Pulling Apart 2–Spheres in 4–Manifolds

Rob Schneiderman and Peter Teichner

Received: November 5, 2012 Revised: May 1, 2014

Communicated by Ursula Hamenstädt

Abstract. An obstruction theory for representing homotopy classes of surfaces in 4–manifolds by immersions with pairwise disjoint images is developed, using the theory of non-repeating Whitney towers. The accompanying higher-order intersection invariants provide a geometric generalization of Milnor’s link-homotopy invariants, and can give the complete obstruction to pulling apart 2–spheres in certain families of 4–manifolds. It is also shown that in an arbitrary simply connected 4–manifold any number of parallel copies of an immersed 2–sphere with vanishing self-intersection number can be pulled apart, and that this is not always possible in the non-simply connected setting. The order 1 intersection invariant is shown to be the complete obstruction to pulling apart 2–spheres in any 4–manifold after taking connected sums with finitely many copies of S^{2} × S^{2}; and the order 2 intersection indeterminacies for quadruples of immersed 2–spheres in a simply-connected 4–manifold are shown to lead to interesting number theoretic questions.

2010 Mathematics Subject Classification: Primary 57M99; Secondary 57M25. Keywords and Phrases: 2–sphere, 4–manifold, disjoint immersion, homotopy invariant, non-repeating Whitney tower.

1 Introduction

We study the question of whether a map A : Σ → X is homotopic to a map A′ such that A′(Σ_{i}) are pairwise disjoint subsets of X, where Σ = ⨿_{i}Σ_{i} is the decomposition into connected components. In this case, we will say that A can be pulled apart.

This question arises as a precursor to the embedding problem – whether or not A is homotopic to an embedding. It also arises in the study of configuration spaces X^{(m)} of m distinct ordered points in X, where elements of π_{n}X^{(m)} are represented by m disjoint maps of n–spheres to X, and one might ask whether or not a given element of the m-fold product ∏^{m} π_{n}X lies in the image of the map π_{n}X^{(m)} → ∏^{m} π_{n}X induced by the canonical projections p_{1}, …, p_{m} : X^{(m)} → X. For example, let Σ = ⨿_{i}S^{n} be a disjoint union of n–spheres and X be a connected 2n–manifold. For n ≥ 2, there are Wall’s well known intersection numbers λ(A_{i}, A_{j}) ∈ ℤ[π_{1}X], where A_{i} : S^{n} → X are the components of A [33]. These are obstructions for representing A by an embedding, and the main geometric reason for the success of surgery theory is that, for n ≥ 3, they are (almost) complete obstructions: The only missing ingredient is Wall’s selfintersection invariant μ, a quadratic refinement of λ. However, for the question of making the A_{i}(S^{n}) disjoint, it is necessary and sufficient that λ(A_{i}, A_{j}) = 0 for i ≠ j. We abbreviate this condition on intersection numbers by writing λ_{0}(A) = 0. As expected, the condition λ_{0}(A) = 0 is not sufficient for pulling apart A if n = 2, but this failure is surprisingly subtle: Given only two maps A_{1}, A_{2} : S^{2} → X^{4} with λ(A_{1}, A_{2}) = 0, one can pull them apart by a clever sequence of finger moves and Whitney moves, see [19] and Section 1.1 below. However, this is not true any more for three (or more) 2–spheres in a 4–manifold. In [30] we defined an additional invariant λ_{1}(A) which takes values in a quotient of ℤ[π_{1}X × π_{1}X] and was shown to be the complete obstruction to pulling apart a triple A = A_{1}, A_{2}, A_{3} : S^{2} → X of 2–spheres mapped into an arbitrary 4–manifold X with vanishing λ_{0}(A). For trivial π_{1}X the analogous obstruction was defined earlier in [25, 34]. In this paper, we extend this work to an arbitrary number of 2–spheres (and other surfaces – see Remark 16) in 4–manifolds. The idea is to apply a variation of the theory of Whitney towers as developed in [3, 4, 5, 6, 29, 30, 31] to address the problem. Before we introduce the relevant material on Whitney towers, we mention a couple of new results that can be stated without prerequisites. Throughout this paper the letter m will usually denote the number of surface components to be pulled apart, and from now on the letters Σ and X will be used to denote surfaces and 4–manifolds, respectively. The distinction between a map of a surface and its image in X will frequently be disregarded in the interest of brevity.

Pulling apart parallel 2–spheres

The following theorem is discussed and proven in Section 6:

Theorem 1 If X is a simply connected 4–manifold and A : ⨿^{m}S^{2} → X consists of m copies of the same map A_{0} : S^{2} → X of a 2–sphere with trivial normal Euler number, then A can be pulled apart if and only if [A_{0}] ∈ H_{2}(X; ℤ) has vanishing homological self-intersection number [A_{0}] · [A_{0}] = 0 ∈ ℤ.

Note that each transverse self-intersection of A_{0} gives rise to m^{2} − m intersections among the m parallel copies A, not counting self-intersections, see Figure 1. As a consequence, there cannot be a simple argument to pull A apart. In fact, the analogous statement fails for non-simply connected 4–manifolds, see Example 6.2.

Figure 1: One self-intersection leads to m^{2} − m intersections among m copies.

Stably pulling apart 2–spheres

We say that surfaces A : Σ → X can be stably pulled apart if A can be pulled apart after taking the connected sum of X with finitely many copies of S^{2} ×S^{2}. The invariants λ_{0} and λ_{1} are unchanged by this stabilization, and in this setting they give the complete obstruction to pulling apart m maps of 2–spheres:

Theorem 2 A : ⨿^{m}S^{2} → X can be stably pulled apart if and only if λ_{0}(A) = 0 = λ_{1}(A).

This result also holds when the stabilizing factors S^{2} × S^{2} are replaced by any simply-connected closed 4-manifolds (other than S^{4}). It also holds when components of A are maps of disks. The invariant λ_{1} is described precisely in sections 2 and 8; and the proof of Theorem 2 is given in section 7.2. (Note that X is not required to be simply connected in Theorem 2.) We remark that the stronger invariant τ_{1}(A) of [30], together with Wall’s self-intersection invariant τ_{0}(A), is the complete obstruction to stably embedding A, see [29, Cor.1].

Remark 3 The question of pulling apart surfaces in 4–manifolds is independent of category. More precisely, any connected 4–manifold can be given a smooth structure away from one point [12] and any continuous map can be approximated arbitrarily closely by a smooth map. As a consequence, we can work in the smooth category and as a first step, we can always turn a map A : Σ^{2} → X^{4} into a generic immersion. We will also assume that surfaces are properly immersed, i.e. A(∂Σ) ⊂ ∂X with the interior of Σ mapping to the interior of X, and that homotopies fix the boundary.

1.1 Pulling apart two 2–spheres in a 4–manifold

To motivate the introduction of Whitney towers into the problem, it is important to understand the basic case of pulling apart two maps of 2–spheres A_{1}, A_{2} : S^{2} → X. Wall’s intersection pairing associates a sign and an element of π_{1}X to each transverse intersection point between the surfaces, and the vanishing of λ(A_{1}, A_{2}) implies that all of these intersections can be paired by Whitney disks. As illustrated in Figures 2 and 3, these Whitney disks can be used to pull apart A_{1} and A_{2} by first pushing any intersection points between A_{2} and the interior of a Whitney disk W_{(1,2)} down into A_{2}, and then using the Whitney disks to guide Whitney moves on A_{1} to eliminate all intersections between A_{1} and A_{2} (details in [19]).

Figure 2: Pushing an intersection between A_{2} and the interior of a Whitney disk W_{(1,2)} down into A_{2} only creates (two) self-intersections in A_{2}.

Figure 3: A Whitney move guided by the Whitney disk of Figure 2. The intersection between A_{1} and the interior of the Whitney disk becomes a pair of self-intersections of A_{1} after the Whitney move.

1.2 Pulling apart three or more 2–spheres

Note that for a triple of spheres one cannot use the method of figures 2 and 3 to eliminate an intersection point between one sphere and a Whitney disk that pairs intersections between the other two spheres. Such “higher-order” intersections were used in [30] to define the invariant λ_{1}(A) discussed above. In this case, the procedure for separating the surfaces involves constructing “second order” Whitney disks which pair the intersections between surfaces and Whitney disks. The existence of these second order Whitney disks allows for an analogous pushing-down procedure which only creates self-intersections and cleans up the Whitney disks enough to pull apart the surfaces by an ambient homotopy. Building on these ideas, we will describe an obstruction theory in terms of non-repeating Whitney towers W built on properly immersed surfaces in X, and non-repeating intersection invariants λ_{n}(W) taking values in quotients of the group ring of (n + 1) products of π_{1}X. The order n of the non-repeating Whitney tower W determines how many of the underlying surfaces at the bottom of the tower can be made pairwise disjoint by a homotopy, and the vanishing of λ_{n}(W) is sufficient to find an order n + 1 non-repeating Whitney tower. Non-repeating Whitney towers are special cases of the Whitney towers defined in [31] (see also [3, 5, 6, 7, 27, 28, 30]). An introduction to these notions is sketched here with details given in Section 2. We work in the smooth oriented category, with orientations usually suppressed.

1.3 Whitney towers and non-repeating Whitney towers

Consider A : Σ = ⨿_{i}Σ_{i}^{2} → X^{4} where the surface components Σ_{i} are spheres or disks (and see Remark 3 for initial clean-ups on A). To begin our obstruction theory, we say that A forms a Whitney tower of order 0, and define the order of each properly immersed connected surface A_{i} : Σ_{i} → X to be zero. If all the singularities (transverse intersections) of A can be paired by Whitney disks then we get a Whitney tower of order 1 which is the union of these order 1 Whitney disks and the order 0 Whitney tower. If we only have Whitney disks pairing the intersections between distinct order 0 surfaces A_{i} : Σ_{i} → X of A, then we get an order 1 non-repeating Whitney tower. If it exists, an order 2 Whitney tower also includes Whitney disks (of order 2) pairing all the intersections between the order 1 Whitney disks and the order 0 surfaces. An order 2 non-repeating Whitney tower only requires second order Whitney disks for intersections between an A_{i} and Whitney disks pairing A_{j} and A_{k}, where i, j and k are distinct. As explained in Section 2, all of this generalizes to higher order, including the distinction between non-repeating and repeating intersection points, however things get more subtle as different “types” of intersections of the same order can appear (parametrized by isomorphism classes of unitrivalent trees). An order n Whitney tower has Whitney disks pairing up all intersections of order less than n, and an order n non-repeating Whitney tower is only required to have Whitney disks pairing all non-repeating intersections of order less than n (sections 2.1 and 2.4). So “order n non-repeating” is a weaker condition than “order n”. The underlying order 0 surfaces A in a Whitney tower W are said to support W, and we say that A admits an order n Whitney tower if A is homotopic (rel boundary) to A′ supporting W of order n.

1.4 Pulling apart surfaces in 4–manifolds

As a first step towards determining whether or not A : Σ → X can be pulled apart, we have the following translation of the problem into the language of Whitney towers. This is the main tool in our theory:

Theorem 4 Let m be the number of components of Σ. Then A : Σ → X can be pulled apart if and only if A admits a non-repeating Whitney tower of order m − 1.

The existence of a non-repeating Whitney tower of sufficient order encodes “pushing down” homotopies and Whitney moves which lead to disjointness, as will be seen in the proof of Theorem 4 given in Section 3. It will be clear from the proof of Theorem 4 that for 1 < n < m the existence of a nonrepeating Whitney tower of order n implies that any n + 1 of the order 0 surfaces A_{i} : Σ_{i} → X can be pulled apart.

1.5 Higher-order intersection invariants

An immediate advantage of this point of view is that the higher-order intersection theory of [31] can be applied inductively to increase the order of a Whitney tower or, in some cases, detect obstructions to doing so. The main idea is that to each unpaired intersection point p in a Whitney tower W on A one can associate a decorated unitrivalent tree t_{p} which bifurcates down from p through the Whitney disks to the order 0 surfaces A_{i} (Figure 4, also Figure 12). The order of p is the number of trivalent vertices in t_{p}. The univalent vertices of t_{p} are labeled by the elements i ∈ {1, …, m} from the set indexing the A_{i}. The edges of t_{p} are decorated with elements of the fundamental group π := π_{1}X of the ambient 4–manifold X. Orientations of A and X determine vertex-orientations and a sign sign(p) ∈ {±} for t_{p}, and the order n intersection invariant τ_{n}(W) of an order n Whitney tower W is defined as the sum

τ_{n}(W) := ∑ sign(p) · t_{p} ∈ T_{n}(π, m)

over all order n intersection points p in W. Here T_{n}(π, m) is a free abelian group generated by order n decorated trees modulo relations which include the usual antisymmetry (AS) and Jacobi (IHX) relations of finite type theory (Figure 5). Restricting to non-repeating intersection points in an order n non-repeating Whitney tower W, yields the analogous order n non-repeating intersection invariant λ_{n}(W):

λ_{n}(W) := ∑ sign(p) · t_{p} ∈ Λ_{n}(π, m) which takes values in the subgroup Λ_{n}(π, m) < T_{n}(π, m) generated by order n trees whose univalent vertices have distinct labels. We refer to Definition 18 for more precise statements. In the following we shall sometimes suppress the number m of components of Σ and just write Λ_{n}(π).

Remark 5 We will show in Lemma 19 that Λ_{n}(π, m) is isomorphic to the direct sum of \binom{m}{n+2}n!-many copies of the integral group ring ℤ[π^{(n+1)}] of the (n + 1)-fold cartesian product π^{(n+1)} = π × π × ⋯ × π. Note that Λ_{n}(π, m) is trivial for n ≥ m − 1 since an order n unitrivalent tree has n + 2 univalent vertices. For π left-orderable, T_{1}(π, m) is computed in [29, Sec.2.3.1]. For π trivial, T_{n}(m) := T_{n}(1, m) is computed in [7] for all n, and in [5] the torsion subgroup of T_{n}(m) (which is only 2-torsion) is shown to correspond to obstructions to “untwisting” Whitney disks in twisted Whitney towers in the 4–ball. The absence of torsion in Λ_{n}(π, m) corresponds to the fact that such obstructions are not relevant in the non-repeating setting since a boundary-twisting operation [12, Sec.1.3] can be used to eliminate non-trivially twisted Whitney disks at the cost of only creating repeating intersections.

In the case n = 0, our notation λ_{0}(A) ∈ Λ_{0}(π) just describes Wall’s Hermitian intersection pairing λ(A_{i}, A_{j}) ∈ ℤ[π] (see section 2.6). For n = 1, we showed in [30] that if λ_{0}(A) = 0 then taking λ_{1}(A) := λ_{1}(W) in an appropriate quotient of Λ_{1}(π) defines a homotopy invariant of A (independent of the choice of non-repeating Whitney tower W). See sections 2.7 and 8.2. The main open problem in this intersection theory is to determine for n ≥ 2 the largest quotient of Λ_{n}(π) for which λ_{n}(W) only depends on the homotopy class of A : Σ → X. Even for n = 1, this quotient will generally depend on A, unlike Wall’s invariants λ_{0}.

1.6 The geometric obstruction theory

In Theorem 2 of [31] it was shown that the vanishing of τ_{n}(W) ∈ T_{n}(π) implies that A admits an order n + 1 Whitney tower. The proof of this result uses controlled geometric realizations of the relations in T_{n}(π), and the exact same constructions (which are all homogeneous in the univalent labels – see Section 4 of [31]) give the analogous result in the non-repeating setting:

Theorem 6 If A : Σ → X admits a non-repeating Whitney tower W of order n with λ_{n}(W) = 0 ∈ Λ_{n}(π), then A admits an order (n+1) non-repeating Whitney tower. □

Combining Theorem 6 with Theorem 4 above yields the following result, which was announced in [31, Thm.3]:

Corollary 7 If Σ has m components and A : Σ → X admits a non-repeating Whitney tower W of order m − 2 such that λ_{m−2}(W) = 0 ∈ Λ_{m−2}(π), then A can be pulled apart. □

Thus, the problem of deciding whether or not any given A can be pulled apart can be attacked inductively by determining the extent to which λ_{n}(W) only depends on the homotopy class of A. The next two subsections describe settings where λ_{n}(W) ∈ Λ_{n}(π) does indeed tell the whole story. For Whitney towers in simply connected 4–manifolds, we drop π from the notation, writing Λ_{n}(m), or just Λ_{n} if the number of order 0 surfaces is understood.

1.7 Pulling apart disks in the 4–ball

A link-homotopy of an m-component link L = L_{1} ∪ L_{2} ∪ ⋯ ∪ L_{m} in the 3– sphere is a homotopy of L which preserves disjointness of the link components, i.e. during the homotopy only self-intersections of the L_{i} are allowed. In order to study “linking modulo knotting”, Milnor [23] introduced the equivalence relation of link-homotopy and defined his (non-repeating) μ-invariants, showing in particular that a link is link-homotopically trivial if and only if it has all vanishing μ-invariants. In the setting of link-homotopy, Milnor’s algebraically defined μ-invariants are intimately connected to non-repeating intersection invariants as implied by the following result, proved in Section 4 using a new notion of Whitney tower-grope duality (Proposition 25).

