Morse Homology Continuation and Comparison

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the smooth-bundle setup. Let $M$ be a closed manifold and let $(f^{-},g^{-})$ and $(f^{+},g^{+})$ be Morse--Smale pairs in the metric sense (Morse--Smale pairs).

A continuation datum from $(f^{-},g^{-})$ to $(f^{+},g^{+})$ is a choice of $S>0$ together with smooth families $f_s:M\to \mathbb{R}$ and of Riemannian metrics $g_s$, $s\in \mathbb{R}$, such that $(f_s,g_s)=(f^{-},g^{-})(s\le -S),(f_s,g_s)=(f^{+},g^{+})(s\ge S)\mathrm{.}$ Its continuation equation is the non-autonomous first-order equation ${\mathrm{\partial }}_{s}u(s)=-{\mathrm{\nabla }}^{g_s}{f}_s(u(s)),s\in \mathbb{R},$ for smooth $u:\mathbb{R}\to M$, read as an equation for curves of the time-dependent field $s\mapsto -{\mathrm{\nabla }}^{g_s}{f}_s$ (Time-dependent vector fields and their evolution operators, The Riemannian gradient is the metric dual of the differential). Solutions are never quotiented by time translation. The right-hand side is generally not translation invariant; constant data are an autonomous exception, for which the same unquotiented convention applies. For $p\in \mathrm{Crit}(f^{-})$ and $q\in \mathrm{Crit}(f^{+})$ set $\mathcal{C}(p,q)=\{u\in C^{\mathrm{\infty }}(\mathbb{R},M):{\mathrm{\partial }}_{s}u=-{\mathrm{\nabla }}^{g_s}{f}_s(u),{\lim }_{s\to -\mathrm{\infty }}u(s)=p,{\lim }_{s\to +\mathrm{\infty }}u(s)=q\}\mathrm{.}$

Fix a smooth torsion-free background connection, for example the Levi--Civita connection of $g^{-}$. Along a solution, put $E_u=C_0^1(\mathbb{R},u^{\ast }TM)$ and $F_u=C_0^0(\mathbb{R},u^{\ast }TM)$, with supremum norms; the subscript means that the section (and its covariant first derivative in $E_u$) tends to zero at both ends. The linearization is $D_u\xi ={\mathrm{\nabla }}_s\xi +{\mathrm{\nabla }}_{\xi }({\mathrm{\nabla }}^{g_s}{f}_s)\mathrm{.}$ The datum is regular at $(p,q)$ if $D_u:E_u\to F_u$ is onto for every $u\in \mathcal{C}(p,q)$; it is regular if this holds for every critical pair. Empty solution spaces satisfy this condition vacuously.

Here is the finite-dimensional description and index calculation. Let ${\mathrm{\Psi }}_{S,-S}$ be evolution across the compact window. Local smooth evolution and compactness of $M$ extend it across every finite time interval: finitely many coordinate neighbourhoods give a common positive local existence time, which can be iterated; backward evolution is its inverse (Time-dependent vector fields have local smooth evolution operators). Evaluation at $-S$ identifies $\mathcal{C}(p,q)$ with the fibre product $\{(x,y)\in W_{-}^u(p)\times W_{+}^s(q):{\mathrm{\Psi }}_{S,-S}(x)=y\}\mathrm{.}$ The end stable/unstable disks and their transported tangent spaces have the expected dimensions and exponentially decaying variations (Stable and unstable manifolds for a Morse gradient with an arbitrary smooth metric). For smooth data the disks are smooth: differentiate the supplier's contraction fixed-point equation repeatedly in its finite-dimensional initial parameter; at every order the unknown derivative has the same linear contraction operator, while its forcing involves already obtained lower derivatives and bounded derivatives of the cut-off smooth vector field. Its Neumann series therefore gives a continuous derivative of each order. Finite-time smooth flow transport gives smooth global immersion charts. In a frame converging on both autonomous tails, $D_u$ becomes a first-order matrix operator with invertible self-adjoint limits. The whole-line operator lemma gives index $\mathrm{ind}(p)-\mathrm{ind}(q)$ and says that surjectivity is equivalent to the two transported decaying initial-value spaces spanning $T_{y}M$ (Fredholm index and range of an asymptotically hyperbolic first-order operator). These spaces are exactly $d{\mathrm{\Psi }}_{S,-S}(T_x{W}_{-}^u(p))$ and $T_y{W}_{+}^s(q)$. Thus regularity is transversality of this fibre product; it makes $\mathcal{C}(p,q)$ a smooth manifold of dimension $\mathrm{ind}(p)-\mathrm{ind}(q)$, empty for negative dimension (Transverse fibre products are embedded submanifolds). No translation quotient is taken, even for the constant datum, for which translations happen to be symmetries.

