Higher-Order Intersections in Low-Dimensional Topology

Higher-order intersections in SPECIAL FEATURE low-dimensional topology

Jim Conanta, Rob Schneidermanb, and Peter Teichnerc,d,1

aDepartment of Mathematics, University of Tennessee, Knoxville, TN 37996-1300; bDepartment of Mathematics and Computer Science, Lehman College, City University of New York, 250 Bedford Park Boulevard West, Bronx, NY 10468; cDepartment of Mathematics, University of California, Berkeley, CA 94720-3840; and dMax Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved February 24, 2011 (received for review December 22, 2010)

We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants W of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with – previously known link invariants such as Milnor, Sato Levine, and Fig. 1. (Left) A canceling pair of transverse intersections between two local Arf invariants. We also define higher-order Sato–Levine and Arf sheets of surfaces in a 3-dimensional slice of 4-space. The horizontal sheet invariants and show that these invariants detect the obstructions appears entirely in the “present,” and the red sheet appears as an arc that to framing a twisted Whitney tower. Together with Milnor invar- is assumed to extend into the “past” and the “future.” (Center) A Whitney iants, these higher-order invariants are shown to classify the exis- disk W pairing the intersections. (Right) A Whitney move guided by W tence of (twisted) Whitney towers of increasing order in the 4-ball. eliminates the intersections. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations ing double points) into geometric information (existence of of string links and 3-dimensional homology cylinders are described. embeddings). It was successfully used in the classification of manifolds of dimension >4, specifically in Smale’s celebrated MATHEMATICS link concordance ∣ trivalent tree ∣ quasi-Lie algebra ∣ k-slice ∣ h-cobordism theorem (3) (implying the Poincaré conjecture) Jacobi identity and Wall’s surgery theory (4). The failure of the Whitney move in dimension 4 is the main reason that, even today, there is espite how it may appear in high school, mathematics is not no classification of 4-dimensional manifolds in sight. To be more Dall about manipulating numbers or functions in more and precise, one needs to distinguish between topological and smooth more complicated algebraic or analytic ways. In fact, one of the 4-manifolds to make correct statements. A topological n-mani- Rn most interesting quests in mathematics is to find a good notion of fold is locally homeomorphic to , whereas a smooth manifold space. It should be general enough to cover many real life situa- is locally diffeomorphic to it (in the given smooth structure). tions and at the same time sufficiently specialized so that one can Casson realized that in the setting of the 4-dimensional h-cobordism theorem, even though Whitney disks cannot always still prove interesting properties about it. A first candidate was 2 2 4 Euclidean n-space Rn, consisting of n-tuples of real numbers. be embedded (because þ ¼ ), they always fit into what is This covers all dimensions n but is too special: The surface of the now called a Casson tower. This is an iterated construction that earth, mathematically modeled by the 2-sphere S2, is 2-dimen- works in simply connected 4-manifolds, where one adds more and sional but compact, so it cannot be R2. However, S2 is locally more layers of disks onto the singularities of a given (immersed) Euclidean: Around every point one can find a neighborhood that Whitney disk (5). In an amazing tour de force, Freedman (6, 7) can be completely described by two real coordinates (but global showed that there is always a topologically embedded disk in a neighborhood of certain Casson towers (originally, one needed coordinates do not exist). This observation was made into the definition of an n-dimen- seven layers of disks, later this was reduced to three). This result sional manifold in 1926 by Kneser: It is a (second countable) implied the topological h-cobordism theorem (and hence the topological Poincaré conjecture) in dimension 4. At the same Hausdorff space that looks locally like Rn. The development time, Donaldson used gauge theory to show that the smooth of this definition started at least with Riemann in 1854, and 4-dimensional h-cobordism theorem fails (8), and both results important contributions were made by Poincaré and Hausdorff were awarded with a Fields medal in 1982. Surprisingly, the at the turn of the 19th century (1). It covers many important phy- smooth Poincaré conjecture is still open in dimension 4—the only sical notions, such as the surface of the earth, the universe, and remaining unresolved case. space-time (for n ¼ 2, 3, and 4, respectively) but is special enough In the nonsimply connected case, even the topological classi- to allow interesting structure theorems. One such statement is fication of 4-manifolds is far from being understood because Whitney’s (strong) embedding theorem: Any n-manifold Mn can be Casson towers cannot always be constructed. See refs. 9–11 for embedded into R2n (for all n ≥ 1). The proof in small dimensions a precise formulation of the problem and a solution for funda- n ¼ 1, 2 is fairly elementary and special, but in all dimensions mental groups of subexponential growth. However, there is a n>2, Whitney (2) found the following beautiful argument: By simpler construction, called a Whitney tower, which can be per- general position, one finds an immersion M → R2n with at worst formed in many more instances (Fig. 2). Here one again adds transverse double points. By adding local cusps, one can assume more and more layers of disks to a given (immersed) Whitney that all double points can be paired up by Whitney disks as in disk; however, one does not control all intersections as in a Cas- Fig. 1, using the fact that R2n is simply connected. Because 2 þ 2 < 2n and n þ 2 < 2n, one can arrange that all Whitney disks are disjointly embedded, framed, and meet the image of M only Author contributions: J.C., R.S., and P.T. performed research; and J.C., R.S., and P.T. wrote on the boundary. Then a sequence of Whitney moves, as shown in the paper. Fig. 1, leads to the desired embedding of M. The authors declare no conflict of interest. The Whitney move, sometimes also called the Whitney trick, This article is a PNAS Direct Submission. remains a primary tool for turning algebraic information (count- 1To whom correspondence should be addressed. E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1018581108 PNAS ∣ May 17, 2011 ∣ vol. 108 ∣ no. 20 ∣ 8131–8138 Downloaded by guest on September 23, 2021 problem in the easiest possible ambient manifold M ¼ B4, the 4-dimensional ball. We start with maps

