Gluing of trajectories

Intuition Beginner

In the previous unit a sequence of downhill streams degenerated, in the limit, into a broken path: one stream running from a high resting point down to a middle resting point, where it pauses forever, and a second stream carrying on from that middle point down to the bottom. Gluing is the reverse move. You start with the broken path — two streams meeting at a middle rest point — and you build honest, unbroken streams nearby.

The recipe is simple to picture. Take the first stream, which spends a long time settling into the middle rest point, and the second stream, which spends a long time leaving it. Cut each one off near the middle point and splice the cut ends together. The splice almost satisfies the downhill rule, except for a small mismatch at the join. You then nudge the spliced path slightly until the mismatch disappears, and what you are left with is a genuine single stream from top to bottom that passes close to the middle rest point.

There is a dial in this construction: how long the spliced path lingers near the middle point before moving on. Turn the dial up and the new stream hugs the middle point longer, looking more and more like the broken path. Turn it down and the stream pulls away. As you turn the dial all the way up, the honest streams you build march straight back to the broken path you started from.

So breaking and gluing are two directions of one story. Breaking says: push a family of streams and they can fall apart at a rest point. Gluing says: every such broken path is the endpoint of a family of honest streams. The two together pin down exactly how the collection of streams sits at its edges.

Visual Beginner

Alt text: An upright torus with a top, a saddle, and a bottom. On the left a broken path made of two streams meeting at the saddle; in the centre the two streams cut near the saddle and spliced by a short bridge of length rho; on the right a single honest stream from top to bottom passing close to the saddle, with a slider showing that turning rho up bends the honest stream back toward the broken path. The picture conveys that gluing rebuilds honest streams from a broken one.

Worked example Beginner

Take the upright doughnut again: a top $t$, a saddle $s$, and a bottom $b$, with the height dropping $t$ to $s$ to $b$. The broken path $t\to s\to b$ is made of one stream that ends at the saddle and one that starts there. We want to see the honest streams that gluing produces.

Picture the first stream slowing to a crawl as it approaches $s$, and the second stream creeping away from $s$ before it speeds downhill. Cut the first one a little before it reaches $s$, cut the second one a little after it leaves, and join the loose ends with a short bridge near the saddle. The result is a single path that dips toward $s$, skims past it, and continues to $b$.

Right now this spliced path has a tiny kink at the join: it does not perfectly obey the downhill rule there. Nudging it smooths the kink and yields an honest stream from $t$ to $b$.

The dial is how close the splice comes to the saddle. A long lingering splice gives a stream that nearly touches $s$; a shorter one gives a stream that gives the saddle a wider berth. As the lingering grows without bound, these honest streams approach the broken path $t\to s\to b$. So near the broken path there sits a whole one-parameter family of honest streams, one for each setting of the dial, all of them sweeping out a single arc that ends at the broken path.

Check your understanding Beginner

Formal definition Intermediate+

Fix a closed Riemannian manifold $(M^n,g)$ and a Morse-Smale pair $(f,g)$, with $\nabla f$ the gradient and $\mu (x)$ the Morse index of a critical point, exactly as in 03.15.01 and 03.15.02. For critical points $x\ne y$ write $\mathcal{M}(x,y)$ for the space of parametrized negative-gradient trajectories and $\hat{\mathcal{M}}(x,y)=\mathcal{M}(x,y)/\mathbb{R}$ for the unparametrized quotient. Trajectories are realized as zeros of the section $F(\gamma )=\dot{\gamma }+\nabla f(\gamma )$ of a Banach bundle over the path space ${\mathcal{P}}^{1,2}(x,y)$, with asymptotically constant linearization $D_{\gamma }F=\frac{d}{dt}+A(t)$, Fredholm of index $\mu (x)-\mu (y)$.

Throughout, fix critical points $x,z,y$ with $\mu (x)-\mu (z)=\mu (z)-\mu (y)=1$, so that $\hat{\mathcal{M}}(x,z)$ and $\hat{\mathcal{M}}(z,y)$ are $0$-dimensional, hence (by compactness) finite sets of isolated trajectories. Choose representatives $v\in \mathcal{M}(x,z)$ and $w\in \mathcal{M}(z,y)$ of points $[v]\in \hat{\mathcal{M}}(x,z)$, $[w]\in \hat{\mathcal{M}}(z,y)$. The pair $(v,w)$ is a once-broken trajectory from $x$ to $y$ breaking at $z$.

