Introduction to Whitney towers

Introduction to Modular Towers

Knot Concordance, Whitney Towers and L2-signatures

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 topological knot concordance group. As special cases of Whitney towers of height less than four, the bottom part of the filtration exhibits all classical conco...

Whitney towers and the Kontsevich integral

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 with relations well known from the 3–dimensional theory of finite type invariants. Surprisingly, the same exact relations arise in 4 dimensions, for example t...

Whitney Towers and Gropes in 4–manifolds

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 co...

3 Whitney Towers and Gropes in 4 – Manifolds

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 co...