Handles are to Manifolds as Cells are to CW-Complexes. If is a Manifold together with a -Sphere embedded in its boundary with a trivial Tubular Neighborhood, we attach a -handle to by gluing the tubular Neighborhood of the -Sphere to the Tubular Neighborhood of the standard -Sphere in the dim()-dimensional Disk.
In this way, attaching a -handle is essentially just the process of attaching a fattened-up -Disk to along the -Sphere . The embedded Disk in this new Manifold is called the -handle in the Union of and the handle.
See also Handlebody, Surgery, Tubular Neighborhood