 Two-handle

A two-handle on an $n$-dimensional manifold with boundary is $D^2\times D^{n-2}$ attached along $S^1\times D^{n-2}$. Its attaching sphere is a circle, and its framing specifies the neighbouring normal disks. In dimension four, this is a framed knot in the three-dimensional boundary. On the fundamental group, the attachment kills the element represented by the attaching circle, together with its normal closure. Different framings can give different resulting manifolds even when the attaching circle is fixed.