The pre-glued curve is built by splicing $v$ and $w$ across a long neck near $z$. Fix a smooth cutoff $\beta :\mathbb{R}\to [0,1]$ with $\beta (t)=0$ for $t\le -1$ and $\beta (t)=1$ for $t\ge 1$, and use the exponential map ${\exp }_z$ of $g$ to write the asymptotics. For a gluing length $\rho >0$, define

$${\gamma }_{\rho }(t)=\{\begin{array}{ll}v(t+\rho ),& t\le -1,\\ {\exp }_z((1-\beta (t)){\xi }_v(t+\rho )+\beta (t){\xi }_w(t-\rho )),& -1\le t\le 1,\\ w(t-\rho ),& t\ge 1,\end{array}$$

where ${\xi }_v(s)={\exp }_z^{-1}(v(s))$ and ${\xi }_w(s)={\exp }_z^{-1}(w(s))$ are the geodesic-normal coordinates of the two ends, defined for $|s|$ large because $v(+\infty )=z=w(-\infty )$. The curve ${\gamma }_{\rho }$ runs along $v$, lingers near $z$ for a time of order $2\rho$, then runs along $w$.

A choice of ${\rho }_0>0$ is admissible when, for all $\rho \ge {\rho }_0$, the operator $D_{{\gamma }_{\rho }}F$ admits a right inverse $Q_{\rho }$ with $\Vert Q_{\rho }\Vert$ bounded independently of $\rho$. The gluing map is the assignment $$ #: [\rho_0, \infty) \longrightarrow \mathcal{M}(x,y), \qquad \rho \longmapsto \widetilde\gamma_\rho = \exp_{\gamma_\rho}(\eta_\rho), $$ where ${\eta }_{\rho }$ is the unique small solution of $F({\exp }_{{\gamma }_{\rho }}(\eta ))=0$ furnished by the construction below. We also write $v{\#}_{\rho }w:={\tilde{\gamma }}_{\rho }$ and $[v\#w]:[{\rho }_0,\infty )\to \hat{\mathcal{M}}(x,y)$ for the induced unparametrized arc.

Counterexamples to common slips

• The pre-glued curve ${\gamma }_{\rho }$ is not a trajectory: $F({\gamma }_{\rho })\ne 0$. It is an approximate trajectory whose defect $\Vert F({\gamma }_{\rho })\Vert$ is small but positive, supported on the neck $|t|\le 1$ where the cutoff interpolates. The honest trajectory is ${\tilde{\gamma }}_{\rho }$, the correction of ${\gamma }_{\rho }$, not ${\gamma }_{\rho }$ itself.

• Gluing requires the consecutive index drops to equal one ($\mu (x)-\mu (z)=\mu (z)-\mu (y)=1$). If $\mu (x)-\mu (z)=2$, the moduli $\hat{\mathcal{M}}(x,z)$ is a $1$-manifold rather than a finite set, and the once-broken trajectories form a positive-dimensional family, not a discrete set of boundary points; the clean one-parameter gluing arc is replaced by a higher-dimensional collar.

• The gluing parameter $\rho$ is a genuine extra dimension only in the unparametrized picture. In $\mathcal{M}(x,y)$ the time-translation $\mathbb{R}$ acts freely, and $\rho$ together with the residual time-shift parametrizes a $1$-parameter family; passing to $\hat{\mathcal{M}}(x,y)$ kills the time-shift and leaves the half-open arc $[{\rho }_0,\infty )\to \hat{\mathcal{M}}(x,y)$.

• The right inverse must be uniform in $\rho$. A right inverse exists for each fixed $\rho$ by transversality, but if $\Vert Q_{\rho }\Vert$ blew up as $\rho \to \infty$ the Newton iteration would not close. The content of the analytic estimate is the $\rho$-independent bound, which rests on the spectral gap of the asymptotic operator $A_{\pm \infty }$ at $z$.

Key theorem with proof Intermediate+

The gluing theorem is the central construction of Schwarz Part II Chapter 2 ^{[Schwarz Part II Ch. 2]}. It inverts the breaking of 03.15.03: every once-broken trajectory is the endpoint of a unique nearby arc of honest trajectories.

