Whitney towers and gropes in 4–manifolds

Abstract

Abstract. Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2-complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4-manifolds and the failure of the Whitney move in terms of iterated 'towers' of Whitney disks. The 'flexibility' of these Whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to n-solvability, k-cobordism and grope concordance. The key observation is that the essential structure of gropes and Whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and Whitney moves. It is shown that a Whitney move in a Whitney tower induces an IHX (Jacobi) relation on the embedded trees.

Embedding obstructions in ${\mathbb R}^d$ from the Goodwillie-Weiss calculus and Whitney disks

“…The other cap is a disk normal to σ i and may be thought of as a fiber of the normal bundle to σ i . A general translation between Whitney towers and capped gropes is discussed in [24]. An advantage of this point of view is the symmetry between the original map of σ j (intersecting σ i in two points, as shown on the left in the figure) and the result of the Whitney move where the two intersections σ i ∩ σ j are eliminated but σ j acquires two intersections with σ k .…”

Clasper Concordance, Whitney towers and repeating Milnor invariants

“…A proof of this result can be found in Lemma 2.18 of [7]; and in the case where all Whitney disks are framed in [25,Lem.3.5] or [26,Lem.13].…”

“…Proof. The (twisted) capped grope concordance between L and the unlink from Theorem 3.4 can be surgered to a (twisted) Whitney tower by [25,Thm.6] and capped off. (The correspondence between twisted caps and twisted Whitney disks is illustrated in e.g.…”

Introduction to Whitney Towers

“…A framed Whitney tower is split if the set of singularities in the interior of any Whitney disk consists of either a single point, or a single boundary arc of a Whitney disk, or is empty. This can always be arranged, as observed in Lemma 13 of [30] (Lemma 3.5 of [26]), by performing finger moves along Whitney disks guided by arcs connecting the Whitney disk boundary arcs (see Figure 10). Implicit in this construction is that the finger moves preserve the Whitney disk twistings (by not twisting relative to the Whitney disk that is being split -see Figure 56).…”