Whitney towers and gropes in 4–manifolds

Abstract

Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2–complexes called <italic>gropes</italic> . 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 <italic>Whitney towers</italic> is used to demonstrate some geometric consequences for knot and link concordance connected to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -solvability, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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.