Trajectory spaces, the Fredholm setup, and transversality

Intuition Beginner

A Morse function $f$ on a curved space records a height. Water released anywhere runs downhill along a unique path, and every such path starts and ends sitting still at a peak, a pass, or a valley floor. The unit before this one named those resting points and counted how many independent downhill directions each has. Now the objects of interest are the running streams themselves: the downhill paths that begin at one resting point $p$ and end at another, lower one $q$.

Collect every such stream from $p$ to $q$ into one set. This set is the trajectory space. Its size is not random. If $p$ has many more downhill directions than $q$, there is a whole family of streams between them; if the gap is small, there may be just a few, or none. The whole machinery of Morse homology comes from understanding the shape of these families.

The key discovery is that, after a tiny generic adjustment of the space's geometry, each such family is itself a smooth shape whose dimension you can predict in advance from the two index numbers. "Generic" means: it holds for almost every choice, and any bad choice can be nudged into a good one.

Visual Beginner

Alt text: An upright torus carrying its height function. The maximum at the top, two saddles around the inner ring, and the minimum at the bottom are labelled with their indices 2, 1, 1, 0. Downhill flow arrows connect higher critical points to lower ones. The picture shows that streams from the top to a saddle come in a one-parameter family, while streams between an index-2 and an index-0 point can break at a saddle in between.

Worked example Beginner

Take the height function on the round sphere, north pole $N$ at the top, south pole $S$ at the bottom. The north pole has $2$ downhill directions and the south pole has $0$.

Count the streams from $N$ to $S$. Starting just below $N$, water runs down a meridian straight to $S$. There is one such meridian through each compass direction, so the streams form a full circle's worth of paths, one for every angle from $0$ to $360$ degrees.

Now notice two of these streams are not really different in the way that matters. Two streams that trace the same meridian but start the clock at different moments are the same path of points, just timed differently. After we agree to ignore the starting time, the number that remains is the number of distinct meridians: a circle of directions.

What this tells us: the raw trajectory space here is two-dimensional (an angle, plus a choice of when the clock starts), and removing the irrelevant clock choice drops it to one dimension, a circle. The prediction "dimension equals difference of the two index numbers" gives $2-0=2$ for the timed count and $2-0-1=1$ for the time-forgotten count. Both match.

Check your understanding Beginner

Formal definition Intermediate+

Fix a closed Riemannian manifold $(M^n,g)$ and a Morse function $f:M\to \mathbb{R}$ with the Morse-Smale property (intersections of unstable and stable manifolds are transverse, as set up in 03.15.01). Write $\nabla f$ for the gradient with respect to $g$ and $\mu (p)$ for the Morse index of a critical point $p$.

A parametrized trajectory from $p$ to $q$ is a smooth curve $\gamma :\mathbb{R}\to M$ solving the negative gradient flow

$$\dot{\gamma }(t)=-\nabla f(\gamma (t)),\underset{t\to -\infty }{\lim }\gamma (t)=p,\underset{t\to +\infty }{\lim }\gamma (t)=q.$$

The trajectory space is $$\mathcal{M}(p,q)=\{\gamma \in C^{\infty }(\mathbb{R},M):\dot{\gamma }=-\nabla f(\gamma ),\gamma (-\infty )=p,\gamma (+\infty )=q\}.$$ Equivalently $\mathcal{M}(p,q)=W^u(p)\cap W^s(q)$ under the map sending $\gamma$ to $\gamma (0)$, where $W^u(p)$ and $W^s(q)$ are the unstable and stable manifolds. Translation in time, $(\tau \cdot \gamma )(t)=\gamma (t-\tau )$, defines a free $\mathbb{R}$-action on $\mathcal{M}(p,q)$ whenever $p\ne q$, because a non-constant solution cannot be fixed by any non-zero shift. The unparametrized trajectory space is the quotient $$\hat{\mathcal{M}}(p,q)=\mathcal{M}(p,q)/\mathbb{R}.$$

