Georgia Institute of Technology College of Sciences

For any smooth manifold, the topology of its tangent bundle determines much about how vector fields behave on it. On a closed, connected, oriented surface, every tangent vector field must vanish somewhere, unless the surface is a torus. The hairy ball theorem gives the familiar example: you can’t comb a hairy sphere without making a cowlick.

The situation changes dramatically in dimension 3. Not only can we always find nonvanishing vector fields on closed, oriented 3-manifolds, but their tangent bundles are entirely trivial! In this talk, we’ll prove this remarkable fact using obstruction theory and introduce several ideas from algebraic topology that help us understand vector fields on manifolds.