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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.

For $n\ge 1$, a topological $n$-manifold with boundary is a Hausdorff second-countable space locally homeomorphic to relatively open sets in the half-space ${\mathbb{H}}^n$. The openness here is in the subspace topology of ${\mathbb{H}}^n$, not necessarily in ${\mathbb{R}}^n$. Its intrinsic boundary $\partial M$ consists of points sent to $x_n=0$; the remaining points have ordinary coordinate charts with images open in Euclidean space. The intrinsic boundary is independent of the chosen chart.^[ref]

The closed interval $[0,1]$ illustrates the distinction: its interior points have open interval charts, while $0$ and $1$ have charts modeled on $[0,\epsilon )\subseteq {\mathbb{H}}^1$, using $x$ and $1-x$. Its intrinsic boundary is $\{0,1\}$. The circle has empty intrinsic boundary even though its boundary of a set as a subset of the Euclidean plane is the whole circle. A topological manifold without boundary is therefore a special case with $\partial M=\varnothing$, and a smooth atlas on the half-space charts gives a manifold with boundary in the smooth sense.

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.