To turn this into an analytic problem, replace smooth curves by a Banach manifold. Choose a smooth reference path ${\gamma }_{pq}$ from $p$ to $q$ that is constant near $\pm \infty$, and let ${\mathcal{P}}^{1,2}(p,q)$ be the space of maps $\gamma :\mathbb{R}\to M$ that agree with ${\gamma }_{pq}$ outside a compact set and whose difference, read in geodesic normal coordinates centred on ${\gamma }_{pq}$, lies in the Sobolev space $W^{1,2}(\mathbb{R},{\mathbb{R}}^n)$. This ${\mathcal{P}}^{1,2}(p,q)$ is a smooth Banach manifold modelled on $W^{1,2}$. Over it sits the Banach bundle $\mathcal{E}\to {\mathcal{P}}^{1,2}(p,q)$ with fibre ${\mathcal{E}}_{\gamma }=L^2(\mathbb{R},{\gamma }^{\ast }TM)$, the vector fields of class $L^2$ along $\gamma$.

The gradient flow defines a smooth section $$F:{\mathcal{P}}^{1,2}(p,q)⟶\mathcal{E},F(\gamma )=\dot{\gamma }+\nabla f(\gamma )\in L^2(\mathbb{R},{\gamma }^{\ast }TM),$$ whose zero set is exactly $\mathcal{M}(p,q)$. The point of the construction is that $F$ is a Fredholm section: its vertical linearization at a zero is a Fredholm operator, so $F^{-1}(0)$ is, under transversality, a finite-dimensional manifold cut out cleanly. The Sobolev exponent $W^{1,2}/L^2$ is the standard choice; weighted Sobolev spaces $W_{\delta }^{1,2}$ with a small exponential weight $e^{\delta |t|}$ give the same Fredholm index and are used when the asymptotic operator is degenerate, which here it never is because the critical points are nondegenerate.

Counterexamples to common slips

• The asymptotic limits are part of the data. A solution of $\dot{\gamma }=-\nabla f(\gamma )$ that does not converge to critical points at both ends is not an element of any $\mathcal{M}(p,q)$; on a closed manifold every bounded gradient trajectory does converge, but the limit set could a priori be larger than a point if $f$ were not Morse.
• The $\mathbb{R}$-action is free only for $p\ne q$. For $p=q$ the only solution is the constant $\gamma \equiv p$, which is fixed by every shift; one never forms $\hat{\mathcal{M}}(p,p)$.
• The dimension formula counts the unparametrized space at one less than the parametrized one. Writing $\dim \mathcal{M}(p,q)=\mu (p)-\mu (q)$ and then also $\dim \hat{\mathcal{M}}(p,q)=\mu (p)-\mu (q)$ is the most common index-bookkeeping error.
• Transversality is a statement about the metric (equivalently the vector field), not about $f$ alone. A fixed Morse $f$ with a carelessly chosen $g$ can fail Morse-Smale; the theorem below says generic $g$ fixes it.

Key theorem with proof Intermediate+

The central analytic statement has two halves: the linearization is Fredholm with a computable index, and transversality holds for generic data.

Theorem (Fredholm index and transversality). Let $(M^n,g)$ be closed Riemannian and $f$ a Morse function. For $p\ne q$ critical:

1. At every $\gamma \in \mathcal{M}(p,q)$ the vertical linearization $D_{\gamma }F:W^{1,2}(\mathbb{R},{\gamma }^{\ast }TM)\to L^2(\mathbb{R},{\gamma }^{\ast }TM)$ is a Fredholm operator of index $\mathrm{ind}{D}_{\gamma }F=\mu (p)-\mu (q)$.
2. For a residual (hence dense) set of metrics $g$, the section $F$ is transverse to the zero section. For such $g$ the space $\mathcal{M}(p,q)$ is a smooth manifold of dimension $\mu (p)-\mu (q)$, and $\hat{\mathcal{M}}(p,q)$ is a smooth manifold of dimension $\mu (p)-\mu (q)-1$.

