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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
24 views

Height function on the sphere is Morse

I'm trying to prove that the height function on the sphere $\mathbb{S}^2$ is a Morse function. Somehow from my onw calculations, I conclude that it's not Morse. Using the embedding with spherical ...
SiS_sos's user avatar
0 votes
0 answers
58 views

How to prove a function is Morse

In general, if I have a function $g:X \rightarrow R$ defined on a manifold $X$, how can I show it is a Morse function? Is the best way to just compute $g \circ \phi$, where $\phi$ is a parametrization ...
Albi's user avatar
  • 29
2 votes
0 answers
69 views

Proving height function is Morse

We have a differentiable manifold $X$ embedded in $\mathbb{R}^{n+1}$ and let $h: X \to \mathbb{R}$ as $h(x_{1},...,x_{n+1})=x_{n+1}$. We need to prove that $h$ is a Morse function. I don't know how to ...
Guilherme N.'s user avatar
0 votes
0 answers
22 views

Morse theory graphic

How can I make the images that appear in Morse theory?. Type Beginner's question about homotopy type in Milnor's Morse Theory, i try in geogebra but i cannot.
GillThunder's user avatar
3 votes
2 answers
103 views

Birth-death : Always more than 1 bifurcation?

Say I have a (smooth) function $f : \mathbb{R}^n \to \mathbb{R}$, and a critical point $x$ (ie, $f'(x) = 0$). I call this point degenerate if $\det \text{Hess}_x f = 0$ (so, equivalently, if the ...
Azur's user avatar
  • 2,194
1 vote
0 answers
17 views

Stationary phase approximation, compact support needed

I am trying to understand the stationary phase approximation as described on Wikipedia https://en.wikipedia.org/wiki/Stationary_phase_approximation. As necessary condition, they mention a compact ...
Sebastian 's user avatar
1 vote
0 answers
54 views

Topology of blow up of a cone

Let $X$ be a real algebraic surface with a unique $A_1$ point at $p \in X$ (that is, the germ of $X$ at $p$ is isomorphic to the germ of $\{x^2+y^2=z^2\}$ at the origin). Let $Y \to X$ be the blow up ...
Serge the Toaster's user avatar
0 votes
0 answers
11 views

Reference request - "cones with multiple components" near higher order critical point

I am looking for information and relevant terminology for a phenomenon related to the geometry of sublevel sets of functions. I believe that the possibly related fields are Morse theory and algebraic ...
GSofer's user avatar
  • 4,313
2 votes
0 answers
17 views

Morse flow: cancelling handle pairs away from deformation retract

Given a smooth manifold (not closed, maybe with boundary) $M$ in $R^n$, take a section with a hyperplane $H$ of some dimension $d$. Assume that $M$ has $M\cap H$ as deformation retract. For example, a ...
MathBug's user avatar
  • 374
1 vote
0 answers
18 views

Why no locally trivial stratification in $C^1$ sense of Whitney's four sheets example

I am reading this paper https://www.math.ias.edu/~goresky/pdf/MatherBio.pdf from Mark Goresky. He talked about an example on page 5 saying that The first example is an algebraic subset of Euclidean ...
Kenneth.K's user avatar
  • 1,405
2 votes
0 answers
31 views

CW complex structure on manifolds with certain dimensional cells

I started reading Morse theory and I learnt one of the fundamental theorem which says that any smooth closed manifold admits a CW complex structure. I had a following question: Given a smooth, closed ...
Alexander93's user avatar
4 votes
1 answer
274 views

Two Morse functions that are equal at their critical points

$f,g$ are Morse functions from $\mathbb{R}^2$ to $\mathbb{R}$ such that if $(x,y)$ is a critical point of $f$ or if $(x,y)$ is a critical point of $g$ implies $f(x,y)=g(x,y)$. Can we say that $f$ and $...
Giovanni Barbarani's user avatar
2 votes
0 answers
66 views

Morse function on solid torus

Update Denote $\mathcal{M}$ by the manifold with boundary (i.e., solid torus). Let $i(x)$ denote the vector field index of the gradient flow $f$ of the Morse function, where $f$ points outward at all ...
Tatan's user avatar
  • 21
3 votes
1 answer
110 views

