2handle

A 2-handle is a specific type of 4-dimensional handle used in the handle decomposition of smooth manifolds. In general, a k-handle is a copy of D^k × D^{n−k} attached to a manifold along ∂D^k × D^{n−k}. For a 4-manifold, a 2-handle has the form D^2 × D^2 and is attached along its boundary, which is S^1 × D^2 ∪ D^2 × S^1, by a chosen attaching map from S^1 × D^2 into the boundary of the existing manifold.

In practice, a 2-handle attachment is often described by a framed knot in the boundary of the manifold prior to attachment. Specifically, one can attach a 2-handle along a framed embedded circle (a knot together with a framing) in the boundary 3-manifold. The framing encodes how the D^2 factor twists as it is glued, and different framings yield non-equivalent attachments. The core of the 2-handle is the disk D^2 × {0}, while the cocore is {0} × D^2; these are the two fundamental disks that reflect the handle’s role in the manifold’s topology.

The addition of a 2-handle changes the topology by introducing a new 2-dimensional homology class and altering the intersection form of the manifold. The framing determines the self-intersection of the core disk. In a handle decomposition, a 2-handle can sometimes be canceled with a 3-handle if the attaching region interacts suitably with existing structures, reflecting a local simplification of the decomposition.

As an example, CP^2 has a standard handle decomposition with one 0-handle, one 2-handle attached along an unknot with framing +1, and one 4-handle, yielding the closed manifold CP^2.

Related topics include Kirby calculus and surgery theory in 4-manifolds.