= Three-handle
{title2=$D^3\times D^{n-3}$}
A <three-handle> on an $n$-dimensional <manifold with boundary>, $n\geq3$, 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.
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