Problem with understanding Morse's Lemma / Function.

https://math.stackexchange.com/a/398282/1257548 In this answer it is said that $f:S→\mathbb{R}, (x,y,z)↦y$ is Morse function but I don't see why. As far as I understand, because function is defined ...
blek's user avatar
  • 33
0 votes
0 answers
56 views

critical points of projection function on low dimensional set

Suppose $S = \{x \in \mathbb{R}^n | P(x) = 0\}$, for some polynomial $P$, and I want to find the critical points of $f: S \rightarrow \mathbb{R}$ which takes $(x_1, \ldots, x_n) \mapsto x_1$. It is ...
Jyothi's user avatar
  • 89
1 vote
0 answers
60 views

Square distance $f_p(x) = \|x-p\|^2$ is a Morse function

Let $M \subset \mathbb R^n$ a smooth submanifold. I've been said that for generic $p$ the function $f_p : x \mapsto \|x-p\|^2$ (euclidean norm) defines a Morse function on M. Here's my attempt : The ...
Kieran McShane's user avatar
0 votes
0 answers
64 views

(Explaination of) Proof of Global Stable Manifold Theorem in Audin & Damian's book

I am reading Michèle Audin and Mihai Damian's book $\textit{Morse Theory and Floer Homology}$ and I stick at one sentence regarding Global Stable Manifold Theorem which states $W^s(a)$(that is, the ...
skt_zheng's user avatar
0 votes
0 answers
49 views

Help understanding Morse theory proof (Milnor)

I'll start with the statement of the theorem and its proof, and I'll end by explaining my difficulty understanding the proof. What follows are not Milnor's original words, but rather my best attempt ...
JMM's user avatar
  • 1,106
1 vote
0 answers
44 views

Role of basins of attraction in the Morse decomposition

Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by $$\dot{x}=F(x(t))$$ An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
NicAG's user avatar
  • 661
2 votes
0 answers
38 views

Handles have the form $D^λ×D^{m−λ}$

I'm studying Matsumoto's An Introduction to Morse Theory. I want to solve a problem on page 76. Context: Let $M$ be a closed $m-$manifold and $f:M\rightarrow \mathbb{R}$ a Morse function. Let $c$ be a ...
3435's user avatar
  • 385
3 votes
0 answers
63 views

What is the finite Morse index solution?

I'm dealing with an elliptic PDE, it depends on the dimension of the domain, for some situations, I highly doubt that there is no $C^2$ solution, so I searched for some papers related to this equation ...
Elio Li's user avatar
  • 567
0 votes
0 answers
33 views

Exercise about Palais-Smale condition

I'm here to ask for some help in solving the following exercise: Exercise. Suppose that $X$ is $\mathbb{R}^N$, a Hilbert space or more in general a $C^2$ complete Hilbert manifold. Suppose that $f:X\...
TeoricodeiNumeri's user avatar
1 vote
0 answers
33 views

Nice Morse chart with respect to a Riemannian metric

Let $(M,g)$ be a smooth Riemannian manifold and let $f: M \to \mathbb{R}$ be a Morse function (i.e. critical points are nondegenerate). The Morse lemma says that around every critical point $p$ of $f$ ...
Cois Monis's user avatar
1 vote
1 answer
56 views

Why is $M_{f\geq 0}$ a manifold with boundary?

I'm studying Morse theory and I found this fact: Let be $f:M\rightarrow \mathbb{R} $ a Morse function on a m-manifold $M$. Suppose $0$ is not a critical value of $f$. Then $M_{f\geq 0}=\lbrace p\in M |...
3435's user avatar
  • 385
1 vote
0 answers
36 views

Induced orientation given a short exact sequence (in Morse homology over $\mathbb{Z}$)

I am new to the field of algebraic topology and am currently studying Morse homology for a project. I read that given a short exact sequence $$ 0 \to A \to B \to C \to 0 $$ if we know the orientation ...
David's user avatar
  • 21
1 vote
1 answer
79 views

Handle body $H_g$ of genus $g$ as a CW complex

Often is the torus (as a $2$-dimensional manifold) given as an example of a CW-complex (see here, for example, for the genus $g$ manifold). However, one can ask how to decompose the $3$-manifold (...
Robert's user avatar
  • 528
3 votes
1 answer
80 views

Examples of degenerate (Floer) Hamiltonian orbits?

