Integral Morse homology does not require orientability of the manifold

Under the Axiom of Choice, the integral Morse complex of The integral Morse differential squares to zero is built from orientations of the unstable manifolds $W^u(p)$ only: it requires neither an orientation of $M$ nor orientability of $M$. Indeed, the co-orientation of $W^s(p)$ used to orient the moduli spaces is induced by the orientation of $T_p{W}^u(p)$ and transported along the flow (Unstable orientations induce orientations of the trajectory moduli spaces, The orientation line of a Morse critical point), and the Morse--Smale condition itself never uses an ambient orientation (Ambient orientability is not required for Morse--Smale transversality).

Consequently a Morse--Smale pair on a nonorientable closed manifold has a well-defined integral Morse complex, even though its moduli spaces receive no orientation induced by an orientation of $M$. The point is that the orientation data live on the unstable manifolds, which are always orientable because they are diffeomorphic to Euclidean spaces, while $M$ itself need not be orientable (Orientable manifolds); the definition of a Morse--Smale pair Morse--Smale pairs imposes only transversality of the stable and unstable manifolds. In particular orientability of $M$ is not among the hypotheses of The integral Morse differential squares to zero. Different choices of unstable orientations multiply each coefficient $n(x,y)$ by the product of the basis signs at $x$ and $y$, by the orientation lemma. The diagonal automorphism $T(p)=s_{p}p$, where $s_p$ records the orientation reversal at $p$, therefore satisfies $\mathrm{\partial }\text{\hspace{0.17em}⁣}\mathrm{\prime }=T\mathrm{\partial }{T}^{-1}$ and induces an isomorphism on kernels modulo images; the mod-two theory of this page needs no orientation choices at all.