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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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Find all points that make $f$ Morse

I have function $f(x)=d(x, x_0)$ in $\mathbb{R^3}$, I have to find all points $x_0$ in $\mathbb{R^3}$ for the standard sphere $S^2$. A Morse Function must have these two properties: $(i)$ All ...
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1answer
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Why are the level sets connected for a Hamiltonian $S^1$-action?

In these lecture notes by Weimin Chen, in Corollary 1.8, it is stated that Let $H : M → \mathbb{R}$ be a moment map of a Hamiltonian $S^1$-action on a compact, connected manifold M. Then each level ...
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1answer
70 views

Morse Theory in Riemannian Geometry

In the introduction of Cheeger and Gromoll's paper On the structure of complete manifolds of nonnegative curvature the following is stated: A point $p \in M$ is called simple if there are no ...
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33 views

Definition approximation of smooth function by Morse function

Hi everyone I'm currently studying Morse Theory on my own and I've come across a proposition where I would need a precise definition of what is meant. Here is the proposition: Let $M$ be a manifold ...
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33 views

Critical points in a simple case of manifold with boundary.

Let $X$ be a smooth, connected, compact manifold and $Y=X\times [a,b]$. Let $f:Y\to\mathbb{R}$ be a Morse function on $Y$ such that $f|_{X\times \{a\}}=a$ and $f|_{X\times \{b\}}=b$. What can we say ...
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1answer
40 views

Deducing topology from a function with no critical values.

Let $X\subset\mathbb{R}^n$ be a compact manifold with a boundary. Let $f:X\to\mathbb{R}$ be a smooth function constant on each component of the boundary of $X$ (not sure if thats important). Assume ...
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55 views

Unit of (Lagrangian) Floer cohomology

Let $(M,\omega)$ be a symplectic manifold and $L \subseteq M$ be a compact Lagrangian in M. My question is what is a geometric/natural representative for the unit of the Floer cohomology $HF(L,L)$? Or ...
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1answer
76 views

Holomorphic Morse functions

For a holomorphic $f:\mathbb{C}^n\rightarrow \mathbb{C}$ and $a=(a_1, \dots, a_n) \in \mathbb{C}^n$, let $f_a:\mathbb{C}^n\rightarrow \mathbb{C}$ be the function $$(z_1, \dots, z_n) \mapsto a_1z_1 + ...
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Discrete Morse Function for $n$-simplex

I am trying to find a "useful" discrete Morse function for the $n$-simplex. According to (https://www.emis.de/journals/SLC/wpapers/s48forman.pdf page 12), a possible discrete Morse function is $f(\...
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37 views

Defining a pseudo-gradient field for a $1$-form

I'm reading Audin and Damian's Morse Theory and Floer Homology; they say there is an analogous way to define nondegenerate critical points for 1-forms as well as pseudo-gradient fields but don't ...
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On morse theory and foliations

Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: ...
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36 views

Poincaré–Hopf and Morse inequalities

Disclaimer: I am not a differential geometer, so maybe this question does not make sense: Let $(M^m,g)$ be a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse function. Since $g$ is pointwise non-...
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gradient flows on Hilbert manifolds

I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded. To be more precise, a ...
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Weak Lefchetz for a quasiprojective variety and a non-generic hyperplane

In the remarks on page 153-154 of Stratified Morse Theory, Goresky and MacPherson make a claim that they say follows from the theorem on that page. It seems to be false and I'm wondering if I'm ...
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39 views

Morse Homology of $\mathbb{R}\mathbb{P}^2$

I'm trying to compute the homology of $\mathbb{R}\mathbb{P}^2$ using the following Morse function. First, consider $f:S^2 \to \mathbb{R}$ taking $(x,y,z) \mapsto y^2 + 2z^2$. This can be shown to be ...
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63 views

Level sets of Morse functions

Suppose we have a compact connected manifold $M$ of dimension $n$ and a Morse function $f$ on the manifold such that there are no critical points of index $n-1$ and of index $1$, how does one see that ...
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Conley index as a subset of an isolated invariant set

In Sec. 7 (p. 60) of Conley's $\textit{Isolated Invariant Sets and the Morse Index (1976)}$, the following passage appears: (In fact any isolated invariant set in $S$ is isolated in $\Phi$. It ...
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1answer
87 views

A difficulty in understanding the solution of problem 1.7.17 in Guillemin and Pollack.(p.47)

The question is given below: And here is exercise (16): And here is the solution to exercise(17) But I have difficulties in understanding the following parts of the solution: 1-Why the codomain of ...
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48 views

Question about the homology groups of the complex projective space $\mathbb{C}P_n$

My question is how we can compute the homology groups of the complex projective space $\mathbb{C}P_n$ by the following Corollary5.4 at page 31 in Milnor's book: Corollary5.4 If $c_{\lambda+1}=c_{\...
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1answer
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What is the free loop space $\mathcal{L}M$ of a manifold of a manifold $M$ for which the energy functional has no critical points?

