In mathematics, 4-manifold is a 4 dimensional topological manifold.

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.

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 <math>Z/2Z<math> 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.