Canonical Morse homology of a closed manifold

Assume the Axiom of Choice (The Axiom of Choice). Let $M$ be a closed smooth manifold and $\mathrm{\Lambda }=\mathbb{Z}\mathrm{/}2$ or $\mathbb{Z}$. Choose an excellent Morse function $f:M\to \mathbb{R}$ (Every compact smooth manifold admits an excellent Morse function) and a Riemannian metric $g$ for which $(f,g)$ is Morse--Smale (Morse--Smale metrics are residual for a fixed Morse function; equivalently use Every Morse function admits a complete downward gradient-like field on a closed manifold and the perturbation theorem for gradient-like fields, Morse--Smale pairs). In the integral branch, also choose a positive ray in the orientation line at each critical point, as required by Morse homology of a Morse--Smale pair. Define the canonical Morse homology $H{M}_k(M;\mathrm{\Lambda }):=H{M}_k(f,g;\mathrm{\Lambda })$ (Morse homology of a Morse--Smale pair).

For two Morse--Smale pairs $(f_0,g_0)$ and $(f_1,g_1)$ the regular continuation maps give isomorphisms $H{M}_k(f_0,g_0;\mathrm{\Lambda })\to H{M}_k(f_1,g_1;\mathrm{\Lambda })$ that are independent of the chosen regular continuation datum (Homotopic continuation data give chain homotopic maps), with identity and composition laws (Composition of continuation maps on homology) and inverses given by the reverse continuation (Reverse continuation is an inverse on Morse homology). Hence the chosen pair determines $H{M}_{\ast }(M;\mathrm{\Lambda })$ only up to a canonical isomorphism, and the notation is unambiguous in that sense; taking homology of a supplied finite complex makes no further choice, while existence of the pairs, the moduli-space finiteness, and the continuation comparisons use the stated Axiom of Choice in both coefficient branches.