… 2 1 → 4 3 A1; ;Am: ðD ;S Þ ðB ;S Þ;

which exhibit a fixed link in the boundary 3-sphere S3. If this link was slice, then the Ai would be homotopic (rel. boundary) to Fig. 2. Part of a Whitney tower in 4-space. disjoint embeddings; and our Whitney tower theory gives obstructions to this situation. In the simplest example discussed above son tower but only pairs of intersections that allow higher-order we have m ¼ 1, and the boundary of A is just a knot K in S3: Whitney disks; see Fig. 3. Thus a Casson tower gives a Whitney tower but not vice versa. Theorem 2. (The easiest case of knots) (14) The first-order intersection The current authors have developed an obstruction theory for τ ∈ T ≅ Z – invariant 1ðA; W iÞ 1 2 can be identified with the Arf such Whitney towers in a sequence of papers (12 20). Even invariant of the knot K. It is thus a well-defined invariant that though the existence of a Whitney tower does not lead to an depends only on ∂A ¼ K. Moreover, it is the complete obstruction to embedded (topological) disk, it is still a necessary condition. finding a Whitney tower of arbitrarily high order ≥2 with boundary K. Hence our obstruction theory provides higher-order (intersection) invariants for the existence of embedded disks, spheres, or There is a very interesting refinement of the theory for knots in surfaces in 4-manifolds. – – The easiest example of our intersection invariant is Wall’s self- the setting of the Cochran Orr Teichner n-solvable filtration: intersection number for disks in 4-manifolds. If A: ðD2; ∂D2Þ → Certain special symmetric Whitney towers of orders that are ðM4; ∂MÞ has a trivial self-intersection number (we say that powers of 2 have a refined measure of complexity called height and are obstructed by higher-order signatures of associated the order zero invariant τ0ðAÞ vanishes), then all self-intersections covering spaces (21). However, there are no known algebraic can be paired up by Whitney disks W i. However, the W i will in “ ” general self-intersect and intersect each other and also the criteria for raising the height of a Whitney tower analogous to τ Theorem 1. original disk A. Our (first-order) intersection invariant 1ðA; W iÞ 1 τ … ⋔ If m> , then the order zero invariant 0ðA1; ;AmÞ is given by counts the transverse intersections A W i and vanishes if they ≔∂ the linking numbers of the components Li Ai of the link all can be paired up by (second-order) Whitney disks W i; j. This ∪m ⊂ 3 τ L ¼ 1Li S that is the boundary of the given disks. Milnor procedure continues with an invariant 2ðA; W i;Wi; jÞ that mea- i¼ ⋔ ⋔ (22, 23) showed in 1954 how to generalize linking numbers μði; jÞ sures both A W i; j and W i W k intersections, and the con- W inductively to higher order. Here we use the order n total Milnor struction of a higher-order Whitney tower if the invariant μ 2 W invariants n, which correspond to all length (n þ ) Milnor num- vanishes. is the union of A (at order 0) and all Whitney disks μ … bers ði1; ;inþ2Þ. W i (order 1), W i; j (order 2), and continuing with higher-order Whitney disks. If A is homotopic (rel. boundary) to an embedding, then these constructions can be continued ad infinitum. Theorem 3. (Milnor numbers as intersection invariants) If a link L The intersection invariants τ ðA; W ;W ;…Þ¼τ ðWÞ take bounds a Whitney tower W of order n, then the Milnor invariants n i i; j n μ values in a finitely generated abelian group T , which is generated k of order k < n vanish. Moreover, the order n Milnor invariants n τ W ∈ T by certain trivalent trees that describe the 1-skeleton of of L can be computed from the intersection invariant nð Þ n. T Compare Theorem 20. a Whitney tower (Fig. 3 and Definition 4). The relations in n correspond to Whitney moves, and quite surprisingly most of these relations can be expressed in terms of the so-called IHX- In the remaining sections, we will make these statements relation that is a geometric incarnation of the Jacobi identity precise and explain how to get complete obstructions for the for Lie algebras. All the relations can be realized by controlled existence of Whitney towers for links. Unlike the case of knots, manipulations of Whitney towers, and as a result we recover these get more and more interesting for increasing order. In the following approximation of the “algebra implies geometry” addition to the above Milnor invariants (higher-order linking principle that is available in high dimensions: numbers), we will need higher-order versions of Sato–Levine and Arf invariants. In a fixed order, these are finitely many Z2-valued Theorem 1. (Raising the order of a Whitney tower) If A supports an invariants, so that, surprisingly, the Milnor invariants already W τ W detect the problem up to this 2-torsion information. order n Whitney tower with vanishing nð Þ, then A is homotopic (rel. boundary) to A0, which supports an order n þ 1 Whitney tower. Compare Theorem 18. Theorem 4. (Classification of Whitney tower concordance) A link L bounds a Whitney tower W of order n if and only if its Milnor As usual in an obstruction theory, the dependence on the invariants, Sato–Levine invariants, and Arf invariants vanish up to lower-order Whitney towers makes it hard to derive explicit order n. Compare Corollary 10. invariants that prevent the original disk A from being homotopic to an embedding. In this paper we discuss how to solve this To prove this classification, we use Theorem 1 to show that the τ W intersection invariant nð Þ leads to a surjective realization map T ↠ W W Rn: n n, where n consist of links bounding Whitney p towers of order n, up to order n þ 1 Whitney tower concordance (see the next section). The Milnor invariant can be translated into μ W → D a homomorphism n: n n, where the latter is a group defined from a free Lie algebra (which can be expressed via rooted trivalent trees modulo the Jacobi identity). The composition η T → W → D n: n n n

Fig. 3. An unpaired intersection-point p among local sheets in a Whitney is hence a map between purely combinatorial objects both given tower (Left), and its associated tree (Right). in terms of trivalent trees. Using a geometric argument (grope