Existence and its qualification. Assume the Axiom of Choice (The Axiom of Choice) for the following genericity assertion. Allow both $f_s$ and $g_s$ to vary, with the two ends fixed. Regular data form a residual set in the smooth path space and can be obtained by arbitrarily small perturbations supported inside $(-S,S)\times M$. To see the required transversality, choose finitely many smooth functions ${\varphi }_j$ whose gradients span every tangent space (coordinate functions times cutoffs on a finite chart cover), and perturb $f_s$ by ${\sum }_j{a}_j\beta (s){\varphi }_j$, where $\beta$ is a nonnegative unit-integral bump supported very near some $s_0\in (-S,S)$. Differentiating evolution with respect to $a_j$ gives the integral of the transported vector $-\beta (s){\mathrm{\nabla }}^{g_s}{\varphi }_j$. As the support shrinks, these vectors converge uniformly in the initial point to $-d{\mathrm{\Psi }}_{S,s_0}{\mathrm{\nabla }}^{g_{s_0}}{\varphi }_j$; hence they span $T_{y}M$ for a sufficiently narrow bump. The universal endpoint map is therefore a submersion. Apply finite-dimensional parametric transversality to its fibre products with the stable/unstable immersion charts (Parametric transversality). Their countable chart covers and the finitely many critical pairs leave a null exceptional parameter set, so arbitrarily small good parameters exist. Transversality on each compact piece of a countable chart exhaustion is open and dense; intersecting these sets gives the residual assertion. This argument permits function variations, including along constant solutions.

For an arbitrary fixed function path, metric variations alone need not give regularity: a point critical for every $f_s$ remains a constant solution for every metric path. If its two end indices have negative difference, the linearization there has negative index and cannot be onto. The generic-metric statement in the autonomous distinct-end supplier The universal metric--trajectory projection is Fredholm does not cover this obstruction. Regularity of a specified datum is a hypothesis below, rather than a consequence of that supplier.

Let $A_s$ and $A_u$ be self-adjoint matrices whose eigenvalues are respectively negative and positive. Suppose the smooth vector field near the origin in ${\mathbb{R}}^l\times {\mathbb{R}}^k$ is $s^{\mathrm{\prime }}=A_{s}s+R_s(s,u),u^{\mathrm{\prime }}=A_{u}u+R_u(s,u),$ where $R(0)=DR(0)=0$ and the coordinate axes are invariant: $R_s(0,u)=0,R_u(s,0)=0.$ There are $r>0$ and $\beta >0$, independent of $T>0$, such that for every $\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid }<2r$ there is a unique solution on $[0,T]$ staying in $\mathrm{\mid }s\mathrm{\mid },\mathrm{\mid }u\mathrm{\mid }\le 4r$ with mixed boundary values $s(0)=a$, $u(T)=b$. It depends smoothly on $(T,a,b)$ for $T>0$. The endpoint maps ${\alpha }_T(a,b)=u(0),{\zeta }_T(a,b)=s(T)$ satisfy $\mathrm{\mid }{\alpha }_T(a,b)\mathrm{\mid }+\mathrm{\mid }{\zeta }_T(a,b)\mathrm{\mid }\le (\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid })e^{-\beta T}\mathrm{.}$ For each nonnegative integer $N$ there is a constant $C_N$, independent of $T,a,b$, such that every coordinate derivative of total order at most $N$ in $(T,a,b)$ of either endpoint map has norm at most $C_N{e}^{-\beta T}$. In particular the first boundary-data derivatives satisfy the corresponding operator-norm bound. The enlarged boundary-data ball permits sections with $\mathrm{\mid }a\mathrm{\mid }$ or $\mathrm{\mid }b\mathrm{\mid }$ near $r$.

Consequently the maps $\widehat{\alpha }(\rho ,a,b)={\alpha }_{1\mathrm{/}\rho }(a,b),\widehat{\zeta }(\rho ,a,b)={\zeta }_{1\mathrm{/}\rho }(a,b)(\rho >0)$ extend by zero to smooth maps for $\rho \ge 0$, and are flat along $\rho =0$: every derivative, including mixed derivatives in $(\rho ,a,b)$, vanishes there. These assertions concern invariant-axis passage coordinates; they do not by themselves assert a moduli-space collar chart.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $(f_s,g_s)$ be a continuation datum from $(f^{-},g^{-})$ to $(f^{+},g^{+})$ on a closed manifold $M$ and let $u:\mathbb{R}\to M$ be a solution of the continuation equation (A regular continuation datum between Morse--Smale pairs). Then the limits ${\lim }_{s\to -\mathrm{\infty }}u(s)=p\in \mathrm{Crit}(f^{-}),{\lim }_{s\to +\mathrm{\infty }}u(s)=q\in \mathrm{Crit}(f^{+})$ exist. Moreover, in the Morse coordinates of Morse lemma at $p$ and $q$ the curve $u$ converges to $p$ (resp. $q$) exponentially fast as $s\to -\mathrm{\infty }$ (resp. $s\to +\mathrm{\infty }$), and $\mathrm{\mid }{\mathrm{\partial }}_{s}u(s){\mathrm{\mid }}_{g_s}$ decays exponentially; in particular the energy integral ${\int }_{\mathbb{R}}\mathrm{\mid }{\mathrm{\partial }}_{s}u{\mathrm{\mid }}^{2}ds$ is finite.

