Introduction to Whitney Towers

TL;DR

This paper introduces Whitney towers in 4-manifolds, explaining their fundamental concepts, classifications, and related invariants, providing a comprehensive overview of their geometric and algebraic properties in four-dimensional topology.

Contribution

It offers an expanded, accessible exposition of Whitney towers, including new insights into their classifications, invariants, and applications in 4-manifold topology.

Findings

Classification of order n twisted Whitney towers in the 4-ball

Development of geometric Jacobi identities for Whitney towers

Analysis of low-order Whitney towers on 2-spheres in 4-manifolds

Abstract

These introductory notes on Whitney towers in 4-manifolds, as developed in collaboration with Jim Conant and Peter Teichner, are an expansion of three expository lectures given at the Winter Braids X conference February 2020 in Pisa, Italy. Topics presented include local manipulations of surfaces in 4-space, fundamental definitions related to Whitney towers and their associated trees, geometric Jacobi identities, the classification of order n twisted Whitney towers in the 4-ball and higher-order Arf invariants, and low-order Whitney towers on 2-spheres in 4-manifolds and related invariants.