What is the free loop space $\mathcal{L}M$ of a manifold $M$ for which $E:LM\to\mathbb{R}$ for $E:\gamma\mapsto\int_{S^1}\|\dot\gamma(t)\|^2dt$ has no non-degenerate critical points? Is it simply the ...
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1answer
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Morse and Conley theory: Proof of finite rank for certain cohomology groups

My question concerns Conley theory for topological flows and its connection to classical Morse theory on compact manifolds. Specifically, I have in mind Conley and Zehnder's seminal paper Morse-type ...
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Handlebody Decomposition of a torus seems to follow Pascal's triangle

I have been studying Morse Theory and found something really cool that I have no idea how to prove. The handlebody decomposition of a 2-torus is a 0-handle, two 1-handles, and one 2-handle. Similarly, ...
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1answer
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Acyclicity of a pair in Morse Theory.

Let $\delta\colon M\times\mathbb{R}^{2N}\to\mathbb{R}$ be a smooth function such that outside a compact set, one has: $$\delta(x,e_1,e_2)=A(e_1)-A(e_2),$$ where $A\colon\mathbb{R}^N\to\mathbb{R}$ is a ...
2
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1answer
135 views

Morse lemma for holomorphic functions

If $f:C^n\to C$ is holomorphic in a neighbourhood of $0$ and $0$ is a nondegenerate critical point, then there is a neighbourhood $U$ of $0$ with a holomorphic local chart, namely a holomorphic ...
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1answer
31 views

Isolated degenerate critical points

Suppose $f:\mathbb{R}^n\rightarrow\mathbb{R}$ does not have any degenerate critical points on a set $S\subset\mathbb{R}^n$ (i.e. the Hessian of $f$ has full rank on $S$). Is it possible to introduce ...
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1answer
43 views

General case of a result in differential geometry

Let $W \subset \Bbb R^n$ open and $F:W \to \Bbb R$, $F$ is $C^{\infty}$ and $a\lt b \in F(W)$ so that $L=F^{-1}([a,b])$ is compact and $\nabla F(p) \ne 0$ for every $p \in L$ By taking the field $X = ...
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99 views

Morse–Palais Lemma and local approximation

Let $M$ be an $n$-dimensional Riemannian manifold and let $E:\Lambda M\to\mathbb{R}$ be the energy functional $E[\gamma]=\int_{S^1}||\dot\gamma(t)||^2dt$ where $\Lambda M$ is the free loop space of $M$...
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critical points relevant to the lowest order non-perturbative correction

I am interested in the Hyperasymptotics of multidimensional integrals of the form $$\mathcal{I}(\lambda) = \int_{\mathbb{R}^n} dz_1 \wedge dz_2 \wedge \dotsi \wedge dz_n \, g(z_1,\dotsi,z_n) \, e^{\...
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1answer
73 views

(Negative) Gradient and Orientability of its flow.

Before asking my question, I put the necessary definitions and some context. If you are used with Morse Theory, you can skip the text within [[[...]]]. [[[Let me first define what I mean by gradient ...
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Computing derivative in second tangent bundle

Suppose $M$ is an $m$-dimensional manifold and $\gamma \colon \mathbb{R} \rightarrow M$ is a path with $\gamma(0) = p_0$ and $\gamma'(0) = v_0 \in T_{p_0}M$. Let $g \colon \mathbb{R} \rightarrow TM$ ...
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1answer
74 views

If $f:S^1 \to \mathbb{R}$ is a Morse function then $f$ has an even number of critical points.

I'm trying to solve the following problem Let $f: S^1 \to \mathbb{R}$ be a smooth Morse function, then $f$ has an even number of critical point. My progress: I was able to prove that if $g:\...
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32 views

Viewing existence of Morse function geometrically

I was reading the existence of Morse function in two variables. In between the proof, it is written that, If a function $z=f(x,y)$ is defined near the origin with $f(0,0)=0$, then there are ...
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Proving $\partial ^ 2 = 0 $ for the case of Morse-Complex with $\mathbb{Z}$ using orientation of the moduli space

I was going through the book Morse theory and Floer homology by Audin-Damian and got stuck where they talk about defining the complex for $\mathbb{Z}$ coefficient. Assume that $a,b,c$ are critical ...
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1answer
47 views

How is a condition on symplectic triviality expressed in Chern classes?