Proof of (1). Trivialize ${\gamma }^{\ast }TM$ over $\mathbb{R}$ by an orthonormal parallel frame, identifying sections with maps $\mathbb{R}\to {\mathbb{R}}^n$. In this frame the linearization of $F(\gamma )=\dot{\gamma }+\nabla f(\gamma )$ is the first-order operator $$D_{\gamma }F=\frac{d}{dt}+A(t),A(t)=(\nabla (\nabla f){)}_{\gamma (t)}=\mathrm{Hess}f(\gamma (t)),$$ a path of symmetric matrices given by the Hessian of $f$ along $\gamma$, read in the parallel frame. Because $\gamma (t)\to p$ and $\gamma (t)\to q$ as $t\to \mp \infty$ and $f$ is Morse, the limits $$A_{-}=\underset{t\to -\infty }{\lim }A(t)=\mathrm{Hess}f(p),A_{+}=\underset{t\to +\infty }{\lim }A(t)=\mathrm{Hess}f(q)$$ exist and are invertible symmetric matrices (nondegeneracy of the critical points). An operator $\frac{d}{dt}+A(t)$ on $W^{1,2}\to L^2$ with asymptotically constant, invertible, self-adjoint limits $A_{\pm }$ is Fredholm; this is the standard asymptotically-hyperbolic Fredholm criterion ^{[Robbin-Salamon §2]}. Its index equals the spectral flow of the path $\{A(t)\}$ from $A_{-}$ to $A_{+}$, i.e. the signed count of eigenvalues of $A(t)$ that cross zero. Since $A_{\pm }$ are invertible the endpoints contribute no kernel, and the Robbin-Salamon identification gives

$$\mathrm{ind}(\frac{d}{dt}+A(t))=\mathrm{sf}(\{A(t){\}}_{t\in \mathbb{R}})=\dim E^{-}(A_{-})-\dim E^{-}(A_{+}),$$

where $E^{-}(\cdot )$ is the negative eigenspace. Now $\dim E^{-}(\mathrm{Hess}f(p))$ is by definition the Morse index $\mu (p)$, and likewise for $q$. Hence $\mathrm{ind}{D}_{\gamma }F=\mu (p)-\mu (q)$. $\square$

Proof of (2). Let $\mathcal{G}$ be the Banach manifold of $C^k$ metrics on $M$ (for $k$ large, completed in a suitable Floer $C^{\epsilon }$-norm to keep it a Banach space and to keep the perturbations dense in the smooth metrics). Form the universal section $$\mathcal{F}:{\mathcal{P}}^{1,2}(p,q)\times \mathcal{G}⟶\mathcal{E},\mathcal{F}(\gamma ,g)=\dot{\gamma }+{\nabla }_{g}f(\gamma ),$$ where ${\nabla }_{g}f$ is the gradient with respect to $g$. The first claim is that $\mathcal{F}$ is transverse to the zero section. At a zero $(\gamma ,g)$ the linearization in the $\mathcal{G}$-direction varies the metric, hence varies ${\nabla }_{g}f$; because $f$ has no critical point along $\gamma$ except at the asymptotic ends, one can produce, by a compactly supported perturbation of $g$ near an interior point of $\gamma$, any prescribed $L^2$ variation transverse to the image of $D_{\gamma }F$. Thus the full linearization $D\mathcal{F}=D_{\gamma }F\oplus D_g\mathcal{F}$ is surjective, so $\mathcal{F}⋔0$ and the universal moduli space $\mathcal{Z}={\mathcal{F}}^{-1}(0)$ is a Banach manifold.

