4-manifold

In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, the topological and smooth categories are not equivalent. There exist some topological 4-manifolds which admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). If a closed 4-manifold M is simply connected, then Poincare duality implies that it can be obtained from a wedge of 2-spheres by attaching a 4-ball along its boundary. It follows that the homotopy type of the 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of Michael Freedman implies that the homeomorphism type of the manifold only depends on this intersection form, and on a Z/2Z invariant called the Kirby-Siebenmann invariant, and moreover that every unimodular nondegenerate form can arise. This implies the 4-dimensional topological Poincare conjecture. By contrast, if the 4-manifold is differentiable, a theorem of Simon Donaldson implies that if the form is positive definite, then it is diagonalizable, thus showing that many topological 4-manifolds do not admit a smooth structure. Note: in higher (i.e. higher than 4) dimensions, similar Kirby-Siebenmann invariants provide the obstruction to the existence of a smooth structure. In dimension 3 and lower, every manifold admits a unique smooth structure. This points to the importance of differential topology in dimension 4.

Handle decomposition

A closed 4-manifold M is usually described by a handle decomposition. A 0-handle is just a ball, and the attaching map is disjoint union. A 1-handle is attached along two disjoint 3-balls. A 2-handle is attached along a solid torus; since this solid torus is embedded in a 3-manifold, there is a relation between handle decompositions on 4-manifolds, and knot theory in 3-manifolds. A pair of handles with index differing by 1, whose cores link each other in a sufficiently simple way can be cancelled without changing the underlying manifold. Similarly, such a cancelling pair can be created. Two different smooth handlebody decompositions of a smooth 4-manifold are related by a finite sequence of isotopies of the attaching maps, and the creation/cancellation of handle pairs.

4-Manifold

Handle decomposition

See also