Handle

Handles are to Manifolds as Cells are to CW-Complexes. If $M$ is a Manifold together with a $(k-1)$-Sphere $\Bbb{S}^{k-1}$ embedded in its boundary with a trivial Tubular Neighborhood, we attach a $k$-handle to $M$ by gluing the tubular Neighborhood of the $(k-1)$-Sphere $\Bbb{S}^{k-1}$ to the Tubular Neighborhood of the standard $(k-1)$-Sphere $\Bbb{S}^{k-1}$ in the dim($M$)-dimensional Disk.

In this way, attaching a $k$-handle is essentially just the process of attaching a fattened-up $k$-Disk to $M$ along the $(k-1)$-Sphere $\Bbb{S}^{k-1}$. The embedded Disk in this new Manifold is called the $k$-handle in the Union of $M$ and the handle.

See also Handlebody, Surgery, Tubular Neighborhood