8132 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1018581108 Conant et al. Downloaded by guest on September 23, 2021 duality), we show that it is simply given by summing over all (n − 1) together with (immersed) Whitney disks pairing all order

choices of a root in a given tree (which is a more precise state- (n − 1) intersection points of W. SPECIAL FEATURE ment of Theorem 3). This map was previously studied by Levine The Whitney disks in a Whitney tower may self-intersect in his work on 3-dimensional homology cylinders (24, 25), where and intersect each other as well as lower-order surfaces, but η he made a precise conjecture about the kernel and cokernel of n. the boundaries of all Whitney disks are required to be disjointly He verified the conjecture for the cokernel in ref. 26, using a embedded. In addition, all Whitney disks are required to be generalized Hall algorithm. framed, as is discussed below. In ref. 15 we prove Levine’s full conjecture via an application 4 of combinatorial Morse theory to tree homology. In particular, Whitney Tower Concordance. We now specialize to the case M ¼ B we show that the kernel of η consists only of 2-torsion. This and also assume that a Whitney tower W has disks for its order 0 n 3 4 2-torsion corresponds to our higher-order Sato–Levine and surfaces that have an m-component link in S ¼ ∂B as their ∂W W Arf invariants and is characterized geometrically in terms of a boundary, denoted .Let n be the set of all framed links framing obstruction for twisted Whitney towers (in which certain ∂W, where W is an order n Whitney tower, and the link framing Whitney disks are not required to be framed). is induced by the order 0 disks in W. This defines a filtration In the above classification of Whitney tower concordance there ⋯ ⊆ W3 ⊆ W2 ⊆ W1 ⊆ W0 ⊆ L of the set of framed m-compo- remains one key geometric question: Although our higher-order nent links L ¼ LðmÞ. Note that W0 consists of links that are evenly framed because a component has even framing if and only if it Arf invariants are well-defined, it is not currently known if they 4 are in fact nontrivial. All potential values are realized by simple bounds a framed immersed disk in B . links, so the question here is whether or not there are any further In order to detect what stage of the filtration a particular geometric relations; see Definition 2. We conjecture that indeed link lies in, it would be convenient to define a set measuring W W all the higher-order Arf invariants are nontrivial, or equivalently, the difference between n and nþ1. Because these are sets ~ T~ → W and not groups, the quotient is not defined. However, we can still that our realization maps Rn: n n are isomorphisms for all n. Here T~ is a certain quotient of T by what we call framing define an associated graded set in the following way: n n W 1 3 × 0 1 relations that come from IHX-relations on twisted Whitney Suppose is an order n þ Whitney tower in M ¼ S ½ ; , ~ … towers. For n ≡ 0, 2, 3 mod 4, we do show that R is an isomorph- where each of the order 0 surfaces A1; ;Am is an annulus with n 3 × 0 3 × 1 ism, implying that in this further quotient the intersection invar- one boundary component in S f g and one in S f g. Then ∂ W 1 iant τ ðWÞ depends only on the link ∂W, and not on the choice of we say that the link 0 is order n þ Whitney tower concordant n ∂ W W Whitney tower W. The higher-order Arf invariants appear when to 1 . This allows us to define the associated graded set n MATHEMATICS W 1 n ¼ 4k − 3, and our conjecture says that the same conclusion as n modulo order n þ Whitney tower concordance. Knots holds in these orders. have a well-defined connected sum operation, but the analogous This conjecture is in turn equivalent to the vanishing of the band-sum operation for links is not well-defined, even up to intersection invariants on all immersed 2-spheres in S4. Of course concordance. This makes the following proposition somewhat all such maps are null-homotopic, and a general goal of the surprising; it follows from Theorem 1: Whitney tower theory is to extract higher-order invariants of representatives of classes in the second homotopy group π2M. Proposition 5. (13) Band sum of links induces a well-defined opera- W This obstruction theory is still being developed, but certain tion that makes each n into a finitely generated abelian group. aspects of it appeared in refs. 12, 19, 20, and 27. The fundamental π T π W group 1M leads to more interesting obstruction groups nð 1MÞ Our goal is to determine these groups n. and a nontrivial π2M leads to more relations to make the intersection invariants dependent only on the order zero surfaces. Free Lie and Quasi-Lie Algebras. Let L ¼ LðmÞ denote the free Lie Z … In this paper, we give a survey of the material needed to under- algebra (over the ground ring ) on generators fX 1;X2; ;Xmg. N L ⊕ L L stand the above results for Whitney towers in the 4-ball. More It is -graded, ¼ n n, where the degree n part n is the details and proofs can be found in our recent series of five papers additive abelian group of length n brackets, modulo Jacobi iden- (13–17) from which we also survey here the following aspects of tities, and the self-annihilation relations ½X;X¼0. The free the theory: quasi-Lie algebra L0 is gotten from L by replacing the self-annihilation relations with the weaker antisymmetry relations ½X;Y¼ • Twisted Whitney towers and their framing obstructions −½Y;X. • Geometrically k-slice links and vanishing Milnor invariants L ⊗ L → L The bracketing map 1 nþ1 nþ2 has a nontrivial kernel, • String links and the Artin representation D denoted n. The analogous bracketing map on the free quasi-Lie • Filtrations of 3-dimensional homology cylinders D0 algebra is denoted n. For later purposes, we now define a homo- ℓ D → Z ⊗ L D morphism s 2n: 2n 2 nþ1. Given an element X in 2n, Whitney Towers L its image under the bracketing map is zero in 2nþ2. However, We work in the smooth oriented category (with discussions of L0 regarding the bracket as being in 2nþ2, we get an element of orientations mostly suppressed), even though all results hold 0 the kernel of the projection L → L2 2. in the locally flat topological category by the basic results on 2nþ2 nþ This kernel is isomorphic to Z2 ⊗ L 1 by ref. 26, and so we get topological immersions in Freedman–Quinn (9). In particular, as nþ an element sℓ2 ðXÞ of Z2 ⊗ L 1 as desired. remarked in ref. 13, our techniques do not distinguish smooth n nþ from locally flat surfaces. The Total Milnor Invariant. Let L be a link where all the longitudes Order n Whitney towers are defined recursively as follows. Γ 1 lie in nþ1, the (n þ )th term of the lower central series of the 3 link group Γ ≔ π1ðS \ LÞ. As a consequence of Stallings’s theo- Γ Definition 1: A surface of order 0 in an oriented 4-manifold M is a nþ1 ≅ Fnþ1 ≅ L rem (32), it follows that Γ nþ1, where F ¼ FðmÞ is the connected oriented surface in M with boundary embedded in the nþ2 Fnþ2 μi ∈ L boundary and interior immersed in the interior of M. A Whitney free group on meridians. Let nðLÞ nþ1 denote the image of μ tower of order 0 is a collection of order 0 surfaces. The order of the ith longitude. The total Milnor invariant nðLÞ of order n is a (transverse) intersection point between a surface of order n defined by and a surface of order m is n þ m. The order of a Whitney disk μ ≔ ⊗ μi ∈ L ⊗ L is (n þ 1) if it pairs intersection points of order n.Forn ≥ 1,a nðLÞ ∑X i nðLÞ 1 nþ1: Whitney tower of order n is a Whitney tower W of order i

