 One-handle

A one-handle on an $n$-dimensional manifold with boundary is $D^1\times D^{n-1}$ attached along $S^0\times D^{n-1}$, two disjoint boundary disks. In dimension two it is a band joined along its two ends; depending on the attachment it can connect components or create a new loop. The core is $D^1\times \{0\}$ and the belt sphere is $\{0\}\times S^{n-2}$. The attachment data, including how the two ends are identified with the old boundary, is part of the construction.