Classifying 4-manifolds

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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, albeit restricted to special classes of manifolds. 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 project aims to improve our understanding of 4-dimensions by classifying 4-manifolds in terms of algebraic invariants. Given two 4-manifolds, we seek computable invariants that can decide whether two 4-manifolds are the same, analogous to the number of holes in a surface in dimension two. I have identified a number of open questions in this direction that I believe are tractable given my expertise. In particular certain 4-manifolds with so-called cyclic fundamental groups are not well understood, but with sufficient work this ought to be possible.Four dimensional manifolds come in two distinct flavours: smooth and topological. Roughly speaking, smooth manifolds admit a description using differentiable functions, whereas topological manifolds can be somewhat wilder. The project focusses on topological manifolds. Often complete results on topological 4-manifolds can be obtained, since they can exhibit a more precise correspondence with algebra, whereas there is no analogous global programme for understanding their smooth cousins.