Theorem 8 Let L be an m-component link in S^{3} bounding D : ⨿^{m}D^{2} → B^{4}. If D admits an order n non-repeating Whitney tower W then λ_{n}(W) ∈ Λ_{n}(m) does not depend on the choice of W. In fact, λ_{n}(W) contains the same information as all non-repeating Milnor invariants of length n + 2 and it is therefore a link-homotopy invariant of L.

We refer to Theorem 24 for a precise statement on how Milnor’s invariants are related to λ_{n}(L) := λ_{n}(W) ∈ Λ_{n}(m). Together with Corollary 7 we get the following result:

Corollary 9 An m-component link L is link-homotopically trivial if and only if λ_{n}(L) vanishes for all n = 0, 1, 2, …, m − 1. □

This recovers Milnor’s characterization of links which are link-homotopically trivial [23], and uses the fact that L bounding disjointly immersed disks into B^{4} is equivalent to L being link-homotopically trivial [13, 14].

Remark 10 A precise description of the relationship between general (repeating) Whitney towers on D and Milnor’s μ-invariants (with repeating indices [24]) for L is given in [6]. Our current discussion is both easier and harder at the same time: We only make a statement about non-repeating Milnor invariants, a subset of all Milnor invariants, but as an input we only use a nonrepeating Whitney tower, an object containing less information then a Whitney tower.

1.8 Pulling apart 2–spheres in special 4–manifolds

The relationship between Whitney towers and Milnor’s link invariants can be used to describe some more general settings where the non-repeating intersection invariant λ_{n}(W) ∈ Λ_{n} of a non-repeating Whitney tower gives homotopy invariants of the underlying order 0 surfaces. Denote by X_{L} the 4–manifold which is gotten by attaching 0-framed 2–handles to the 4–ball along a link L in the 3–sphere. The following theorem is proved in Section 5:

Theorem 11 If a link L bounds an order n Whitney tower on disks in the 4–ball, then:

(i) Any map A : ⨿^{m}S^{2} → X_{L} of 2–spheres into X_{L} admits an order n Whitney tower.

(ii) For any order n non-repeating Whitney tower W supported by A : ⨿^{m}S^{2} → X_{L}, the non-repeating intersection invariant λ_{n}(A) := λ_{n}(W) ∈ Λ_{n}(m) is independent of the choice of W.

Note that the number m of 2–spheres need not be equal to the number of components of the link L. Using the realization techniques for Whitney towers in the 4–ball described in [5, Sec.3], examples of such A realizing any value in Λ_{n}(m) can be constructed.

Corollary 12 If a link L bounds an order n Whitney tower on disks in the 4–ball, then:

(i) A : ⨿^{m}S^{2} → X_{L} admits an order n + 1 non-repeating Whitney tower if and only if λ_{n}(A) = 0 ∈ Λ_{n}(m).

(ii) In the case m = n + 2, we have that A : ⨿^{m}S^{2} → X_{L} can be pulled apart if and only if λ_{m−2}(A) = 0 ∈ Λ_{m−2}(m). □

The “if” parts of the statements in Corollary 12 follow from Theorem 6 and Corollary 7 above. The “only if” statements follow from the fact that the second statement of Theorem 11 implies that λ_{n}(A) := λ_{n}(W) ∈ Λ_{n}(m) only depends on the homotopy class of A, as explained in Proposition 14 below. In this setting, Kojima [20] had identified (via Massey products) the first non-vanishing Milnor invariant μ_{L}(123 ⋯ m) of an m-component link L as an obstruction to pulling apart the collection of m 2–spheres determined by L in X_{L}.

1.9 Indeterminacies from lower-order intersections.

The sufficiency results of Theorem 6 and Corollary 7 show that the groups Λ_{n}(π) provide upper bounds on the invariants needed for a complete obstruction theoretic answer to the question of whether or not A : Σ → X can be pulled apart. And as illustrated by Theorem 8 and Theorem 11 above, there are settings in which λ_{n}(W) ∈ Λ_{n} only depends on the homotopy class (rel boundary) of A, sometimes giving the complete obstruction to pulling A apart.

In general however, more relations are needed in the target group to account for indeterminacies in the choices of possible Whitney towers on a given A. In particular, for Whitney towers in a 4–manifold X with non-trivial second homotopy group π_{2}X, there can be indeterminacies which correspond to tubing the interiors of Whitney disks into immersed 2–spheres. Such INT intersection relations are, in principle, inductively manageable in the sense that they are determined by strictly lower-order intersection invariants on generators of π_{2}X. For instance, the INT_{1} relations in the target groups of the order 1 invariants τ_{1} and λ_{1} of [25, 30] are determined by the order 0 intersection form on π_{2}X. However, as we describe in Section 8, higher-order INT relations can be nonlinear, and if one wants the resulting target to carry exactly the obstruction to the existence of a higher-order tower then interesting subtleties already arise in the order 2 setting. It is interesting to note that these INT indeterminacies are generalizations of the Milnor-invariant indeterminacies in that they may involve intersections between 2–spheres other than the A_{i}. The Milnor link-homotopy invariant indeterminacies come from sub-links because there are no other essential 2spheres in X_{L}. For instance, the proof of Theorem 2 exploits the hyperbolic summands of the stabilized intersection form on π_{2}. We pause here to note another positive consequence of the intersection indeterminacies before returning to further discussion of the well-definedness of the invariants.

1.9.1 Casson’s separation lemma

The next theorem shows that in the presence of algebraic duals for the order 0 surfaces A_{i}, all our higher-order obstructions vanish. This recovers the following result of Casson (proved algebraically in the simply-connected setting [2]) and Quinn (proved using transverse spheres [10, 26]):

Theorem 13 If λ(A_{i}, A_{j}) = 0 for all i ≠ j, and there exist 2–spheres B_{i} : S^{2} → X such that λ(A_{i}, B_{j}) = δ_{ij} for all i, then A_{i} can be pulled apart.

Here λ denotes Wall’s intersection pairing with values in ℤ[π], and δ_{ij} ∈ {0, 1} is the Kronecker delta. Note that there are no restrictions on intersections among the dual spheres B_{i}. Theorem 13 is proved in section 7.1.

1.9.2 Homotopy invariance of higher-order intersection invariants

Our proposed program for pulling apart 2–spheres in 4–manifolds involves refining Theorem 6 by formulating (and computing) the relations INT_{n}(A) ⊂ Λ_{n}(π) so that λ_{n}(A) := λ_{n}(W) ∈ Λ_{n}(π)/INT_{n}(A) is a homotopy invariant of A (independent of the choice of order n non-repeating Whitney tower W) which represents the complete obstruction to the existence of an order n + 1 non-repeating tower supported by A. Via Theorem 4 this would provide a procedure to determine whether or not A can be pulled apart. The following observation clarifies what needs to be shown:

Proposition 14 If for a fixed immersion A the value of λ_{n}(W) ∈ Λ_{n}(π)/INT_{n}(A) does not depend the choice of order n non-repeating Whitney tower W supported by A, then λ_{n}(A) := λ_{n}(W) ∈ Λ_{n}(π)/INT_{n}(A) only depends on the homotopy class of A. □

To see why this is true, observe that, up to isotopy, any generic regular homotopy from A to A′ can be realized as a sequence of finitely many finger moves followed by finitely many Whitney moves. Since any Whitney move has a finger move as an “inverse”, there exists A′′ which differs from each of A and A′ by only finger moves (up to isotopy). But a finger move is supported near an arc, which can be assumed to be disjoint from the Whitney disks in a Whitney tower, and the pair of intersections created by a finger move admit a local Whitney disk; so any Whitney tower on A or A′ gives rise to a Whitney tower on A′′ with the same intersection invariant. Thus, the problem is to find INT_{n}(A) relations which give independence of the choice of W for a fixed immersion A, and can be realized geometrically so that λ_{n}(W) ∈ INT_{n}(A) implies that A bounds an order n + 1 non-repeating Whitney tower. We conjecture that all these needed relations do indeed correspond to lower-order intersections involving 2–spheres, and hence deserve to be called “intersection” relations. Although such INT_{n}(A) relations are completely understood for n = 1 (see 8.2 below), a precise formulation for the n = 2 case already presents interesting subtleties. We remark that for maps of higher genus surfaces there can also be indeterminacies (due to choices of boundary arcs of Whitney disks) which do not come from 2–spheres; see [29] for the order 1 invariants of immersed annuli. Useful necessary and sufficient conditions for pulling apart four or more 2– spheres in an arbitrary 4–manifold are not currently known. In Section 8 we examine the intersection indeterminacies for the relevant order 2 non-repeating intersection invariant λ_{2} in the simply connected setting, and show how they can be computed as the image in Λ_{2}(4) ≅ ℤ^{2} of a map whose non-linear part is determined by certain Diophantine quadratic equations which are coupled by the intersection form on π_{2}X (see section 8.3.6). Carrying out this computation in general raises interesting number theoretic questions, and has motivated work of Konyagin and Nathanson in [21]. We’d like to pose the following challenge: Formulate the INT_{n}(A) relations for n ≥ 2 which make the following conjecture precise and true:

Conjecture 15 A : ⨿^{m}S^{2} → X can be pulled apart if and only if λ_{n}(A) := λ_{n}(W) vanishes in Λ_{n}(π)/INT_{n}(A) for n = 0, 1, 2, 3, …, m − 2.

2 Whitney towers

This section contains a summary of relevant Whitney tower notions and notations as described in more detail in [3, 5, 6, 27, 28, 29, 30, 31]. Recall our blurring of the distinction between a map A : Σ → X and its image, which leads us to speak of A as a “collection” of immersed connected surfaces in X.

Remark 16 Although this paper focuses on pulling apart A in the case where the components Σ_{i} of Σ are spheres and/or disks, much of the discussion is also relevant to the π_{1}-null setting; i.e. the Σ_{i} are compact connected surfaces of arbitrary genus and the component maps A_{i} : Σ_{i} → X induce trivial maps π_{1}Σ_{i} → π_{1}X on fundamental groups.

2.1 Whitney towers

The following formalizes the discussion from the introduction by inductively defining Whitney towers of order n for each non-negative integer n.

Definition 17

• A surface of order 0 in a 4–manifold X is a properly immersed connected compact surface (boundary embedded in the boundary of X and interior immersed in the interior of X). A Whitney tower of order 0 in X is a collection of order 0 surfaces.

• The order of a (transverse) intersection point between a surface of order n_{1} and a surface of order n_{2} is n_{1} + n_{2}.

• The order of a Whitney disk is n + 1 if it pairs intersection points of order n.

• For n ≥ 0, a Whitney tower of order n+1 is a Whitney tower W of order n together with Whitney disks pairing all order n intersection points of W. These order n + 1 Whitney disks are allowed to intersect each other as well as lower-order surfaces.

The Whitney disks in a Whitney tower are required to be framed [5, 12, 30] and have disjointly embedded boundaries. Each order 0 surface in a Whitney tower is also required to be framed, in the sense that its normal bundle in X has trivial (relative) Euler number. Interior intersections are assumed to be transverse. A Whitney tower is oriented if all its surfaces (order 0 surfaces and Whitney disks) are oriented. Orientations and framings on any boundary components of order 0 surfaces are required to be compatible with those of the order 0 surfaces. A based Whitney tower includes a chosen basepoint on each surface (including Whitney disks) together with a whisker (arc) for each surface connecting the chosen basepoints to the basepoint of X.

We will assume our Whitney towers are based and oriented, although whiskers and orientations will usually be suppressed from notation. The collection A of order 0 surfaces in a Whitney tower W is said to support W, and we also say that W is a Whitney tower on A. A collection A of order 0 surfaces is said to admit an order n Whitney tower if A is homotopic (rel boundary) to A′ supporting an order n Whitney tower.

2.2 Trees for Whitney disks and intersection points.

In this paper, a tree will always refer to a finite oriented unitrivalent tree, where the (vertex) orientation of a tree is given by cyclic orderings of the adjacent edges around each trivalent vertex. The order of a tree is the number of trivalent vertices. Univalent vertices will usually be labeled from the set {1, 2, 3, …, m} indexing the order 0 surfaces, and we consider trees up to isomorphisms preserving these labelings. A tree is non-repeating if its univalent labels are distinct. When X is not simply connected, edges will be oriented and labeled with elements of π_{1}X. A root of a tree is a chosen univalent vertex (usually left un-labeled). We start by considering the case where X is simply connected: Formal non-associative bracketings of elements from the index set are used as subscripts to index surfaces in a Whitney tower W ⊂ X, writing A_{i} for an order 0 surface (dropping the brackets around the singleton i), W_{(i,j)} for an order 1 Whitney disk that pairs intersections between A_{i} and A_{j}, and W_{((i,j),k)} for an order 2 Whitney disk pairing intersections between W_{(i,j)} and A_{k}, and so on, with the ordering of the bracket components determined by an orientation convention described below (2.3). When writing W_{(I,J)} for a Whitney disk pairing intersections between W_{I} and W_{J}, the understanding is that if a bracket I is just a singleton i then the surface W_{I} = W_{i} is just the order zero surface A_{i}. Note that both Whitney disks and order 0 surfaces are referred to as “surfaces in W”. Via the usual correspondence between non-associative brackets and rooted trees, this indexing gives a correspondence between surfaces in W and rooted trees: To a Whitney disk W_{(I,J)} we associate the rooted tree corresponding to the bracket (I, J). We use the same notation for rooted trees and brackets, so the bracket operation corresponds to the rooted product of trees which glues together the root vertices of I and J to a single vertex and sprouts a new rooted edge from this vertex. With this notation the order of a Whitney disk W_{K} is equal to the order of (the rooted tree) K. The rooted tree (I, J) associated to W_{(I,J)} can be considered to be a subset of W, with its root edge (including the root edge’s trivalent vertex) sitting in the interior of W_{(I,J)}, and its other edges bifurcating down through lowerorder Whitney disks. The unrooted tree t_{p} associated to any intersection point p ∈ W_{(I,J)} ∩W_{K} is the inner product t_{p} = ⟨ (I, J), K ⟩ gotten by identifying the roots of the trees (I, J) and K to a single non-vertex point. Note that t_{p} also can be considered as a subset of W, with the edge of t_{p} containing p a sheetchanging path connecting the basepoints of W_{(I,J)} and W_{K} (see Figure 4). If X is not simply connected, then the edges of the just-described trees are decorated by elements of π_{1}X as follows: Considering the trees as subsets of W, each edge of a tree is a sheet-changing path connecting basepoints of adjacent surfaces of W. Choosing orientations of these sheet-changing paths determines elements of π_{1}X (using the whiskers on the surfaces) which are attached as labels on the correspondingly oriented tree edges.

Note that the notation for trees is slightly different in the older papers [27, 31], where the rooted tree associated to a bracket I is denoted t(I), and the rooted and inner products are denoted by ∗ and · respectively. The notation of this paper agrees with the more recent papers [4, 5, 6, 7, 8, 29].

Figure 4: A local picture of the tree t_{p} = ⟨(I, J), K⟩ associated to p ∈ W_{(I,J)} ∩ W_{K} near a trivalent vertex adjacent to the edge of t_{p} passing through an unpaired intersection point p in a Whitney tower W. On the left t_{p} is pictured as a subset of W, and on the right as an abstract labeled vertex-oriented tree. In a non-simply connected 4–manifold X the edges of t_{p} would also be oriented and labeled by elements of π_{1}X (as in Figure 5 below).

2.3 Orientation conventions

Thinking of the tree I associated to a Whitney disk W_{I} as a subset of W, it can be arranged that the trivalent orientations of I are induced by the orientations of the corresponding Whitney disks: Note that the pair of edges which pass from a trivalent vertex down into the lower-order surfaces paired by a Whitney disk determine a “corner” of the Whitney disk which does not contain the other edge of the trivalent vertex. If this corner contains the negative intersection point paired by the Whitney disk, then the vertex orientation and the Whitney disk orientation agree. Our figures are drawn to satisfy this convention. This “negative corner” convention (also used in [5, 6]), which differs from the positive corner convention used in [3, 31], turns out to be compatible with the usual commutator conventions, for instance in the setting of Milnor invariants (see Figure 13).

2.4 Non-repeating Whitney towers

Whitney disks and intersection points are called non-repeating if their associated trees are non-repeating. This means that the univalent vertices are labeled by distinct indices (corresponding to distinct order 0 surfaces, i.e. distinct connected components of A). A Whitney tower W is an order n non-repeating Whitney tower if all non-repeating intersections of order (strictly) less than n are paired by Whitney disks. In particular, if W is an order n Whitney tower then W is also an order n non-repeating Whitney tower. In a non-repeating Whitney tower repeating intersections of any order are not required to be paired by Whitney disks.

