The continuation chain map

Assume the Axiom of Choice (The Axiom of Choice). Let $(f_s,g_s)$ be a regular continuation datum from $(f^{-},g^{-})$ to $(f^{+},g^{+})$ on a closed manifold $M$ (A regular continuation datum between Morse--Smale pairs, Morse--Smale pairs).

Over $\mathbb{Z}\mathrm{/}2$, for every $k$ define ${\mathrm{\Phi }}_k:C{M}_k(f^{-},g^{-};\mathbb{Z}\mathrm{/}2)\to C{M}_k(f^{+},g^{+};\mathbb{Z}\mathrm{/}2),{\mathrm{\Phi }}_k(p)=\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\sum }_{q\in {\mathrm{Crit}}_k(f^{+})}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{\#}\mathcal{C}(p,q)\cdot q,$ extended linearly (The mod-two Morse chain group). The coefficient sum is finite because for $\mathrm{ind}(p)=\mathrm{ind}(q)$ the moduli space $\mathcal{C}(p,q)$ is compact and zero-dimensional, hence finite (Continuation trajectories are compact up to breaking, part 3), and each critical set is finite (A Morse function on a compact manifold has finitely many critical points, Nondegenerate critical points, nullity, index, and coindex). Only index-matched pairs are counted: negative index difference gives an empty space, while positive index difference may give a nonempty positive-dimensional space, which has no zero-dimensional count in this definition. Thus ${\mathrm{\Phi }}_k$ preserves degree by its displayed formula.

Over $\mathbb{Z}$ fix an orientation (a positive ray in the orientation line) at every critical point of both pairs (The orientation line of a Morse critical point). Define, with the signs $\tau$ of Orientation lines orient the continuation moduli spaces compatibly with gluing, ${\mathrm{\Phi }}_k:C{M}_k(f^{-},g^{-};\mathbb{Z})\to C{M}_k(f^{+},g^{+};\mathbb{Z}),{\mathrm{\Phi }}_k(p)=\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\sum }_{q\in {\mathrm{Crit}}_k(f^{+})}({\sum }_{u\in \mathcal{C}(p,q)}\tau (u))q,$ extended linearly (The signed Morse differential over the integers). Here too the inner sum is finite by compactness and the dimension formula, and the outer sum is finite by finiteness of the critical set. The modules are over $\mathbb{Z}$ with the scalar action of The integers as equivalence classes of pairs of naturals and over $\mathbb{Z}\mathrm{/}2=\mathbb{Z}\mathrm{/}2\mathbb{Z}$ with the scalar action of The congruence class $[a{]}_n$ and the quotient set $\mathbb{Z}\mathrm{/}n$ (Unital left and right modules over a ring; unqualified module means left module).

In both cases $\mathrm{\Phi }=({\mathrm{\Phi }}_k{)}_k$ is a well-defined homomorphism of graded modules, called the continuation map of the datum. It records the datum, not only its two ends, and it is not induced by a time-translation quotient. The continuation equation is generally not translation invariant; for constant data it is autonomous, and the same unquotiented counting convention applies. That $\mathrm{\Phi }$ is a chain map is proved separately on this page (Orientation lines orient the continuation moduli spaces compatibly with gluing is used only for the signs in the integral branch).

The metric-end complexes in these formulas are supplied by Arbitrary metric Morse--Smale end counts form finite Morse chain complexes. That lemma extends the same finite count and ordered sign conventions to arbitrary metric ends; it does not assume a normalized Morse-coordinate form for their gradients.