Dual handle decomposition

Let $(W;M_0,M_1)$ be a smooth cobordism triad and let a finite handle decomposition of $W$ relative to $M_0$ be given, with handles of indices $k_1,\dots ,k_r$ in that order and handle bodies $D^{k_i}\times D^{n-k_i}$ (Handle decomposition relative to the incoming boundary, K handle core cocore attaching region and belt sphere). The dual decomposition is the following handle presentation of the reversed triad $(W;M_1,M_0)$ relative to $M_1$: its handle bodies are the same products, read with the two disk factors exchanged as $D^{n-k_i}\times D^{k_i}$, and they are attached in the reverse order, the $(r+1-i)$-th handle of the dual presentation being the $i$-th handle of the original one with the factors exchanged.

Under this reading a $k$-handle becomes an $(n-k)$-handle, the attaching region $S^{k-1}\times D^{n-k}$ of the original handle is the outgoing region $D^{n-k}\times S^{k-1}$ of the dual handle, and the attaching sphere $S^{k-1}\times \{0\}$ of the original handle is the belt sphere $\{0\}\times S^{k-1}$ of the dual handle; conversely the belt sphere of the original handle becomes the attaching sphere of the dual handle. In particular the attaching sphere of a dual handle is the belt sphere of the original handle, and conversely.

Endpoint cases are included: a $0$-handle of the original presentation becomes an $n$-handle of the dual presentation and conversely, and an $n$-handle becomes a $0$-handle. The dual decomposition is a presentation of the same manifold $W$; the assertion that it is the decomposition induced by negating a Morse function adapted to the original triad is a theorem, not part of this definition.