# 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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### 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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### 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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 \$...