Theorem (gluing). Let $(f,g)$ be a Morse-Smale pair on a closed Riemannian manifold $(M^n,g)$ and let $x,z,y$ be critical points with $\mu (x)-\mu (z)=\mu (z)-\mu (y)=1$. For each once-broken trajectory $(v,w)$ with $[v]\in \hat{\mathcal{M}}(x,z)$, $[w]\in \hat{\mathcal{M}}(z,y)$ there is ${\rho }_0>0$ and a smooth embedding $$ v # w : [\rho_0, \infty) \longrightarrow \widehat{\mathcal{M}}(x,y), \qquad \rho \longmapsto [v #\rho w], $$ such that $v #\rho w \to (v,w)$geometricallyas$\rho \to \infty$.Distinctbrokentrajectoriesgivearcsthatareeventuallydisjoint,andeverysequencein$\widehat{\mathcal{M}}(x,y)$convergingto$(v,w)$lies,forlargeindex,intheimageof$v # w$.Thus$v#w$isalocalhomeomorphismontoahalf-opencollarof$\overline{\widehat{\mathcal{M}}}(x,y)$at$(v,w)$,and$(v,w)$isaboundarypointofthecompactified$1$-manifold$\overline{\widehat{\mathcal{M}}}(x,y)$.

Proof. Write $E_{\rho }=D_{{\gamma }_{\rho }}F:W^{1,2}\to L^2$ for the linearization at the pre-glued curve. Three estimates drive the argument.

(1) Small defect. The cutoff in ${\gamma }_{\rho }$ is supported on $|t|\le 1$, where ${\gamma }_{\rho }$ interpolates between the two ends $v(t+\rho )$ and $w(t-\rho )$. Each end converges to $z$ exponentially: $|{\xi }_v(t+\rho )|\le C{e}^{-\lambda (t+\rho )}$ and $|{\xi }_w(t-\rho )|\le C{e}^{-\lambda (\rho -t)}$, where $\lambda >0$ is the smallest absolute value of an eigenvalue of $\mathrm{Hess}f(z)$ — the spectral gap of $A_{\pm \infty }=\mathrm{Hess}f(z)$, which is invertible by nondegeneracy. On $|t|\le 1$ both quantities are bounded by $C{e}^{-\lambda (\rho -1)}$, and $F({\gamma }_{\rho })$, which vanishes off the neck because ${\gamma }_{\rho }$ solves the flow there, satisfies $\Vert F({\gamma }_{\rho }){\Vert }_{L^2}\le C^{\prime }{e}^{-\lambda \rho }$.

(2) Uniform right inverse. Each of $D_{v}F$ and $D_{w}F$ is surjective with kernel $\mathbb{R}\dot{v}$, $\mathbb{R}\dot{w}$ (the index-one transversality of 03.15.02). A linear-gluing argument splices their right inverses: cutting off a section, applying the inverse on each piece, and correcting the overlap on the neck produces a right inverse $Q_{\rho }$ of $E_{\rho }$ whose norm is bounded by a constant independent of $\rho$, again because the overlap error is controlled by $e^{-\lambda \rho }$ via the spectral gap. So $\Vert Q_{\rho }{\Vert }_{L^2\to W^{1,2}}\le C_0$ for $\rho \ge {\rho }_0$.

(3) Quadratic estimate. The nonlinear map ${\mathcal{F}}_{\rho }(\eta )=F({\exp }_{{\gamma }_{\rho }}(\eta ))$ has ${\mathcal{F}}_{\rho }(0)=F({\gamma }_{\rho })$, derivative $E_{\rho }$ at $0$, and remainder controlled by $\Vert {\mathcal{F}}_{\rho }(\eta )-{\mathcal{F}}_{\rho }(0)-E_{\rho }\eta \Vert \le c\Vert \eta {\Vert }^2$ uniformly in $\rho$, since the curvature terms in the second derivative of $\exp$ are bounded on the compact $M$.

