Morse homology of a Morse--Smale pair

Assume the Axiom of Choice (The Axiom of Choice). Let $(f,X)$ be a Morse--Smale pair on a closed manifold $M$ (Morse--Smale pairs), so that the critical set is finite (A Morse function on a compact manifold has finitely many critical points). Let $\mathrm{ind}$ denote the Morse index (Nondegenerate critical points, nullity, index, and coindex).

Over $\mathbb{Z}\mathrm{/}2$: the mod-two Morse complex is the chain complex of The mod-two Morse chain group with differential The mod-two Morse differential, which satisfies ${\mathrm{\partial }}_{k-1}\circ {\mathrm{\partial }}_k=0$ by The mod-two Morse differential squares to zero (Chain complex in an abelian category). Its mod-two Morse homology is $H{M}_k(f,X;\mathbb{Z}\mathrm{/}2):=H_k(C{M}_{\ast }(f,X;\mathbb{Z}\mathrm{/}2),\mathrm{\partial })=\ker {\mathrm{\partial }}_k\mathrm{/}\mathrm{im}{\mathrm{\partial }}_{k+1}$ (Homology object of a chain complex, The congruence class $[a{]}_n$ and the quotient set $\mathbb{Z}\mathrm{/}n$); this branch needs no orientation or ambient orientability. The choice hypothesis is inherited from the finiteness and squaring-to-zero suppliers; taking homology of an already supplied finite complex makes no further choice.

Over $\mathbb{Z}$: fix an orientation $o{r}_p$, that is, a positive ray in the orientation line of $W^u(p)$, for every critical point $p$ (The orientation line of a Morse critical point). The integral Morse complex is the free $\mathbb{Z}$-module of The signed Morse differential over the integers with basis $\mathrm{Crit}(f)$ and differential the signed trajectory count, which satisfies ${\mathrm{\partial }}_{k-1}\circ {\mathrm{\partial }}_k=0$ by The integral Morse differential squares to zero; its integral Morse homology is $H{M}_k(f,X;\mathbb{Z}):=\ker {\mathrm{\partial }}_k\mathrm{/}\mathrm{im}{\mathrm{\partial }}_{k+1}$ (The integers as equivalence classes of pairs of naturals, Unital left and right modules over a ring; unqualified module means left module). Reversing a chosen orientation ray multiplies the corresponding basis element by $-1$ and conjugates the differential by the diagonal isomorphism of the complex (The orientation line of a Morse critical point), so the isomorphism class of $H{M}_k(f,X;\mathbb{Z})$ is independent of the orientation choices.

The notation records $X$ because the complex depends on the trajectory moduli of the field; that the resulting homology depends only on $M$ and not on the choice of $f$ and $X$ is proved later on this page by continuation, and the relative notation $H{M}_{\ast }(W,M_0;\mathrm{\Lambda })$ is introduced with the relative complex of an adapted cobordism.

For an arbitrary Morse--Smale metric pair $(f,g)$, use instead the finite metric-end complex of Arbitrary metric Morse--Smale end counts form finite Morse chain complexes and write $H{M}_k(f,g;\mathrm{\Lambda })$ for its homology. Its critical basis and chosen orientation-ray conventions are the same; the new supplier proves finiteness and the squared-zero differential for the actual gradient without requiring normalized local coordinates. When that gradient also satisfies the normalized field convention, the two complexes and their homology agree.