Say I have a periodic hamiltonian $H: S^1 \times M \longrightarrow M$ defined on a symplectic manifold $M$. Then, a $1$-periodic Hamiltonian orbit of $H$ is the same thing as a fixed point of the ...
Azur's user avatar
  • 2,194
1 vote
1 answer
70 views

What is the image of the Homology group of the boundary of a space under the map induced by inclusion?

Suppose $M$ is a compact manifold of dimension $n+1$ and $\delta M$ its boundary. What can we say about the image of $H_n(\delta M)$ under the inclusion $H_n(\delta M)\to H_n(M)$? I was originally ...
LGu's user avatar
  • 458
3 votes
1 answer
63 views

Orbit Space of Moduli Space in Morse Homology

If I have a group $G$ acting on a topological space $X$, and this action is free (no fixed points), what can I say about the orbit space $X/G$? I am aware that freeness plus proper discontinuity leads ...
contingent's user avatar
2 votes
0 answers
24 views

Is there any version of (discrete) Morse Theory for subcomplexs?

I am now working on the (discrete) Morse theory and its application about simplices and cell complexes. However, I cannot find any statement about the property on closed subcomplexes / submanifolds of ...
Erus Izumi's user avatar
3 votes
1 answer
282 views

Placement of critical points of Morse function on the sphere

On a similar note to my previous question, I am still thinking about functions on the sphere. Assume I have a $C^2$ function on the sphere $ f : S^{d-1} \mapsto \mathbb{R} $ which is Morse (i.e. all ...
Andreea M's user avatar
  • 350
4 votes
2 answers
141 views

Can a real-valued function on the sphere have exactly 2 critical points which are not antipodal?

I came across this question while working on something quite different: Can we build a $C^2$ function on the sphere of radius one, i.e. $f : S^{d-1} \mapsto \mathbb{R}, d \geq 3 $ such that it has ...
Andreea M's user avatar
  • 350
1 vote
0 answers
55 views

Real analytic periodic function $f$ such that $\nabla f(x)=0 \Rightarrow \nabla^2f(x)=0$.

Is there any non-constant real analytic periodic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\{x\in\mathbb{R}^n\mid\nabla f(x)=0 \}\subset\{x\in\mathbb{R}^n\mid\nabla^2 f(x)=0 \}? $$ ...
Jianxing's user avatar
  • 141
0 votes
0 answers
23 views

Exercise 12 Chapter 1 - Computational topology for DA by Krishna Dey & Wang

I am solving problem 12 (chapter 1) in the book Computational topology for DA by Krishna Dey & Wang. I have a question about notation. It is asked to prove that $f^{(-\infty,0]}\cap S^{2}$ and $f^{...
Faye3's user avatar
  • 121
1 vote
0 answers
122 views

Critical points a Morse function on SO(3)

Let $f_A: SO(3) \to \mathbb{R}$ and is given by $R\mapsto \frac{1}{2}|| A - R ||_F$, where $A \in \mathbb{R}^{3\times 3}$, and the norm is the Frobenius norm. As indicated in [1], the gradient of $f_A$...
a-deel's user avatar
  • 91
1 vote
0 answers
43 views

Local Modification of Morse Functions

If $f_0 , f_1$ are two Morse functions defined on some smooth manifold $M$ with a common critical point $p\in M$, the same value $f_0 (p)=f_1 (p)$ and with the same stable discs $W^s_{f_1} (p)=W^s_{...
TheWildCat's user avatar
1 vote
0 answers
36 views

Diffeomorphism of level sets of functions depending on a parameter

Let $H : \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function with arguments $(x, \alpha)$, where $\alpha$ is a parameter. Fix a constant $c$ and suppose the set $S_{\alpha_0} :=...
Cauchy's Sequence's user avatar
1 vote
0 answers
95 views

