 Three-handle

A three-handle on an $n$-dimensional manifold with boundary, $n\ge 3$, is $D^3\times D^{n-3}$ attached along $S^2\times D^{n-3}$. Its core is $D^3\times \{0\}$ and its attaching sphere is two-dimensional. In dimension three it caps a boundary sphere with a three-ball; in dimension four its attaching region is $S^2\times D^1$. Viewed from the opposite end of a handle decomposition, it becomes an index-$(n-3)$ handle. This dual viewpoint explains why handles of different indices appear in complementary pairs.