Related papers: Geometric Filtrations of Classical Link Concordanc…
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…
The first part of this paper completes the classification of Whitney towers in the 4-ball that was started in three related papers. We provide an algebraic framework allowing the computations of the graded groups associated to geometric…
We show that for each $k\in\mathbb{N}$, a link $L\subset S^3$ bounds a degree $k$ Whitney tower in the 4-ball if and only if it is \emph{$C_k$-concordant} to the unlink. This means that $L$ is obtained from the unlink by a finite sequence…
We show that the Artin representation on concordance classes of string links induces a well-defined epimorphism modulo order n twisted Whitney tower concordance, and that the kernel of this map is generated by band sums of iterated…
This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks…
We show how to measure the failure of the Whitney trick 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…
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…
In a groundbreaking work A. Levine proved the surprising result that there exist knots in homology spheres which are not smoothly concordant to any knot in $S^3$, even if one allows for concordances in homology cobordisms. Since then…
We present complete classifications of links in the 3-sphere modulo framed and twisted Whitney towers in a rational homology 4-ball. This provides a geometric characterization of the vanishing of the Milnor invariants of links in terms of…
We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional…
We introduce a notion of symmetric Whitney tower cobordism between bordered 3-manifolds, aiming at the study of homology cobordism and link concordance. It is motivated by the symmetric Whitney tower approach to slicing knots and links…
We consider the Grope filtration of the classical knot concordance group that was introduced in a paper of Cochran, Orr and Teichner. Our main result is that successive quotients at each stage in this filtration have infinite rank. We also…
We study the structure of the exteriors of gropes and Whitney towers in dimension 4, focusing on their fundamental groups. In particular we introduce a notion of unknottedness of gropes and Whitney towers in the 4-sphere. We prove that…
We continue to develop an obstruction theory for embedding 2-spheres into 4-manifolds in terms of Whitney towers. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees, and…
The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order…
In his study of the group of homology cylinders, J. Levine made the conjecture that a certain homomorphism eta': T -> D' is an isomorphism. Here T is an abelian group on labeled oriented trees, and D' is the kernel of a bracketing map on a…
We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball bounded by the knot. We also give a new 3-dimensional…
Given a finite CW complex $K$, we use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embedding $K$ into a Euclidean space $\mathbb{R}^d$. For $2$-dimensional complexes in $\mathbb{R}^4$, a geometric analogue…
The problem of interpreting a set of ${\cal W}$-algebra constraints constructed in terms of an arbitrarily twisted scalar field as the recursion relations of a topological theory is addressed. In this picture, the conventional models of…
A geometric characterization of the Arf invariant of a knot in the 3-sphere is given in terms of two kinds of 4-dimensional bordisms, half-gropes and Whitney towers. These types of bordisms have associated complexities class and order which…