(p_i\)–based loop \(\gamma\) on \(\Sigma_i\), and Theorem 31 follows.

We stress that the paths \(\gamma,\gamma'\) in the statement may have arbitrary endpoints, and are in particular not required to start at the region \(R_i\).

6.4 Chen–Milnor presentation for reduced quotients: proof of Theorem 67↩︎

The presentation for R\(G(\mathcal{C})\) given in Theorem 67 is obtained in a completely similar way as Theorem 31 in the previous subsection. Hence we just sketch the argument below, stressing only the new ingredients.

We first note that the Chen homomorphisms \(\eta_q\) induce well-defined homomorphisms \(\textrm{R}\bar{F} \rightarrow \textrm{R}F\). The arguments proving Lemma 93 and Proposition 94 then apply verbatim to show that, for any \(q\ge 1\), the \(q\)-th nilpotent quotient of \(\textrm RG(\mathcal{C})\) has the following presentation: \[\label{eq:SingPresentation} N_q \textrm RG(\mathcal{C}) \cong \left\langle R_1,\ldots,R_n\;\;\left| \begin{array}{l} F_q\\[.0cm] [R_i,R_i^g]\textrm{ for all i and all g\in F}\\[.0cm] [R_i,\eta_q(w_\gamma)] \textrm{ for all i and all p_i--based generic loop \gamma on \Sigma_i} \end{array} \right.\right\rangle.\tag{8}\]

From this point, we can use an induction on \(q\) to rework the presentation so that the third type of relations are substituted by commutation relations of the form \(\big[R_i,\eta_q(w_{ij})\big]\), as in the proof of Theorem 31. The only new ingredient here is that move \(\textrm H_2\) from Lemma 14 may now involve a \(\circ\) endpoint, that is, a terminal cut-arc which is not labeled by the region supporting the move, but by any region in the same connected component. Suppose hence that two arcs \(\gamma\) and \(\gamma'\) on \(\Sigma_i\) differ by such a move \(\textrm H_2\), with \(\gamma\) corresponding to the left-hand side of the move (in the figure of Lemma 14). By the labeling condition 2’) of Definition 61, the cut-arc involved in this move is labeled by some region \(A\) of \(\Sigma_i\). We then have \[w_{\gamma}= B^{s}\widetilde{w}_{\gamma_1}\widetilde{w}_{\gamma_2} \quad\textrm{and}\quad w_{\gamma'}=B^{s-\varepsilon}\widetilde{w}_{\gamma_1}A^{\varepsilon}\widetilde{w}_{\gamma_2},\] where \(\gamma_1,\gamma_2\) are paths such that \(\gamma=\gamma_1.\gamma_2\), \(s\in {\mathbb{Z}}\), \(\varepsilon={\pm 1}\), and \(B\) is the region of \(\Sigma_i\) where \(\gamma_1\) starts. As in the proof of Proposition 96, denote by \(v_R\), for any region \(R\), the word associated to the road of \(\alpha\) running to the region \(R\). We have, by the definition of the Chen homomorphisms: \[\label{ouiche} \eta_{q+1}(w_{\gamma'}) = \left(R_{i}^{-\varepsilon}\right)^{\eta_{q}(v_B)} \left( R_i^{\varepsilon}\right)^{\eta_{q}(v_A) \eta_{q+1}(\widetilde{w}_{\gamma_1}^{-1}B^{-s})}\eta_{q+1}(\widetilde{w}_{\gamma}).\tag{9}\] Hence \(\eta_{q+1}(w_{\gamma'})\) and \(\eta_{q+1}(w_{\gamma})\) differ by a product of conjugates of \(R_i\). This implies that \([R_i,\eta_{q+1}(w_{\gamma'})]\) and \([R_i,\eta_{q+1}(w_{\gamma})]\) are equivalent modulo the normal closure of the relations \([R_i,R_i^g]\) for all \(i\) and all \(g\in F\).

Now, as already noted in Section 4.2, Habegger–Lin’s result [28] implies that \(\textrm R G(\mathcal{C})\) is a nilpotent group of order at most \(n\). Picking any \(q\geq n\), it follows that (8 ) provides a presentation for \(\textrm RG(\mathcal{C})\cong N_q \textrm RG(\mathcal{C})\), and also that the relations involving \(q\)-th iterated commutators are trivially satisfied. We thus obtain the desired presentation and the proof is complete.

6.5 Chen homomorphisms and basepoint change↩︎

We now consider the situation where we pick a new set of basepoints \(\{p'_i\}\) for \(\Sigma\). This induces a new set of meridians \(\{R'_i\}\), which provides a new presentation for \(N_qG(\mathcal{C})\) by Theorem 31. The purpose of this subsection is to understand explicitly how a given word in the alphabet \(\big\{R_i^{\pm 1}\big\}\), representing some element of \(N_qG(\mathcal{C})\), translates into a word in this new alphabet \(\big\{{R'_i}^{\pm 1}\big\}\). This is done in Corollary 99, where the Chen homomorphisms provide the desired dictionary between the two presentations of \(N_qG(\mathcal{C})\). We will use the following.

Now, let \(\alpha\) be a road network for \(\mathcal{C}\) based at \(\{p_i\}\), and let \(\alpha'\) be another choice of road network, based at \(\{p'_i\}\). Consider in \(\alpha'\) the road \(a_i\) running from \(p'_i\) to the basepoint \(p_i\) of \(\alpha\), and set \(v_i:=\widetilde{w}_{a_i}\), the associated (unnormalized) word. The maps \(\eta^\alpha_q\) take values in the free group \(F=\langle R_i \rangle\), while the maps \(\eta^{\alpha'}_q\) take values in the free group \(F'=\langle R'_i \rangle\), and both can be seen as taking values in the free group \(\bar{F}=\langle R_{ij} \rangle\) generated by all regions.

Recall that \(W\) is the normal subgroup of \(\bar{F}\) generated by all Wirtinger relations, see Notation 12. Then for any \(w\in \bar{F}\), one can see \(\eta^\alpha_q(w)\) as an element of \(\leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F_q\!\cdot\!\eta^\alpha_q(W) }\), while \(\eta^{\alpha'}_q(w)\) is seen as an element of \(\leavevmode \kern.1em \raise .25ex \the\scriptfont 0 F'\kern-.1em \big/ {\kern-.15em \lower .5ex \the\scriptfont 0 F'_q\!\cdot\!\eta^{\alpha'}_q(W) }\).

This lemma indeed provides the desired dictionary between our two presentations, as follows.

References↩︎

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1. Note that this definition coincides for link maps with the definition usually used in the litterature [5], [17], [18]. ↩︎

2. Note that we made here a counterclockwise choice of orientation, but other cases are completely similar.↩︎

3. This word is not to be confused with the normalized word \(w_\gamma\), defined in Section 2.2, which is a word in the larger alphabet \(\{R_{kj}^{\pm 1}\}\).↩︎

4. Points (ii) and (iii) of Claim 39 are not used in the present proof, but will be needed later.↩︎

5. See Corollary 99 for an explicit formula.↩︎

6. Here we use the usual convention that \(\gcd(0,0)=0\).↩︎

7. When the self-singular cut-diagram is obtained from a diagram of a surface-link map, this is precisely the sign of the corresponding singularity.↩︎