2.5 Intersection invariants

For a group π, denote by T_{n}(m, π) the abelian group generated by order n (decorated) trees modulo the relations illustrated in Figure 5.

Figure 5: The relations in T_{n}(π, m): IHX (Jacobi), AS (antisymmetry), OR (orientation), HOL (holonomy). These are ‘local’ pictures, meaning that the unlabeled univalent vertices extend to fixed decorated subtrees in each equation. For instance, in the right-hand term of the HOL relation only the three visible edge decorations are multiplied by the element g, corresponding to a change of whisker on a Whitney disk at the indicated trivalent vertex. All vertex-orientations are induced from a fixed orientation of the plane; in particular, the two terms in the AS relation only differ by the orientation at the indicated trivalent vertex, where the two edges extending to the subtrees I and J have been interchanged.

Note that when π is the trivial group, the edge decorations (orientations and π-labels) disappear, and the relations reduce to the usual AS antisymmetry and IHX Jacobi relations of finite type theory (compare also the decorated graphs of [15]). All the relations are homogeneous in the univalent labels, and restricting the generating trees to be non-repeating order n trees defines the subgroup Λ_{n}(m, π) < T_{n}(m, π). (See sections 2.1 and 3 of [31] for explanations of these relations.)

Definition 18 For an order n (oriented) Whitney tower W in X, the order n intersection invariant τ_{n}(W) is defined by summing the signed trees ±t_{p} over all order n intersections p ∈ W:

τ_{n}(W) := ∑ sign(p) · t_{p} ∈ T_{n}(π).

Here π = π_{1}X; and sign(p) = ±, for p ∈ W_{I} ∩ W_{J}, is the usual sign of an intersection between the oriented Whitney disks W_{I} and W_{J}. If W is an order n non-repeating Whitney tower, the order n non-repeating intersection invariant λ_{n}(W) is analogously defined by

λ_{n}(W) := ∑ sign(p) · t_{p} ∈ Λ_{n}(π)

where the sum is over all order n non-repeating intersections p ∈ W.

2.6 Order 0 intersection invariants

The order 0 intersection invariants τ_{0} and λ_{0} for A : ⨿^{m}S^{2} → X carry the same information as Wall’s [33] Hermitian intersection form μ, λ: The generators in τ_{0}(A) ∈ T_{0}(π, m) with both vertices labeled by the same index i correspond to Wall’s self-intersection invariant μ(A_{i}). For μ(A_{i}) to be a homotopy (not just regular homotopy) invariant, one must also mod out by a framing relation which kills order 0 trees labeled by the trivial element in π (see [5] for higherorder framing relations). Wall’s homotopy invariant Hermitian intersection pairing λ(A_{i}, A_{j}) ∈ ℤ[π] for i ≠ j corresponds to λ_{0}(A) ∈ Λ_{0}(π, m). The vanishing of these invariants corresponds to the order 0 intersections coming in canceling pairs (after perhaps a homotopy of A), so A admits an order 1 Whitney tower if and only if τ_{0}(A) = 0 ∈ T_{0}(π, m), and admits an order 1 non-repeating Whitney tower if and only if λ_{0}(A) = 0 ∈ Λ_{0}(π, m).

2.7 Order 1 intersection invariants

It was shown in [30], and for π_{1}X = 1 and m = 3 in [25, 34], that for A : ⨿^{m}S^{2} → X admitting an order 1 Whitney tower (resp. non-repeating Whitney tower) W, the order 1 intersection invariant τ_{1}(A) := τ_{1}(W) (resp. order 1 nonrepeating intersection invariant λ_{1}(A) := λ_{1}(W)) is a homotopy invariant of A, if taken in an appropriate quotient of T_{1}(π, m) (resp. Λ_{1}(π, m)). The relations defining this quotient are determined by order 0 intersections between the A_{i} and immersed 2–spheres in X. These are the order 1 intersection relations INT_{1} which are described in [30] (in slightly different notation) and below in Section 8 (for λ_{1}). As remarked in the introduction, for τ_{1} there are also framing relations, but there are no framing relations for λ_{n} (for all n) because Whitney disks can always be framed by the boundary-twisting operation [12, Sec.1.3] which creates only repeating intersections.

From [30], we have that A admits an order 2 Whitney tower (resp. order 2 nonrepeating Whitney tower) if and only if τ_{1}(A) (resp. λ_{1}(A)) vanishes. In particular, λ_{1}(A_{1}, A_{2}, A_{3}) ∈ Λ_{1}(π, 3)/INT_{1} is the complete obstruction to pulling apart three order 0 surfaces with vanishing λ_{0}(A_{1}, A_{2}, A_{3}).

2.8 Order n intersection invariants

As was shown in Theorem 2 of [31], for A admitting a Whitney tower W of order n, if τ_{n}(W) = 0 ∈ T_{n}(π) then A admits a Whitney tower of order n + 1. The proof of this result proceeds by geometrically realizing the relations in the target group of the intersection invariant in a controlled manner, so that one can convert “algebraic cancellation” of pairs of trees to “geometric cancellation” of pairs of points (paired by next-order Whitney disks). The exact same arguments work restricting to the non-repeating case to prove Theorem 6 of the introduction: For A admitting a non-repeating Whitney tower W of order n, if λ_{n}(W) = 0 ∈ Λ_{n}(π) then A admits a non-repeating Whitney tower of order n + 1. Beyond this “sufficiency” result, it is not known for n ≥ 2 what additional relations INT_{n} ⊂ Λ_{n}(π) would also make the vanishing of λ_{n}(W) in the quotient a necessary condition for A to admit a non-repeating Whitney tower of order n + 1, as discussed in 1.9.2 of the introduction.

2.9 The groups Λ_{n}

The groups Λ_{n}(π, m) provide upper bounds for the order n non-repeating obstruction theory, and hence by Corollary 7 also for the obstructions to pulling apart surfaces. The following result describes the structure of Λ_{n}(π, m):

Lemma 19 Λ_{n}(π, m) is isomorphic (as an additive abelian group) to the \binom{m}{n+2}n!-fold direct sum of the integral group ring ℤ[π^{n+1}] of the (n + 1)-fold cartesian product π^{n+1} = π × π × ⋯ × π.

Proof: First consider the case where π is trivial. Since the relations in Λ_{n}(m) are all homogenous in the univalent labels, Λ_{n}(m) is the direct sum of subgroups Λ_{n}(n + 2) over the \binom{m}{n+2} choices of n + 2 of the m labels. (As noted in the introduction, Λ_{n}(π, m) is trivial for n ≥ m−1 since an order n unitrivalent tree has n + 2 univalent vertices.) We will show that each of these subgroups has a basis given by the n! distinct simple non-repeating trees shown in Figure 6 (ignoring the edge decorations for the moment), where an order n tree is simple if it contains a geodesic of edge-length n + 1. For a given choice of n + 2 labels, placing a root at the minimal-labeled vertex of each order n tree gives an isomorphism from Λ_{n}(n + 2) to the subgroup of non-repeating length n+1 brackets in the free Lie algebra (over ℤ) on the other labels (with AS and IHX relations going to skew-symmetry relations and Jacobi identities). This “reduced” free Lie algebra (see also 4.1 below) is known to have rank n!, as explicitly described in [22, Thm.5.11] (also implicitly contained in [23, Sec.4–5]), so the trees in Figure 6 are linearly independent if they span.

To see that the trees in Figure 6 form a spanning set, first observe that for a given choice of n+2 labels, each order n non-repeating tree t has a distinguished geodesic edge path T_{t} from the minimal-label univalent vertex to the maximallabel univalent vertex. For an orientation-inducing embedding of t in the plane, it can be arranged that all the sub-trees of t emanating from T_{t} lie on a preferred side of T_{t} by applying AS relations at the trivalent vertices of T_{t} as needed. Then, by repeatedly applying IHX relations (replacing the left-most I-tree by the difference of the H-tree and X-tree in the IHX relation of Figure 5) at trivalent vertices of distinguished geodesics to reduce the order of the emanating sub-trees one eventually gets a linear combination of simple non-repeating trees as in Figure 6 which is uniquely determined by t. (To see how the IHX relation reduces the order of subtrees emanating from a distinguished geodesic, observe that if the central edge of the I-tree in an IHX relation is the first edge of such a subtree, then the corresponding emanating subtrees in the H-tree and X-tree both have order decreased by one.) In the case of non-trivial π, the group elements decorating the edges of the simple trees can always be (uniquely) normalized to the trivial element on all but n + 1 of the edges as shown in Figure 6 (by applying HOL relations from Figure 5 and working from the minimal towards the maximal vertex label). □

Figure 6: A simple order n tree with minimal- and maximal-labeled vertices connected by a length n+1 geodesic; 1 ≤ i_{min} < i_{max} ≤ m, and i_{min} < i_{k} < i_{max} for 1 ≤ k ≤ n. Vertex orientations are induced by the planar embedding. By the HOL relations, all but n+1 of the edge decorations can be set to the trivial element in π (indicated by the ‘empty-labeled’ edges in the figure).

2.10 Some properties of Whitney towers

For future reference, we note here some elementary properties of Whitney towers and their intersection invariants. Let A : Σ → X support an order n Whitney tower W ⊂ X, where Σ has m connected components Σ_{i}. We will consider the effects on τ_{n}(W) of changing the order 0 surfaces A_{i} : Σ_{i} → X of A by the operations of re-indexing, including parallel copies, taking internal sums, switching orientations, and deletions; all of which preserve the property that A supports an order n Whitney tower. We will focus on the case where X is simply connected, which will be used in Section 5. (Analogous properties hold in the non-simply connected setting, although when taking internal sums (2.10.3) some care would be needed in keeping track of the effect on the edge decorations due to choices of arcs guiding the sums.)

2.10.1 Re-indexing order 0 surfaces

For A : Σ → X the natural indexing of the order 0 surfaces of W is by π_{0}Σ. In practice, we fix an identification of π_{0}Σ with the label set {1, 2, …, m}, and the effect of changing this identification is given by the corresponding permutation of the univalent labels on all the trees representing τ_{n}(W).

2.10.2 Parallel Whitney towers

Suppose A is extended to A′ by including a parallel copy A_{m+1} of the last order 0 surface A_{m} of A. Recall from Definition 17 that order 0 surfaces have trivial (relative) normal Euler numbers, so each self-intersection of A_{m} will give rise to a single self-intersection of A_{m+1} and a pair of intersections between A_{m+1} and A_{m}; and each intersection between A_{m} and any A_{i}, for i ≠ m, will give rise to a single intersection between A_{m+1} and A_{i}; and no other intersections in A′ will be created. By the splitting procedure of [31, Lem.13] (also [27, Lem.3.5]) it can be arranged that all Whitney disks in W are embedded and contained in standard 4–ball thickenings of their trees. Since the Whitney disks are all framed, W can be extended to an order n Whitney tower W′ on A′ by including parallel copies of the Whitney disks in W as illustrated by Figure 7. This new Whitney tower W′ can be constructed in an arbitrarily small neighborhood of W, and the intersection invariants are related in the following way. Define δ: T_{n}(m) → T_{n}(m + 1) to be the homomorphism induced by the map which sends a generator t having r-many m-labeled univalent vertices to the 2^{r}- term sum over all choices of replacing the label m by the label (m + 1). Then τ_{n}(W′) = δ(τ_{n}(W)). (In the non-simply connected setting, group elements decorating the edges would be preserved by taking parallel whiskers.) Via re-indexing, the effect of including a parallel copy of any ith order 0 surface can be described by analogous relabeling maps δ_{i}, and iterating this procedure constructs an order n Whitney tower near W on any number of parallel copies of any order 0 surfaces of A, with the resulting change in τ_{n}(W) described by compositions of the δ_{i} maps.

2.10.3 Internal sums

Suppose A′ is formed from A by taking the ambient connected sum of A_{m−1} with A_{m} in X (or by joining ∂A_{m−1} to ∂A_{m} with a band in ∂X), so that A′ has m−1 components. Since it may be assumed that the interior of the arc guiding the sum is disjoint from W, it is clear that A′ bounds an order n Whitney tower W′ all of whose Whitney disks and singularities are identical to W. Then τ_{n}(W′) = σ(τ_{n}(W)) ∈ T_{n}(m − 1), where the map σ : T_{n}(m) → T_{n}(m − 1) is induced by the relabeling map on generators which changes all m-labeled Figure 7: Extending a Whitney tower using parallel Whitney disks: An unpaired intersection point p ∈ W_{(I,J)} ∩ W_{K} in a Whitney tower on A, where each of I, J and K contain exactly one occurrence of the index m, gives rise to eight unpaired intersections in four Whitney disks in the new Whitney tower on A′, formed from A by including a parallel copy of A_{m}. The dotted oval loops (left) bound neighborhoods of the Whitney arcs in the opaque I′ sheet which have been perturbed into a nearby time coordinate along with the two corresponding translucent Whitney disks (right).

univalent vertices to (m − 1)-labeled univalent vertices. (In the non-simply connected setting, group elements decorating all edges would be preserved if the guiding arc together with the whiskers on A_{m−1} and A_{m} formed a nullhomotopic loop.) Via re-indexing, the effect of summing any A_{i} with any A_{j} (j ≠ i) is described by the analogous map σ_{ij}, and for iterated internal sums the resulting intersection invariant is described by compositions of the σ_{ij} maps.

2.10.4 Switching order 0 surface orientations

As explained in [31, Sec.3], the orientation of A determines the vertexorientations of the trees representing τ_{n}(W) up to AS relations, via our above convention (2.3). The effect on τ_{n}(W) of switching the orientation of an order 0 surface A_{i} of A is described as follows. Define s_{i}: T_{n}(m) → T_{n}(m) to be the automorphism induced by the map which sends a generator t to (−1)^{i(t)}t, where again i(t) denotes the multiplicity of the univalent label i in t. Then if W′ is a reorientation of W which is compatible with a reversal of orientation of A_{i}, then we have τ_{n}(W′) = s_{i}(τ_{n}(W)). The effect on the intersection invariant of reorienting any number of order 0 surfaces of A is described by compositions of the s_{i} maps.

2.10.5 Deleting order 0 surfaces

The result A′ of deleting the last order 0 surface A_{m} of A supports an order n Whitney tower W′ formed by deleting those Whitney disks from W which involve A_{m}; that is, deleting any Whitney disk whose tree has at least one univalent vertex labeled by m. We have τ_{n}(W′) = e(τ_{n}(W)), where the homomorphism e: T_{n}(m) → T_{n}(m − 1) is induced by the map which sends a generator t to zero if m appears as a label in t, and is the identity otherwise. Via re-indexing, the effect of deleting any A_{i} can be described by analogous maps e_{i}, and the change in τ_{n}(W) due to multiple deletions of order 0 surfaces is described by compositions of the e_{i}.

2.10.6 Canceling parallels

We note here the following easily-checked lemma, which will be used in Section 5:

Lemma 20 The composition σ_{ji′} ∘ σ_{i′i′′} ∘ s_{i′′} ∘ δ_{i′} ∘ δ_{i} is the identity map on T_{n}(m). □

Lemma 20 describes the effect on the intersection invariant that corresponds to including two parallel copies A′_{i} and A′′_{i} of A_{i}, switching the orientation on A′′_{i}, then recombining A′_{i} and A′′_{i} by an internal sum into a single i′th component, and then internal summing this combined i′th component into any jth component of A. (Note that applying the analogous sequence of operations to a link obviously preserves the isotopy class of the link.)

3 Proof of Theorem 4

We want to show that m connected surfaces A_{i} : Σ_{i} → X can be pulled apart if and only if they admit an order m − 1 non-repeating Whitney tower. Proof: The “only if” direction follows by definition, since disjoint order 0 surfaces form a non-repeating Whitney tower of any order. So let W be a nonrepeating Whitney tower of order m − 1 on A_{1}, A_{2}, …, A_{m}. If W contains no Whitney disks, then the A_{i} are pairwise disjoint. In case W does contain Whitney disks, we will describe how to use finger moves and Whitney moves to eliminate the Whitney disks of W while preserving the non-repeating order m − 1. First note that W contains no unpaired non-repeating intersections: All nonrepeating intersections of order < m−1 are paired by definition; and since trees of order ≥ m − 1 have ≥ m + 1 univalent vertices, all intersections of order greater than or equal to m − 1 in any Whitney tower on m order 0 surfaces must be repeating intersections. Now consider a Whitney disk W_{(I,J)} in W of maximal order. If W_{(I,J)} is clean (the interior of W_{(I,J)} contains no singularities) then do the W_{(I,J)}-Whitney move on either W_{I} or W_{J}. This eliminates W_{(I,J)} (and the corresponding canceling pair of intersections between W_{I} and W_{J}) while creating no new intersections, hence preserves the order of the resulting non-repeating Whitney tower which we continue to denote by W.

