Research project: Homotopy groups of manifolds

Manifolds are central objects in modern geometry and physics. The algebraic topology of manifolds is a classical topic that has engaged researchers for decades, and continues to do so. Often this takes the form of calculating the cohomology of the manifold, which leads to very concrete and manipulable invariants. Much less has been done to consider the homotopy groups of manifolds. The n ^th homotopy group of a space X is the group of homotopy classes of maps from an n-sphere into X. An enormous amount of research has been done to understand the homotopy groups of spheres, but very little has been done to understand the homotopy groups of manifolds. The aim of this project is to gain a much better understanding of the homotopy groups of certain classes of manifolds by using and enhancing a powerful body of techniques from unstable homotopy theory.