Structural stability of Morse-Smale diffeomorphisms

I'm trying to wrap my head around why Morse-Smale diffeomorphisms are structurally stable, using the real line as a toy example. Say $f$ is a Morse-Smale diffeomorphism. Proofs for more generalized ...
bluecheese's user avatar
0 votes
0 answers
60 views

On the definition of the second-order derivative of a Morse function

In Morse Theory and Floer Homology by Michele Audin and Mihai Damian (Morse theory and Floer Homology) they give the following definition to the second-order derivative at a critical point x. $$(d^2f)...
Bessel's user avatar
  • 120
1 vote
0 answers
50 views

Morse-Bott inequalities in $\mathbb{CP}^n$

I want to prove the following statement (which I heavily believe is true): Let $g:\mathbb{CP}^n\to \mathbb{R}$ be a non-constant Morse-Bott function and denote by $\text{Icrit}(g)$ the set of ...
kvicente's user avatar
  • 369
0 votes
1 answer
175 views

Poincaré-Hopf theorem and Morse theory on three-dimensional torus

I am asking this question to clarify a comment that appears below Eq.(32) of this paper, which applies Morse theory to classify van Hove singularities in energy bands of crystalline solids. These ...
Tomáš Bzdušek's user avatar
0 votes
1 answer
112 views

Morse functions with minimal number critical points

Is it true that $RP^n$ has a Morse function with n critical points,and dont have Morse function with n-1 critical points
VadimStacheff's user avatar
0 votes
0 answers
57 views

Construction of Morse function on $RP^1$

I wanna construct Morse function on $RP^1$ with one critical point,how can i do it. P.S. i wanna construct such function on every $RP^n$,but at first wanna understand it on the simplest example.
VadimStacheff's user avatar
2 votes
0 answers
52 views

Issue with proof of theorem 2.1.11 in Audin-Damian

The theorem in question is as follows: Let $V$ be a closed smooth manifold and $f: V\to \mathbb{R}$ a Morse function. Let $a$ be a critical point of $f$ with index $k$ and $\alpha=f(a)$. Suppose that ...
J.V.Gaiter's user avatar
  • 2,492
0 votes
1 answer
50 views

Attaching a handle at a local maximum of Morse function

I am studying Morse theory and I am bit confused about what happens to a manifold when going through a critical value of a Morse function. Here is the setting: $M$ is a compact manifold and $f : M \to ...
fresh's user avatar
  • 341
2 votes
1 answer
99 views

How to show there is an $S^4$ included in a simplicial complex?

A $\mathbb{Z}_2$-space is a pair $(T, \nu)$, where $T$ is a topological space and $\nu: T \rightarrow T$, called the $\mathbb{Z}_2$-action, is a homeomorphism such that $\nu \circ \nu= id_{T}$ . If $(...
Inez's user avatar
  • 393
1 vote
0 answers
23 views

Signature property of the Maslov (Conley-Zhender) index

In the proof of the signature property of Maslov index, one uses the fact that if $S$ is a symmetric matrix whose norm is less than $2\pi$, then the matrix $\exp(J_0S)$ does not admit $1$ as ...
Rubi's user avatar
  • 11
1 vote
0 answers
60 views

a manifold on which every smooth function has at least two local minima

I am wondering whether there exists such a compact and connected manifold such that every smooth function on it has at least two local minima. Because it is compact, so there is at least one local ...
poisson's user avatar
  • 1,005
1 vote
0 answers
45 views

Does a self-indexing Morse function gives information about the Heegaard genus?

This question comes up when I am working on an exercise finding the Heegaard genus of the 3-torus $S^1\times S^1\times S^1$. By definition of Heegaard genus, it is the minimal possible genus of the ...
Ivan So's user avatar
  • 797
7 votes
0 answers
65 views

How to find a discrete Morse function on this simplicial complex

I have a graph called partition graph. This graph gives rise to a simplicial complex called box complex $B_{edge}$. Since this simplicial complex is too big and studying the topological features of it ...
Inez's user avatar
  • 393

1
2 3 4 5
9