--- date: 2022-02-23 18:45 modification date: Friday 1st April 2022 09:48:07 title: "Morse Theory" aliases: [Morse theory, Morse, Morse index, Morse homology] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags - #projects/notes/reading #geomtop - Refs: - [Extremely good review](https://webusers.imj-prg.fr/~alexandru.oancea/BAMS-review-Stein.pdf) #resources/notes - #resources/videos - [Video of conformal flows](https://youtu.be/mIUi1zIUQJw?t=42) - [Gradient Descent](https://youtu.be/vWFjqgb-ylQ?t=5) - - Hiro-Lurie - Links: - [Unsorted/handle decomposition](Unsorted/handle%20decomposition.md) - [Stein](Unsorted/Stein.md) --- # Morse Theory The standard procedure: - Show that $D$ is a Fredholm operator - Show that $D$ is surjective, so $\Ind D = \dim \ker D$ - Show a moduli space is the intersection of some section $s$ of a bundle with the zero section. - Show that this intersection is transverse, i.e. $Ds$ is surjective. - Vary the Riemannian metric and use a [second category](Unsorted/second%20category.md) theorem to get the Morse-Smale condition. - Apply the infinite-dimensional inverse function theorem - Show that a Frechet manifold is in fact a Banach manifold and apply a version of [Sard's Theorem](Sard's%20Theorem) ![](attachments/Pasted%20image%2020220401094806.png) ![](attachments/Pasted%20image%2020220401094918.png) # Motivations - Can be used to prove the high dimensional case of the generalized [Unsorted/Poincare conjectures](Unsorted/Poincare%20conjectures.md) # Results **Theorem**: Every compact smooth manifold admits a Morse function. **Theorem**: [Morse function](Morse%20functions) are **generic**: given any smooth function $f: X\to Y$, there is an arbitrarily small perturbation of $f$ that is Morse. See [Morse lemma](Morse%20lemma.md) **Theorem 3**: If $(W; M_0, M_1) \to I$ is Morse with no critical points then $W \cong_{\Diff} I \cross M_0$ **Theorem**: If $X$ is closed and admits a Morse function with exactly 2 critical points, $X$ is homeomorphic to $S^n$. Possibly used in Milnor's [Unsorted/parahoric](Unsorted/parahoric.md) 7-sphere (show a diffeomorphism invariant differs but admits such a Morse function) **Theorem**: $M$ is homotopy equivalent to a CW complex with one cell of dimension $k$ for each critical point of $f$ of [index of a Morse function](index%20of%20a%20Morse%20function) $k$. ![attachments/Pasted image 20210501235532.png](attachments/Pasted%20image%2020210501235532.png) ![](attachments/Pasted%20image%2020220422101431.png) ![](attachments/Pasted%20image%2020220422101514.png) ![](attachments/Pasted%20image%2020220422101532.png) ![](attachments/Pasted%20image%2020220422101540.png) # Gradients ![](attachments/Pasted%20image%2020220422121526.png) ![](attachments/Pasted%20image%2020220422121549.png) ![](attachments/Pasted%20image%2020220422122055.png) # Energy ![](attachments/Pasted%20image%2020220422121652.png) ![](attachments/Pasted%20image%2020220422121714.png) # Critical points Idea: the number of linearly independent direction you can move for which the function *decreases*. ![](attachments/Pasted%20image%2020220422120052.png) ![](attachments/Pasted%20image%2020220422120201.png) # Morse chain complex ![](attachments/Pasted%20image%2020220422120306.png) ![](attachments/Pasted%20image%2020220422120319.png) # Morse inequalities ![](attachments/Pasted%20image%2020220422120836.png) # Broken trajectories ![](attachments/Pasted%20image%2020220422121840.png) # Moduli space of flow lines ![](attachments/Pasted%20image%2020220422122307.png) ![](attachments/Pasted%20image%2020220422122322.png) ![](attachments/Pasted%20image%2020220422122513.png) # Zero set of a section ![](attachments/Pasted%20image%2020220422123136.png) See [vertical and horizontal subspace](Unsorted/vertical%20and%20horizontal%20subspace.md) ![](attachments/Pasted%20image%2020220422123254.png) # Sard ![](attachments/Pasted%20image%2020220422123700.png) # Pictures ![](attachments/Pasted%20image%2020220422120347.png) # Examples ![](attachments/Pasted%20image%2020220422120415.png) ![](attachments/Pasted%20image%2020220422120442.png) ![](attachments/Pasted%20image%2020220422120518.png) ![](attachments/Pasted%20image%2020220422120603.png) ![](attachments/Pasted%20image%2020220422120616.png) ![](attachments/Pasted%20image%2020220422120637.png) ![](attachments/Pasted%20image%2020220422120712.png) ![](attachments/Pasted%20image%2020220422120725.png) # Notes ## Dave's Videos - Historic note: Morse wanted to know not the diffeomorphism type of $M$, but rather the homotopy type. - Definition: critical values and critical points - Definition: [critical point](critical%20point) - Theorem (Smale, [h-cobordism theorem](h-cobordism%20theorem) - If $X^n$ is a smooth [cobordism](cobordism.md). - Corollary (High-Dimensional [Unsorted/Poincare conjectures](Unsorted/Poincare%20conjectures.md) - If $X_1^n, X_2^n \cong_{\diff} S^n$, then there exists an [h-cobordism](h-cobordism.md) between them. - Proof: use algebraic topology to eliminate (cancel) critical points. - Definition: [index of a Morse function](index%20of%20a%20Morse%20function) - Look at coordinate-free def? - Standard form at critical points - Alternatively: Hessian is non-singular at every critical point. - $f\inv\bd Y) = \bd X$ - Definition: Stable and generic - Definition: [cobordism](cobordism.md) - Example: (pair of pants) - Category: Objects are manifolds, morphisms are cobordisms between them - Consequence of theorem 3: $M_0 \cong_{\text{Diff}} M_1$ is a diffeomorphism, useful to show two things are diffeomorphic, used in higher-dimensional Poincare. - Recall that this is proved by constructing a [vector field](vector%20field) $V$ on $W$, then using a diffeomorphism $\phi:I \cross M_0 \to W$ by flowing along $V$. - Can we do gradient flow in the presence of a [metric](metric.md)? #todo/questions ## Intro Video [https://www.youtube.com/watch?v=78OMJ8JKDqI](https://www.youtube.com/watch?v=78OMJ8JKDqI) Morse theory: handles nice singularities. Can have worse ones, covered by [dynamical systems](catastrophe theory](dynamical%20systems](catastrophe%20theory) (dynamical systems). Importance of CW complexes: [triangulation](triangulation) of surfaces. See [Morse lemma](Morse%20lemma.md) **Morse Theorem 1:** If there are no critical points, $M_A \homotopic M_B$. ![attachments/Pasted image 20210501235559.png](attachments/Pasted%20image%2020210501235559.png) Stable vs unstable manifolds: ![attachments/Pasted image 20210501235734.png](attachments/Pasted%20image%2020210501235734.png) Consider height function on torus. Circles are index 0 critical points, triangle is index 1. ![attachments/Pasted image 20210501235700.png](attachments/Pasted%20image%2020210501235700.png) Cancellation: ![attachments/Pasted image 20210501235757.png](attachments/Pasted%20image%2020210501235757.png) ![attachments/Pasted image 20210501235820.png](attachments/Pasted%20image%2020210501235820.png) Can use persistent homology to measure "importance" of critical points. ## Unsorted [https://youtu.be/mIUi1zIUQJw?t=42](https://youtu.be/mIUi1zIUQJw?t=42) ![attachments/Pasted image 20210501235429.png](attachments/Pasted%20image%2020210501235429.png) ![attachments/Pasted image 20210501235456.png](attachments/Pasted%20image%2020210501235456.png) - Diffeomorphism type depends on [isotopy](isotopy) classes of attaching maps. See [handle decomposition](handle decomposition.md) # More Notes Historic note: Morse wanted to know not the diffeomorphism type of $M$, but rather the homotopy type. - Theorem (Smale, h-cobordism) - If $X^n$ is a smooth cobordism, $n\geq 6$, $\pi_1(X) = 0$, and $X$ "looks like" a product in algebraic topology, then $X$ is a product cobordism. - Corollary (High-Dim Poincare) - If $X_1^n, X_2^n \cong_{\diff} S^n$, then there exists an $h\dash$cobordism between them. - Proof: use algebraic topology to eliminate (cancel) critical points. - Theorem: Every compact manifold has a Morse function. - Theorem: Morse functions are generic (given any smooth function $f: X\to Y$, there's an arbitrarily small perturbation of $f$ that is Morse). - Theorem (Morse Lemma): If $p\in \RR^n$ is a critical point of $f: \RR^n \to \RR$ such that the Hessian $H_f(p)$ is a non-degenerate bilinear form, then $f$ is locally Morse (standard form). - Theorem: If $(W; M_0, M_1) \to I$ is Morse with no critical points then $W \cong_{\diff} I \cross M_0$ - Consequence: $M_0 \cong_{\text{Diff}} M_1$ is a diffeomorphism, useful to show two things are diffeomorphic, used in higher-dimensional Poincare. - Theorem: If $X$ is closed and admits a Morse function with exactly 2 critical points, $X$ is homeomorphic to $S^n$. - Possibly used in Milnor's exotic 7-sphere (show a diffeomorphism invariant differs but admits such a Morse function) - Diffeomorphism type depends on isotopy classes of attaching maps. ## Morse Theory Goal: handlebody decomposition, or for the purposes of the above theorems, retractions onto a CW complex. Decomposing a cobordism into a sequence of elementary cobordisms (admit a Morse function with a single critical point). Fact: since $\phi$ is Morse, $M^{2n}$ can be retracted onto a complex of dimension $d\leq n$, since all critical points will have index $\leq n$. > Note: this immediately implies the Lefschetz Hyperplane theorem for affine manifolds $N$, i.e. that they are entirely determined by the homology and homotopy of $N\intersect H$ for any hyperplane. Very strong! Setting up notation/definitions: - $V$ will be a smooth $n\dash$manifold - $W$ an $n\dash$dimensional cobordism - $\phi: V\to \RR$ a smooth function - $p$ a critical point of $\phi$ (i.e. the derivative $d_p \phi$ vanishes) - $H_p\phi = ({\del^2 \phi \over \del x_i^2 \del x_j^2})$ the Hessian matrix - $\null_\phi(p)$ the *nullity* of $\phi$ at $p$ is $\dim \ker H_p$, regarding $H_p\phi$ as a symmetric bilinear form on $T_p V$ - $p$ is *nondegenerate* iff $\null_\phi(p) = 0$. - The *Morse index* at $p$ is the dimension of the maximal subspace on which the associated *quadratic* form $H_p \phi$ is negative definite. Theorem (Morse Lemma) : Near a nondegenerate critical point $p$ of $\phi$ of index $k$ there exists a smooth coordinate chart $U$ and coordinates $\vector x \in \RR^n$ such that $\phi$ has the form $$\phi(\vector x) = \phi(p) + \vector x^t A \vector x$$ where $A = \diag(-1, \cdots, -1, 1,\cdots 1)$. Toward a generalization, we can also write $\RR = \RR^{k} \cross \RR^{n-k}$ and $$ \phi(\vector x_1, \vector x_2) = \phi(p) - \norm{\vector x_1}^2 + \norm{\vector x_2}^2 $$ Lemma (The nondegenerate directions can be split off) : If $\null_\phi(p) = \ell$ then we can instead write $\RR = \RR^{n-k-\ell} \cross \RR^k \cross \RR^\ell$ and $$ \phi(\vector x, \vector y, \vector z) = \norm{\vector x}^2 - \norm{\vector y}^2 + \psi(\vector z) $$ where $\psi: \RR^\ell \to \RR$ is some smooth function. Definition : A degenerate critical point is *embryonic* iff $\null_\phi(p) = 1$ and writing $L = \ker H_p\phi = \spanof_\RR{\vector v}$, the third directional derivative $D^3_{\vector v}\phi$ (?) is nonzero. We now consider homotopies of Morse functions $\phi: I \cross V \to \RR$, where we can partially apply the $I$ factor to get a 1-parameter family $\theset{\phi_t \suchthat t\in I}$. Definition : A homotopy $\Phi: V\cross I \to \RR$ of Morse functions has a *birth-death type* critical point at $p$ at $t=t_0$ iff $p$ is embryonic for $\phi_0$ and $(t_0, p)$ is a nondegenerate critical point of $\Phi$. > Recall what a Cerf diagram/profile is -- I don't Theorem (Whitney) : In three parts: 1. Near an embryonic critical point $p$ of $\phi$ of index $k$ there exist coordinate $(\vector x, \vector y, z) \in \RR^{n-k-1} \oplus \RR^{k} \oplus \RR$ such that $\phi$ has the form $$ \phi(\vector x, \vector y, z) = \phi(p) + \norm{\vector x}^2 - \norm{\vector y}^2 + z^3 $$ 2. If $p$ is birth-death type for $\Phi$ of index $k$, then up to conjugating $\phi_t$ by a (uniform in $t$) family of diffeomorphisms, each $\phi_t$ is of the form $$ \phi(\vector x, \vector y, z) = \phi(p) + \norm{\vector x}^2 - \norm{\vector y}^2 + z^3 \pm tz $$ 3. Any two homotopies $\Phi, \Phi'$ with points $(p, 0)$ and $(p', 0)$ with the same index and Cerf profile differ only by a diffeomorphism, i.e. there is a family of diffeomorphisms $h_t$ such that $\phi'_t \circ h_t = \phi_t$ for every $t$. 