Conant et al. PNAS ∣ May 17, 2011 ∣ vol. 108 ∣ no. 20 ∣ 8133 Downloaded by guest on September 23, 2021 μ ∈ D “ ” It turns out that, in fact, nðLÞ n [by cyclic symmetry (28)]. slightly modified version of the Whitney tower filtration would μ The invariant nðLÞ is a convenient way of packaging all Milnor put the Milnor invariants all in the right order, with no more need invariants of length n þ 2 in one piece. for the Sato–Levine invariants. In this section we discuss how this corresponds to the geometric notion of twisted Whitney towers. Theorem 6. (16) For all n ∈ N, the total Milnor invariant is a well- μ W → D Twisted Whitney Disks. The normal disk-bundle of a Whitney disk defined homorphism n: n n such that W↬M is isomorphic to D2 × D2 and comes equipped with a μ D0 D i. For even n, n is a monomorphism with image n < n. canonical nowhere-vanishing Whitney section over the boundary μ Kμ ii. For odd n, n is an epimorphism; denote its kernel by n. given by pushing ∂W tangentially along one sheet and normally μ So n is an algebraic obstruction for L bounding a Whitney along the other. tower of order n þ 1, which is a complete invariant in half The Whitney section determines the relative Euler number the cases. In the other half, we need the following additional ωðWÞ ∈ Z, which represents the obstruction to extending the invariants: Whitney section across W. It depends only on a choice of orientation of the tangent bundle of the ambient 4-manifold restricted to – ∈ W Higher-Order Sato Levine Invariants. Suppose L 2n−1 represents the Whitney disk, i.e., a local orientation. Following traditional Kμ μ 0 an element of 2n−1. Because 2n−1ðLÞ¼ , the longitudes lie in terminology, when ωðWÞ vanishes W is said to be framed. (Be- Γ μ ∈ D 2 − 1 – 2 2 2n,so 2nðLÞ 2n is defined. Define the order n Sato cause D × D has a unique trivialization up to homotopy, this ℓ ∘ μ ℓ Levine invariant by SL2n−1ðLÞ¼s 2n 2nðLÞ, where s 2n is terminology is only mildly abusive.) If ωðWÞ¼k, we say that defined above. W is k-twisted, or just twisted if the value of ωðWÞ is not specified (Fig. 4). Theorem 7. (16) For all n, the Sato–Levine invariant gives a well-de- In the definition of an order n Whitney tower given above, all Kμ ↠Z ⊗ L fined epimorphism SL2n−1: 2n−1 2 nþ1. Moreover, it is an Whitney disks are required to be framed (0-twisted). It turns out isomorphism for even n. that the natural generalization to twisted Whitney towers involves allowing nontrivially twisted Whitney disks only in at least “half The case SL1 is the original Sato–Levine (29) invariant of a 2- the order” as follows: component classical link, and we describe in ref. 16 (and below) “ ” 2 how the SL2n−1 are obstructions to untwisting an order n Definition 3: A twisted Whitney tower of order ð2n − 1Þ is just twisted Whitney tower. a (framed) Whitney tower of order ð2n − 1Þ as in Defini- tion 1 above. Higher-Order Arf Invariants. We saw above that the structure of A twisted Whitney tower of order 2n is a Whitney tower having W ≡ 0 the groups n is completely determined for n , 2, 3 mod 4 all intersections of order less than 2n paired by Whitney disks, by Milnor and higher-order Sato–Levine invariants. with all Whitney disks of order less than n required to be framed, but Whitney disks of order at least n allowed to be k-twisted for KSL Theorem 8. (16) Let 4k−3 be the kernel of SL4k−3. Then there is an any k. α Z ⊗ L ↠KSL epimorphism k: 2 k 4k−3. Note that, for any n, an order n (framed) Whitney tower is also an order n twisted Whitney tower. We may sometimes refer α Conjecture 9. k is an isomorphism. to a Whitney tower as a framed Whitney tower to emphasize the distinction, and we will always use the adjective “twisted” in the This conjecture is true when k ¼ 1, and indeed the inverse map setting of Definition 3. −1 α : W1 → Z2 ⊗ L1 is given by the classical Arf invariant of each 1 W∾ Twisted Whitney Tower Concordance. Let n be the set of framed component of the link. 3 α links in S , which are boundaries of order n twisted Whitney Regardless of whether or not Conjecture 9 is true, k induces 4 an isomorphism α¯ on ðZ2 ⊗ L Þ∕Ker α . towers in B , with no requirement that the link framing is induced k k k W∾ W by the order 0 disks. Notice that 2n−1 ¼ 2n−1. Although not immediately obvious, it is true that this defines a filtration Definition 2: The higher-order Arf invariants are defined by ∾ ∾ ∾ ∾ ⋯ ⊆ W3 ⊆ W2 ⊆ W1 ⊆ W0 ¼ L. As in the framed setting W∾ W∾ 1 −1 SL above, letting n be the set n modulo order (n þ ) twisted Arf ≔ ðα¯ Þ : K → ðZ2 ⊗ L Þ∕Ker α : k k 4k−3 k k Whitney tower concordance yields a finitely generated abelian group. Any of the Arfk that are nontrivial would be the only possible remaining obstructions to a link bounding a Whitney tower of order 4k − 2, following the Milnor and Sato–Levine invariants: Theorem 11. (14, 16) The total Milnor invariants give epimorphisms μ W∾↠D ≡ 0 n: n n, which are isomorphisms for n ; 1, 3 mod 4. More- W Corollary 10. (16) The associated graded groups n are classified by j μ 4 − 3 Dj n, SLn if n is odd, and, for n ¼ k , Arfk. W(i,j)