Project $\pi :\mathcal{Z}\to \mathcal{G}$, $(\gamma ,g)\mapsto g$. Surjectivity of $D_{\gamma }F$ at $(\gamma ,g)$ is equivalent to $g$ being a regular value of $\pi$ at that point, and $D\pi$ is Fredholm of the same index $\mu (p)-\mu (q)$ as $D_{\gamma }F$ because the $\mathcal{G}$-directions are split off. The Sard-Smale theorem ^{[Smale 1965]} states that for a $C^k$ Fredholm map between second-countable Banach manifolds with $k>\max (\mathrm{ind},0)$, the set of regular values is residual. Applying it to $\pi$ yields a residual set ${\mathcal{G}}_{\mathrm{reg}}(p,q)\subset \mathcal{G}$ of metrics for which every $\gamma \in \mathcal{M}(p,q)$ is a transverse zero. Intersecting the countably many residual sets over all pairs $(p,q)$ of critical points (finite in number since $M$ is closed) keeps the intersection residual. For $g$ in this intersection, $F$ is transverse, so by the regular-value theorem on Banach manifolds (implicit function theorem) $\mathcal{M}(p,q)=F^{-1}(0)$ is a smooth manifold of dimension $\mathrm{ind}{D}_{\gamma }F=\mu (p)-\mu (q)$.

Finally the $\mathbb{R}$-action by time-translation is smooth, free (as $p\ne q$), and proper, so the quotient $\hat{\mathcal{M}}(p,q)=\mathcal{M}(p,q)/\mathbb{R}$ is a smooth manifold of dimension $\mu (p)-\mu (q)-1$. A standard upgrade from $C^k$ to $C^{\infty }$ metrics (Taubes' argument: the smooth regular metrics are a countable intersection of open dense sets) gives a residual set within the smooth metrics with the same conclusion. $\square$

The formal definition and this theorem follow Schwarz ^{[Schwarz Part I Ch. 2-3]}, with the Fredholm-index computation organised around Robbin-Salamon spectral flow ^{[Robbin-Salamon §2]} and the genericity argument around Sard-Smale ^{[Smale 1965]}.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib does not yet carry the infinite-dimensional differential topology this unit needs (Banach manifolds of paths, the asymptotically-constant Fredholm operator $\frac{d}{dt}+A(t)$, Sard-Smale), so there is no compiling Lean module. What can be stated against current Mathlib is the finite-dimensional skeleton of the index computation: that the relevant linearization is Fredholm and that its index equals a difference of negative-eigenspace dimensions. The following is illustrative pseudo-Lean for the index statement, not a compiling proof.

-- Illustrative only; not wired into the Lean build (lean_status: none).
-- The asymptotically-constant operator d/dt + A(t) with invertible,
-- self-adjoint limits A_-, A_+ is Fredholm with index given by spectral flow.
variable {n : ℕ} (A : ℝ → Matrix (Fin n) (Fin n) ℝ)

-- Hypotheses: A t symmetric for all t; A has invertible limits A_neg, A_pos.
-- Conclusion (target, currently unformalizable end-to-end in Mathlib):
--   Fredholm (fun u => deriv u + A · u) ∧
--   index (...) = negEigenspaceDim A_neg - negEigenspaceDim A_pos
-- where negEigenspaceDim M = Morse index of the corresponding critical point.

The gap is recorded in Mathlib gap analysis; each named missing piece (path Banach manifolds, the Fredholm criterion for $\frac{d}{dt}+A(t)$, the Robbin-Salamon spectral-flow index, Sard-Smale) is a concrete Mathlib contribution target.

Advanced results Master

The construction above admits several refinements that the compactness and complex-building units depend on.

Exponential decay and the weighted picture. A trajectory $\gamma \in \mathcal{M}(p,q)$ converges to its endpoints at an exponential rate governed by the spectral gaps of $A_{\pm }$. Precisely, if ${\lambda }_{-}>0$ is the smallest absolute value of an eigenvalue of $\mathrm{Hess}f(q)$ and $0<\delta <{\lambda }_{-}$, then in normal coordinates centred at $q$ one has $|\gamma (t)-q|+|\dot{\gamma }(t)|=O(e^{-\delta t})$ as $t\to +\infty$, with the symmetric statement at $-\infty$. This is why $\dot{\gamma }\in L^2$ in the first place, and it makes the weighted Sobolev spaces $W_{\delta }^{1,2}$ (norm $\Vert e^{\delta |t|}\cdot {\Vert }_{W^{1,2}}$) interchangeable with $W^{1,2}$ for the Fredholm theory when the endpoints are nondegenerate: shifting the weight by $\delta$ smaller than the spectral gap changes neither kernel nor cokernel, hence not the index. The weighted spaces become essential only when one allows degenerate (Morse-Bott) ends, where the asymptotic operator has kernel and the unweighted operator ceases to be Fredholm.