If any maximal order Whitney disk W_{(I,J)} in W is not clean, then the singularities in the interior of W_{(I,J)} are exactly a finite number of unpaired intersection points, all of which are repeating. (Since W_{(I,J)} is of maximal order, the interior of W_{(I,J)} contains no Whitney arcs; and W contains no unpaired non-repeating intersections, as noted above.) So, for any p ∈ W_{(I,J)} ∩ W_{K}, at least one of (I, K) or (J, K) is a repeating bracket. Assuming that (I, K), say, is repeating, push p off of W_{(I,J)} down into W_{I} by a finger move (Figure 8). This creates only a pair of repeating intersections between W_{I} and W_{K}. After pushing down all intersections in the interior of W_{(I,J)} by finger moves in this way, do the clean W_{(I,J)}-Whitney move on either W_{I} or W_{J}. Repeating this procedure on all maximal order Whitney disks eventually yields the desired order m − 1 non-repeating Whitney tower (with no Whitney disks) on order 0 surfaces A′_{i}. The A′_{i} are regularly homotopic to the A_{i}; the pushing-down finger moves will have created pairs of self-intersections in the pairwise disjointly immersed A′_{i}. □

Figure 8:

4 Proof of Theorem 8

Consider a link L = L_{1} ∪ L_{2} ∪ ⋯ ∪ L_{m} ⊂ S^{3} that bounds an order n non-repeating Whitney tower W on immersed disks in the 4–ball. We will prove Theorem 8 by relating λ_{n}(W) to Milnor’s length n + 2 link-homotopy μ-invariants of L in Theorem 24, showing in particular that λ_{n}(L) := λ_{n}(W) ∈ Λ_{n}(m) only depends on the link-homotopy class of L (and not on the Whitney tower W). The essential idea is that W can be used to compute the link longitudes as iterated commutators in Milnor’s nilpotent quotients of the fundamental group of the link complement. The proof uses a new result, Whitney tower-grope duality, which describes certain class n + 2 gropes that live in the complement of an order n Whitney tower in any 4–manifold (Proposition 25). After fixing notation for the first-non-vanishing Milnor invariants of L in section 4.1, we give the explicit identification of them with λ_{n}(W) in Theorem 24 of section 4.2.

4.1 Milnor’s link-homotopy μ-invariants

This subsection briefly reviews and fixes notation for the first non-vanishing non-repeating μ-invariants of a link. See any of [1, 16, 17, 23] for details. For a group G normally generated by elements g_{1}, g_{2}, …, g_{m}, the Milnor group of G (with respect to the g_{i}) is the quotient of G by the subgroup normally generated by all commutators between g_{i} and g_{i}^{h} := hg_{i}h^{−1}, so we kill the elements

[g_{i}, g_{i}^{h}] = g_{i}g_{i}^{h}g_{i}^{−1}g_{i}^{−h}

for 1 ≤ i ≤ m, and all h ∈ G. One can prove (e.g. [16, Lem.1.3]) by induction on m that this quotient is nilpotent and (therefore) generated by g_{1}, …, g_{m}. The Milnor group M(L) of an m-component link L is the Milnor group of the fundamental group of the link complement π_{1}(S^{3} ∖ L) with respect to a generating set of meridional elements. Specifically, M(L) has a presentation

M(L) = ⟨x_{1}, x_{2}, …, x_{m} | [ℓ_{i}, x_{i}], [x_{j}, x_{j}^{h}]⟩

where each x_{i} is represented by a meridian (one for each component), and the ℓ_{i} are words in the x_{i} determined by the link longitudes. The Milnor group M(L) is the largest common quotient of the fundamental groups of all links which are link homotopic to L. Since M(L) only depends on the conjugacy classes of the meridional generators x_{i}, it only depends on the link L (and no base-points are necessary). A presentation for the Milnor group of the unlink (or any link-homotopically trivial link) corresponds to the case where all ℓ_{i} = 1, and Milnor’s μ-invariants (with non-repeating indices) compare M(L) with this free Milnor group M(m) by examining each longitudinal element in terms of the generators corresponding to the other components. Specifically, mapping x_{i}^{±1} to ±X_{i} induces a canonical isomorphism

M(m)_{(n)}/M(m)_{(n+1)} ≅ RL_{n}(m)

from the lower central series quotients to the reduced free Lie algebra RL(m) = ⊕_{n=1}^{m}RL_{n}(m), which is the quotient of the free ℤ-Lie algebra on the X_{i} by the relations which set an iterated Lie bracket equal to zero if it contains more than one occurrence of a generator. This isomorphism takes a product of length n commutators in distinct x_{i} to a sum of length n Lie brackets in distinct X_{i}. In particular, RL_{n}(m) = 0 for n > m. Let M^{i}(L) denote the quotient of M(L) by the relation x_{i} = 1. If the element in M^{i}(L) determined by the longitude ℓ_{i} lies in the (n + 1)th lower central subgroup M^{i}(L)_{(n+1)} for each i, then we have isomorphisms:

M(L)_{(n+1)}/M(L)_{(n+2)} ≅ M(m)_{(n+1)}/M(m)_{(n+2)} ≅ RL_{(n+1)}(m).

Via the usual identification of non-associative bracketings and binary trees, RL_{(n+1)}(m) can be identified with the abelian group on order n rooted nonrepeating trees modulo IHX and antisymmetry relations as in Figure 5 (with π trivial). This identification explains the subscripts in the following definition:

Definition 21 The elements μ_{n}^{i}(L) ∈ RL_{(n+1)}^{i}(m) determined by the longitudes ℓ_{i} are the non-repeating Milnor-invariants of order n. Here RL^{i}(m) is the reduced free Lie algebra on the m − 1 generators X_{j}, for j ≠ i.

This definition of non-repeating μ-invariants was originally given by Milnor [23]. He later expressed the elements μ_{n}^{i}(L) in terms of integers μ_{L}(i, k_{1}, …, k_{n+1}), which are the coefficients of X_{k_{1}} ⋯ X_{k_{n+1}} in the Magnus expansion of ℓ_{i}. We note that our order n corresponds to the originally used length n+2 (of entries in μ_{L}). By construction, these non-repeating μ-invariants depend only on the linkhomotopy class of the link L. We have only defined order n μ-invariants assuming that the lower-order μ-invariants vanish, which will turn out to be guaranteed by the existence of an order n non-repeating Whitney tower.

4.2 Mapping from trees to Lie brackets

For each i, define a map

η_{n}^{i} : Λ_{n}(m) → RL_{(n+1)}^{i}(m)

by sending a tree t which has an i-labeled univalent vertex v_{i} to the iterated bracketing determined by t with a root at v_{i}. Trees without an i-labeled vertex are sent to zero. For example, if t is an order 1 Y -tree with univalent labels 1, 2, 3, and cyclic vertex orientation (1, 2, 3), then η_{1}^{1}(t) = [X_{2}, X_{3}], and η_{1}^{3}(t) = [X_{1}, X_{2}], and η_{1}^{2}(t) = [X_{3}, X_{1}]. Note that the IHX and AS relations in Λ_{n}(m) go to the Jacobi and skew-symmetry relations in RL_{(n+1)}^{i}(m), so the maps η_{n}^{i} are well-defined.

Lemma 22

∑_{i=1}^{m} η_{n}^{i} : Λ_{n}(m) ⟶ ⊕_{i=1}^{m}RL_{(n+1)}^{i}(m)

is a monomorphism.

Proof: Putting an i-label in place of the root in a tree corresponding to a Lie bracket in RL_{(n+1)}^{i}(m) gives a left inverse to η_{n}^{i}. In fact, for the top degree n + 2 = m, this is an inverse because every index i appears exactly once in a tree t of order n = m − 2. For arbitrary n, it is easy to check that composing the sum of these left inverse maps with ∑_{i=1}^{m} η_{n}^{i} is just multiplication by n + 2 on Λ_{n}(m). Since Λ_{n}(m) is torsion-free by Lemma 19, it follows that ∑_{i=1}^{m} η_{n}^{i} is injective. □

Remark 23 The monomorphism ∑_{i=1}^{m} η_{n}^{i} fits into the bottom row of a commutative diagram:

[figure: T_{n}(m); η_{n}; ⊕_{i=1}^{m}L_{(n+1)}(m); Λ_{n}(m); ⊕_{i=1}^{m}RL_{(n+1)}^{i}(m)]

Here the upper row is relevant for repeating Milnor invariants as explained in [4, 5]. The injectivity of the top horizontal map η_{n}, defined by Jerry Levine, is much harder to show and is the central result of [7] (implying that T_{n}(m) has at most 2-torsion). The two vertical projections simply set trees with repeating labels to zero.

The maps η_{n}^{i} correspond to tree-preserving geometric constructions which desingularize an order n Whitney tower to a collection of class n + 1 gropes, as described in detail in [27], and sketched in section 4.3 below. Gropes are 2-complexes built by gluing together compact orientable surfaces, and this correspondence will be used in the proof of the following theorem:

Theorem 24 If a link L ⊂ S^{3} bounds a non-repeating Whitney tower W of order n on immersed disks D = ⨿^{m}D^{2} → B^{4}, then for each i the longitude ℓ_{i} lies in M^{i}(L)_{(n+1)}, and

η_{n}^{i}(λ_{n}(W)) = μ_{n}^{i}(L) ∈ RL_{(n+1)}^{i}(m)

Since the sum of the η_{n}^{i} is injective, this will prove Theorem 8: The intersection invariant λ_{n}(W) ∈ Λ_{n}(m) does not depend on the Whitney tower W and is a link homotopy invariant of L, denoted by λ_{n}(L). For L bounding an honest order n Whitney tower, one can deduce this theorem from the main result in [6, Thm.5] (and the diagram in Remark 23 above); but here we only have a non-repeating order n Whitney tower as an input. Proof: We start by giving an outline of the argument, introducing some notation that will be clarified during the proof:

(i) First the Whitney tower will be cleaned up, including the elimination of all repeating intersections of positive order and all repeating Whitney disks, to arrive at an order n non-repeating Whitney tower W bounded by L such that all unpaired intersection points of positive order have nonrepeating trees (so the only repeating intersections are self-intersections in the order 0 disks D_{j}).

(ii) Then the preferred order 0 disk D_{i} (and all Whitney disks involving D_{i}) will be resolved to a grope G_{i} of class n+1 bounded by L_{i}, such that G_{i} is in the complement B^{4}∖W^{i}, where W^{i} is the result of deleting D_{i} and the Whitney disks used to construct G_{i} from W. The grope G_{i} will display the longitude ℓ_{i} in π_{1}(B^{4} ∖ W^{i}) as a product of (n + 1)-fold commutators of meridians to the order 0 surfaces D^{i} := ∪_{j≠i}D_{j} of W^{i} corresponding to putting roots at all i-labeled vertices of the trees representing λ_{n}(W). This is the same formula as in the definition of the map η_{n}^{i}, so it only remains to show that μ_{n}^{i}(L) can be computed in π_{1}(B^{4} ∖ W^{i}).

(iii) This last step is accomplished by using Whitney tower-grope duality (Proposition 25) and Dwyer’s theorem [9] to show that the inclusion S^{3} ∖ ∂D^{i} → B^{4} ∖ W^{i} induces an isomorphism on the Milnor groups modulo the (n + 2)th terms of the lower central series.

Step (i): Let W be an order n non-repeating Whitney tower on D → B^{4} bounded by L ⊂ S^{3}. As described in [27, Lem.3.5] (or [31, Lem.13]), W can be split, so that each Whitney disk of W is embedded, and the interior of each Whitney disk contains either a single unpaired intersection p or a single boundary arc of a higher-order Whitney disk, and no other singularities. This splitting process does not change the trees representing λ_{n}(W), and results in each tree t_{p} associated to an order n intersection p being contained in a 4-ball thickening of t_{p}, with all these 4–balls pairwise disjoint. Splitting simplifies combinatorics, and facilitates the use of local coordinates for describing constructions. Also, split Whitney towers correspond to dyadic gropes (whose upper stages are all genus one), and dyadic gropes are parametrized by trivalent (rooted) trees. We continue to denote the split order n non-repeating Whitney tower by W, and will keep this notation despite future modifications. In the following, further splitting may be performed without mention. If W contains any repeating intersections of positive order, then by following the pushing-down procedure described in the proof of Theorem 4 given in section 3, all these repeating intersections can be pushed-down until they create (many) pairs of self-intersections in the order 0 disks. Then all repeating Whitney disks are clean, and by doing Whitney moves guided by these clean Whitney disks it can be arranged that W contains no repeating Whitney disks and no repeating intersections of positive order. Step (ii): Consider now the component L_{i} bounding D_{i}. We want to convert D_{i} into a class n + 1 grope displaying the longitude ℓ_{i} as a product of (n + 1)-fold iterated commutators in meridians to the D_{j≠i} using the tree-preserving Whitney tower-to-grope construction of [27, Thm.5]. This construction is sketched roughly below in section 4.3, and a simple case is illustrated in Figure 12. Actually, the resulting grope G_{i} comes with caps, which in this setting are embedded normal disks to the other D_{j} which are bounded by essential circles called tips on G_{i}. For our purposes the caps only serve to show that these tips are meridians to the D_{j}. The trees associated to gropes are rooted trees, with the root vertex corresponding to the bottom stage surface, and the other univalent vertices corresponding to the tips (or to the caps). Since W was split, the upper surface stages of G_{i} will all be genus one, so the collection of order n unitrivalent trees t(G_{i}) associated to G_{i} will contain one tree for each dyadic branch of upper stages, with each trivalent vertex of a tree corresponding to a genus one surface in a branch. In this setting the class of G_{i} is equal to n + 1, the number of non-root univalent vertices in each tree in the collection t(G_{i}) (see e.g. [27, Sec.2.3]). Applying the construction of [27, Thm.5] to D_{i} converts D_{i} and all the Whitney disks of W corresponding to trivalent vertices in trees containing an i-label into a class n + 1 grope G_{i}. This grope G_{i} (without the extra caps provided by [27, Thm.5]) is disjoint from W^{i} ⊂ W, where the order n non-repeating Whitney tower W^{i} consists of the order 0 immersed disks D^{i} := ∪_{j≠i}D_{j} together with the Whitney disks of W whose trees do not have an i-labeled vertex. In the present setting, any self-intersections of D_{i} will give rise to self-intersections in the bottom stage surface of G_{i} (which is bounded by L_{i}), but all higher stages of G_{i} will be embedded. At the level of trees, this construction of G_{i} corresponds to replacing each ilabeled vertex on a tree representing λ_{n}(W) with a root [27, Thm.5(v)], which is the same formula as the map on generators defining η_{n}^{i} (signs and orientations are checked in [6, Lem.31] and [6, Sec.4.2] in the setting of repeating Milnor invariants; see also sketch in section 4.3 below). So we have shown that, as an element in π_{1}(B^{4} ∖ W^{i}), the ith longitude ℓ_{i} is represented by the iterated commutators in meridians to the D_{j} that correspond to the image of η_{n}^{i}(λ_{n}(W)) if the inclusion S^{3} ∖ ∂D^{i} → B^{4} ∖ W^{i} induces an isomorphism on the quotients of the Milnor groups by the (n + 2)th terms of the lower central series. Step (iii): To finish the proof of Theorem 24 we will use Dwyer’s Theorem [9] and a new notion of Whitney tower-grope duality to check that the inclusion S^{3} ∖ ∂D^{i} → B^{4} ∖ W^{i} does indeed induce the desired isomorphism on the quotients of the Milnor groups by the (n+2)th terms of the lower central series. It is easy to check that the inclusion induces an isomorphism on first homology, so by [9] the kernel of the induced map on π_{1} is generated by the attaching maps of the 2-cells of surfaces generating the (integral) second homology group H_{2}(B^{4} ∖ W^{i}). The order 0 self-intersections D_{j} ∩ D_{j} only contribute Milnor relations, coming from the attaching maps of the 2-cells of the Clifford tori around the self-intersections. If the D_{j} were pairwise disjoint, then by introducing (more) self-intersections as needed (by finger moves realizing the Milnor relations, see e.g. [18, XII.2]), it could be arranged that π_{1}(B^{4} ∖ D^{i}) was in fact isomorphic to the free Milnor group. Since the D_{j} will generally intersect each other, we have to use the fact that W^{i} is a non-repeating Whitney tower of order n to show that any new relations coming from (higher-order) intersections are trivial modulo (n + 2)-fold commutators. Since H_{2}(B^{4} ∖ W^{i}) is Alexander dual to H_{1}(W^{i}, ∂D^{i}), the proof of Theorem 24 is completed by applying the following general duality result to W^{i} ⊂ B^{4}, which shows that the other generating surfaces extend to class n + 2 gropes. □

Proposition 25 (Whitney tower-grope duality) If V is a split Whitney tower on A : Σ = ⨿_{j}Σ_{j}^{2} → X, where each order 0 surface A_{j} is a sphere S^{2} → X or a disk (D^{2}, ∂D^{2}) → (X, ∂X), then there exist dyadic gropes G_{k} ⊂ X ∖ V such that the G_{k} are geometrically dual to a generating set for the relative first homology group H_{1}(V, ∂A). Furthermore, the tree t(G_{k}) associated to each G_{k} is obtained by attaching a rooted edge to the interior of an edge of a tree t_{p} associated to an unpaired intersection p of V.