The first unknown Arf invariant is Arf2: W5 → Z2 ⊗ L2, which i Di in the case of 2-component links would be a Z2-valued invariant,

evaluating nontrivially on the Bing double of any knot with non- j Dj trivial classical Arf invariant. Evidence supporting the existence Dj W of nontrivial Arfk is provided by the fact that such links are known (i,j) to not be slice (30). All cases for k>1 are currently unknown, i Di but if Arf2 is trivial, then all higher-order Arfk would also be trivial (14). Fig. 4. Pushing into the 4-ball from left to right: An i-andj-labeled twisted Bing double of the unknot bounds disks Di and Dj , which support a 2-twisted Twisted Whitney Towers W Whitney disk ði; jÞ. The Whitney section is indicated by the dotted red The order n Sato–Levine invariants are defined as a certain loop (Bottom Center), and the intersections between its extension and the projection of order n þ 1 Milnor invariants, suggesting that a Whitney disk (Bottom Right).

8134 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1018581108 Conant et al. Downloaded by guest on September 23, 2021 K∾ μ KSL Σ0 over, the kernel 4k−2 of 4k−2 is isomorphic to the kernel 4k−3 of the i. a basis of curves on i bound disjointly embedded framed gropes

Σ≔∪ Σ SPECIAL FEATURE Sato–Levine map from the previous section. Gij of class k in the complement of i i, Σ00 ii. a basis of curves on i bound immersed disks in the complement ∾ Σ∪ K ≅ Z ⊗ L of G, where G is the union of the gropes Gij. Conjecture 9 hence says that 4k−2 2 k and our Arf invariants Arf represent the only remaining obstruction to a link This is an enormous geometric strengthening of Igusa and k ’ bounding an order 4k − 1 twisted Whitney tower: Orr s result, which under the same assumption on the vanishing of Milnor invariants, shows the existence of a surface Σ with a ∾ basis of curves bounding maps of class k gropes, with no control Corollary 12. The groups W are classified by μ and, for n n on their intersections and self-intersections. Our proof uses the n ¼ 4k − 2, Arf . k full power of the obstruction theory for twisted Whitney towers, whereas they do a sophisticated computation of the third homology of the groups F∕F2 . Gropes and k-Slice Links. Roughly speaking, a link is said to be k “ ” “ k-slice if it is the boundary of a surface that looks like a collec- String Links and the Artin Representation. Let L be a string link with tion of slice disks modulo k-fold commutators in the fundamental 2 × 0 1 ’ ” ⊂ 3 m strands embedded in D ½ ; . By Stallings s theorem (32), group of the complement of the surface. Precisely, L S is the inclusions D2 \ m points × i ↪ D2 × 0;1 \ L for i 0, Σ ⊂ 4 ð f gÞ f g ð ½ Þ ¼ k-slice if L bounds an embedded orientable surface B such 1 induce isomorphisms on all lower central quotients of the fun- that π0ðLÞ → π0ðΣÞ is a bijection and there is a push-off homo- 4 damental groups. In fact, the induced automorphism of the lower morphism π1ðΣÞ → π1ðB \ ΣÞ whose image lies in the kth term of 2 central quotients F∕F of the free group F ¼ π1ðD \ fm pointsgÞ π 4 \ Σ n the lower central series ð 1ðB ÞÞk. Igusa and Orr proved the is explicitly characterized by conjugating the meridional genera- following “k-slice conjecture” in ref. 31: ∕ tors of F by longitudes. Let Aut0ðF FnÞ consist of those auto- morphisms of F∕F , which are defined by conjugating each μ 0 n Theorem 13. (31) A link L is k-slice if and only if iðLÞ¼ for generator and which fix the product of generators. This leads ≤ 2 − 2 SL → ∕ SL all i k . to the Artin representation Aut0ðF Fnþ2Þ, where is the set of concordance classes of pure framed string links. A k-fold commutator in π1X has a nice topological model in The set of string links has an advantage over links in that it terms of a continuous map G → X, where G is a grope of class k. has a well-defined monoid structure given by stacking. Indeed, Such 2-complexes G (with specified “boundary” circle) are recur- modulo concordance, it becomes a (noncommutative) group. MATHEMATICS sively defined as follows. A grope of class 1 is a circle. A grope of Whitney tower filtrations can also be defined in this context, SW SW∾ SL class 2 is an orientable surface with one boundary component. A giving rise to filtrations n and n of this group . grope of class k is formed by attaching to every dual pair of basis ∾ curves on a class 2 grope a pair of gropes whose classes add to k.A Theorem 17. (17) The sets SW and SW are normal subgroups of 1 n n curve γ: S → X in a topological space X is a k-fold commutator if SL, which are central modulo the next order. We obtain nilpotent SL∕SW SL∕SW∾ and only if it extends to a continuous map of a grope of class k. groups n and n , and the associated graded groups Thus one can ask whether being k-slice implies there is a basis of are isomorphic to our previously defined groups: curves on Σ that bound disjointly embedded gropes of class k in 4 \ Σ SW ∕SW ≅ W SW∾∕SW∾ ≅ W∾ B . Call such a link geometrically k-slice. n nþ1 n and n nþ1 n :