The orientation of the determinant line. The Fredholm operator $D_{\gamma }F$ carries a determinant line $\det (D_{\gamma }F)={\Lambda }^{\max }\ker D_{\gamma }F\otimes ({\Lambda }^{\max }\mathrm{coker}{D}_{\gamma }F{)}^{\ast }$. These lines fit into a real line bundle over $\mathcal{M}(p,q)$ whose orientation, made coherent across gluings, is what later assigns the integer signs $n(\gamma )\in \{\pm 1\}$ to isolated trajectories. The Fredholm setup of this unit is the precondition: without the asymptotically-constant operator and its index there is no determinant line to orient. The coherent-orientation theory itself is developed downstream in the orientation unit of this chapter.

Spectral flow as the index, intrinsically. The identity $\mathrm{ind}(\frac{d}{dt}+A(t))=\mathrm{sf}(\{A(t)\})$ is not an accident of the trivialization. Robbin-Salamon prove it for any path of self-adjoint Fredholm operators with invertible ends, with the spectral flow counting eigenvalue crossings of zero with sign. In the finite-dimensional Morse case the ends are honest invertible symmetric matrices and the spectral flow collapses to $\dim E^{-}(A_{-})-\dim E^{-}(A_{+})=\mu (p)-\mu (q)$. This is the finite-dimensional shadow of the instanton spectral-flow grading: in the gauge-theoretic setting the same operator $\frac{d}{dt}+A(t)$ appears with $A(t)$ the Hessian of the Chern-Simons functional, and the index lives in $\mathbb{Z}/8$ rather than $\mathbb{Z}$.

Why $W^{1,2}$ and not $C^1$. The Banach-manifold structure needs a model space in which the implicit function theorem applies and in which the linearization is Fredholm. Hölder spaces $C^{k,\alpha }$ also work and are used by some authors; the $W^{1,2}/L^2$ pair is preferred here because the inner-product structure makes the cokernel a genuine orthogonal complement, simplifying the surjectivity arguments and matching the Hilbert-bundle framing that the Floer-theoretic analogue requires.

Full proof set Master

Exponential convergence. Let $\gamma \in \mathcal{M}(p,q)$ and work near the end $t\to +\infty$, in normal coordinates centred at $q$ in which $f(x)=f(q)+\frac{1}{2}⟨Hx,x⟩+O(|x{|}^3)$ with $H=\mathrm{Hess}f(q)$ invertible symmetric. Set $x(t)=\gamma (t)-q$. Then $\dot{x}=-Hx+R(x)$ with $|R(x)|=O(|x{|}^2)$. Let ${\lambda }_{-}=\min |\mathrm{spec}(H)|>0$. Because $x(t)\to 0$, for $t\ge T$ large the nonlinear term is dominated and the energy $E(t)=\frac{1}{2}|x(t){|}^2$ satisfies $\dot{E}=⟨x,\dot{x}⟩=-⟨x,Hx⟩+⟨x,R(x)⟩\le -{\lambda }_{-}|x{|}^2+C|x{|}^3\le -\delta |x{|}^2=-2\delta E$ for any fixed $\delta <{\lambda }_{-}$ once $|x|$ is small enough. Grönwall gives $E(t)\le E(T)e^{-2\delta (t-T)}$, hence $|x(t)|=O(e^{-\delta t})$. Then $|\dot{x}|=|-Hx+R(x)|=O(e^{-\delta t})$ as well. The same argument at $-\infty$ uses that $-\nabla f$ flows away from $p$, so reversing time gives exponential decay toward $p$ as $t\to -\infty$. In particular $\dot{\gamma }\in L^2(\mathbb{R})$, justifying $F(\gamma )\in L^2$ and $\dot{\gamma }\in W^{1,2}$. $\square$

