Handle decomposition

Remarks

The first step attaches a $0$-handle $D^0 \times D^n = D^n$ to the empty set, whose attaching region is empty. A relative version starts from $\partial_0 M \times [0, 1]$ for a part $\partial_0 M$ of the boundary instead of the empty set; this is the form used for cobordisms. Example: the sphere $S^n$ has a handle decomposition with one $0$-handle and one $n$-handle: the northern and southern hemispheres, glued along the equator $\partial D^n \times D^0 = S^{n-1}$. Non-example: gluing a second disc $D^2$ to $D^2$ along a single arc of its boundary circle is not a handle attachment. In dimension $2$ the attaching region of a $1$-handle is $\partial D^1 \times D^1$, two disjoint arcs. That of a $2$-handle is the whole circle $\partial D^2 \times D^0$, and that of a $0$-handle is empty; none of them is a single arc. If $f \colon M \to [a, b]$ is a Morse function with critical values in $(a, b)$, then each critical point of index $k$ gives one $k$-handle in a handle decomposition of $M$ relative to $f^{-1}(a)$.