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.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the smooth-bundle setup. Let $(f_s,g_s)$ be a continuation datum from $(f^{-},g^{-})$ to $(f^{+},g^{+})$ on a closed manifold $M$, and let $u$ be a solution of the continuation equation with limits $p\in \mathrm{Crit}(f^{-})$, $q\in \mathrm{Crit}(f^{+})$ (A regular continuation datum between Morse--Smale pairs, Continuation solutions have critical limits and exponential decay). Write $E(u)={\int }_{\mathbb{R}}\mathrm{\mid }{\mathrm{\partial }}_{s}u(s){\mathrm{\mid }}_{g_s}^{2}ds$ and $({\mathrm{\partial }}_s{f}_s)(x)=\frac{\mathrm{\partial }{f}_s}{\mathrm{\partial }s}(x)$. Then $E(u)=f^{-}(p)-f^{+}(q)+{\int }_{\mathbb{R}}({\mathrm{\partial }}_s{f}_s)(u(s))ds\mathrm{.}$ Since ${\mathrm{\partial }}_s{f}_s=0$ on $M\times ((-\mathrm{\infty },-S]\cup [S,\mathrm{\infty }))$ and $({\mathrm{\partial }}_s{f}_s)(u(s))$ is integrable by smoothness and compact support, in particular $0\le E(u)\le f^{-}(p)-f^{+}(q)+2S{\max }_{(s,x)\in [-S,S]\times M}\mid ({\mathrm{\partial }}_s{f}_s)(x)\mid \mathrm{.}$ For a constant datum ($f_s=f$, $g_s=g$ for all $s$) the identity reduces to the classical energy identity ${\int }_{\mathbb{R}}\mathrm{\mid }\dot{u}{\mathrm{\mid }}^2=f(p)-f(q)$ of A negative-gradient trajectory satisfies the energy identity.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the smooth-bundle setup. 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), let $p\in \mathrm{Crit}(f^{-})$, $q\in \mathrm{Crit}(f^{+})$ (Morse--Smale pairs), and let $\mathcal{C}(p,q)$ be the continuation moduli spaces of the datum.

A broken continuation trajectory from $p$ to $q$ consists of nonconstant autonomous tail pieces and one continuation solution (which may be constant) $({\gamma }_1^{-},\dots ,{\gamma }_r^{-};v;{\gamma }_1^{+},\dots ,{\gamma }_s^{+}),r,s\ge 0,$ together with critical points $p=p_0^{-},p_1^{-},\dots ,p_r^{-}$ of $f^{-}$ and $q=p_0^{+},p_1^{+},\dots ,p_s^{+}$ of $f^{+}$ such that ${\gamma }_i^{-}\in {\mathcal{M}}^{-}(p_{i-1}^{-},p_i^{-}),v\in \mathcal{C}(p_r^{-},p_s^{+}),{\gamma }_j^{+}\in {\mathcal{M}}^{+}(p_j^{+},p_{j-1}^{+})$ for all $i,j$ (Unparametrized Morse trajectory moduli space, A Morse trajectory from one critical point to another). The pieces are read in the temporal order ${\gamma }_1^{-},\dots ,{\gamma }_r^{-},v,{\gamma }_s^{+},{\gamma }_{s-1}^{+},\dots ,{\gamma }_1^{+}$; the tuple displays the pieces of the two ends in increasing order of their index-drop chains. The space of broken continuation trajectories from $p$ to $q$ is denoted $\bar{\mathcal{C}}(p,q)$; the locus $r=s=0$ is $\mathcal{C}(p,q)$ itself.