Weight invariance of the index. For $0<\delta <{\lambda }_{-}=\min (\min |\mathrm{spec}(A_{-})|,\min |\mathrm{spec}(A_{+})|)$, conjugating $D=\frac{d}{dt}+A(t)$ by the multiplication $M_{\delta }:u\mapsto e^{\delta |t|}u$ produces an operator $M_{\delta }D{M}_{\delta }^{-1}=\frac{d}{dt}+A(t)-\delta \mathrm{sgn}(t)\mathrm{Id}$ (away from $t=0$) acting $W^{1,2}\to L^2$, whose asymptotic matrices $A_{\pm }\mp \delta \mathrm{Id}$ are still invertible because $\delta <{\lambda }_{-}$ leaves the spectral sign pattern unchanged. The number of negative eigenvalues of $A_{\pm }\mp \delta \mathrm{Id}$ equals that of $A_{\pm }$, so the spectral flow, and hence the index, is unchanged. Thus the weighted operator on $W_{\delta }^{1,2}$ and the unweighted operator on $W^{1,2}$ have the same Fredholm index $\mu (p)-\mu (q)$. $\square$

Properness of the $\mathbb{R}$-action. For $p\ne q$ and $\gamma$ non-constant, the action $\mathbb{R}\times \mathcal{M}(p,q)\to \mathcal{M}(p,q)$, $(\tau ,\gamma )\mapsto \tau \cdot \gamma$, is free because $\tau \cdot \gamma =\gamma$ forces $\gamma$ periodic, impossible for a gradient trajectory with distinct ends (the function $f\circ \gamma$ is strictly decreasing). It is proper: if ${\tau }_k\cdot {\gamma }_k\to {\gamma }_{\infty }$ and ${\gamma }_k\to {\gamma }_{\infty }^{\prime }$ in $\mathcal{M}(p,q)$, then fixing a regular value $c$ of $f$ with $f(q)<c<f(p)$, each trajectory crosses the level set $f^{-1}(c)$ exactly once, at a time pinned by the convergence; the crossing times of ${\tau }_k\cdot {\gamma }_k$ and ${\gamma }_k$ differ by ${\tau }_k$, and both converge, so ${\tau }_k$ converges. Hence the quotient $\hat{\mathcal{M}}(p,q)$ is a smooth manifold. $\square$

The surjectivity of the universal linearization claimed in the proof of the main theorem is the one place where geometry enters: the variation $\delta g$ of the metric changes ${\nabla }_{g}f$ by a term supported wherever one chooses, and on the (non-empty, since $p\ne q$) interior of $\gamma$ the value $\nabla f(\gamma (t))\ne 0$, so $\delta g$ can be steered to hit any direction in the finite-dimensional cokernel of $D_{\gamma }F$. A complete account of this density argument is in Schwarz ^{[Schwarz Part I Ch. 3]}; the abstract perturbation-of-section version, which replaces metric perturbations by perturbations of the gradient vector field through a Banach space of compactly supported sections, is in Floer ^{[Floer 1988]}.

Connections Master

• 03.15.01 (gradient flow, stable/unstable manifolds, the Morse-Smale condition) is the geometric input: $\mathcal{M}(p,q)=W^u(p)\cap W^s(q)$, and the transversality theorem proved here is precisely what makes that intersection clean and of the expected dimension. This unit re-reads the Morse-Smale condition as the analytic surjectivity of a Fredholm section.
• 03.02.30 (Morse functions, the Morse lemma, and the Morse index) supplies the nondegeneracy of critical points that makes the asymptotic operators $A_{\pm }=\mathrm{Hess}f(p),\mathrm{Hess}f(q)$ invertible — without which $\frac{d}{dt}+A(t)$ would not be Fredholm — and identifies $\dim E^{-}(\mathrm{Hess}f)$ with the Morse index appearing in the index formula.
• 03.15.03 (compactness: broken trajectories) consumes this unit's output directly: once $\hat{\mathcal{M}}(p,q)$ is a smooth manifold of dimension $\mu (p)-\mu (q)-1$, compactness asks how non-compact it is, and answers that the missing ends are broken trajectories. The exponential-decay estimates proved here are the technical input to that compactification.
• 03.09.06 (Fredholm operators) is the abstract theory specialized here to the asymptotically-constant first-order operator; the index, kernel, and cokernel notions used throughout are its content.
• The infinite-dimensional analogue is the instanton spectral flow of 03.07.19, where the identical operator $\frac{d}{dt}+A(t)$ — now with $A(t)$ the Hessian of the Chern-Simons functional — gives the $\mathbb{Z}/8$-valued Floer grading; this finite-dimensional unit is its template.

