Symmetries of 4-manifolds

A manifold is a topological space that is locally euclidean, that is in every small neighbourhood looks like euclidean space
R^n, for some n. The number n is the dimension of the manifold. One of the most fundamental questions in topology is to
classify manifolds. In order to make the question more manageable, we often restrict to compact, connected manifolds;
those that roughly speaking are of bounded size, and every two points has a path between them. Every compact,
connected 1-dimensional manifold is equivalent, or homeomorphic, to a circle. Surfaces, or 2-dimensional manifolds, were
classified in the 19th century. We have the orientable surfaces with some nonnegative number of holes, obtained from the
sphere by adding handles, and nonorientable surfaces obtained by adding Möbius bands to the sphere instead.
Remarkably, manifolds of dimension 3 have been understood rather well in the last 50 years, with important breakthroughs
due to Thurston, Perelman and Agol. On the other hand the h-cobordism theorem of Smale, exotic spheres of Kervaire-
Milnor, and the surgery programme of Browder-Novikov-Sullivan-Wall, led to a likewise deep understanding of manifolds of
dimension at least 5. This work helped Smale, Milnor, Novikov, Sullivan, and Thurston win Fields medals.

Manifolds of dimension 4 occupy a curious middle ground, at the confluence of high and low dimensional manifold
topology. Many techniques from both high and low dimensional manifolds partially extend to dimension four, but thus far
never conclusively.
As a result, outstanding mysteries abound. For example, the smooth Poincaré conjecture that every homotopy 4-sphere is
diffeomorphic to the 4-sphere, the Schoenflies problem that every smooth embedding of the 3-sphere in the 4-sphere is
isotopic to the standard equatorial embedding remain open.
On the other hand there are a wealth of techniques for studying 4-manifolds, coming from low dimensional geometric methods
such as knot theory, high dimensional surgery theory, group theory and mathematical physics, as well as techniques special
to dimension 4. In particular the Fields medal work of Freedman and Donaldson opened up the world of 4-manifolds.

The aim of this project is to understand symmetries of 4-manifolds: a symmetry of a manifold is a self-map that preserves the structure.
These are called homeomorphisms, or in the case of smooth manifolds, they are called diffeomorphisms. To avoid repeating myself,
let me just discuss homeomorphisms from now on; everything I say has an analogue for diffeomorphims. The set of homeomorphisms
from a manifold to itself form a group, and they also form a topological space in a natural way. This means that one can study the set
of homeomorphisms from the point of view of group theory and of algebraic topology.

The most basic question is to determine when two homeomorphisms are isotopic, meaning that one map can be continuously deformed
until it agrees with the other map. The isotopy classes of homeomorphisms of a manifold also form a group, called the mapping class
group of the manifold. Studying these groups for surfaces is both an old, beautiful topic, and the subject of significant current research.
It is an exciting new area to investigate the analogous question for 4-dimensional manifolds. The principal goal of this project is to
develop new machinery and techniques with which to do so, and to make new computations of 4-dimensional mapping class groups.