Now solve ${\mathcal{F}}_{\rho }(\eta )=0$ by Newton-Picard. Seek $\eta =Q_{\rho }\zeta$ in the image of the right inverse; the equation becomes $\zeta +{\mathcal{N}}_{\rho }(\zeta )=-F({\gamma }_{\rho })$, where ${\mathcal{N}}_{\rho }(\zeta )={\mathcal{F}}_{\rho }(Q_{\rho }\zeta )-{\mathcal{F}}_{\rho }(0)-E_{\rho }{Q}_{\rho }\zeta$ obeys $\Vert {\mathcal{N}}_{\rho }(\zeta )\Vert \le c{C}_0^2\Vert \zeta {\Vert }^2$ and a matching Lipschitz bound. For $\rho$ large the inhomogeneity $\Vert F({\gamma }_{\rho })\Vert \le C^{\prime }{e}^{-\lambda \rho }$ lies in the contraction radius, so the map $\zeta \mapsto -F({\gamma }_{\rho })-{\mathcal{N}}_{\rho }(\zeta )$ is a contraction on a small ball and has a unique fixed point ${\zeta }_{\rho }$ with $\Vert {\zeta }_{\rho }\Vert \le 2C^{\prime }{e}^{-\lambda \rho }$. Set ${\eta }_{\rho }=Q_{\rho }{\zeta }_{\rho }$ and ${\tilde{\gamma }}_{\rho }={\exp }_{{\gamma }_{\rho }}({\eta }_{\rho })\in \mathcal{M}(x,y)$. Uniqueness of the fixed point in the slice $\mathrm{im}{Q}_{\rho }$ makes $\rho \mapsto {\tilde{\gamma }}_{\rho }$ smooth (implicit-function theorem with $\rho$ a parameter), and $\Vert {\eta }_{\rho }\Vert \to 0$ forces ${\tilde{\gamma }}_{\rho }\to {\gamma }_{\rho }$, hence ${\tilde{\gamma }}_{\rho }\to (v,w)$ geometrically as $\rho \to \infty$.

Injectivity for large $\rho$ and the surjectivity onto a neighborhood (every trajectory near the broken one is some ${\tilde{\gamma }}_{\rho }$) follow from the same implicit-function setup applied to the compactness of 03.15.03: a sequence converging to $(v,w)$ has, for each large index, a unique pre-glued model and a unique correction, which is ${\tilde{\gamma }}_{\rho }$. Hence $v\#w$ is a homeomorphism onto a half-open collar, exhibiting $(v,w)$ as a boundary point. $\square$

Bridge. The gluing theorem builds toward 03.15.06, where it supplies the boundary identification that forces ${\partial }^2=0$: putting these together with the compactness of 03.15.03, the compactified moduli $\overline{\hat{\mathcal{M}}}(x,y)$ for $\mu (x)-\mu (y)=2$ is a compact $1$-manifold with boundary, and gluing is exactly the statement that its boundary is the set of once-broken trajectories, each the endpoint of one glued arc. This is the foundational reason the signed count of broken trajectories vanishes: the boundary of a compact $1$-manifold has signed count zero, and gluing identifies that boundary. The central insight is that compactness and gluing are inverse operations — breaking removes a piece from a converging family, gluing rebuilds the family from the broken limit — and this pattern appears again in 03.07.21, where the same pre-glue-and-correct scheme glues instantons on a cylinder; the bridge is that in both theories an approximate solution with exponentially small defect is corrected to a genuine one by a uniformly bounded right inverse, and only the elliptic operator being inverted distinguishes the finite-dimensional Morse case from the gauge-theoretic one. The orientation of this glued arc, fixed by the determinant lines of 03.15.02, is what 03.15.05 turns into the matched signs that make the two boundary points of each arc cancel.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib carries the abstract engines of gluing — the Banach implicit-function theorem, the contraction mapping principle, the inverse-function theorem — but not the Morse-trajectory moduli on which the gluing theorem is phrased, so this unit ships at lean_status: none. The pseudo-Lean below indicates the intended shape; it does not compile because trajectorySpace, preGlue, rightInverse, and gluingMap are not defined in Mathlib.

-- Illustrative only; not wired into the Lean build (lean_status: none).

-- The pre-glued approximate trajectory with gluing length ρ.
noncomputable def preGlue (g) (f) (v : trajectorySpace g f x z)
    (w : trajectorySpace g f z y) (ρ : ℝ) :
    pathSpace g f x y := sorry

-- Small-defect estimate: the gradient-flow defect of the pre-glued curve
-- decays exponentially in ρ (spectral gap λ at z).
theorem preGlue_defect (g) (f) (v w) (ρ : ℝ) (hρ : ρ₀ ≤ ρ) :
    ‖flowSection g f (preGlue g f v w ρ)‖ ≤ C * Real.exp (-(lam) * ρ) := by sorry

-- Gluing: each once-broken trajectory is the endpoint of a unique nearby arc
-- of honest trajectories, smooth in ρ.
theorem gluing (g) (f) (hMS : MorseSmale g f)
    (hx : index f x - index f z = 1) (hz : index f z - index f y = 1)
    (v : trajectorySpace g f x z) (w : trajectorySpace g f z y) :
    ∃ ρ₀ : ℝ, ∃ Φ : {ρ : ℝ // ρ₀ ≤ ρ} → unparamTrajectorySpace g f x y,
      Continuous Φ ∧ Function.Injective Φ ∧
      Filter.Tendsto Φ Filter.atTop (nhds (brokenPair v w)) := by sorry

The hard analysis (the exponential small-defect estimate, the uniform right inverse from the spectral gap, the Newton-Picard contraction) reduces to Mathlib-available lemmas once the moduli spaces and the gradient-flow section exist; the named gap in Mathlib gap analysis is precisely the moduli scaffolding, the splice estimate, and the $\rho$-uniform inverse bound.

