Handle decomposition

A handle decomposition builds an $n$-dimensional manifold by successively adjoining $D^h\times D^{n-h}$ along $S^{h-1}\times D^{n-h}$. The attaching data comprise an attaching sphere and its framing of an embedded sphere. Reading a decomposition backwards replaces index $h$ by index $n-h$ and interchanges its attaching spheres and belt spheres.

A handle slide replaces the framed attaching sphere of one handle by a band sum with a parallel copy of another handle of the same index. It changes the attaching data while preserving the diffeomorphism type of the resulting manifold. On the handle incidence matrix it performs an elementary row operation, or a column operation when applied to the dual decomposition.

In the intermediate-index range, if an attaching sphere $A_1$ meets a belt sphere $B_1$ at exactly one transverse point, handle slides can remove the intersections of every other attaching sphere with $B_1$. The dual slides then remove the intersections of $A_1$ with the other belt spheres. Each move uses a band to the unique pivot sheet and a local isotopy removing the old and new intersections. If $m_{ji}=A_j\cdot B_i$ and $\delta =m_{11}=\pm 1$, the remaining incidence block is $m_{ji}-m_{j1}{m}_{1i}/\delta$, the Schur complement of the pivot. Extreme-index cases need separate hypotheses.

The belt sphere of an index-$h$ handle is $\{0\}\times S^{n-h-1}$ in the boundary after attachment. It is the attaching sphere of the dual index-$(n-h)$ handle. Its intersections with the attaching spheres of later handles detect passages through this handle.

The attaching sphere of an index-$h$ handle $D^h\times D^{n-h}$ is $S^{h-1}\times \{0\}$ in the previous boundary. Its framing of an embedded sphere identifies a neighbourhood with $S^{h-1}\times D^{n-h}$ and specifies the handle attachment.