Here geometrically dual means that the bottom stage surface of each G_{k} bounds a 3–manifold which intersects exactly one generating curve of H_{1}(V, ∂A) transversely in a single point, and is disjoint from the other generators. In particular, there are as many gropes G_{k} as free generators of H_{1}(V, ∂A). Note that it follows from the last sentence of the proposition that if V is order n, then each G_{k} is class n + 2.

Proof: Since the A_{j} are simply-connected, the group H_{1}(V, ∂A) is generated by sheet-changing curves in V which pass once through a transverse intersection (and avoid all other transverse intersections in V). Such curves either pass through an unpaired intersection or a paired intersection. First we consider a sheet-changing curve through an unpaired intersection p ∈ W_{I} ∩ W_{J} (so t_{p} = ⟨I, J⟩). The Clifford torus T around p is geometrically dual to the curve, and the dual pair of circles in T represent meridians to W_{I} and W_{J}, respectively (recall our convention that if, say, J = j is order 0, Then W_{J} = A_{j} is an order 0 surface). The next lemma shows that the circles on T bound branches of the desired grope G_{(I,J)}, with t(G_{(I,J)}) = (I, J).

Figure 9: The normal circle bundle T_{I} to W_{I_{1}} and W_{(I_{1},I_{2})} over the dotted circle and arc on the left is shown on the right.

Lemma 26 Any meridian to a Whitney disk W_{(I_{1},I_{2})} in a Whitney tower V ⊂ X bounds a grope G_{(I_{1},I_{2})} ⊂ X ∖ V such that t(G_{(I_{1},I_{2})}) = (I_{1}, I_{2}).

Proof: As illustrated in Figure 9, such a meridian bounds a punctured Clifford torus T_{I} around one of the intersections paired by W_{(I_{1},I_{2})}. Each of a symplectic pair of circles on T_{I} is a meridian to one of the Whitney disks W_{I_{i}} paired by W_{(I_{1},I_{2})}, so iterating this construction until reaching meridians to order 0 surfaces yields the desired grope G_{(I_{1},I_{2})} with bottom stage T_{I}. □ Now we consider the sheet-changing curves through intersection points that are paired by Whitney disks. Let W_{(I,J)} be a Whitney disk, and consider the boundary γ of a neighborhood of a boundary arc of W_{(I,J)} in one of the sheets paired by W_{(I,J)}, as illustrated in the left-hand side of Figure 10. We call such a loop γ an oval of the Whitney disk. Clearly, an oval intersects once with a sheetchanging curve that passes once through one of the two intersections paired by W_{(I,J)}. So the normal circle bundle to the sheet over an oval is geometrically dual to such a sheet-changing curve. The following lemma completes the proof of Proposition 25. □

Lemma 27 Let W_{(I,J)} be a Whitney disk in a split Whitney tower V such that W_{(I,J)} contains a trivalent vertex of a tree t_{p} = ⟨(I, J), K⟩ associated to an Figure 10: Left: A (dotted) oval γ ⊂ W_{I}. Right: A (dotted) parallel γ′ to the oval bounds a grope in a nearby ‘time’ coordinate. Not shown is the dual branch of the grope attached to the meridian of W_{J} (as per Lemma 26).

unpaired intersection point p ∈ V. If γ ⊂ W_{I} is an oval of W_{(I,J)} ⊂ V; then the normal circle bundle T to W_{I} over γ is the bottom stage of a dyadic grope G ⊂ (X ∖ V), such that t(G) = (I, (J, K)).

Proof: The torus T contains a symplectic pair of circles, one of which is a meridian to W_{I}, while the other is a parallel γ′ of γ. By Lemma 26, the meridian to W_{I} bounds a grope G_{I} with t(G_{I}) = I, so we need to check that γ′ bounds a grope G_{(J,K)} with tree (J, K). As shown in Figure 10, γ′ bounds a grope whose bottom stage contains a symplectic pair of circles, one of which is a meridian to W_{J}; while the other is either parallel to an oval in W_{(I,J)} around the boundary arc of a higher-order Whitney disk W_{((I,J),K_{1})} for K = (K_{1}, K_{2}) (as shown in the figure), or is a meridian to W_{K} if W_{(I,J)} contains the unpaired intersection p = W_{(I,J)} ∩ W_{K} (since V is split, these are the only two possible types of singularities in W_{(I,J)}). By Lemma 26, the meridian to W_{J} bounds a grope G_{J}; and inductively the oval-parallel circle, or again by Lemma 26 the meridian to W_{K}, bounds a grope G_{K}; so the grope bounded by γ′ does indeed have tree (J, K). □

4.3 The Whitney tower-to-grope construction

This subsection briefly sketches the Whitney tower-to-grope construction used above in Step (ii) of the proof of Theorem 24. In [6] this procedure of converting Whitney towers to capped gropes in order to read off commutators determined by link longitudes is covered in detail in the setting of repeating Milnor invariants. The analogous computation of repeating Milnor invariants from capped gropes described there is trickier in that meridians to a given link component L_{i} can also contribute to the same longitude ℓ_{i}. Hence the computation of ℓ_{i} uses a push-off G′_{i} of the grope body G_{i}, and there may be intersections Figure 11: Moving radially into B^{4} from left to right, a link L ⊂ S^{3} bounds an order 2 (non-repeating) Whitney tower W: The order 0 disk D_{1} consists of a collar on L_{1} together with the indicated embedded disk on the right. The other three order 0 disks in W consist of collars on the other link components which extend further into B^{4} and are capped off by disjointly embedded disks. The order 1 Whitney disk W_{(1,2)} pairs D_{1} ∩ D_{2}, and the order 2 Whitney disk W_{((1,2),3)} pairs W_{(1,2)} ∩ D_{3}, with p = W_{((1,2),3)} ∩ D_{4} the only unpaired intersection point in W. See Figure 12 for the tree-preserving resolution of W to a grope.

between the bottom stage of G′_{i} and caps on G_{i}^{c} which correspond to repeating indices on the associated tree.

Here in the non-repeating setting, ℓ_{i} can be computed as described above directly from the body G_{i} of the capped grope G_{i}^{c}, by throwing away the caps and just remembering how the tips of G_{i} are meridians to the other components corresponding to the univalent labels on t(G_{i}^{c}). See Figures 11 and 12 for the local model near a tree.

A typical 0-surgery which converts a Whitney disk W_{(I,J)} into a cap c_{(I,J)} is illustrated in Figure 13, which also shows how the signed tree is preserved. The sign associated to the capped grope is the product of the signs coming from the intersections of the caps with the bottom stages, which corresponds to the sign of the un-paired intersection point in the Whitney tower; (surgering along the other boundary arc of the Whitney disk, and the other sign cases are checked similarly). If either of the J- and K-labeled sheets is a Whitney disk, then the corresponding cap will be surgered after a Whitney move which turns the single cap-Whitney disk intersection into a cancelling pair of intersections between the cap and a surface sheet that was paired by the Whitney disk, as described in Section 4.2 of [27] (with orientations checked in Lemma 14, Figures 10 and 11 of [31]).

Figure 12: Both sides of this figure correspond to the slice of B^{4} shown in the right-hand side of Figure 11. The tree t_{p} = ⟨((1, 2), 3), 4⟩ ⊂ W is shown on the left. Replacing this left-hand side by the right-hand side illustrates the construction of a class 3 (capped) grope G_{1}^{c} bounded by L_{1}, shown (partly translucent) on the right, gotten by surgering D_{1} and W_{(1,2)}. Each of the disks D_{2}, D_{3} and D_{4} has a single intersection with a cap of G_{1}^{c}, with G_{1} displaying the longitude of L_{1} as the triple commutator [x_{2}, [x_{3}, x_{4}]] in π_{1}(B^{4}∖W^{1}), where W^{1} = D_{2} ∪D_{3} ∪D_{4}. This simple case illustrates the local picture of the general computation of η_{n}^{i}(λ_{n}(W)): For a more complicated L = ∂W this construction would be carried out in a 4–ball neighborhood of each tree containing an ilabeled vertex, and W^{i} would consist of other Whitney disks as well as the D_{j≠i}.

Figure 13: Resolving a Whitney tower to a capped grope preserves the associated oriented trees. The boundary of the I-labeled sheet represents the commutator [x_{J}, x_{K}], up to conjugation, of the meridians x_{J} and x_{K} to the Jand K-labeled sheets.

5 Proof of Theorem 11

Let L ⊂ S^{3} bound an order n Whitney tower in B^{4}, and let X_{L} be the 4– manifold gotten by attaching 0-framed 2–handles to L. We need to show:

(i) Any map A : ⨿^{m}S^{2} → X_{L} of 2–spheres into X_{L} admits an order n Whitney tower.

(ii) For any order n non-repeating Whitney tower W supported by A : ⨿^{m}S^{2} → X_{L}, the non-repeating intersection invariant λ_{n}(A) := λ_{n}(W) ∈ Λ_{n}(m) is independent of the choice of W.

The first statement of Theorem 11 follows from the observations in section 2.10: Any A : ⨿^{m}S^{2} → X_{L} is homotopic to the union of band sums of parallel copies of cores of the 2–handles of X_{L} with 0-framed immersed disks bounded by a link L′ formed from L by the operations of adding parallel components, switching orientations, taking internal band sums and deleting components. Any order n Whitney tower on immersed disks in the 4–ball bounded by L can be modified to give an order n Whitney tower on immersed disks bounded by L′ as described in subsection 2.10. Then the union of the Whitney tower bounded by L′ with the 2–handle cores forms an order n Whitney tower supported by A. To prove the second statement of Theorem 11 we will use the following consequence of Theorem 8: If V is any order n non-repeating Whitney tower on a collection of m immersed 2–spheres in the 4–sphere, then the order n nonrepeating intersection invariant λ_{n}(V) must vanish in Λ_{n}(m). Otherwise, the 2–spheres supporting V could be tubed into disjointly embedded 2–disks in the 4–ball bounded by an unlink U in the 3–sphere to create an order n Whitney tower W_{U} in B^{4} = B^{4}#S^{4} with λ_{n}(U) = λ_{n}(W_{U}) = λ_{n}(V) ≠ 0 ∈ Λ_{n}(m). We start with the case where L is an m-component link, and A = ⨿_{i=1}^{m}A_{i} : S^{2} → X_{L} is such that each A_{i} goes geometrically once over the 2–handle of X_{L} attached to the ith component L_{i} of L, and is disjoint from all other 2–handles. We assume that the orientations of A and L are compatible. In this case, the union of an order n Whitney tower W_{L} bounded by L with the cores of the 2–handles forms an order n Whitney tower W on A, with λ_{n}(W) = λ_{n}(L) := λ_{n}(W_{L}) ∈ Λ_{n}(m). If W′ is any other order n non-repeating Whitney tower on A′, with A′ homotopic to A, then we will show that λ_{n}(W′) = λ_{n}(W) ∈ Λ_{n}(m) by exhibiting the difference λ_{n}(W) − λ_{n}(W′) as λ_{n}(V), where V is an order n non-repeating Whitney tower on a collection of immersed 2–spheres in the 4–sphere. To start the construction, let W′ ⊂ X_{L} = B ∪ H_{1} ∪ H_{2} ∪ ⋯ ∪ H_{m} be an order n non-repeating Whitney tower on A′, with A′ homotopic to A. Here B is the 4–ball, and the H_{i} are the 0-framed 2–handles. Any singularities of W′ which are contained in the H_{i} can be pushed off by radial ambient isotopies, so that W′ may be assumed to only intersect the H_{i} in disjointly embedded disks which are parallel copies of the handle cores. These embedded disks lie in the order 0 2–spheres and the interiors of Whitney disks of W′. It also may be assumed that the trees representing λ_{n}(W′) are disjoint from all the H_{i}. The intersection W′∩∂B is a link L′ in S^{3}, such that L′ is related to L by adding some parallel copies of components and switching some orientations. Note that since each A_{i} goes over H_{i} algebraically once, L′ contains L as a sublink. Write L′ as the union L′ = L^{0} ∪ L^{1} of two links where the components of L^{0} bound handle core disks in the order 0 2–spheres of W′, and the components of L^{1} bound handle core disks in the Whitney disks (surfaces of order at least 1) of W′. For each i, the components of L^{0} which are parallel to L_{i} ⊂ L must come in oppositely oriented pairs except for one component which is oriented the same as L_{i}. The components of L^{1} can be arbitrary parallels of components of L. Now delete the H_{i} from X_{L}, and form S^{4} by gluing another 4–ball B^{+} to B along their 3–sphere boundaries. Since L bounds the order n Whitney tower W_{L} in B^{+}, an order n Whitney tower W^{+} ⊂ B^{+} bounded by L′ can be constructed using parallel order 0 disks and Whitney disks of W_{L} as in section 2.10 above. The union of W^{+} together with W̊′ := W′ ∩ B is an order n non-repeating Whitney tower V := W^{+} ∪ W̊′ on m immersed 2–spheres in S^{4}. Figure 14 gives a schematic illustration of V.

Figure 14: The non-repeating Whitney tower V = W^{+} ∪W̊′ ⊂ S^{4} = B ∪_{S^{3}} B^{+}: The links L ⊂ L′ ⊂ S^{3} are shown in the horizontal middle part of the figure. The components of L are black; the component of L^{1} ⊂ L′ is blue, and an oppositely oriented pair of components in L^{0} ⊂ L′ are shown in green. The lower part of the figure shows W̊′ ⊂ B, and the upper part shows W^{+} ⊂ B^{+}. The tree shown involving the blue L^{1}-component passes down through L^{1} into a Whitney disk of W̊′ and then down into a pair of order 0 disks in W̊′, so the tree is of order greater than n. The pair of trees each having a univalent vertex on a green order 0 disk in W^{+} have opposite signs due to the opposite orientations on the green disks.

We will check that λ_{n}(V) = λ_{n}(W_{L}) − λ_{n}(W′) ∈ Λ_{n}(m), which will complete the proof in this case by the opening observation that λ_{n}(V) vanishes. We take the orientation of (B^{+}, ∂B^{+}) in S^{4} to be the standard orientation of (B^{4}, S^{3}), and that of (B, ∂B) to be the opposite. Since W̊′ ⊂ B contains all the trees representing λ_{n}(W′), these trees contribute the term −λ_{n}(W′) to λ_{n}(V).

Consider next the trees corresponding to intersections in W^{+} involving components of L^{1} (i.e. the trees that intersect at least one order 0 disk of W^{+} bounded by a component of L^{1}). In V these trees are subtrees of trees (for the same intersections) which pass down through L^{1} into the Whitney disks of W̊′ until reaching the order 0 disks in W̊′. Any such tree is of order strictly greater than n, since it contains an order n proper subtree (the part of the tree in W^{+})

– see Figure 14. Such higher-order trees do not contribute to λ_{n}(V). Consider now the remaining trees in V which only involve the components of L^{0}. These trees represent λ_{n}(L^{0}) = λ_{n}(W^{0}), where W^{0} ⊂ W^{+} is the order n Whitney tower in B^{+} bounded by L^{0} ⊂ ∂B^{+}; but we claim that in V these trees contribute exactly λ_{n}(L), which completes the proof in this case. To see the claim, recall that L^{0} consists of L together with oppositely oriented pairs of parallel components of L. Denote by +L_{i}^{j}, −L_{i}^{j} such a pair which is parallel to the ith component L_{i} of L, and which bounded oppositely oriented handlecores +H_{i} and −H_{i} in the jth component A′_{j} of A′. The univalent labels on trees representing λ_{n}(L^{0}) = λ_{n}(W^{0}) which correspond to +L_{i}^{j} and −L_{i}^{j} when considered as trees in V are labeled by the same label j. Such re-labelings correspond exactly to the operations of Lemma 20 in section 2.10.6, which implies that all trees involving such pairs of components contribute trivially to λ_{n}(V), verifying the claim. The proof of Theorem 11 in the general case follows the argument just given with essentially only notational differences: An arbitrary A is represented by the union of a linear combination of cores of the H_{i} with immersed disks in B^{4} bounded by a link L_{A} formed from L by the operations of adding parallel components, switching orientations, taking internal sums and deleting components. Since L bounds an order n Whitney tower, so does L_{A} by section 2.10.6. Hence A supports an order n Whitney tower W with λ_{n}(W) = λ_{n}(L_{A}) ∈ Λ_{n}(m). One shows that λ_{n}(W′) = λ_{n}(L_{A}) for any non-repeating W′ on A′ homotopic to A by proceeding as above with L_{A} taking the place of L.