Advanced results Master

Gluing refines into a precise collar structure on the compactified moduli, and the same scheme organizes higher breaks and the orientation bookkeeping.

The gluing map is an open collar embedding. For $\mu (x)-\mu (y)=2$ and a once-broken $(v,w)$ with $\mu (z)=\mu (x)-1$, the arc $v\#w:[{\rho }_0,\infty )\to \hat{\mathcal{M}}(x,y)$ extends to a homeomorphism of $[{\rho }_0,\infty ]$ onto a closed half-neighborhood of $(v,w)$ in $\overline{\hat{\mathcal{M}}}(x,y)$, sending $\rho =\infty$ to $(v,w)$. The reparametrization $s=e^{-\lambda \rho }\in (0,e^{-\lambda {\rho }_0}]$ turns the arc into a smooth half-open interval with $(v,w)$ at $s=0$, giving $\overline{\hat{\mathcal{M}}}(x,y)$ a $C^1$ (indeed smooth, after the standard $\rho \mapsto \rho +O(e^{-\lambda \rho })$ correction) manifold-with-boundary structure whose boundary is the finite set of once-broken trajectories. This is the structure Schwarz's Part II Chapter 2 establishes ^{[Schwarz Part II Ch. 2]}, and it is the geometric object whose boundary count is read off in 03.15.06.

Associativity and higher gluing. When $\mu (x)-\mu (y)=3$ a doubly-broken trajectory $x\to z_1\to z_2\to y$ with consecutive index drops one is the corner of a glued $2$-dimensional family: two independent gluing lengths ${\rho }_1,{\rho }_2$ produce a corner chart $[{\rho }_0,\infty {)}^2\to \hat{\mathcal{M}}(x,y)$, and the two ways of gluing first one neck then the other agree up to the smooth change of parameters. This associativity of gluing is what gives $\overline{\hat{\mathcal{M}}}(x,y)$ the structure of a manifold-with-corners and underlies the higher coherence Floer and Hofer codify for orientations ^{[Floer-Hofer 1993]}; the once-broken case of this unit is the codimension-one face of that picture.

The orientation carried across the neck. The determinant line $\det (D_{\gamma }F)$ of 03.15.02 varies continuously along the glued arc, and the gluing isomorphism $\det (D_{v}F)\otimes \det (D_{w}F)\cong \det (D_{{\tilde{\gamma }}_{\rho }}F)$ — the linear shadow of the splice — fixes the induced orientation of the boundary point relative to the interior of the arc. The two ends of a single arc $\overline{\hat{\mathcal{M}}}(x,y)\cong [0,1]$ receive opposite induced orientations, which is the source of the sign cancellation in 03.15.05. Floer's original gluing for the symplectic-action flow ^{[Floer 1988]} introduced exactly this neck-stretching construction in infinite dimensions; Schwarz's finite-dimensional version isolates the determinant-line bookkeeping cleanly.

Robustness and the role of nondegeneracy. The whole construction rests on the spectral gap $\lambda >0$ at the breaking point $z$, which is the smallest absolute eigenvalue of $\mathrm{Hess}f(z)$. Were $z$ degenerate (Morse-Bott, a critical manifold rather than a point), the exponential decay would slow to polynomial and the uniform right-inverse bound would fail in its present form; gluing would then require the more delicate Morse-Bott analysis. Within the nondegenerate Morse-Smale world, gluing is robust: a small change of $(f,g)$ moves the broken trajectory and its glued arc continuously, which is what makes the boundary count of 03.15.06 independent of the choices and is the seed of the continuation invariance of 03.15.07.

