Questions tagged [morse-theory]

For questions about Morse theory, which is a branch of differential topology to analyze topology of manifolds by studying differentiable functions on them.

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947 views

Critical points for a smooth function defined on a manifold - Milnor, Morse theory

I need some help with a statement from Milnor's Morse theory book! While studying the proof of a theorem I got stuck. That's what we know: Let $f$ be a differentiable function on a manifold $M$ ...
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2answers
436 views

Is any vector field without periodic orbits a gradient field?

Background: I have read that vector fields which are the gradients of some scalar field cannot have periodic orbits. See, e.g., (1)(2). This probably expresses the fact from vector calculus that $\...
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42 views

Difference between 'similar looking' Morse functions

Let's say $f$ and $g$ are two Morse functions on same manifold $M$, and have same domain, same range, same critical points, same critical values at those points and same index at those critical points....
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1answer
94 views

Equivariant deformation of Morse functions.

Let $f$ be a Morse function on closed smooth manifold $M$, (one can also suppose that this is a hyperbolic manifold) G is a group which acts effectively and smooth on $M$, f and G (hyperbolic metric) ...
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1answer
144 views

Piecewise smooth vector field

I am reading Milnor's book Morse Theory on p.67 he defines tangent space. "By the tangent space of $\Omega$ (which is the path space) at a path $\omega$ will be meant the vector space consisting of ...
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64 views

Use the Poincaré-Bendixson theorem to construct a Morse function

I am trying to understand a passage in Gromov paper "Singularities, expanders and the topology of maps", p. 12 (Section 2.1). At some point he constructs a "generic" non-vanishing non-exact 1-form $...
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1answer
378 views

Morse Function (Critical Point)

Consider the sphere $\mathbb S^2$ embedded in $\mathbb R^3$ and take the height $z$-function. This function has critical points at $(0,0,1)$ and $(0,0,-1)$. Sorry for the simple question but why that ...
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609 views

Symplectic gradient

If $f$ is a smooth function on a symplectic manifold $(M, \omega)$ we can define its symplectic gradient : this is a vector field $X_f$ such that $\iota_{X_f} \omega = - df$. My question is the ...
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1answer
66 views

How to use a smooth function on manifold to get its decomposition?

Let manifold $M$, and let a Morse function $f:M \to \mathbb{R}$ the answer to my question follows from Morse theory. For fixed $f$ the manifold then decomposes as for example here. Now, what happens ...
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1answer
184 views

A problem about diffeomorphism of two components of the boundary of a manifold.

My geometry professor said that the following statement is true: Let $M$ be a compact smooth manifold such that $\partial M = M_0 \cup M_1$. Suppose that there exist a smooth function $f:M \to \mathbb ...
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1answer
259 views

A basic theorem on Morse Theory: diffeomorphism between two manifolds with boundary induced by a map on a manifold

I'm trying to understand the following theorem from "Morse Theory" by John Milnor: Theorem 3.1 Let f be a smooth real valued function on a manifold M. Let $a<b$ and suppose that the set $f^{-1}...
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1answer
54 views

Getting rid of square root via integration

How do we prove, for positive $D$, this result? $$ e^{-2\sqrt D} \sqrt{\pi} = \int_0^\infty s^{-1/2} e^{-(s+D/s)} ds $$
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212 views

Morse theory of knot complements

Suppose $K^1 \subset S^3$ is a (connected) knot. First, I am wondering is there a Morse function $f$ on $S^3$ such that $f$ restricted to a neighborhood $D^2 \times S^1$ of $K^1$ have standard form, i....
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1answer
223 views

Problem in Milnor's proof of h-cobordism theorem

Milnor's 'Lectures on h-cobordism theorem' Theorem 7.6 ("Basis Theorem") reads Suppose $(W;V,V')$ is a triad of dimension $n$ (ie. a cobordism $W$ between $V$ and $V'$) possessing a Morse function $...
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209 views

A result on the minimum number of critical points of a Morse function

At 8:22 in this video the professor cites a result that, for a simply connected symplectic manifold of dimension at least 6, the minimum number of critical points of a Morse function is the sum of the ...
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1answer
179 views

Morse Theory and Electrostatic Problem: Why a finite number of critical points?