6 Pulling apart parallel 2–spheres

In this section we prove Theorem 1 of the introduction, which states that for a map A_{0} : S^{2} → X of a 2–sphere in a simply connected 4–manifold X with vanishing normal Euler number, the homological self-intersection number [A_{0}] · [A_{0}] vanishes if and only if any number of parallel copies of A_{0} can be pulled apart. Note that since the Euler number e(A) of the normal bundle of a map A : S^{2} → X of a 2-sphere in a 4–manifold X can be changed by ±2 by performing a cusp homotopy of A, the condition e(A) = 0 can be arranged if and only if the second Stiefel-Whitney class ω_{2} ∈ H^{2}(X; ℤ_{2}) vanishes on [A] (see e.g. [12, Sec.1.3A]). On the other hand, if ω_{2}([A]) ≠ 0, then [A] · [A] is odd and hence not even two copies of A can be pulled apart. The proof of Theorem 1 includes a geometric proof that boundary links in the 3–sphere are link-homotopically trivial (Proposition 28 below). We also give an example (6.2) illustrating that the “only if” direction of Theorem 1 is not generally true in non-simply connected 4–manifolds.

6.1 Proof of Theorem 1.

We drop the subscript from notation and consider a map A : S^{2} → X with vanishing normal Euler number e(A) = 0 and X simply connected. From the relationship [A] · [A] = e(A) + λ(A, A′), and the hypothesis that e(A) = 0, we have that [A] · [A] is equal to the Wall pairing λ(A, A′) which counts signed intersections between A and any transverse parallel copy A′ (a generic normal section). (Since X is simply connected, the Wall pairing is just the usual algebraic intersection number in ℤ.) So the “if” direction of Theorem 1 is clear, since λ(A, A′) obstructs pulling apart any two copies of A. For the other direction, start by observing that the vanishing of [A] · [A] = λ(A, A′) implies that A supports an order 1 Whitney tower W: The intersections between the parallels A and A′ correspond in pairs to self-intersections of A, so λ(A, A′) is equal to twice the sum of signed self-intersections of A. These self-intersections must come in oppositely signed pairs, which admit Whitney disks since X is simply connected. First consider the case where A also has vanishing order 1 intersection invariant (section 2.7): If A is characteristic, then τ_{1}(A) := τ(W) = 0 ∈ T_{1}(1)/INT_{1}(A) ≅ ℤ_{2} = ℤ/2ℤ; or if A is not characteristic, then T_{1}(1)/INT_{1}(A) is trivial (see [30, Sec.1]). By Theorem 2 of [30] the vanishing of τ_{1}(A) implies that A admits an order 2 Whitney tower, so by Lemma 3 of [28] for any m ∈ {3, 4, 5, …}, A admits a Whitney tower W of order m. (The fact that A is connected and X is simply connected is crucial here, since under these hypotheses Lemma 3 of [28] shows that higher-order Whitney towers can be built using a Whitney disk boundary-twisting construction.) Now, taking parallel copies of the Whitney disks in W yields an order m Whitney tower on m + 1 parallel copies of A, as observed above in 2.10.2. In particular, we get an order m non-repeating Whitney tower so, by Theorem 4, the m + 1 parallel copies of A can be pulled apart. Consider now the case where τ_{1}(A) = τ_{1}(W) is the non-trivial element in ℤ_{2}. We will first isolate (to a neighborhood of a point) the obstruction to building an order 2 Whitney tower, and then combine the previous argument away from this point with an application of Milnor’s Theorem 4 of [23] (which we will also prove geometrically in Proposition 28 below). As illustrated in Figure 15, a trefoil knot in the 3–sphere bounds an immersed 2– disk in the 4–ball which supports an order 1 Whitney tower containing exactly one Whitney disk whose interior contains a single order 1 intersection point. It follows that the square knot, which is the connected sum of a right- and a lefthanded trefoil knot, bounds an immersed disk D in the 4–ball which supports a Whitney tower V containing exactly two first order Whitney disks, each of which contains a single order 1 intersection point with D. Being a well-known slice knot, the square knot also bounds an embedded 2–disk D′ in the 4–ball, Figure 15: Moving into B^{4} from left to right: A trefoil knot in S^{3} (left) bounds an immersed disk having a single pair of self-intersections p^{±} admitting a Whitney disk W containing a single order 1 intersection point (center). An unknotted ‘slice’ of the immersed disk is shown on the right. The rest of the immersed disk is described by a null isotopy further into B^{4} (not shown) of this unknot.

and by gluing together two 4–balls along their boundary 3–spheres we get an immersed 2–sphere S = D ∪ D′ in the 4–sphere having the square knot as an “equator” and supporting the obvious order 1 Whitney tower W_{S} consisting of S together with the two Whitney disks from V pairing the intersections in D ⊂ S. Now take W_{S} in a (small) 4–ball neighborhood of a point in X (away from A), and tube (connected sum) A into S. This does not change the (regular) homotopy class of A, so we will still denote this sum by A. Note that by construction there is a (smaller) 4–ball B^{4} such that the intersection of the boundary ∂B^{4} of B^{4} with A is a trefoil knot (one of the trefoils in the connected sum decomposition of the square knot), and B^{4} contains one of the two Whitney disks of W_{S}. Denote by X^{∘} the result of removing from X the interior of B^{4}, and denote by A^{∘} the intersection of A with X^{∘} (so A^{∘} is just A minus a small open disk). Since the order 1 intersection point in the Whitney disk of W_{S} which is not contained in B^{4} now cancels the obstruction τ_{1}(W) ∈ ℤ_{2}, we have that A^{∘} admits an order 2 Whitney tower in X^{∘}, and hence again by Lemma 3 of [28], A^{∘} admits a Whitney tower of any order in X^{∘}. As before, it follows that parallel copies of A^{∘} can be pulled apart by using parallel (non-repeating) copies of the Whitney disks in a high order Whitney tower on A^{∘}. The parallel copies of A^{∘} restrict on their boundaries to a link of 0-parallel trefoil knots in ∂B^{4}, and the proof of Theorem 1 is completed by the following lemma which implies that these trefoil knots bound disjointly immersed 2–disks in B^{4}. □

Proposition 28 If the components L_{i} of a link L = ∪L_{i} ⊂ S^{3} are the boundaries of disjointly embedded orientable surfaces F_{i} ⊂ S^{3} in the 3–sphere, then the L_{i} bound disjointly immersed 2–disks in the 4–ball.

This proposition first appeared as Milnor’s Theorem 4 of [23], and is a special case of the general results of [32] which are proved using symmetric surgery.

Proof: Choose a symplectic basis of simple closed curves on each F_{i} bounding properly immersed 2–disks into the 4–ball. We shall refer to these disks as caps. These caps may intersect each other, but the interiors of these caps lie in the interior of B^{4} and so are disjoint from ∪_{i}F_{i} ⊂ S^{3}. The proof proceeds inductively by using half of these caps to surger each F_{i} to an immersed disk F_{i}^{0}, while using the other half of the caps to construct Whitney disks which guide Whitney moves to achieve disjointness. We start with F_{1}. Let D_{1r} and D_{1r}^{∗} denote the caps bounded by the symplectic circles in F_{1}, with ∂D_{1r} geometrically dual to ∂D_{1r}^{∗} in F_{1}. Step 1: Using finger moves, remove any interior intersections between the D_{1r}^{∗} and any D_{1s} by pushing the D_{1s} down into F_{1} (Figure 16).

Figure 16:

Step 2: Surger F_{1} along the D_{1r} (Figure 17). The result is a properly immersed 2–disk F_{1}^{0} in the 4–ball bounded by L_{1} in S^{3}. The self-intersections in F_{1}^{0} come from intersections and self-intersections in the surgery disks D_{1r}, and any intersections between the D_{1r} and F_{1} created in Step 1, as well as any intersections created by taking parallel copies of the D_{1r} during surgery. We don’t care about any of these self-intersections in F_{1}^{0}, but we do want to eliminate all intersections between F_{1}^{0} and any of the disks D_{j} on the other F_{j}, j ≥ 2. These intersections between F_{1}^{0} and the disks on the other F_{j} all occur in cancelling pairs, with each such pair coming from an intersection between a D_{1r} and a D_{j}. Each of these cancelling pairs admits a Whitney disk W_{1r}^{∗} constructed by adding a thin band to (a parallel copy of) the dual disk D_{1r}^{∗} as illustrated in Figure 17. Note that by Step 1 the interiors of the D_{1r}^{∗} are disjoint from F_{1}^{0}, hence the interiors of the W_{1r}^{∗} are also disjoint from F_{1}^{0}. The interiors of the W_{1r}^{∗} may intersect the D_{j}, but we don’t care about these intersections. Step 3: Do the W_{1r}^{∗} Whitney moves on the D_{j}. This eliminates all intersections between F_{1}^{0} and the disks D_{j} on all the other F_{j} (j ≥ 2). Note that any interior

Figure 17:

intersections the W_{1r}^{∗} may have had with the D_{j} only lead to more intersections among the D_{j}, so these three steps may be iterated, starting next by applying Step 1 to F_{2}. □

6.2 Example

If π_{1}X is non-trivial, then the conclusion of Theorem 1 may not hold, as we now illustrate. Let X be a 4–manifold with π_{1}X ≅ ℤ, such that π_{2}X has trivial order 0 intersection form; and let A_{1} be an immersed 0-framed 2–sphere admitting an order 1 Whitney tower W in X with a single order 1 intersection point p such that τ_{1}(A_{1}) = t_{p} ∈ T_{1}(ℤ, 1) is represented by the single Y -tree Y (e, g, h) = t_{p} having one edge labeled by the trivial group element e, and the other edges labeled by non-trivial elements g ≠ h, all edges oriented towards the trivalent vertex. Such examples are given in [30], and can be easily constructed by banding together Borromean rings in the boundary of B^{3}×S^{1} and attaching a 0-framed two handle. If A_{2} and A_{3} are parallel copies of A_{1}, then the order 1 non-repeating intersection invariant λ_{1}(A_{1}, A_{2}, A_{3}) takes values in Λ_{1}(ℤ, 3) (since the vanishing of the order 0 intersections means that all INT_{1} relations are trivial). By normalizing the group element decorating the edge adjacent to the 1-label to the trivial element using the HOL relations, Λ_{1}(ℤ, 3) is isomorphic to ℤ[ℤ × ℤ]. Using six parallel copies of the Whitney disk in W, we can compute that λ_{1}(A_{1}, A_{2}, A_{3}) is represented by the sum of six Y -trees Y (e, g, h), where the univalent vertex labels vary over the permutations of {1, 2, 3} (see [30, Thm.3.(iii)]). This element corresponds to the element

(g, h)−(h, g)+(hg^{−1}, g^{−1})−(g^{−1}, hg^{−1})+(gh^{−1}, h^{−1})−(h^{−1}, gh^{−1}) ∈ ℤ[ℤ×ℤ]

which is non-zero if (and only if) g and h are distinct non-trivial elements of ℤ. Since λ_{1}(A_{1}, A_{2}, A_{3}) ∈ Λ_{1}(ℤ, 3) is a well-defined homotopy invariant [30, Thm.3], the A_{i} can not be pulled apart whenever g and h are distinct and non-trivial.

7 Dual spheres and stablizations

This section contains proofs of Theorem 2 and Theorem 13, both of which involve using low-order intersections to kill higher-order obstructions.

7.1 Proof of Theorem 13

We need to show that surfaces A_{i} with pairwise vanishing Wall intersections can be pulled apart if they have algebraic duals B_{i} : S^{2} → X. Proof: Wall’s intersection pairing λ(A_{i}, A_{j}) ∈ ℤ[π] is defined when the A_{k} : Σ_{k} → X are maps of simply connected surfaces Σ_{k}, or more generally when the A_{k} are π_{1}-null (Remark 16). The pairwise vanishing of Wall’s invariant gives an order 1 non-repeating Whitney tower on the A_{i} (2.6). Assuming inductively for 1 ≤ n < m − 2 the existence of an order n non-repeating Whitney tower W on the A_{i}, it is enough to show that it can be arranged that λ_{n}(W) = 0 ∈ Λ_{n}(π, m), which allows us to find an order n + 1 non-repeating Whitney tower by Theorem 6, and then to apply Theorem 4 when n = m − 2. By performing finger moves to realize the rooted product, any order n Whitney tower W ⊂ X can be modified (in a neighborhood of a 1-complex) to have an additional clean order n Whitney disk W_{J} whose decorated tree corresponds to any given bracket J, with edges labeled by any given elements of π := π_{1}X. If J is non-repeating and does not contain the label i, then tubing the 2–sphere B_{i} into W_{J} will change λ_{n}(W) exactly by adding the order n generator ±⟨J, i⟩_{g}, where the sign can be chosen by the choice of orientation on B_{i}, and the element g ∈ π decorating the i-labeled edge is determined by the choice of arc guiding the tubing (together with the whiskers on W_{J} and A_{i}): Since W_{J} is order n, any intersections between B_{i} and other Whitney disks in W will only contribute intersections of order strictly greater than n; and since λ(A_{j}, B_{i}) = δ_{ij} ∈ ℤ[π], any other intersections between A_{j} and B_{i} contribute only canceling pairs of order n intersections. For 1 ≤ n, any generator of Λ_{n}(π, m) can be realized as ⟨J, i⟩_{g}, so the just-described tubing procedure can be used to modify W until λ_{n}(W) = 0 ∈ Λ_{n}(π, m). □

7.2 Proof of Theorem 2

We need to show that λ_{1}(A) = 0 ∈ Λ_{1}(π, m)/INT_{1}(A) if and only if A : ⨿^{m}S^{2} → X can be pulled apart stably. Note that the vanishing of the homotopy invariant λ_{0}(A) is implied by λ_{1}(A) being defined. Proof: The “if” direction is immediate since λ_{1}(A) only depends on the homotopy class of A (by [30]), and any 2–spheres carried by the stabilization contribute trivially to INT_{1}(A). (See section 8.2 below for details on the INT_{1}(A) relations in the cases m = 3, 4 and X simply connected.) For the “only if” direction, observe first that the vanishing of λ_{1}(A) gives an order 2 non-repeating tower supported by A (by Theorem 6). Assuming inductively for 2 ≤ n < m − 2 the existence of an order n non-repeating Whitney tower W on A, it is enough to show that it can be arranged that λ_{n}(W) = 0 ∈ Λ_{n}(π, m), which allows us to find an order n + 1 non-repeating Whitney tower by Theorem 6, and then to apply Theorem 4 when n = m − 2. For n ≥ 2, any generator of Λ_{n}(π, m) can be written as ⟨I, J⟩_{g} where I and J are both of order greater than or equal to 1, and g ∈ π decorates the edge where the roots of I and J are joined. As in the above proof of Theorem 13, any order n non-repeating Whitney tower W on A can be modified to have new clean Whitney disks W_{I} and W_{J}, without affecting λ_{n}(W). Stabilizing the ambient 4–manifold by S^{2} × S^{2} and tubing W_{I} and W_{J} into a pair of dual 2–spheres coming from the stablization creates an intersection realizing the generator ⟨I, J⟩_{g}, where the element g ∈ π is determined by the choices of whiskers on W_{I} and W_{J} and the tubes into the dual spheres. By realizing generators in this way it can be arranged that λ_{n}(W) = 0. By Poincaré duality, the same holds for any closed simply connected 4–manifold other than S^{4}. For instance, for stablization by ℂℙ^{2} (or \overline{ℂℙ}^{2}), where the dual 2–spheres are copies of ℂℙ^{1}, the framings on W_{I} and W_{J} can be recovered by boundary-twisting [12, Sec.1.3], which only creates repeating intersections. □ We remark that some control on the number of stablizations needed in Theorem 2 can be obtained in terms of m when X is simply connected (so that Λ_{n}(m) is finitely generated). For instance, a single stablization realizes k times a generator by tubing W_{I} or W_{J} into k copies (tubed together) of one of the dual spheres.

