 Handle core

The core of an index-$k$ handle $D^k\times D^{n-k}$ is the disk $D^k\times \{0\}$. Its boundary $S^{k-1}\times \{0\}$ is the attaching sphere. The complementary disk $\{0\}\times D^{n-k}$ is the cocore, whose boundary is the belt sphere.

The core records the new $k$-dimensional direction introduced by handle attachment. Core and cocore meet transversely at the single center point, with complementary dimensions $k$ and $n-k$. This intersection explains their dual roles in handle descriptions and cancellation arguments.

For a three-dimensional one-handle $D^1\times D^2$, the core is an arc joining the two attaching disks, while the cocore is a cross-sectional two-disk. Cutting across the cocore and tracing along the core therefore describe different geometric operations. The full handle is an $n$-dimensional thickening of its core, not just the core disk itself.