So it states in Guillemin and Pollack: Let $x_1, \ldots x_4$ be points in general position $R^3$ (that is they all don't lie in a plane.) Let $q_1, \ldots, q_4$ be electric charges placed at these ...
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97 views

Inverse image of a Morse function is smooth

Suppose $M$ is a smooth manifold and $f:M\rightarrow\mathbb{R}$ a Morse function. Let $$M_a := \{p\in M\,|\,f(p)<a\}.$$ Then $\overline{M_a} = \{p\in M\,|\,f(p)\leq a\}$ and $\partial\overline{M_a} ...
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2answers
2k views

Geometric Intuition of Eigenvalues of Hessian Matrix

I have a very simple question, which I suspect speaks more to my lack of intuitive understanding of parts of linear algebra than anything calculus related. I have come across this statement (or ...
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1answer
310 views

What is a generic (genetic/geometric) map? (In the study of manifolds)

At 29:30 in his lecture on Youtube, Mikhail Gromov talks about how one only gets a manifold from the zero set of the equation $f(0)=0$ if the map $f$ is "generic" (or genetic or geometric -- I mostly ...
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1answer
117 views

How do I construct pairs of pants and morse functions on them?

I'm learning about morse theory and one of the pictures that keeps popping up is that of a pair of pants. Unfortunately, these pairs of pants are only a doodle for me; I have no idea how to model ...
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1answer
99 views

Subadditivity of Betti number for relative homology

This question stems from page 28 of the "Morse Inequalities" chapter of Milnor's Morse Theory. Given a pair of topological spaces, $Y \subset X$ and any field $F$, define $R_\lambda(X,Y)$, or the $\...
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138 views

Introduction to morse theory with applications to optimization

I am wondering if there are any easy-to-read introduction materials on morse theory (especially with applications to nonconvex optimization) for people with non-math background.
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1answer
168 views

Relation between tangent spaces of (un)stable manifolds in Morse theory

After asking this question about signs in the Morse complex, I realised that my confusion is really about how tangent spaces to different (un)stable manifolds are related. So suppose we have a Morse ...
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1answer
136 views

Explicit verification of signs in Morse complex

I'm trying to check by hand that the signs in the Morse complex, defined via choices of orientations on the unstable manifolds, lead to $\partial^2=0$. The books I've looked in seem to say either ...
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1answer
195 views

Such a manifold is homeomorphic to a sphere

I recently read that if a compact differentiable manifold admits a real function with only two critical points, then it is homeomorphic to a sphere. If the function is Morse, this follows from ...
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1answer
78 views

Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible.

Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible. The problem is from the following material. It contends that the result is by standard Morse theory....
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1answer
138 views

Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
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48 views

Is there $f: U \to \mathbb{R}^{n}$ injective such that…

Let $f: U \to \mathbb{R}^{n}$ $C^{1}$ injective where $U$ is a open in $\mathbb{R}^{n}$ (so $f$ is open by invariance domain theorem). a) Is there exist $f$ such that dim $ker(df_{x}) >$ dim $...
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2answers
138 views

Doubt in definition of symmetric continuous function and norm in Kupka's paper

In this article "Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds" of I. Kupka has the following passage: For $H=l^{2}$ "Let $H^{*}$ be the dual of $H$. A base ...
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1answer
350 views

Potential proof for the Slice-Ribbon conjecture (may be wrong).

Let $f:(D^2,S^1)\to(D^4,S^3)$ be a smooth embedding (so called a slice disk), and we set $M:=f(D^2)$. Then, is the restriction map $C^{\infty}(D^4)\to C^{\infty}(M)$ open map with relative to $C^2$ ...
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222 views

Integral Curves of Gradient-like Vector Fields

If $X$ is a gradient-like vector field of a Morse function $f\colon M\to \mathbb{R}$, then the integral curve $c_p(t)$ starting at an arbitrary point $p$ approaches critical points as $t\to \pm \...
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96 views

Regular CW complex arising from a Morse decomposition

Suppose $(M,g)$ is a Riemannian manifold equipped with a Morse function $f: M \rightarrow \mathbb R$. It's been shown that $f$ gives rise to a CW decomposition homeomorphic to $M$ under the generic ...
3
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1answer
309 views

Use of implicit function theorem in showing that $f(x) \leq a$ is a submanifold with boundary

This question comes from a statement in John Milnor's "Morse Theory" on page 4. Let $f: M \to \mathbb{R}$ be a smooth function on a manifold $M$. Milnor claims that if $a$ is not a critical value of $...
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1answer
118 views