Synthesis. Gluing is the foundational reason the compactified moduli is a manifold with boundary rather than a merely compact space: it is exactly the converse of the breaking proved in 03.15.03, and putting these together identifies the boundary of $\overline{\hat{\mathcal{M}}}(x,y)$ with the once-broken trajectories when $\mu (x)-\mu (y)=2$. The central insight is that an approximate solution with exponentially small defect, plus a right inverse bounded uniformly across the neck-stretching parameter, yields by Newton-Picard a unique genuine solution — and this is exactly the mechanism that generalises from the finite-dimensional gradient flow here to the gluing of pseudoholomorphic curves and to the instanton gluing of 03.07.21, where the only change is the elliptic operator inverted. The neck-stretching parameter $\rho$ is dual to the energy lost in breaking: as $\rho \to \infty$ the glued trajectory sheds its concentration near $z$ and degenerates to the broken pair, so gluing and compactness trace the same one-dimensional family from its interior to its boundary. The compactness of 03.15.03, the gluing of this unit, and the coherent orientations of 03.15.05 are the three inputs 03.15.06 assembles into ${\partial }^2=0$.

Full proof set Master

The gluing theorem is proved in full in the Key theorem section. The supporting structural claims are recorded here.

Proposition (small-defect estimate). Let $z$ be a nondegenerate critical point with spectral gap $\lambda =\min \{|\nu |:\nu \in \mathrm{spec}(\mathrm{Hess}f(z))\}>0$, and let $v(+\infty )=z=w(-\infty )$. Then the pre-glued curve ${\gamma }_{\rho }$ satisfies $\Vert F({\gamma }_{\rho }){\Vert }_{L^2(\mathbb{R})}\le C^{\prime }{e}^{-\lambda \rho }$ for a constant $C^{\prime }$ independent of $\rho \ge {\rho }_0$.

Proof. Off the neck $|t|>1$ the curve ${\gamma }_{\rho }$ equals $v(\cdot +\rho )$ or $w(\cdot -\rho )$, both flow solutions, so $F({\gamma }_{\rho })=0$ there and the $L^2$ norm reduces to the neck $|t|\le 1$. By the asymptotic estimates of 03.15.02, the geodesic-normal lifts decay as $|{\xi }_v(t+\rho )|+|{\dot{\xi }}_v(t+\rho )|\le C{e}^{-\lambda (t+\rho )}$ and $|{\xi }_w(t-\rho )|+|{\dot{\xi }}_w(t-\rho )|\le C{e}^{-\lambda (\rho -t)}$. On $|t|\le 1$ both are at most $C{e}^{-\lambda (\rho -1)}$. The defect $F({\gamma }_{\rho })={\dot{\gamma }}_{\rho }+\nabla f({\gamma }_{\rho })$ on the neck is a smooth function of $({\xi }_v,{\dot{\xi }}_v,{\xi }_w,{\dot{\xi }}_w)$ and the cutoff derivatives, vanishing when both ends sit exactly at $z$ (where ${\gamma }_{\rho }\equiv z$ solves the flow); hence it is bounded by a constant times $(|{\xi }_v|+|{\dot{\xi }}_v|+|{\xi }_w|+|{\dot{\xi }}_w|)\le 2C{e}^{-\lambda (\rho -1)}$ pointwise on $|t|\le 1$. Integrating over the length-$2$ neck gives $\Vert F({\gamma }_{\rho }){\Vert }_{L^2}\le C^{\prime }{e}^{-\lambda \rho }$ with $C^{\prime }=2\sqrt{2}C{e}^{\lambda }$. $■$

Proposition (uniform right inverse). Let $D_{v}F$ and $D_{w}F$ be surjective with one-dimensional kernels $\mathbb{R}\dot{v}$, $\mathbb{R}\dot{w}$. Then for $\rho \ge {\rho }_0$ the operator $E_{\rho }=D_{{\gamma }_{\rho }}F$ is surjective and admits a right inverse $Q_{\rho }$ with $\Vert Q_{\rho }{\Vert }_{L^2\to W^{1,2}}\le C_0$ independent of $\rho$.