Historical & philosophical context Master

The trajectory-space viewpoint crystallized over four decades. Thom (1949) and Smale, in his work on gradient dynamical systems (Smale, Annals of Mathematics 74, 1961), introduced the stable and unstable manifolds and the transversality condition now bearing Smale's name, recovering the cell structure of a manifold from a Morse function. The functional-analytic reading — flow lines as zeros of a section whose linearization is Fredholm — is due to the period after Witten's 1982 Journal of Differential Geometry paper reframed the Morse complex through supersymmetric quantum mechanics, and Floer's 1988 papers carried the picture into infinite dimensions for Lagrangian intersections and instantons.

The two analytic theorems used here predate their Morse-homological application. Smale's infinite-dimensional Sard theorem (Smale, American Journal of Mathematics 87, 1965) extended Sard's measure-zero-critical-values statement to Fredholm maps between Banach manifolds, and is the device that converts "generic" from a heuristic into a residual-set theorem. The identification of the Fredholm index of $\frac{d}{dt}+A(t)$ with spectral flow was given its definitive form by Robbin and Salamon (Bulletin of the London Mathematical Society 27, 1995), unifying the Morse, Maslov, and Conley-Zehnder indices under one analytic invariant.

Schwarz's 1993 Birkhäuser monograph (from his 1992 ETH dissertation under Zehnder and Salamon) is the first source to assemble all of this into a single rigorous finite-dimensional construction, explicitly as a rehearsal for Floer theory: the trajectory spaces, their Fredholm description, and the Sard-Smale transversality argument occupy its Part I, on which the entire Morse complex of Part II is built.

Bibliography Master

• M. Schwarz, Morse Homology. Progress in Mathematics 111. Basel: Birkhäuser, 1993. (Part I: trajectory spaces, the Fredholm section, transversality.)
• S. Smale, "An infinite-dimensional version of Sard's theorem." American Journal of Mathematics 87 (1965), 861–866.
• S. Smale, "On gradient dynamical systems." Annals of Mathematics 74 (1961), 199–206.
• J. Robbin and D. Salamon, "The spectral flow and the Maslov index." Bulletin of the London Mathematical Society 27 (1995), 1–33.
• A. Floer, "Morse theory for Lagrangian intersections." Journal of Differential Geometry 28 (1988), 513–547.
• A. Floer, "The unregularized gradient flow of the symplectic action." Communications on Pure and Applied Mathematics 41 (1988), 775–813.
• M. Audin and M. Damian, Morse Theory and Floer Homology. Universitext. London: Springer, 2014. (Chapters 2–3, the trajectory-space and transversality account.)
• A. Banyaga and D. Hurtubise, Lectures on Morse Homology. Kluwer Texts in the Mathematical Sciences 29. Dordrecht: Kluwer, 2004.

@book{schwarz1993morse, author = {Schwarz, Matthias}, title = {Morse Homology}, series = {Progress in Mathematics}, volume = {111}, publisher = {Birkh"auser}, address = {Basel}, year = {1993} }

@article{smale1965sard, author = {Smale, Stephen}, title = {An infinite-dimensional version of {S}ard's theorem}, journal = {American Journal of Mathematics}, volume = {87}, pages = {861--866}, year = {1965} }

@article{robbinsalamon1995spectral, author = {Robbin, Joel and Salamon, Dietmar}, title = {The spectral flow and the {M}aslov index}, journal = {Bulletin of the London Mathematical Society}, volume = {27}, pages = {1--33}, year = {1995} }