Finding a domain of an integral curve of a vector field

Studying Morse theory, I am stuck on some problem. Let $M$ be a compact smooth manifold, and $f$ is a smooth real-valued function on $M$. Choose a Riemannian metric $g$ on $M$, let $X$ be the vector ...
2
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1answer
733 views

Self-indexing Morse function on a torus which is a height function

A Morse function $f: \Bbb T^2\to [0,2]$ is called self-indexing if $f^{-1}(n)$ is the set of critical points of index $n$. It is relatively easy to see that on any compact manifold, any Morse function ...
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1answer
159 views

Construct a smooth function on $T^2$ that has exactly three critical points

By the results in Morse theory, a smooth function on $T^2$ has at least three critical points, and at least one of them is degenerate. I'm asked to construct a smooth function that has exactly three ...
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1answer
235 views

Morse functions and connected sum

My question is closely related to this post but it is slightly different. Let $M_1$ and $M_2$ be two smooth closed $n$-manifolds such that there is a Morse function $f_i:M_i\rightarrow \mathbb R$ for ...
4
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1answer
465 views

Hints for an exercise on Morse theory

Exercise: Let $M$ be a $3$-dimensional smooth manifold with boundary $\partial M$ which is a surface of genus $g$. Moreover let $f:M\longrightarrow [0,1]$ be a Morse function with the following ...
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2answers
311 views

Does the Morse homology depend on the orientation?

Before asking my question I need to define some objects. I will follow the book "M. Audin, M.Damian - Morse theory and Floer homology", but the terminology is quite standard: Let $M$ be a smooth ...
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1answer
332 views

Problem in Morse Theory: critical points on the exotic sphere

I've taken a course on Morse theory a couple of years ago, but I have no idea about how to solve the following problem. Could you give some hints? Problem: Let $M$ be a smooth manifold homeomorphic ...
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82 views

Extending Morse-Smale pair from submanifolds?

The following proposition is extracted from Audin & Damian's Morse Theory and Floer Homology, Proposition 4.6.3: Let $(f,X)$ be a Morse-Smale pair on $V$ (a submanifold of $W$). Then there exists ...
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1answer
98 views

Equivalency of h-cobordisms

I'm reading lectures on the h-cobordism theorem by Milnor and I have a little problem understanding some basic points. I can't understand this theorem , not the theorem , not the proof. I appreciate ...
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1answer
489 views

Boundary of a manifold is a submanifold?

I was reading in the book Morse Theory and Floer Homology by Audin and Damian (translated in english) that the boundary of a manifold is not always a submanifold. I cannot see why that is true. Any ...
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643 views

Index of zero of a gradient vector field at a critical point

Let $M$ be a Riemannian manifold with a Morse function $f: M \to \mathbb{R}$. The zeroes of the gradient vector field of $f$ are the critical points of $f$. How do you show that a critical point of ...
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1answer
148 views

Does the Morse-Bott index of a critical point depend on the choice of metric?

By the Morse lemma there exists a coordinate chart $(x_1,...,x_n)$ in the neighbourhood of a critical point $p$ of a Morse function $f : M^n \to \mathbb{R}$ such that \begin{equation*} f(x) = f(p) - \...
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0answers
125 views

Heegaard splitting via a Morse function - twisted union or not?

Let $M$ be a smooth, closed, connected, oriented 3-manifold and let $f: M \rightarrow \mathbb{R}$ be a self-indexing Morse function. Since $\frac{3}{2}$ is a regular value of $f$ it follows from Morse ...
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1answer
513 views

Proof of the last part of the Reeb theorem

I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. Suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both non-degenerate),...
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233 views

Manifold in Milnors Morse Theory

While reading "Morse Theory" by Milnor, I noticed that certain arguments would not work, if the considered manifolds have nonempty boundary. Example: Proof of 3.5 I could not find the definition ...
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0answers
80 views

What can be said about the space of vector fields for which a given, say $C^1$, function is a Lyapunov function?

I am learning Morse homology and I have been thinking about the following observation. One way of doing, say finite-dimensional, Morse theory is by fixing a Morse function $f\in C^{\infty}(M)$, where $...
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2answers
975 views

How does Morse theory on non-compact manifolds differ from compact manifolds?

What is the Morse homology of a non-compact manifold? When is it, as in the compact case, isomorphic to singular homology of the underlying manifold? What other constructions can be identified with ...

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