8 Second order intersection indeterminacies

It is an open problem to give necessary and sufficient algebraic conditions for determining whether or not an arbitrary quadruple A : ⨿^{4}S^{2} → X of 2–spheres in a 4–manifold can be pulled apart. The vanishing of λ_{0}(A) and λ_{1}(A) is of course necessary, and is equivalent to A admitting an order 2 non-repeating Whitney tower. As explained in the introduction (1.9), refining the sufficiency statement provided by Corollary 7 requires the introduction of intersection relations INT_{2}(A) in the target of λ_{2}(A) which correspond to order 0 and order 1 intersections involving 2-spheres which can be tubed into the Whitney disks of any Whitney tower W supported by A. With an eye towards stimulating future work, the goals of this section are to present some relevant details, describe some partial results, and introduce a related number theoretic problem, while formulating order 2 intersection relations which make the following conjecture precise:

Conjecture 29 If a quadruple of immersed 2–spheres A : ⨿^{4}S^{2} → X in a simply connected 4–manifold X admits an order 2 non-repeating Whitney tower W, then A can be pulled apart if and only if λ_{2}(A) := λ_{2}(W) vanishes in Λ_{2}(4)/INT_{2}(A).

This section is somewhat technical, so we begin by providing an outline: After quickly recalling in 8.1 the lack of indeterminacies in the order 0 non-repeating invariant λ_{0}, the intersection relations INT_{1} in the target of the order 1 nonrepeating invariant λ_{1} are examined in detail for triples and then quadruples of 2–spheres, including notation and examples intended to clarify and motivate the introduction of the intersection relations INT_{2}(A) in the target of the order 2 non-repeating invariant λ_{2}. These INT_{2}(A) relations, which are determined by λ_{0} and λ_{1} on π_{2}X, are discussed throughout section 8.3. Section 8.3.1 observes that if A has any non-trivial order 0 intersections with any other 2– spheres in X, then the target Λ_{2}(4)/INT_{2}(A) of λ_{2}(A) must be finite; and presents two related results, Proposition 30 and Proposition 31, which give sufficient conditions for pulling apart A in the setting where λ_{2}(A) is defined. Section 8.3.3 describes the INT_{2} relations as the image in Λ_{2} ≅ ℤ ⊕ ℤ of a linear map determined by λ_{1} on π_{2}X in the setting where λ_{0} vanishes on π_{2}X, as motivation for the discussion in section 8.3.5 on how non-trivial values of λ_{0} away from A can affect the INT_{2} relations. Section 8.3.6 shows how the INT_{2} relations can be computed as the image of a map whose non-linear part is determined by Diophantine quadratic equations coupled by the order 0 intersection form λ_{0} on π_{2}X, leading naturally to some relevant number theoretic questions. Throughout the rest of this section we assume that the ambient 4–manifold X is simply connected. For brevity we suppress the domains of the components of A from notation and consider collections A = A_{1}, …, A_{m} ↬ X of immersed 2–spheres.

8.1 Order 0 intersection invariants

Recall (2.6) that the order 0 non-repeating intersection invariant λ_{0}(A_{1}, …, A_{m}) = ∑ sign(p) · i − j ∈ Λ_{0}(m) on 2–spheres immersed in a simply connected 4–manifold X carries exactly the same information as the integral homological intersection form on H_{2}(X), with the sum of the coefficients of the i−j corresponding to the usual homological intersection number [A_{i}] · [A_{j}] ∈ ℤ. There are no intersection indeterminacies in this order 0 setting, and A_{1}, …, A_{m} admits an order 1 non-repeating Whitney tower if and only if λ_{0}(A_{1}, …, A_{m}) vanishes in Λ_{0}(m) (which is isomorphic to a direct sum of \binom{m}{2} copies of ℤ, one for each (unordered) pair of distinct indices i, j).

8.2 Order 1 intersection relations.

The order 1 intersection relations INT_{1} are described by order 0 intersections λ_{0}. These INT_{1} relations are examined here in detail for triples and quadruples of 2–spheres, as notational and motivational preparation for describing the order 2 intersection relations.

8.2.1 Order 1 triple intersections

For a triple of immersed 2–spheres A_{1}, A_{2}, A_{3} ↬ X with λ_{0}(A_{1}, A_{2}, A_{3}) = 0, the order 1 non-repeating intersection invariant λ_{1}(A_{1}, A_{2}, A_{3}) is a sum of order 1 Y -trees in Λ_{1}(3) ≅ ℤ modulo the INT_{1}(A_{1}, A_{2}, A_{3}) intersection relations:

_{i}^{j} >−λ_{0}(S_{(i,j)}, A_{k}) = 0

where S_{(i,j)} ranges over π_{2}X, and (i, j) ranges over the three choices of pairs from {1, 2, 3}. (Here the notation _{i}^{j} >−λ_{0}(S_{(i,j)}, A_{k}) indicates the sum of trees gotten by attaching the root of (i, j) to the (i, j)-labeled univalent vertices in λ_{0}(S_{(i,j)}, A_{k}) corresponding to S_{(i,j)}.) Geometrically, these relations correspond to tubing any Whitney disk W_{(i,j)} into any 2–sphere S_{(i,j)}. Via the identification Λ_{1}(3) ≅ ℤ, the quotient Λ_{1}(3)/INT_{1}(A_{1}, A_{2}, A_{3}) is isomorphic to ℤ_{d} = ℤ/dℤ, where d is the greatest common divisor of all the λ_{0}(S_{(i,j)}, A_{k}). This invariant λ_{1}(A_{1}, A_{2}, A_{3}) ∈ Λ_{1}(3)/INT_{1}(A_{1}, A_{2}, A_{3}) is the Matsumoto triple [25] which vanishes if and only if A_{1}, A_{2}, A_{3} admit an order 2 nonrepeating Whitney tower (and hence can be pulled apart [34]). Examples: In the 4–manifold X_{L} gotten by attaching 0-framed 2-handles to the Borromean rings L = L_{1} ∪ L_{2} ∪ L_{3} ⊂ S^{3} = ∂B^{4}, all INT_{1} relations are trivial, and the triple A_{1}, A_{2}, A_{3} of 2–spheres determined (up to homotopy) by the link components can not be pulled apart since λ_{1}(A_{1}, A_{2}, A_{3}) is equal to (±) the generator _{1}^{2} >− 3 of Λ_{1}(3) ≅ ℤ. If X_{L} is changed to X′_{L} by attaching another 2-handle along a meridional circle to L_{3}, then INT_{1}(A_{1}, A_{2}, A_{3}) = Λ_{1}(3) since _{1}^{2} >−λ_{0}(S_{(1,2)}, A_{3}) = (±)_{1}^{2} >− 3, where S_{(1,2)} is the new 2–sphere which is dual to A_{3}. Now A_{1}, A_{2}, A_{3} ↬ X′_{L} can be pulled apart since λ_{1}(A_{1}, A_{2}, A_{3}) takes values in the trivial group.

8.2.2 Computing the INT_{1}(A_{1}, A_{2}, A_{3}) intersection relations

Since each element of π_{2}X can affect the non-repeating order 1 indeterminacies in three independent ways (by tubing 2–spheres into Whitney disks W_{(1,2)}, W_{(1,3)}, and W_{(2,3)}) the INT_{1}(A_{1}, A_{2}, A_{3}) relations can be computed as the image of a linear map ℤ^{r} ⊕ ℤ^{r} ⊕ ℤ^{r} ⟶ ℤ, with r the rank of the ℤ-module π_{2}X modulo torsion. Specifically, let S^{α} be a basis for π_{2}X (mod torsion), and define integers a_{ij}^{α} := λ_{0}(S_{(i,j)}^{α}, A_{k}) for S_{(i,j)}^{α} ranging over the basis, and i, j, k distinct. Then, identifying Λ_{1}(3) ≅ ℤ, the INT_{1}(A_{1}, A_{2}, A_{3}) intersection relations can be described as

∑_{α}(x_{12}^{α}a_{12}^{α} + x_{31}^{α}a_{31}^{α} + x_{23}^{α}a_{23}^{α}) = 0

with the coefficients x_{ij}^{α} ranging (independently) over ℤ.

Using integer vector notation, this map can be written concisely as:

(x_{12}, x_{31}, x_{23}) ⟼ x_{12} · a_{12} + x_{31} · a_{31} + x_{23} · a_{23}.

with “ · ” denoting the dot product in ℤ^{r}.

8.2.3 Order 1 quadruple intersections

For a collection A of four immersed 2–spheres A = A_{1}, A_{2}, A_{3}, A_{4} ↬ X with vanishing λ_{0}(A), the order 1 non-repeating intersection invariant λ_{1}(A) takes values in Λ_{1}(4)/INT_{1}(A), with the INT_{1}(A) relations given by

_{i}^{j} >− λ_{0}(S_{(i,j)}, A_{k}, A_{l}) = 0

where S_{(i,j)} ranges over π_{2}X, and (i, j) ranges over the six choices of distinct pairs from {1, 2, 3, 4}. Each such relation corresponds to tubing the 2–sphere S_{(i,j)} into a Whitney disk W_{(i,j)}. Here Λ_{1}(4) ≅ ℤ ⊕ ℤ ⊕ ℤ ⊕ ℤ, and each generator of the rank r ℤ-module π_{2}X modulo torsion gives six relations, so the target group Λ_{1}(4)/INT_{1}(A) of λ_{1}(A) is the quotient of Z^{4} by the image of a linear map from ℤ^{6r}. The invariant λ_{1}(A) vanishes in Λ_{1}(4)/INT_{1}(A) if and only if A admits an order 2 non-repeating Whitney tower. Example: Note that each of the four copies of ℤ in Λ_{1}(4) corresponds to a target of a Matsumoto triple (a choice of three distinct indices), but the vanishing of the all the triples is not sufficient to get an order 2 non-repeating Whitney tower on the A because of “cross-terms” in the INT_{1} relations; the simplest example is the following: Consider a five component link L = L_{1} ∪ ⋯ ∪ L_{5} ⊂ S^{3} = ∂B^{4} such that L_{1}∪L_{2}∪L_{3} forms a Borromean rings which is split from the component L_{4}, and L_{5} is a band sum of (positive) meridians to L_{3} and L_{4}. In the 4–manifold gotten by attaching 0-framed 2-handles to B^{4} along L, let A_{i} denote the immersed 2–sphere determined (up to homotopy) by the core of the 2-handle attached to L_{i}. Now any three of the quadruple A_{1}, A_{2}, A_{3}, A_{4} will have vanishing first order triple λ_{1}(A_{i}, A_{j}, A_{k}) in Λ_{1}(3)/INT_{1}(A_{i}, A_{j}, A_{k}) for any choice of distinct i, j, k: Since A_{5} is dual to A_{3}, the generator _{1}^{2} >− 3 of Λ_{1}(3) is killed by INT_{1}(A_{1}, A_{2}, A_{3}); and for the other choices of 1 ≤ i < j < k ≤ 4 it is clear that λ_{1}(A_{i}, A_{j}, A_{k}) vanishes since L_{i} ∪ L_{j} ∪ L_{k} is a split link (so A_{i}, A_{j}, A_{k} can be pulled apart). But the first order quadruple λ_{1}(A_{1}, A_{2}, A_{3}, A_{4}) = _{1}^{2} >− 3 = − _{1}^{2} >− 4 is non-zero in Λ_{1}(4)/INT_{1}(A_{1}, A_{2}, A_{3}, A_{4}) ≅ ℤ^{3}, where the only non-trivial INT_{1}(A_{1}, A_{2}, A_{3}, A_{4}) relation is

_{1}^{2} >− λ_{0}(A_{5}, A_{3}, A_{4}) = _{1}^{2} >− 3 + _{1}^{2} >− 4 = 0.

Geometrically, any order 1 intersection in a Whitney tower on A_{1} ∪ A_{2} ∪ A_{3} can be killed by tubing A_{5} into a Whitney disk pairing intersection between A_{1} and A_{2} to create a canceling order 1 intersection, but this also creates an order 1 intersection between the Whitney disk and A_{4}.

8.2.4 Computing the INT_{1}(A_{1}, A_{2}, A_{3}, A_{4}) relations

Choose a basis S^{α} for π_{2}X (mod torsion), and define integers a_{ij,k}^{α} := λ_{0}(S_{(i,j)}^{α}, A_{k}). Then each element of the subgroup INT_{1}(A_{1}, A_{2}, A_{3}, A_{4}) < Λ_{1}(4) can be written

\begin{aligned}_{i}^{j} >− λ_{0}(∑_{α} x_{ij}^{α}S_{(i,j)}^{α}, A_{k}, A_{l}) & = & (∑_{α} x_{ij}^{α}a_{ij,k}^{α}) _{i}^{j} >− k + (∑_{α} x_{ij}^{α}a_{ij,l}^{α}) _{i}^{j} >− l \\ & = & (x_{ij} · a_{ij,k}) _{i}^{j} >− k + (x_{ij} · a_{ij,l}) _{i}^{j} >− l \end{aligned}

where the coefficients in the last expression are dot products of vectors in ℤ^{r}, with i, j, k, l distinct, and r the rank of π_{2}X (mod torsion). Using the basis

{ _{2}^{1} >− 3 , _{2}^{1} >−4 , _{3}^{1} >−4 , _{3}^{2} >−4 }

for Λ_{1}(4), the subgroup INT_{1}(A_{1}, A_{2}, A_{3}, A_{4}) is the image of the linear map ℤ^{6r} ⟶ ℤ^{4}:

(\begin{matrix}x_{12} \\ x_{13} \\ x_{41} \\ x_{23} \\ x_{24} \\ x_{34} \end{matrix}) ⟼ (\begin{matrix}a_{12,3} & −a_{13,2} & 0 & a_{23,1} & 0 & 0 \\ a_{12,4} & 0 & a_{41,2} & 0 & a_{24,1} & 0 \\ 0 & a_{13,4} & a_{41,3} & 0 & 0 & a_{34,1} \\ 0 & 0 & 0 & a_{23,4} & −a_{24,3} & a_{34,2} \end{matrix}) (\begin{matrix}x_{12} \\ x_{13} \\ x_{41} \\ x_{23} \\ x_{24} \\ x_{34} \end{matrix})

where the multiplication of entries is the vector dot product in ℤ^{r}.

8.3 Order 2 intersection relations.

Now consider a quadruple of immersed 2–spheres A = A_{1}, A_{2}, A_{3}, A_{4} ↬ X in a simply connected 4–manifold X, such that λ_{1}(A) = 0 ∈ Λ_{1}(4)/INT_{1}(A), so that A supports an order 2 non-repeating Whitney tower W ⊂ X. Recall that we want to describe order 2 intersection relations INT_{2}(A) which account for changes in the choice of Whitney tower on A and define the target of λ_{2}(A) ∈ Λ_{2}(4)/INT_{2}(A). Note that Λ_{2}(4) is isomorphic to ℤ ⊕ ℤ, generated, for instance, by the elements

t_{1} := _{1}^{2} >−< _{4}^{3} and t_{2} := _{1}^{3} >−< _{4}^{2},

with the IHX relation giving:

_{1}^{4} >−< _{3}^{2} = t_{1} + t_{2}.

We will mostly be concerned with the case that A is in the radical of λ_{0} on π_{2}X, so that for each i ∈ {1, 2, 3, 4} the order 0 pairing λ_{0}(S, A_{i}) vanishes for any immersed 2–sphere S, but first we make some quick general observations related to Theorems 2 and 13 above.

8.3.1 Tubing order 2 Whitney disks into spheres

Let i, j, k, l be distinct indices from {1, 2, 3, 4}. As already observed in the proof of Theorem 13, W can be modified to have an additional clean order 2 Whitney disk W_{((i,j),k)} without creating any new unpaired intersections. If S_{((i,j),k)} is any immersed 2–sphere, then tubing W_{((i,j),k)} into S_{((i,j),k)} preserves the order of the Whitney tower and changes λ(W) by a_{ijk} · ⟨((i, j), k), l⟩, where a_{ijk} = λ_{0}(S_{((i,j),k)}, A_{l}) ∈ ℤ (since any intersections between the new Whitney disk W_{((i,j),k)}#S_{((i,j),k)} and A_{i}, A_{j}, A_{k} are repeating intersections). Letting S_{((i,j),k)} vary over a basis S^{α} for π_{2}X (mod torsion) for distinct triples i, j, k in {1, 2, 3, 4}, this construction generates a subgroup of Λ_{2}(4) isomorphic to dℤ ⊕ dℤ, where d is the greatest common divisor of λ_{0}(S^{α}, A_{i}) over all S^{α} and i. In particular, if these order 0 intersections are relatively prime, then the target Λ_{2}(4)/INT_{2}(A) of λ_{2}(A) is trivial:

Proposition 30 If A = A_{1}, A_{2}, A_{3}, A_{4} admits an order 2 non-repeating Whitney tower and if gcd({λ_{0}(S^{α}, A_{i})}_{α,i}) = 1, then A can be pulled apart. □

8.3.2 Tubing order 1 Whitney disks into spheres

Again as in the proof of Theorem 13, for any choice of distinct indices W can be modified to have two additional clean order 1 Whitney disks W_{(i,j)} and W_{(k,l)}. Tubing either of these Whitney disks into an arbitrary 2–sphere might create unpaired order 1 non-repeating intersections (between A and the 2–sphere) and hence not preserve the order of W, however tubing into 2–spheres created by stabilization does indeed preserve the order (since intersections among order 1 Whitney disks are of order 2). In fact, a single stablization is all that is needed to kill any obstruction to pulling apart the quadruple A_{1}, A_{2}, A_{3}, A_{4}:

Proposition 31 If A = A_{1}, A_{2}, A_{3}, A_{4} admit an order 2 non-repeating Whitney tower, then A can be pulled apart in the connected sum of X with a single S^{2} × S^{2} (or a single ℂℙ^{2}, or a single \overline{ℂℙ}^{2}).

