A regular two-parameter continuation datum

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the smooth-bundle setup. Let $(f_s^0,g_s^0)$ and $(f_s^1,g_s^1)$ be two regular continuation data from the same Morse--Smale pair $(f^{-},g^{-})$ to the same pair $(f^{+},g^{+})$ on a closed manifold $M$ (A regular continuation datum between Morse--Smale pairs, Morse--Smale pairs).

A two-parameter continuation datum between them is a smooth family $(f_s^{\lambda },g_s^{\lambda }{)}_{(\lambda ,s)\in [0,1]\times \mathbb{R}}$ such that for each $\lambda$ the pair $(f_s^{\lambda },g_s^{\lambda }{)}_{s\in \mathbb{R}}$ is a continuation datum from $(f^{-},g^{-})$ to $(f^{+},g^{+})$, the family is independent of $\lambda$ near $\lambda =0$ and $\lambda =1$ (where it equals the two given data), and there is one $S>0$ such that all members equal the same ends $(f^{\pm },g^{\pm })$ for $s\le -S$ and $s\ge S$ (Smooth families of maps and their evaluation maps). Its parametrized moduli space is $\mathcal{P}(p,q)=\{(\lambda ,u):\lambda \in [0,1],u\in {\mathcal{C}}^{\lambda }(p,q)\}\subset [0,1]\times C^{\mathrm{\infty }}(\mathbb{R},M),$ where ${\mathcal{C}}^{\lambda }(p,q)$ is the continuation moduli space of the $\lambda$-member.

The datum is regular if at every solution $(\lambda ,u)$ the augmented linearization $\mathbb{R}\oplus E_u⟶F_u,(a,\xi )⟼D_u^{\lambda }\xi +a{\mathrm{\partial }}_{\lambda }({\mathrm{\nabla }}^{g_s^{\lambda }}{f}_s^{\lambda })(u)$ is surjective, with the spaces and connection convention of A regular continuation datum between Morse--Smale pairs. At the parameter endpoints the vertical linearization is onto because the family equals a given regular datum near each endpoint. The augmented operator has index $\mathrm{ind}(p)-\mathrm{ind}(q)+1$: adding one domain dimension increases the index by one, and its parameter term has finite rank. The finite-dimensional endpoint fibre product, now with the parameter included, therefore gives $\mathcal{P}(p,q)$ the structure of a smooth manifold with boundary of that dimension. Its parameter boundary is ${\mathcal{C}}^0(p,q)\bigsqcup {\mathcal{C}}^1(p,q)$; negative dimension means empty.

Regularity is a property of the specified family, not of an unspecified perturbation. Under the Axiom of Choice (The Axiom of Choice), a family can be perturbed arbitrarily little in the interior, fixing both parameter ends and the common autonomous tails, to become regular: the function perturbations and endpoint-map transversality argument in A regular continuation datum between Morse--Smale pairs apply with an additional bump in $\lambda$, on countably many interior parameter charts. Parametric transversality gives dense good parameters, and compact chart exhaustions give residuality (Parametric transversality, Nowhere dense, meagre, residual, and comeagre subsets of a topological space).

Under the same choice hypothesis, for a regular family Sard's theorem applied to the projections $\mathcal{P}(p,q)\to [0,1]$ gives a null set of parameter values at which a vertical linearization fails to be onto (Morse-Sard for smooth manifolds). It does not imply that this set is finite. In particular a solution with $\mathrm{ind}(p)-\mathrm{ind}(q)=-1$ is a rogue trajectory: its fixed-parameter operator cannot be onto, but the augmented operator can be, giving a zero-dimensional parametrized moduli space. Parametrized compactness, gluing and orientation of these augmented operators are separate requirements for the chain-homotopy argument; they are not supplied by regularity of the individual members or by the fixed-datum orientation lemma.