We provide new information about the structure of the abelian group of topological concordance classes of knots in S3. One consequence is that there is a subgroup of infinite rank consisting entirely… (More)

We present new results, announced in [T], on the classical knot concordance group C. We establish the nontriviality at all levels of the ‘n-solvable’ filtration · · · ⊆ Fn ⊆ · · · ⊆ F1 ⊆ F0 ⊆ C… (More)

The technical lemma underlying the 5-dimensional topological s-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating ~-null… (More)

We explain the notion of a grope cobordism between two knots in a 3-manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in… (More)

In 1933, anticipating formal cohomology theory, van Kampen [5] gave a slightly rough description of an obstruction o(K) ∈ H Z/2(K , Z) which vanishes if and only if an n-dimensional simplicial… (More)

This is the beginning of an obstruction theory for deciding whether a map f : S ! X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual… (More)

In the early 1980’s Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of… (More)

We explain how the usual algebras of Feynman diagrams behave under the grope degree introduced in [CT]. We show that the Kontsevich integral rationally classifies grope cobordisms of knots in 3-space… (More)

The main result of this paper is a four-dimensional stable version of Kneser’s conjecture on the splitting of three-manifolds as connected sums. Namely, let M be a topological respectively smooth… (More)