Disk bundle ($D(E)$)

For a real vector bundle $E\to B$ with a fiber metric, the disk bundle isIts fiberwise boundary is the sphere bundle $S(E)=\{\Vert v\Vert =1\}$. The radial homotopy $H_t(v)=(1-t)v$ gives a deformation retraction onto the zero section, so $D(E)$ is homotopy equivalent to $B$.

If $B$ is a smooth manifold without boundary and the rank is positive, the total-space boundary is exactly $S(E)$. When $B$ is a manifold with boundary, the base boundary also contributes: the boundary faces are $S(E)$ and $D(E){|}_{\partial B}$, meeting along $S(E){|}_{\partial B}$. For the trivial vector bundle of rank one over $[0,1]$, the disk bundle is the rectangle $[0,1]\times [-1,1]$. Its top and bottom sides form the sphere bundle; its left and right sides come from the base boundary. The intersections are corners.

Plumbing identifies a base disk times a fiber disk in one oriented rank-two disk bundle with such a neighborhood in another, exchanging base and fiber coordinates. Their zero sections then intersect transversely once. For disk bundles over $S^2$ arranged along a chain with Euler numbers $-a_i$, the intersection form has diagonal entries $-a_i$, adjacent entries one and other entries zero. The same manifold has a Kirby diagram consisting of adjacent Hopf-linked unknots with framings $-a_i$. Its boundary is described by the corresponding negative continued fraction, with the orientation fixed by the surgery convention.