4. A generic $\Phi$ has only nondegenerate and birth-death type singularities. Definition : A singularity is *birth type* if the sign on $t$ is positive, and *death type* if negative. Fact : Embryonic critical points are isolated, near a birth-type singularity two nondegenerate critical points of indices $k, k-1$ emerge, and near a death type they merge and disappear. > Pretty vague -- I know there are pictures here that make this more obvious, but I couldn't find much. Definition : A *cobordism* is a triple $(W; M_+, M_-)$ where $W$ is an oriented compact smooth manifold with cooriented boundary $\bd W = M_+ \disjoint M_- = \bd_- W \disjoint \bd_+ W$, where the coorientation is provided by the inward (resp. outward) normal vector field (???). We'll usually just denote this as $W$. Definition : A *Lyapunov cobordism* is a triple $(W, \phi , X)$ where - $W$ is a usual cobordism, - $\phi: W\to \RR$ is a smooth functional that is constant and has no critical points when restricted to $\bd W$, - $X$ is a gradient-like vector field for $\phi$ which points inward along $\bd_- W$ and outward along $\bd_+ W$. Definition : Such a cobordism is *elementary* iff there exist no $X\dash$trajectories between distinct critical points of $\phi$. Theorem (Smale, h-cobordism) : Let $W$ be a cobordism of dimension $W\geq 6$ such that $W, \bd_{\pm}W$ are simply connected, and $H_*(W, \bd_- W; \ZZ) = 0$. Then $W$ admits a Morse function without critical points which is constant on $\bd_\pm W$. In particular, $W \cong I\cross M$ is diffeomorphic to the trivial product cobordism, and $M\cong N$ are diffeomorphic. Proof (Sketch) Goal: find a handle decomposition with *no* handles, then integrate along the gradient vector field of a Morse function $\phi$ to get a diffeomorphism. - Find a Morse function and induce a handle decomposition - Rearrange handles so that lower index handles are attached first - Define a chain complex as free $\ZZ\dash$module on handles with boundary given in terms of intersection numbers of attaching spheres $k$ and belt $k-1$ spheres - Find $k\dash$handles, create a pair of $k+1, k+2$ handles such that the $k+1$ handle cancels/fills in the $k\dash$handle (not sure why the $k+2$ is needed here) - End up with nothing but an $n\dash$handle and an $n-1\dash$handle -- turn "upside down" and repeat process with $-\phi$ to remove them. Proof (Sketch) - Pick $\phi: W\to \RR$ Morse such that $\bd_\pm W$ are regular level sets. - Make $\phi$ self-intersecting (uses a transversality argument) - Partition manifold into regular level sets $L_k \definedas \phi\inv(k - \frac 1 2)$ for each $k\in \NN$. - Letting $\theset{p_i}$ be the critical points in $L_k$ and $\theset{q_j}$ the critical points in $L_{k-1}$, form the matrix $A$ of intersection numbers $S_{p_i}^- \smile S_{q_j}^+$ between the stable sphere of $p_i$ and the unstable sphere of $q_j$. - Goal: since homology can be read off $SNF(A)$, and we know $H_* = 0$ here, we try to reduce $A$ to SNF with geometric operations - Handle slides: Add row $j$ to row $i$ by moving $p_i$ to $L_{k+j}, ~j\geq 1$, deform $X$ to produce a trajectory $p_j \to p_i$, then "the stable manifold of $p_i$ slides over the stable manifold of $p_j$" (?) replacing $[S_i^-]$ with $[S_i^-] + [S_j^-]$ in homology. - This makes $A = [I, 0; 0, 0]$ a block matrix with the identity in the top-left. - Handle Cancellation: Take two transverse intersection points $z_+, z_-$ with local intersection indices $1, -1$, connect via two paths: one in $S_i^-$, one in $S_j^+$. This yields a map $S^1 \injects L_k$, use the Whitney trick to fill with an embedded disc $\Delta$, then push $S_i^-$ over $\Delta$ eliminates $z_\pm$. - This leaves a collection $S_i^-, S_i^+$ for $i=1,\cdots, r$ intersecting in a single point $z_0$, then (lemma) there are unique trajectories $q_i \to p_i$ for each $I$ and thus they can be eliminated. - Do this in $L_k$; we now have a Morse function with no critical points except possibly of index $0, 1, n-1$, or $n$. - Use "Smale's trick": trades in an index $k$ critical point for one of index $k+1$ and one of index $k+2$, such that $k, k+1$ cancel. Trade index $1$ for index $2, 3$ and cancel index $3$ as before. - Eliminate $0, n$ with a lemma (unclear)