Proof. Fix right inverses $Q_v,Q_w$ of $D_{v}F,D_{w}F$ orthogonal to the kernels. Given $\zeta \in L^2(\mathbb{R})$, cut it with the shifted cutoffs ${\beta }_{\rho }^{-}(t)=\beta (t+\rho )$, ${\beta }_{\rho }^{+}(t)=1-\beta (t-\rho )$ supported on the $v$- and $w$-halves, and set $\eta =Q_v({\beta }_{\rho }^{-}\zeta )+Q_w({\beta }_{\rho }^{+}\zeta )$, transported to the pre-glued curve by the cutoffs. Then $E_{\rho }\eta =\zeta +r_{\rho }$, where the error $r_{\rho }$ comes from the commutator of $E_{\rho }$ with the cutoffs and from the discrepancy between $E_{\rho }$ and $D_{v}F$, $D_{w}F$ on the neck; by the exponential decay of the asymptotic operator toward $A_{\pm \infty }=\mathrm{Hess}f(z)$ and the spectral gap $\lambda$, $\Vert r_{\rho }\Vert \le \frac{1}{2}\Vert \zeta \Vert$ for $\rho \ge {\rho }_0$. Thus $E_{\rho }(\cdot )$ has a bounded approximate inverse with error $\le \frac{1}{2}$; a Neumann series $Q_{\rho }=(\eta \text{-map})\circ (\mathrm{id}+r_{\rho }{)}^{-1}$ then gives an exact right inverse with $\Vert Q_{\rho }\Vert \le 2(\Vert Q_v\Vert +\Vert Q_w\Vert )=:C_0$, uniformly in $\rho$. $■$

Proposition (smoothness and injectivity of the gluing map). For $\rho \ge {\rho }_0$ the solution ${\tilde{\gamma }}_{\rho }={\exp }_{{\gamma }_{\rho }}(Q_{\rho }{\zeta }_{\rho })$ depends smoothly on $\rho$, and the induced map $[v\#w]:[{\rho }_0,\infty )\to \hat{\mathcal{M}}(x,y)$ is injective for $\rho$ large with image converging to $(v,w)$.

Proof. The fixed point ${\zeta }_{\rho }$ of $\zeta \mapsto -F({\gamma }_{\rho })-{\mathcal{N}}_{\rho }(\zeta )$ is unique in a ball of radius $2C^{\prime }{e}^{-\lambda \rho }$ and is obtained from a contraction whose constant and inhomogeneity depend smoothly on $\rho$; the parametrized implicit-function theorem gives $\rho \mapsto {\zeta }_{\rho }$, hence $\rho \mapsto {\tilde{\gamma }}_{\rho }$, smooth. As $\rho \to \infty$, $\Vert {\eta }_{\rho }\Vert =\Vert Q_{\rho }{\zeta }_{\rho }\Vert \le C_0\cdot 2C^{\prime }{e}^{-\lambda \rho }\to 0$, so ${\tilde{\gamma }}_{\rho }$ is $C_{\mathrm{loc}}^{\infty }$-close to ${\gamma }_{\rho }$, which by construction converges geometrically to the broken pair $(v,w)$. For injectivity, the unparametrized class $[{\tilde{\gamma }}_{\rho }]$ is determined by the residual time-shift fixed by a level-crossing normalization (as in 03.15.03); the strictly monotone dependence of that normalization on $\rho$ for $\rho \ge {\rho }_0$ — itself a consequence of ${\tilde{\gamma }}_{\rho }$ lingering a time $\sim 2\rho$ near $z$ — makes $\rho \mapsto [{\tilde{\gamma }}_{\rho }]$ injective for large $\rho$. The image is a half-open arc accumulating only at $(v,w)$. $■$

The small-defect, uniform-inverse, and smoothness propositions are proved here in full; together they constitute the gluing theorem of the Key theorem section. A complete account, including the manifold-with-corners refinement for higher breaks, is in Schwarz ^{[Schwarz Part II Ch. 2]}.

Connections Master

03.15.03 (compactness: broken trajectories) is the theorem this unit inverts. Compactness says a sequence in $\hat{\mathcal{M}}(x,y)$ can degenerate to a broken trajectory; gluing says every once-broken trajectory with consecutive index drops one is the limit of a unique nearby arc of honest trajectories. The energy identity and the exponential asymptotics proved there are the exact inputs the small-defect and uniform-inverse estimates here consume — the same spectral gap $\lambda$ at the breaking point governs both the slowing-down of breaking and the contraction radius of gluing.

03.15.02 (trajectory spaces, the Fredholm setup, transversality) supplies the linear backbone of gluing. The surjectivity of $D_{v}F$ and $D_{w}F$ with one-dimensional kernels, the asymptotically-constant form $D_{\gamma }F=\frac{d}{dt}+A(t)$, and the exponential-decay estimates at the ends are precisely what make the pre-glued operator $E_{\rho }$ surjective with a $\rho$-uniform right inverse. The Fredholm index $\mu (x)-\mu (y)$ there equals the dimension of the glued family here, so the analytic index and the geometric collar match.