The following is taken from Audin, Damian: Morse Theory and Floer Homology: My questions about this: Question 1: I understand "exists a symplectic trivialization" as: There exists a symplectic ...
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Infimum set of a convex function

In the paper Greene, R.E.; Shiohama, K., Convex functions on complete noncompact manifolds: Topological structure, Invent. Math. 63, 129-157 (1981). ZBL0468.53033, there is a theorem (Theorem A) ...
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2answers
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The Euler characteristic of a manifold

The Euler characteristic of a manifold is the alternating sum of the number of critical points of a Morse function on it. $$ \chi(X) := \sum_{k=0}^{n} (-1)^k b_k = \sum_{k=0}^{n} (-1)^k c_k. $$ My ...
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1answer
306 views

Morse lemma via Moser's trick

In Abraham and Marsden's Foundations of Mechanics, they prove Morse lemma via Moser's trick. They are able to reduce the proof so that it suffices to find a smooth family of vector fields $Z_t$ such ...
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1answer
83 views

Equivalence between function being Morse and $df$ being transversal to zero section.

The proof I know of the fact that $f:M \to \mathbb{R}$ is Morse iff $df:M \to T^*M$ is transversal to the zero section uses local coordinates heavily. I would like to know if there is an ...
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1answer
228 views

A Morse function on a compact manifold has finitely many critical points

We still have a problem with the Morse lemma. Let $u$ be a non-degenerate critical point of the function $f : \mathbb{M} \to \mathbb{R}.$ There are local coordinate with $u = (0, \dots, 0)$ such ...
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1answer
70 views

Codimension-1 submanifold as a inverse image of regular value. [closed]

Let $M$ be a manifold and $N\subset M$ be a codimension-1 submanifold. Is it possible to find a function $H: M\rightarrow \mathbb{R}$ such that $N\subset H^{-1}(a)$ for some regular value a of $H$?. ...
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1answer
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A closed manifold has closed geodesics of at most countably many lengths

In Introduction to Arithmetic Groups by Dave Morris, I read the comment "Since a single closed surface has closed geodesics of only countably many different lengths..." which in context is ...
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1answer
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Understanding Morse's Lemma

Just to provide some context, I'm reading a proof of Morse's Lemma in a book called Topology and Geometry for Physicists, and it's not too difficult of a proof, but I don't understand one tiny part. ...
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Morse index of minimal surfaces in $\mathbb{R}^3$

I am wondering where can I find the Morse index of the most famous examples of minimal surfaces in $\mathbb{R}^3$, such as the cathenoid, the helicoid, etc. Is there any general standard technique to ...
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1answer
197 views

Given a smooth function $f:M\to [a,b]$, $f^{-1}(a)$ and $f^{-1}(b)$ are (immersed) submanifolds

Suppose that $(M,g)$ is a compact Riemannian manifold and $f:M\to [a,b]$ a smooth function such that $\|\nabla f\|$ is constant along each level set. Assume that $\forall c,d \in [a,b]$ and $\forall ...
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1answer
118 views

Morse theory: the loop space of $S^n$

I am currently learning Morse theory through Milnor's book on the subject. I am trying to understand the result given in p.96 that tells us the cell decomposition of the loop space of $S^n$: ...
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1answer
64 views

Morse theory problem II

I keep reading the book of Milnor, Morse theory and i have a problem. It exactly this one. At the very end he says that is clearly that $\varphi_{b-a} $ takes $M^a$ diffeo to $M^b$ and for me is not ...
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1answer
145 views

Extensions of vector fields and the Hessian

Im reading book of Milnor, Morse theory and at the very beginning, he defines the hessian as in the paragraph above. My question is, what is the precise definition of extension to a vector fiels of ...
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How many height function on a given compact oriantable manifold there exist up to $SO(n)$ rotations?

How many height function on a given compact oriantable manifold there exist up to $SO(n)$ rotations? It seems to me that there is only one height function, but I have no Idea how to show it.
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Energy functional is locally constant on moduli space

Let $\alpha \in \Omega^1(M)$ be a closed $1$-form on a closed Riemannian manifold $(M,g)$. Denote by $X$ the corresponding dual vector field. Consider a flow line $$\gamma \colon \mathbb{R} \to M$$ ...
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1answer
97 views

How to calculate second -order derivative of height function?

I quote a paragraph of "Morse Theory and floer homology" The critical points of the height function on the sphere are nondegenerate. Indeed, in the neighborhood of the point $(0, 0, \varepsilon) \...