The signed Morse differential over the integers

Let $(f,X)$ be Morse--Smale on a closed manifold, fix an orientation $o{r}_s$ of $W^u(s)$ for every critical point $s$ (The orientation line of a Morse critical point), and let $ϵ(\gamma )\in \{\pm 1\}$ be the comparison sign of Unstable orientations induce orientations of the trajectory moduli spaces. The integral Morse chain group $C{M}_k(f,X;\mathbb{Z})$ is the free $\mathbb{Z}$-module with basis ${\mathrm{Crit}}_k(f)$ (The integers as equivalence classes of pairs of naturals, Unital left and right modules over a ring; unqualified module means left module; the basis is finite by A Morse function on a compact manifold has finitely many critical points), and the signed Morse differential is the $\mathbb{Z}$-linear map ${\mathrm{\partial }}_k:C{M}_k(f,X;\mathbb{Z})\to C{M}_{k-1}(f,X;\mathbb{Z})$ defined on basis elements by ${\mathrm{\partial }}_{k}p:={\sum }_{q\in {\mathrm{Crit}}_{k-1}(f)}({\sum }_{\gamma \in \mathcal{M}(p,q)}ϵ(\gamma ))q\mathrm{.}$ Both sums are finite — the outer one because the critical set is finite and the inner one by Index-one trajectory moduli spaces are finite — and ${\mathrm{\partial }}_k$ depends on the orientation choices $o{r}_s$. Reducing all coefficients modulo two recovers The mod-two Morse differential.

Choice hypotheses. The inner sums are indexed by the zero-dimensional moduli spaces $\mathcal{M}(p,q)$ with $\lambda (p)-\lambda (q)=1$ (Unparametrized Morse trajectory moduli space), whose finiteness Index-one trajectory moduli spaces are finite is established under the Axiom of Choice (The Axiom of Choice), and the comparison signs are supplied under ${\mathrm{A}\mathrm{C}}_{\omega }$ by Unstable orientations induce orientations of the trajectory moduli spaces (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). The bridge AC implies DC implies countable choice is therefore part of the hypothesis package: assuming AC, as the finiteness corollary does, also supplies the ${\mathrm{A}\mathrm{C}}_{\omega }$ used by the orientation lemma. For a fixed $p$ the definition counts only index-$k-1$ critical points; larger positive index drops may carry trajectories but are not counted; the outer sum is over the finite critical set, and each inner sum is finite, so the displayed coefficient of $q$ is an integer. The free-module property determines the linear extension uniquely. Replacing an orientation $o{r}_s$ by its opposite changes the signs by the reversal clause of the orientation lemma, so the integral differential is not canonical without the orientation data; reducing modulo two makes all signs $+1$ and recovers the mod-two differential of The mod-two Morse differential, which independently of orientations counts the same finite sets.