The mod-two Morse chain group

Let $f:M\to \mathbb{R}$ be a Morse function on a closed manifold and let $X$ be a downward gradient-like field for $f$ (Downward gradient-like vector fields for a Morse function, Morse functions and excellent Morse functions), and let ${\mathrm{Crit}}_k(f)$ be the set of critical points of index $k$ (Nondegenerate critical points, nullity, index, and coindex). The mod-two Morse chain group $C{M}_k(f,X;\mathbb{Z}\mathrm{/}2)$ is the free $\mathbb{Z}\mathrm{/}2$-module with basis ${\mathrm{Crit}}_k(f)$ (The congruence class $[a{]}_n$ and the quotient set $\mathbb{Z}\mathrm{/}n$, Unital left and right modules over a ring; unqualified module means left module): its elements are the formal sums ${\sum }_{p\in {\mathrm{Crit}}_k(f)}{a}_{p}p$ with $a_p\in \mathbb{Z}\mathrm{/}2$, added coefficientwise. It is well defined because a Morse function on a closed manifold has finitely many critical points (A Morse function on a compact manifold has finitely many critical points), and $C{M}_k(f,X;\mathbb{Z}\mathrm{/}2)=0$ unless $0\le k\le \dim M$. The notation records $X$ although the group depends only on $f$; no orientation of $M$ or of any unstable manifold is used.

Concretely, $\mathbb{Z}\mathrm{/}2$ is the field ${\mathbb{F}}_2$ (For every prime $p$, the two operations on $\mathbb{Z}\mathrm{/}p$ make it a field), and the free module on the set ${\mathrm{Crit}}_k(f)$ is the direct sum ${⨁}_{p\in {\mathrm{Crit}}_k(f)}\mathbb{Z}\mathrm{/}2$ of The free module on a set and its standard basis: its elements are the coefficient functions $a:{\mathrm{Crit}}_k(f)\to \mathbb{Z}\mathrm{/}2$, added pointwise and scaled by the unique $\mathbb{Z}\mathrm{/}2$-action, the basis element $p$ corresponding to the standard basis vector $e_p$. When ${\mathrm{Crit}}_k(f)$ is empty this direct sum is the zero module; when it has $m$ elements the group has $2^m$ elements.

The index of a critical point of a Morse function on an $n$-manifold lies in $\{0,1,\dots ,n\}$ (Nondegenerate critical points, nullity, index, and coindex), so ${\mathrm{Crit}}_k(f)$ is empty outside that range and the chain group vanishes there. The set ${\mathrm{Crit}}_k(f)$ and hence the group depend only on $f$, not on the vector field $X$; the symbol $X$ is retained because the differential defined on these groups will use the trajectories of $X$.