Proposition 14. (14) A link L is geometrically k-slice if and only Finally, the Artin representation induces a well-defined epimorphism ∾ ∾ ∈ W Artin : SL∕SW ↠Aut0 F∕F 2 whose kernel is generated by if L 2k−1. n n ð nþ Þ internal band sums of iterated Bing doubles of the figure-eight knot. This is proven using a construction from ref. 18 that allows one to freely pass between class n gropes and order n − 1 Whitney The Artin representation is thus an invariant on the whole SL∕SW∾ towers. So the higher-order Arf-invariants Arf detect the differ- group n , not just on the associated graded groups as in k μ ence between k-sliceness and geometric k-sliceness. It turns out the case of links. It packages the total Milnor invariants k, 0 … that every Arfk value can be realized by (internal) band summing k ¼ ; ;non string links together into a group homomorphism. iterated Bing doubles of the figure-eight knot. Every Bing double (See ref. 17 for Bing-doubling string links.) is a boundary link, and one can choose the bands so that the sum remains a boundary link. This implies the following: Higher-Order Intersection Invariants Proofs of the above results depend on two essential ideas: The higher-order intersection theory of Whitney towers comes with Theorem 15. (14) A link L has vanishing Milnor invariants of all ≤2 − 2 an obstruction theory whose associated invariants take values orders k if and only if it is geometrically k-slice after con- in abelian groups of (unrooted) trivalent trees. And by mapping nected sums with internal band sums of iterated Bing doubles of to rooted trees, which correspond to iterated commutators, the the figure-eight knot. obstruction theory for Whitney towers in B4 can be identified with algebraic invariants of the bounding link in S3. A critical connec- Here (and in Theorem 17 below), the figure-eight knot can be tion between these ideas is provided by the resolution of the replaced by any knot with nontrivial (classical) Arf invariant. Levine conjecture (see below), which says that this map is an The added boundary links in the above theorem bound disjoint 3 4 isomorphism. surfaces in S that clearly allow immersed disks in B bounded by In fact, it can be arranged that all singularities in a Whitney curves representing a basis of first homology. In ref. 14 we will tower are contained in 4-ball neighborhoods of the associated show that this implies: trivalent trees, which sit as embedded “spines,” and all relations among trees in the target group are realized by controlled manip- Theorem 16. (14) A link has vanishing Milnor invariants of all orders ulations of the Whitney disks. Mapping to rooted trees corre- ≤2k − 2 if and only if its components bound disjointly embedded sponds geometrically to surgering Whitney towers to gropes, Σ ⊂ 4 surfaces i B , with each surface a connected sum of two surfaces and these determine iterated commutators of meridians of the Σ0 Σ00 i and i such that Whitney tower boundaries as in Fig. 5.

Conant et al. PNAS ∣ May 17, 2011 ∣ vol. 108 ∣ no. 20 ∣ 8135 Downloaded by guest on September 23, 2021 ∈ T Δ ≔∑ tors t n−1 by ðtÞ v∈ thiðvÞ;ðTvðtÞ;TvðtÞÞi, where TvðtÞ denotes the rooted tree gotten by replacing v with a root, and the sum is over all univalent vertices of t, with iðvÞ the original label of the univalent vertex v. The obstruction theory works as follows: Fig. 5. (Left to Right) An unpaired intersection in a Whitney tower, (part of) its associated tree, and the result of surgering to a grope. τ W Definition 4: The order n intersection tree nð Þ of an order n Whitney tower W is defined to be Trees and Intersections. All trees are unitrivalent, with cyclic order- ings of the edges at all trivalent vertices, and univalent vertices τ W ≔ ϵ ∈ T~ nð Þ ∑ p · tp n; labeled from an index set f1;2;3;…;mg.Arooted tree has one unlabeled univalent vertex designated as the root. Such rooted ϵ 1 trees correspond to formal nonassociative bracketings of ele- where the sum is over all order n intersections p, with p ¼ the ments from the index set. The rooted product ðI;JÞ of rooted trees usual sign of a transverse intersection point (via certain orienta- I and J is the rooted tree gotten by identifying the root vertices of tion conventions; see, e.g., ref. 13). I and J to a single vertex v and sprouting a new rooted edge at v. T~ All relations in n can be realized by controlled manipulations This operation corresponds to the formal bracket, and we identify of Whitney towers, and further maneuvers allow algebraically rooted trees with formal brackets. The inner product hI; Ji of canceling pairs of tree generators to be converted into intersec- rooted trees I and J is the unrooted tree gotten by identifying tion-point pairs admitting Whitney disks. As a result, we get the the roots of I and J to a single nonvertex point. Note that all following partial recovery of the “algebraic cancelation implies the univalent vertices of hI; Ji are labeled. geometric cancellation” principle available in higher dimensions: The order of a tree, rooted or unrooted, is defined to be the number of trivalent vertices, and the following associations of trees to Whitney disks and intersection points respects the notion Theorem 18. (13) If a collection A of properly immersed surfaces in a of order given in Definition 1. simply connected 4-manifold supports an order n Whitney tower W τ W 0 ∈ T~ ∂ 0 To each order zero surface Ai is associated the order zero with nð Þ¼ n, then A is homotopic (rel. )toA , which rooted tree consisting of an edge with one vertex labeled by i, supports an order n þ 1 Whitney tower. ∈ ∩ and to each transverse intersection p Ai Aj is associated ≔ the order zero tree tp hi; ji consisting of an edge with vertices labeled by i and j. The order 1 rooted Y-tree ði; jÞ, with a single Intersection Trees for Twisted Whitney Towers. For any rooted tree J trivalent vertex and two univalent labels i and j, is associated to we define the corresponding ∾-tree (“twisted-tree”), denoted by ∾ any Whitney disk W ði; jÞ pairing intersections between Ai and Aj. J , by labeling the root univalent vertex with the symbol “∾” This rooted tree can be thought of as an embedded subset of M, (which will represent a “twist” in a Whitney disk normal bundle): ∾ with its trivalent vertex and rooted edge sitting in W ði; jÞ, and its J ≔∾−−J. two other edges descending into Ai and Aj as sheet-changing paths. ∾ ~ Definition 5: The group T is the quotient of T2 −1 by the Recursively, the rooted tree ðI;JÞ is associated to any Whitney 2n−1 n boundary-twist relations: disk W ðI; JÞ pairing intersections between W I and W J (see the left- hand side of Fig. 6); with the understanding that if, say, I is just a J singleton i, then W I denotes the order zero surface Ai. To any hði; JÞ;Ji¼i −< ¼ 0. ∈ ∩ J transverse intersection p W ðI; JÞ W K between W ðI; JÞ and ≔ any W K is associated the unrooted tree tp hðI;JÞ;Ki (see − 1 the right-hand side of Fig. 6). Here J ranges over all order n rooted trees (and the first equality is just a reminder of notation). T∾ T~ T Intersection Trees for Whitney Towers. The group T (for each The group 2n is gotten from 2n ¼ 2n by including order n ∾ n ¼ 0;1;2;…) is the free abelian group on (unitrivalent labeled n -trees as new generators and introducing the following new vertex-oriented) order n trees, modulo the usual AS (antisymme- relations (in addition to the IHX and antisymmetry relations try) and IHX (Jacobi) relations: on non-∾ trees):