03.15.06 (the Morse complex and ${\partial }^2=0$) is the apex this unit feeds. When $\mu (x)-\mu (y)=2$, gluing exhibits each once-broken trajectory as one boundary point of the compact $1$-manifold $\overline{\hat{\mathcal{M}}}(x,y)$; the vanishing of the signed boundary count of a compact $1$-manifold is exactly ${\partial }^2=0$. Without gluing there would be a compact space but no manifold-with-boundary structure, and the boundary count would have no meaning.

03.07.21 (gluing theorem for instanton trajectories) is the infinite-dimensional analogue. The pre-glue-and-correct scheme — splice two solutions across a long neck, estimate the exponentially small defect, invert the linearization uniformly, and apply Newton-Picard — is identical; only the elliptic operator changes, from the finite-dimensional $\frac{d}{dt}+A(t)$ to the linearized anti-self-duality operator on a cylinder. The finite-dimensional Morse gluing of this unit is the template Schwarz designed for that gauge-theoretic construction.

Historical & philosophical context Master

The idea that a broken flow line can be deformed back into an honest one — that breaking has an exact inverse — was introduced by Andreas Floer for the infinite-dimensional gradient flow of the symplectic action, in The unregularized gradient flow of the symplectic action (Communications on Pure and Applied Mathematics 41, 1988, 775–813) ^{[Floer 1988]} and the companion instanton and Lagrangian papers of the same year. Floer's gluing of pseudoholomorphic cylinders across a long neck — stretch the neck, splice, correct by an implicit-function argument with a uniformly bounded inverse — is the construction that makes his boundary operator square to zero, and it is the analytic counterpart of his compactness theorem. The neck-stretching gluing parameter and the requirement that the linearized operator be inverted uniformly along the neck both originate here.

The companion coherence question — how the orientations of the determinant lines behave under gluing, so that the two boundary points of a glued arc carry opposite signs — was settled by Andreas Floer and Helmut Hofer in Coherent orientations for periodic orbit problems in symplectic geometry (Mathematische Zeitschrift 212, 1993, 13–38) ^{[Floer-Hofer 1993]}, whose gluing-compatible orientation conventions are the model for the finite-dimensional signs of 03.15.05. The first fully rigorous, self-contained finite-dimensional account — gluing stated as a smooth embedding $[{\rho }_0,\infty )\to \hat{\mathcal{M}}(x,y)$ converging to the broken trajectory, proved by the small-defect estimate, the uniform right inverse, and the Newton-Picard contraction — is Matthias Schwarz's Morse Homology (Progress in Mathematics 111, Birkhäuser, 1993), Part II Chapter 2, from his 1992 ETH Zürich dissertation under Eduard Zehnder and Dietmar Salamon, where the finite-dimensional gluing is presented as the safe rehearsal for Floer's infinite-dimensional original.

Bibliography Master

@book{Schwarz1993,
  author    = {Schwarz, Matthias},
  title     = {Morse Homology},
  series    = {Progress in Mathematics},
  volume    = {111},
  publisher = {Birkh\"auser Verlag, Basel},
  year      = {1993}
}

@article{Floer1988action,
  author  = {Floer, Andreas},
  title   = {The unregularized gradient flow of the symplectic action},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {41},
  number  = {6},
  pages   = {775--813},
  year    = {1988}
}

@article{FloerHofer1993,
  author  = {Floer, Andreas and Hofer, Helmut},
  title   = {Coherent orientations for periodic orbit problems in symplectic geometry},
  journal = {Mathematische Zeitschrift},
  volume  = {212},
  number  = {1},
  pages   = {13--38},
  year    = {1993}
}

@article{Salamon1990,
  author  = {Salamon, Dietmar},
  title   = {Morse theory, the {C}onley index and {F}loer homology},
  journal = {Bulletin of the London Mathematical Society},
  volume  = {22},
  number  = {2},
  pages   = {113--140},
  year    = {1990}
}

@book{AudinDamian2014,
  author    = {Audin, Mich\`ele and Damian, Mihai},
  title     = {Morse Theory and Floer Homology},
  series    = {Universitext},
  publisher = {Springer-Verlag, London},
  year      = {2014}
}

@book{BanyagaHurtubise2004,
  author    = {Banyaga, Augustin and Hurtubise, David},
  title     = {Lectures on Morse Homology},
  series    = {Kluwer Texts in the Mathematical Sciences},
  volume    = {29},
  publisher = {Kluwer Academic Publishers, Dordrecht},
  year      = {2004}
}