The mod-two Morse differential

Let $(f,X)$ be Morse--Smale on a closed manifold and $k\in \mathbb{Z}$. The mod-two Morse differential is the $\mathbb{Z}\mathrm{/}2$-linear map ${\mathrm{\partial }}_k:C{M}_k(f,X;\mathbb{Z}\mathrm{/}2)\to C{M}_{k-1}(f,X;\mathbb{Z}\mathrm{/}2)$ (The mod-two Morse chain group) defined on a basis element $p\in {\mathrm{Crit}}_k(f)$ by ${\mathrm{\partial }}_{k}p:={\sum }_{q\in {\mathrm{Crit}}_{k-1}(f)}{n}_2(p,q)q,n_2(p,q):=\mathrm{\#}\mathcal{M}(p,q)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2\in \mathbb{Z}\mathrm{/}2,$ and extended linearly. Each sum is finite by Index-one trajectory moduli spaces are finite, and the prescribed degree $-1$ restricts the sum to critical points of index $k-1$, so the definition does not depend on any enumeration order. It uses no orientation data.

The objects entering the definition are the finite free module $C{M}_k(f,X;\mathbb{Z}\mathrm{/}2)$ of The mod-two Morse chain group and the unparametrized moduli spaces of Unparametrized Morse trajectory moduli space. For $q\in {\mathrm{Crit}}_{k-1}(f)$ the index drop is $\lambda (p)-\lambda (q)=1$, so $\mathcal{M}(p,q)$ is a zero-dimensional discrete manifold (Index-one trajectory spaces are zero-dimensional) and is finite by Index-one trajectory moduli spaces are finite; its cardinality modulo two is therefore an element of $\mathbb{Z}\mathrm{/}2$ (The congruence class $[a{]}_n$ and the quotient set $\mathbb{Z}\mathrm{/}n$). Trajectories of larger positive index drop may exist, but they are not counted by this degree-one differential. The critical set is finite, so only finitely many coefficients are nonzero; hence ${\mathrm{\partial }}_{k}p$ is a well-defined element of $C{M}_{k-1}(f,X;\mathbb{Z}\mathrm{/}2)$. A $\mathbb{Z}\mathrm{/}2$-linear map out of a free module is determined by its values on a basis, so the linear extension to $C{M}_k(f,X;\mathbb{Z}\mathrm{/}2)$ is unique.

Choice hypothesis. The finiteness input Index-one trajectory moduli spaces are finite is proved under the Axiom of Choice (The Axiom of Choice), and the present definition inherits that hypothesis; no orientation of $M$ or of any unstable manifold is used, so the differential is independent of the orientation choices used for the integral theory.