Handle decomposition relative to the incoming boundary

Let $(W;M_0,M_1)$ be a smooth cobordism triad with its fixed collars (Smooth cobordism triad for Morse theory). A finite handle decomposition of $(W;M_0,M_1)$ relative to $M_0$ is a finite ordered list of indices $k_1,\dots ,k_r$, together with, for each $i$, an embedding of the attaching region $S^{k_i-1}\times D^{n-k_i}$ of a standard $k_i$-handle (K handle core cocore attaching region and belt sphere) into the outgoing boundary of the manifold built so far (away from the retained $M_0$), such that $W$ is diffeomorphic, relative to $M_0$, to the manifold obtained from the collar $M_0\times [0,\epsilon ]$ by successively attaching the handles in the given order with corners rounded (Attaching a smooth handle with corner rounding).

The stages of the decomposition are the manifolds $W_0:=M_0\times [0,\epsilon ]$ and $W_i$, the result of attaching the first $i$ handles; the last stage $W_r$ is diffeomorphic to $W$ relative to $M_0$. The ordered indices are the indices of the handles; the number of handles of index $k$ is the multiplicity of $k$ in the list.

The endpoint cases are part of the definition, not exceptions.

• If $M_0=\mathrm{\varnothing }$ the initial stage is the empty manifold, and the first handle of any presentation whose later stages are nonempty is a $0$-handle, which is a disjoint copy of $D^n$ attached along $S^{-1}\times D^n=\mathrm{\varnothing }$.
• A $k$-handle with $k=0$ is a disjoint $n$-disk; a $k$-handle with $k=n$ attaches along its whole boundary sphere $S^{n-1}\times D^0=S^{n-1}$ and has empty outgoing region.
• The empty list $r=0$ is allowed: it presents the collar $M_0\times [0,\epsilon ]$ itself, and, when $M_0=\mathrm{\varnothing }$, the empty manifold.

The definition fixes the geometric data only up to the corner-rounding convention of the cited attachment definition. Existence, uniqueness up to diffeomorphism and the correspondence with Morse functions are not asserted here; they are the content of later items of this page.