J∾ ¼ð− JÞ∾ I∾ ¼ H∾ þ X ∾ − hH; Xi 2 · J∾ ¼hJ; Ji:

The left-hand symmetry relation corresponds to the fact that the framing obstruction on a Whitney disk is independent of its T~ ≔ T T~ In even orders we define 2n 2n, and in odd orders 2n−1 orientation; the middle twisted IHX relations can be realized by T is defined to be the quotient of 2n−1 by the framing relations. a Whitney move near a twisted Whitney disk, and the right-hand These framing relations are defined as the image of homomorph- interior twist relations can be realized by cusp-homotopies in isms Δ2 −1: Z2 ⊗ T −1 → T2 −1, which are defined for genera- n n n Whitney disk interiors. As described in ref. 16, the twisted groups ∾ T2 can naturally be identified with a universal quadratic refine- W p n K T W(I,J) ment of the 2n-valued intersection pairing h · ; · i on framed Whitney disks. Recalling from Definition 3 that twisted Whitney disks occur only in even order twisted Whitney towers, intersection trees for W W I I twisted Whitney towers are defined as follows: WJ WJ ∾ W τ W Fig. 6. Local pictures of the trees associated to a Whitney disk ðI; JÞ, and an Definition 6: The order n intersection tree n ð Þ of an order n p ∈ W ∩ W intersection point ðI; JÞ K . twisted Whitney tower W is defined to be

8136 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1018581108 Conant et al. Downloaded by guest on September 23, 2021 1 τ∾ W ≔ ϵ ω ∾ ∈ T∾ ∾ n ð Þ p · tp þ ðW J Þ · J n ; η ðtÞ ≔ X ℓ ⊗ B ðtÞ and η ðJ Þ ≔ η ðh J;JiÞ; ∑ ∑ n ∑ ðvÞ v n 2 n SPECIAL FEATURE v∈ t

where the first sum is over all order n intersections p and the where the first sum is over all univalent vertices v of t, and the ∕2 L ⊗ L second sum is over all order n Whitney disks W J with twisting second expression lies in 1 nþ1 because the coefficient of ω ∈ Z η ðW J Þ (computed from a consistent choice of local orien- nðh J; JiÞ is even. tations). The proof of the following theorem (which implies Theorem 11 By “splitting” the twisted Whitney disks (13), it can be ar- above) shows that the map η corresponds to a construction that ω ≤ 1 ϵ ranged that j ðW J Þj , leading to signs like p (or zero coeffi- converts Whitney towers into embedded gropes (18), via the cients). The obstruction theory also holds for twisted Whitney grope duality of ref. 36: towers: Theorem 20. (14) If L bounds a twisted Whitney tower W of order n, μ Theorem 19. (13) If a collection A of properly immersed surfaces in a then the total Milnor invariants kðLÞ vanish for k < n, μ η ∘ τ∾ W ∈ D simply connected 4-manifold supports an order n twisted Whitney and nðLÞ¼ n n ð Þ n. W τ∾ W 0 ∈ T∾ ∂ tower with n ð Þ¼ n , then A is homotopic (rel. )to 0 1 η A , which supports an order n þ twisted Whitney tower. Thus one needs to understand the kernel of n before the obstruction theory can proceed. This is accomplished by resolving (15) a closely related conjecture of Levine (25), as discussed next. Remark on the Framing Relations. The framing relations in the T~ groups 2n−1 correspond to the twisted IHX relations among The Levine Conjecture and Its Implications. The bracket map kernel ∾ T∾ D -trees in 2n via a geometric boundary-twist operation that con- n turns out to be relevant to a variety of topological settings (see, verts an order n ∾-tree ði; JÞ∾ to an order 2n − 1 (non-∾) e.g., the introduction to ref. 15) and was known to be isomorphic T Q ’ tree hði; JÞ;Ji. to n after tensoring with , when Levine s study of the cobordism groups of 3-dimensional homology cylinders (24, 25) led him T Realization Maps. In ref. 13 we describe how to construct surjective to conjecture that n is, in fact, isomorphic to the quasi-Lie ~ T~ ↠W ∾ T∾↠W∾ D0 η0 realization maps Rn: n n and Rn : n n by applying bracket map kernel n, via the analogous map n, which sums the operation of iterated Bing doubling. This construction is over all choices of roots (as in the left formula for η above). MATHEMATICS essentially the same as Cochran’s realization method for Milnor Levine made progress in refs. 25 and 26, and in ref. 15 we invariants (33, 34) and Habiro’s clasper-surgery (35), extended to affirm his conjecture: twisted Bing doubling (Figs. 4 and 7). To prove the realization η0 T → D0 maps are well-defined, we need to use Theorems 18 and 19, Theorem 21. (15) n: n n is an isomorphism for all n. respectively. The above Conjecture 9 on the nontriviality of the higher- The proof of Theorem 21 uses techniques from discrete Morse order Arf invariants can be succinctly rephrased as the assertion theory on chain complexes, including an extension of the theory ~ ∾ that the realization maps Rn and Rn are isomorphisms for all n. to complexes containing torsion. A key idea involves defining Progress toward confirming this assertion—namely complete combinatorial vector fields that are inspired by the Hall basis answers in 3∕4 of the cases and partial answers in the remaining algorithm for free Lie algebras and its generalization by Levine cases, as described by the above-stated results—has been accom- to quasi-Lie algebras. plished by identifying intersection trees with Milnor invariants, as As described in ref. 16, Theorem 21 has several direct applica- we describe next. tions to Whitney towers, including the completion of the calcula- W∾ tion of n in three out of four cases: Intersection Trees and Milnor’s Link Invariants. The connection ∾ between intersection trees and Milnor invariants is via a surjec- Theorem 22. (16) η : T → D are isomorphisms for n ≡ 0,1,3 η T∾ → D n n n tive map n: n n, which converts trees to rooted trees mod 4. As a consequence, both the total Milnor invariants (interpreted as Lie brackets) by summing over all ways of choos- μ W∾ → D ∾ T∾ → W∾ n: n n and the realization maps Rn : n n are iso- ing a root: morphisms for these orders. For v a univalent vertex of an order n (un-rooted non-∾) tree, denote by B ðtÞ ∈ L 1 the Lie bracket of generators v nþ The consequences listed in the second statement follow from X 1;X2;…;X determined by the formal bracketing of indices m the fact that η is the composition which is gotten by considering v to be a root of t. n ∾ Denoting the label of a univalent vertex v by R μ ℓ ∈ 1 2 … η T∾ → L ⊗ L μ T∾↠n W∾↠n D ðvÞ f ; ; ;mg, the map n: n 1 nþ1 is defined on n: n n n: generators by Theorem 21 is also instrumental in determining the only W∾ possible remaining obstructions to computing 4k−2:

Proposition 23. (16) The map sending a rooted tree J to ∾ ∈ T∾ ðJ;JÞ 4k−2 induces an isomorphism Z ⊗ L ≅ η 2 k Kerð 4k−2Þ:

These symmetric ∾-trees ðJ;JÞ∾ correspond to twisted Whitney disks and determine the higher-order Arf-invariants Arfk. All of our above conjectures are equivalent to the statement that W∾ D ⊕ Z ⊗ L Fig. 7. Realizing an order 2 tree in a Whitney tower by Bing-doubling. 4k−2 is isomorphic to 4k−2 ð 2 kÞ via these maps.

Conant et al. PNAS ∣ May 17, 2011 ∣ vol. 108 ∣ no. 20 ∣ 8137 Downloaded by guest on September 23, 2021 Theorem 22 and Proposition 23 imply Theorem 11 and tower of order 2n. In fact (16), the above-defined higher-order – Z ⊗ L Z ⊗ Corollary 12 above, and ref. 16 describes analogous implications Sato Levine invariants detect the quotient 2 nþ1 of 2 L0 ↦ of the above-described results in the framed setting (Theorems 6, nþ1. Levine (25) showed that the squaring map X ½X;X 7, 8, and Corollary 10). induces an isomorphism

Framed Versus Twisted Whitney Towers 0 Z2 ⊗ L ≅ KerðZ2 ⊗ L2 ↠Z2 ⊗ L2 Þ; This section describes how the higher-order Sato–Levine and Arf k k k invariants can be interpreted as obstructions to framing a twisted which leads to our proposed higher-order Arf invariants Arf . Whitney tower. The starting point is the following surprisingly k It is interesting to note that the case n ¼ 0 leads to the predic- simple relation between twisted and framed Whitney towers of ∾ m tion CokðW0 → W Þ ≅ Z2 ⊗ L1 ≅ ðZ2Þ . This is indeed the various orders: 0 group of framed m-component links modulo those with even ∈ N framings! In fact, the consistency of this computation was the mo- Proposition 24. (13, 16) For any n , there is a commutative dia- tivating factor to consider filtrations of the set of framed links L, gram of exact sequences rather than just oriented links. Filtrations of Homology Cylinders H Garoufalidis and Levine (37) studied the group g of homology cylinders over the compact orientable surface of genus g with one boundary component, modulo homology cobordism. It carries the J – Johnson (relative weight) filtration n and the Goussarov Habiro Y Moreover, there are isomorphisms (clasper) filtration n. We improve results on the comparison of J Y the associated graded groups n and n. T → T∾ ≅ Z ⊗ L0 ≅ T~ → T∾ Cokð 2n 2nÞ 2 nþ1 Kerð 2n−1 2n−1Þ:

In the first row, all maps are induced by the identity on the Theorem 25. (17) For all k ≥ 1, there are exact sequences set of links. To see the exactness, observe that there is a natural W ⊆ W∾ W∾ W i. 0 → Y2 → J2 → Z2 ⊗ L 1 → 0, inclusion n n , and by definition 2 −1 ¼ 2n−1. One then k k kþ ∾ ∾ n 0 → Z ⊗ L → Y → J → 0 W ⊆ W ii. 2 2kþ1 4k−1 4k−1 , needs to show that indeed 2n 2n−1, which is accomplished 0 → KY → Y → J → 0 in ref. 13, and then the exact sequence in Proposition 24 follows iii. 4 −3 4k−3 4k−3 , W ≔ W ∕W W∾ ≔ W∾∕W∾ k because n n nþ1 and n n nþ1. ak Y iv. Z2 ⊗ L →K → Z2 ⊗ L2 → 0. If our above conjectures hold, then for every n the various k 4k−3 k (vertical) realization maps in the above diagram are isomorphisms, which would lead to a computation of the cokernel and ACKNOWLEDGMENTS. This paper was written while the first two authors were W → W∾ visiting the third author at the Max-Planck-Institut für Mathematik (MPIM). kernel of the map n n . As a consequence, we would obtain They all thank MPIM for its stimulating research environment and generous Z ⊗ L0 new concordance invariants with values in 2 nþ1 and defined support. P.T. was also supported by National Science Foundation Grants W∾ on 2n, as the obstructions for a link to bound a framed Whitney DMS-0806052 and DMS-0757312.

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8138 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1018581108 Conant et al. Downloaded by guest on September 23, 2021