Rob Schneiderman

Higher order intersection numbers of 2-spheres in 4-manifolds

This is the beginning of an obstruction theory for deciding whether a map f : S 2 ! X 4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The rst obstruction is Wall’s self-intersection number (f ) which tells the whole story in higher dimensions. Our second order obstruction (f ) is dened if (f ) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of 1X modulo S3 -symmetry (rather then just one copy modulo S2 -symmetry). It generalizes to the non-simply connected setting the Kervaire-Milnor invariant dened in [2] and [12] which corresponds to the Arf-invariant of knots in 3-space. We also give necessary and sucient conditions for moving three maps f1;f 2;f 3 :S 2 !X 4 to a position in which they have disjoint images. Again the obstruction (f1;f 2;f 3) generalizes Wall’s intersection number (f1;f 2) which answers the same question for two spheres but is not sufcient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in dimension 3, our new invariant corresponds to the Milnor invariant (1; 2; 3), generalizing the Matsumoto triple [10] to the non simply-connected setting.

Higher-order intersections in low-dimensional topology

We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants 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 Arf invariants. We also define higher-order Sato–Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.

Whitney tower concordance of classical links

This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato‐Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4‐ball bounded by a link in the 3‐sphere. Applications include computation of the grope filtration and new geometric characterizations of Milnor’s link invariants. 57M25, 57M27, 57Q60; 57N10

Jacobi identities in low-dimensional topology

The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it has also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX relation). In addition, this relation was recently found to arise naturally in a theory of embedding obstructions for 2-spheres in 4-manifolds in terms of Whitney towers. This paper contains the first proof of the four-dimensional version of the Jacobi identity. We also expose the underlying topological unity between the three- and four-dimensional IHX relations, deriving from a beautiful picture of the Borromean rings embedded on the boundary of an unknotted genus 3 handlebody in 3-space. This picture is most naturally related to knot and 3-manifold invariants via the theory of grope cobordisms.

Tree homology and a conjecture of Levine

In his study of the group of homology cylinders, J Levine [23] made the conjecture that a certain group homomorphism 0 W T ! D 0 is an isomorphism. Both T and D 0 are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of Out.Fn/. In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain T and range D 0 of Levine’s map. The isomorphism 0 is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [6; 5]. In this paper and [3] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders. 57M27, 57M25; 57N10

Stable concordance of knots in 3–manifolds

Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnors triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with

Universal quadratic forms and Whitney tower intersection invariants

S^2\times S^2

Cochran’s βi-invariants via twisted Whitney towers

. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orientable surface with the circle, and stable link concordance for many links of null-homotopic knots in orientable 3-manifolds.

Whitney towers and gropes in 4–manifolds

A general algebraic theory of quadratic forms is developed and then specialized from the non-commutative to the commutative to, finally, the symmetric settings. In each of these contexts we construct universal quadratic forms. We then show that the intersection invariant for twisted Whitney towers in the 4‐ball is such a universal symmetric refinement of the framed intersection invariant. As a corollary, we obtain a short exact sequence, Theorem 11, that has been essential in a sequence of papers by the authors on the classification of Whitney towers in the 4‐ball. 57N13; 11E16 Dedicated to Mike Freedman, on the occasion of his 60th birthday

Whitney towers and the Kontsevich integral

We show that Tim Cochrans invariants