Proof: S^{2} × S^{2}, or ℂℙ^{2}, or \overline{ℂℙ}^{2}. We will show how to change λ_{2}(W) by any integral linear combination a_{1}t_{1} + a_{2}t_{2} of the above generators t_{1}, t_{2} of Λ_{2}(4): To create a_{1}t_{1}, first modify W to have two additional clean Whitney disks W_{(1,2)} and W_{(3,4)}, then tube W_{(1,2)} into S, and tube W_{(3,4)} into |a_{1}|-many copies of S′ (where the sign of a_{1} corresponds to the orientations of the copies of S′). Note that in case of stabilization by ℂℙ^{2} or \overline{ℂℙ}^{2}, the extra intersections coming from taking |a_{1}| copies of ℂℙ^{1} are all repeating intersections, so that λ_{2}(W) is indeed only changed by a_{1}t_{1}. Now, to further create a_{2}t_{2} proceed in the same way starting with two additional clean Whitney disks W_{(1,3)} and W_{(2,4)}, which are tubed into a parallel copy of the same S and |a_{2}|-many copies of S′. This will also create intersections with the previous copies of S and S′, but these extra intersections will all be repeating intersections. (Any Whitney disks tubed into copies of ℂℙ^{1} can be framed as in the proof of Theorem 2, see section 7.2.) □

Remark 32 By Poincaré duality the statement of Proposition 31 holds for a single stabilization by taking the connected sum of X with any simply connected closed 4–manifold other than S^{4}.

From the observations just before Proposition 30, the existence of any nontrivial order 0 intersections between any A_{i} and any 2–spheres in X implies that the obstruction to pulling apart the A_{i} lives in a finite quotient of Λ_{2}(4). Returning to our goal of defining the INT_{2} relations which clarify Conjecture 15, we will consider settings where the target for λ_{2}(A) is potentially infinite.

8.3.3 Linear INT_{2} relations

Assume first that all order 0 non-repeating intersections λ_{0} on π_{2}X vanish. Let i, j, k, l denote distinct indices in {1, 2, 3, 4}. Suppose that W_{(i,j)} is an order 1 Whitney disk in W, and that W′_{(i,j)} is a different choice of order 1 Whitney disk with the same boundary as W_{(i,j)} such that all intersections W′_{(i,j)} ∩ A_{k} and W′_{(i,j)} ∩ A_{l} are paired by order 2 Whitney disks. Then replacing W_{(i,j)} by W′_{(i,j)}, and replacing the order 2 Whitney disks supported by W_{(i,j)} with those supported by W′_{(i,j)}, changes W to another order 2 non-repeating Whitney tower W′ on A. The union of W_{(i,j)} with W′_{(i,j)} along their common boundary is a 2–sphere S_{(i,j)} = W_{(i,j)} ∪ W′_{(i,j)} with λ_{0}(S_{(i,j)}, A_{k}, A_{l}) = 0 ∈ Λ_{0}((i, j), k, l) as pictured (schematically) in Figure 18.

Figure 18: Changing the interior of W_{(i,j)} to W′_{(i,j)} corresponds to tubing W_{(i,j)} into a 2–sphere S_{(i,j)}. Only intersections which contribute to the difference λ_{2}(W) − λ_{2}(W′) ∈ Λ_{2}(4) are shown.

Via the map Λ_{1}((i, j), k, l) → Λ_{2}(4) induced by sending (i, j) −< _{l}^{k} to _{i}^{j} >−< _{l}^{k}, the corresponding change λ_{2}(W) − λ_{2}(W′) ∈ Λ_{2}(4) is equal to the image of the order 1 non-repeating intersection invariant λ_{1}(S_{(i,j)}, A_{k}, A_{l}), which is defined in Λ_{1}((i, j), k, l) since the vanishing of λ_{0} means that all INT_{1} relations are trivial. Similarly changing the interiors of any number of the order 1 Whitney disks in W leads to the following definition, which makes Conjecture 29 precise in this setting:

Definition 33 For a quadruple of 2–spheres A = A_{1}, A_{2}, A_{3}, A_{4} ↬ X with λ_{1}(A) = 0, with X simply connected and λ_{0} vanishing on π_{2}X, define the order 2 intersection relations INT_{2}(A) < Λ_{2}(4) to be the subgroup generated by

_{i}^{j} >− λ_{1}(S_{(i,j)}, A_{k}, A_{l})

over all choices of distinct i, j, k, l and all representatives S_{(i,j)} of π_{2}X.

This definition of INT_{2}(A) describes all possible changes in the order 2 intersections due to choices of Whitney disks for fixed choices of boundaries of order 1 Whitney disks (up to isotopy), so by Proposition 14 what remains to be done to confirm Conjecture 29 in this case is to show that λ_{2}(W) ∈ Λ_{2}(4)/INT_{2}(A) is independent of the choice of order 1 Whitney disk boundaries.

8.3.4 Computing the linear INT_{2} relations

In this setting (where λ_{0} vanishes on π_{2}X), the subgroup INT_{2}(A) can be computed as follows: For a basis S^{α} for the rank r ℤ-module π_{2}X (mod torsion), and integers a_{ij}^{α} defined by

λ_{1}(S_{(i,j)}^{α}, A_{k}, A_{l}) = a_{ij}^{α} (i, j) −< _{l}^{k},

the INT_{2}(A) relations are described as the image of the linear map ℤ^{6r} → ℤ^{2} given in the basis {t_{1}, t_{2}} = { _{1}^{2} >−< _{4}^{3} , _{1}^{3} >−< _{4}^{2} } by:

(\begin{matrix}x_{12} \\ x_{34} \\ x_{13} \\ x_{24} \\ x_{14} \\ x_{23} \end{matrix}) ↦ (\begin{matrix}a_{12} & a_{34} & 0 & 0 & a_{14} & a_{23} \\ 0 & 0 & a_{13} & a_{24} & a_{14} & a_{23} \end{matrix}) (\begin{matrix}x_{12} \\ x_{34} \\ x_{13} \\ x_{24} \\ x_{14} \\ x_{23} \end{matrix})

where the multiplication of entries is vector inner product. Examples in this setting realizing any coefficient matrix can be constructed by attaching 2-handles to B^{4} along 0-framed links in S^{3} with vanishing linking matrix.

8.3.5 Quadratic INT_{2} relations

Now assume that the A_{i} represent elements in the radical of λ_{0} on π_{2}X, but that λ_{0} may otherwise be non-trivial. We continue to investigate changes in order 2 intersections due to choices of interiors of Whitney disks in W supported by A. Changing the interior of a Whitney disk W_{(i,j)} to W′_{(i,j)} along their common boundary again leads to a 2– sphere S_{(i,j)} = W_{(i,j)} ∪W′_{(i,j)} whose order 1 intersections with A_{k}, A_{l} determine λ_{2}(W) − λ_{2}(W′) ∈ Λ_{2}(4), but the order 1 invariant λ_{1}(S_{(i,j)}, A_{k}, A_{l}) that we want to use to measure this change may now itself have indeterminacies coming from non-trivial order 0 intersections between S_{(i,j)} and any 2–spheres in X. Specifically, λ_{1}(S_{(i,j)}, A_{k}, A_{l}) takes values in Λ_{1}((i, j), k, l) modulo INT_{1}(S_{(i,j)}, A_{k}, A_{l}), where the INT_{1}(S_{(i,j)}, A_{k}, A_{l}) relations are:

_{(i,j)}^{k} >− λ_{0}(S_{((i,j),k)}, A_{l}) = 0  (1)
_{l}^{(i,j)} >− λ_{0}(S_{(l,(i,j))}, A_{k}) = 0  (2)
_{k}^{l} >− λ_{0}(S_{(k,l)}, S_{(i,j)}) = 0.  (3)

Note that the first two relations are empty by our assumption that the A_{i} have vanishing order 0 intersections with all 2–spheres. The third relation corresponds to indeterminacies in λ_{1}(S_{(i,j)}, A_{k}, A_{l}) due to the choice of interiors of order 1 Whitney disks pairing A_{k}∩A_{l}, so computing with the order 1 Whitney disks W_{(k,l)} in W determines a lift λ_{1}^{W}(S_{(i,j)}, A_{k}, A_{l}) ∈ Λ_{1}((i, j), k, l). Mapping (i, j) −< _{l}^{k} to _{i}^{j} >−< _{l}^{k}, we have:

λ_{2}(W) − λ_{2}(W′) = _{i}^{j} >− λ_{1}^{W}(S_{(i,j)}, A_{k}, A_{l}) ∈ Λ_{2}(4).

Now consider changing both W_{(i,j)} to W′_{(i,j)}, and some W_{(k,l)} to W′_{(k,l)} in as illustrated in Figure 19 (recall that i, j, k, l are distinct). The resulting change Δ^{W}(S_{(i,j)}, S_{(k,l)}) := λ_{2}(W) − λ_{2}(W′) ∈ Λ_{2}(4) can be expressed as Δ^{W}(S_{(i,j)}, S_{(k,l)}) =

_{i}^{j} >− λ_{1}^{W}(S_{(i,j)}, A_{k}, A_{l}) + _{i}^{j} >− λ_{0}(S_{(i,j)}, S_{(k,l)}) −< _{l}^{k} + λ_{1}^{W}(A_{i}, A_{j}, S_{(k,l)}) −< _{l}^{k}

Here the 2–sphere S_{(k,l)} determined by W_{(k,l)} and W′_{(k,l)} contributes the righthand term λ_{1}^{W}(A_{i}, A_{j}, S_{(k,l)}) just as discussed above for S_{(i,j)}, but now there is also a “cross-term” coming from order 0 intersections between S_{(i,j)} and S_{(k,l)}. As in the previous paragraph λ_{1}^{W}(S_{(i,j)}, A_{k}, A_{l}) and λ_{1}^{W}(A_{i}, A_{k}, S_{(k,l)}) are lifts of the corresponding order 1 invariants. The three homotopy invariants λ_{1}(S_{(i,j)}, A_{k}, A_{l}), λ_{1}(A_{i}, A_{j}, S_{(k,l)}), and λ_{0}(S_{(i,j)}, S_{(k,l)}) are independent, so the given expression for Δ^{W}(S_{(i,j)}, S_{(k,l)}) only depends on W and the homotopy classes of S_{(i,j)} and S_{(k,l)}. Observe that, since the intersection invariants sum over contributions from the Whitney disks, this entire discussion applies word for word to changing all the Figure 19: A schematic illustration of how order 0 intersections between S_{(i,j)} and S_{(k,l)} can contribute order 2 indeterminacies. (Only relevant intersections are shown.)

first order Whitney disks W_{(i,j)} on A_{i} and A_{j}, and all the first order Whitney disks W_{(k,l)} on A_{k} and A_{l}; with the 2–spheres S_{(i,j)} and S_{(k,l)} interpreted as sums (geometrically: unions) of the 2–spheres determined by each pair of Whitney disks.

Definition 34 For a quadruple of 2–spheres A = A_{1}, A_{2}, A_{3}, A_{4} ↬ X with X simply connected and A in the radical of λ_{0} on π_{2}X, define the order 2 intersection relations INT_{2}^{W}(A) < Λ_{2}(4) to be the subset

INT_{2}^{W} := ⋃{−Δ^{W}(S_{(1,2)}, S_{(3,4)}) − Δ^{W}(S_{(1,3)}, S_{(2,4)}) − Δ^{W}(S_{(1,4)}, S_{(2,3)})}

⊂ Λ_{2}(4).

where (i, j), (k, l) vary over the pair-choices (1, 2), (3, 4), and (1, 3), (2, 4) and (1, 4), (2, 3); and where S_{(i,j)} and S_{(k,l)} vary over all (homotopy classes of) 2–spheres in X.

Note that, as defined, INT_{2}^{W} is only a subset of Λ_{2}(4). Since the above construction can be carried out simultaneously for the three pair-choices, it follows that if λ_{2}(W) ∈ INT_{2}^{W}, then it can be arranged that the A_{i} support W′ with λ_{2}(W′) = 0 ∈ Λ_{2}(4), so the A_{i} can be pulled apart. Since INT_{2}^{W} always contains the zero element of Λ_{2}(4), the statement of Conjecture 29 makes sense, with INT_{2}^{W} taking the place of INT_{2}(A). It would be desirable to have a formulation of the general INT_{2} relations just in terms of A, rather than W.

In the case that all order 0 intersections vanish on π_{2}X, then INT_{2}^{W} reduces to the subgroup INT_{2}(A) < Λ_{2}(4) of Definition 33.

8.3.6 Computing the quadratic INT_{2} relations

In this setting, INT_{2}^{W} can be computed as follows: For a basis S^{α} for the rank r ℤ-module π_{2}X (mod torsion), let Q = q^{αβ} = λ_{0}(S^{α}, S^{β}) denote the intersection matrix. For integers a_{ij}^{α} defined by

λ_{1}^{W}(S_{(i,j)}^{α}, A_{k}, A_{l}) = a_{ij}^{α} (i, j) −< _{l}^{k}

we have the formula

Δ^{W} (∑_{α} x_{ij}^{α}S_{(i,j)}^{α}, ∑_{β} x_{kl}^{β}S_{(k,l)}^{β}) =
= ∑_{α} x_{ij}^{α}a_{ij}^{α} + ∑_{β} x_{kl}^{β}a_{kl}^{β} + ∑_{α} ∑_{β} x_{ij}^{α}x_{kl}^{β}q^{αβ}
= x_{ij} · a_{ij} + x_{kl} · a_{kl} + x_{ij}Qx_{kl}^{T}

where the x_{uv} and a_{uv} are vectors in ℤ^{r}. Using the basis {t_{1}, t_{2}} = { _{1}^{2} >−< _{4}^{3} , _{1}^{3} >−< _{4}^{2} } for Λ_{2}(4), computing INT_{2}^{W} amounts to determining the image of the map ℤ^{6r} → ℤ^{2}:

(\begin{matrix}x_{12} \\ x_{34} \\ x_{13} \\ x_{24} \\ x_{14} \\ x_{23} \end{matrix}) ↦ (\begin{matrix}a_{12} & a_{34} & 0 & 0 & a_{14} & a_{23} \\ 0 & 0 & a_{13} & a_{24} & a_{14} & a_{23} \end{matrix}) (\begin{matrix}x_{12} \\ x_{34} \\ x_{13} \\ x_{24} \\ x_{14} \\ x_{23} \end{matrix}) + (\begin{matrix}x_{12}Qx_{34}^{T} + x_{14}Qx_{23}^{T} \\ x_{13}Qx_{24}^{T} + x_{14}Qx_{23}^{T} \end{matrix})

where the multiplication of entries is vector inner product. For example, in the easiest case where just a single 2–sphere generator S has non-trivial self-intersection number λ_{0}(S, S′) = q ≠ 0 ∈ ℤ, we have that INT_{2}^{W} is the image of the map ℤ^{6} → ℤ^{2} given by:

(\begin{matrix}x_{12} \\ x_{34} \\ x_{13} \\ x_{24} \\ x_{14} \\ x_{23} \end{matrix}) ↦ (\begin{matrix}a_{12} & a_{34} & 0 & 0 & a_{14} & a_{23} \\ 0 & 0 & a_{13} & a_{24} & a_{14} & a_{23} \end{matrix}) (\begin{matrix}x_{12} \\ x_{34} \\ x_{13} \\ x_{24} \\ x_{14} \\ x_{23} \end{matrix}) + q (\begin{matrix}x_{12}x_{34} + x_{14}x_{23} \\ x_{13}x_{24} + x_{14}x_{23} \end{matrix})

Examples in this setting realizing any coefficient matrix can be constructed by attaching 2-handles to B^{4} along links in S^{3}, and the following questions arise: Is this image always a subgroup of ℤ ⊕ ℤ? Konyagin and Nathanson have shown in [21, Thm.3] that the image always projects to subgroups in each ℤ-summand. And under what conditions will the image be all of ℤ ⊕ ℤ? This would imply that the A_{i} can be pulled apart. What about analogous questions in the general case where the equations are coupled by the intersection matrix Q?

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Rob Schneiderman Department of Mathematics and Computer Science, Lehman College, City University of New York, New York NY 10468 USA robert.schneiderman@ lehman.cuny.edu Peter Teichner Max-Planck Institut für Mathematik Vivatsgasse 7, 53111 Bonn Germany teichner@mac.com