The pieces carry the index drops $d_i:=\mathrm{ind}(p_{i-1}^{-})-\mathrm{ind}(p_i^{-})\ge 1,e_j:=\mathrm{ind}(p_j^{+})-\mathrm{ind}(p_{j-1}^{+})\ge 1,m:=\mathrm{ind}(p_r^{-})-\mathrm{ind}(p_s^{+})\ge 0,$ which are nonnegative by the emptiness of the corresponding spaces for nonpositive drop (No Morse--Smale trajectories for nonpositive index drop, A regular continuation datum between Morse--Smale pairs), and the index identity $\mathrm{ind}(p)-\mathrm{ind}(q)=r+s+m+{\sum }_{i=1}^r(d_i-1)+{\sum }_{j=1}^s(e_j-1)$ holds by telescoping the drops along the two chains (Nondegenerate critical points, nullity, index, and coindex). In particular $r+s\le \mathrm{ind}(p)-\mathrm{ind}(q)$, and equality holds precisely when every tail piece connects critical points of consecutive index and the middle piece has index difference $m=0$. The number of tail pieces is bounded by the index drop, exactly as for the broken Morse trajectories of a Morse--Smale pair (Breaking length is bounded by the index drop, Broken Morse trajectories).

A sequence $u_n\in \mathcal{C}(p,q)$ converges geometrically to such a configuration if $u_n\to v$ in $C^{\mathrm{\infty }}$ on compact time intervals without shifting the middle solution, and there are shifts $a_n^i\to -\mathrm{\infty }$ and $b_n^j\to +\mathrm{\infty }$ such that $u_n(\cdot +a_n^i)\to {\gamma }_i^{-},u_n(\cdot +b_n^j)\to {\gamma }_j^{+}$ in $C^{\mathrm{\infty }}$ on compact intervals, for chosen parametrized representatives of the autonomous orbit classes. Temporal order requires $a_n^{i+1}-a_n^i\to +\mathrm{\infty }$ and $b_n^{j-1}-b_n^j\to +\mathrm{\infty }$. For the middle and each shifted autonomous tail, $C^0$ convergence on compact intervals implies convergence of all derivatives by differentiating the corresponding smooth ODE; the shifted equation on every fixed compact tail interval is eventually the fixed autonomous end equation (The topology of compact convergence on $C(X,Y)$ for metric $X$ and $Y$: uniform convergence on each compact subset of $X$, Geometric convergence to a broken trajectory).

The topology on $\bar{\mathcal{C}}(p,q)$ uses compact-time tests on the unshifted middle solution and ordered regular-level transversal tests on the autonomous components. A neighbourhood specifies compact intervals and open neighbourhoods of the middle curve on them, and open transversal neighbourhoods along each tail; configurations may smooth some breaks but must pass these tests in the stated order. Shrinking the tests gives a neighbourhood basis, with the compact-convergence topology on the unbroken locus and the corresponding broken-end topology on each stratum. The middle solution's time coordinate is fixed: translating it generally changes the continuation equation. Every middle solution has critical limits by Continuation solutions have critical limits and exponential decay.

Assume the Axiom of Choice. Consider a finite broken trajectory with autonomous Morse--Smale metric-gradient pieces and zero or one regular continuation middle, or one smooth compact finite-dimensional parameter family of continuation middles. In the family case the parameter space is a smooth manifold with corners, and the augmented linearization using the tangent space of each parameter face is onto at every solution on that face, including the interior. This is facewise augmented regularity; at a zero-dimensional face it requires vertical regularity. A regular two-parameter datum as in A regular two-parameter continuation datum satisfies these conditions. The ends of each continuation window are fixed independently of the parameter. At a configuration with $N$ autonomous breaks, the geometric compactification has a neighbourhood parametrized by $[0,\delta {)}^N\times B,$ where $B$ is a sufficiently small chart of the broken stratum, retaining its parameter faces. The positive neck coordinates are ${\rho }_j=1\mathrm{/}{T}_j$, where $T_j$ is the actual passage time between fixed local sections at an autonomous-to-autonomous junction, or between a local section and a fixed-time anchor adjacent to the continuation middle. Zero neck coordinates retain the corresponding break. The parametrization is a homeomorphism onto a neighbourhood, is smooth on each stratum, and $(0,\delta {)}^N\times B$ parametrizes precisely the unbroken solutions nearby, including those on parameter faces. Its full corner-chart interior additionally excludes the parameter boundary. The continuation middle stays at its fixed time origin. Changes of the chosen critical charts and sections are smooth in these corner coordinates, with positive normal derivatives at the corresponding faces.

The same local matching construction applies to two regular continuation pieces joined through a Morse--Smale intermediate pair, when the two finite windows are placed on opposite sides of a variable autonomous plateau. The plateau length is the extra parameter; at infinite length its broken stratum is the product of the two regular continuation spaces. The finite plateau chart retains the pair $(T,u)$: the plateau datum parameter is not forgotten, even when constant windows produce an identical constant curve for different $T$. No global compactness statement is included.