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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27
votes
1answer
3k views

When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
20
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1answer
508 views

Reconstructing a manifold from critical points

I am teaching theoretical calculus this semester, and on the last discussion section we were discussing critical points of functions. I explained the idea of Morse theory, and a student of mine asked ...
18
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2answers
1k views

Is this surface diffeomorphic to a 2-sphere?

Let $f:\mathbb{R}^3\to \mathbb{R}$ be defined by $f(x,y,z)=x^4+y^6+z^8$. Let $M=f^{−1}(1)$. Is $M$ is diffeomorphic to a sphere $S^2$? I tried to solve this problem, but I realized that I have no ...
16
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1answer
536 views

Question about the proof of the index theorem appearing in Milnor's Morse Theory

I am trying to get through the proof of the index theorem. The background: I have been stuck for quite a while on the following point which Milnor says is evident: Let $\gamma: [0,1]\rightarrow M$ be ...
14
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0answers
553 views

Proof of Reeb's theorem without using Morse Lemma

I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. That is, suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both non-...
13
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1answer
1k views

Gradient-like vector fields

Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function. My goal is to better understand gradient-like vector fields for $f$. Question: Do any two ...
11
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0answers
143 views

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 ...
9
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2answers
318 views

Is Gauss curvature a Morse function?

Given a Gauss map $\nu: M \rightarrow S^k$ of a orientable, compact manifold, we define the shape operator $S_p = -d \nu: T_p M \rightarrow T_{\nu(p)} S^k$ to be the negative differential. Define the ...
9
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0answers
196 views

how to obtain a generalized Morse function out of a fiber bundle?

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth ...
8
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2answers
538 views

Morse theory Vs degree theory

I have this paragraph from K.C. Chang Infinite dimensional Morse theory In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in estimating ...
8
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2answers
442 views

History of Morse theory.

How can I get good references which give many information about history of Morse theory? Now I am interested in how and who found that Hessian have a lot of data. Thank you for your helping!!
8
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1answer
742 views

Prove Poincare duality theorem with Morse theory.

First let us consider a smooth n-manifold. And find a Morse function $f$. Now let's consider $-f$. A singular point of $f$ with index $k$ is a singular point of -f with index n-k. Thus we have a ...
8
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1answer
499 views

How to check whether a vector field is Morse-Smale?

Setup and notation: Let $f:M\to \mathbb{R}$ be a Morse-function on the compact $m$-dimensional manifold $M$ and let $X$ be a gradient-like vector field for the function $f$. Denote the unstable ...
8
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1answer
418 views

Morse homology of $P^2$

I have seen and worked through the explicit computation of the Morse homology of the sphere and the torus (with signs and all). But trying it for $\mathbb{R}\mathbb{P}^2$ has lead me to dead ends. Is ...
8
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1answer
184 views

Smooth classification of vector bundles

I am trying to understand the smooth classification of $n$-disk bundles over $S^n$. As vector bundles, these are classified by $\pi_{n-1}(SO(n))$ via the clutching construction but I am interested in ...
8
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2answers
511 views

Why is the moduli space of gradient flow lines $\widehat{\mathcal M}(p,q) = \mathcal M(p,q) / \mathbb R$ a smooth manifold?

Let $(M,g)$ be a compact smooth orientable riemannian manifold, and let $f: M \to \mathbb R$ be a Morse function. All functions here are assumed to be smooth. We will be considering the integral ...
7
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2answers
672 views

Cup product in Morse cohomology

Dualizing the Morse complex, we obtain the Morse cohomology, which is isomorphic to the usual singular cohomology and thus admits a cup product. Does anybody know how this cup product would look like ...
7
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1answer
1k views

Background for reading Milnor's Morse Theory book

I wish to study the book 'Morse Theory' by J. Milnor, but I am not sure whether I have the necessary prerequisites. I know basic point set topology, real analysis (limits, continuity, differentiation, ...
7
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1answer
156 views

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 ...
7
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2answers
151 views

Family of Morse functions made constant

I'm looking for a proof of the following theorem: Let $f_t$ be a family of real-valued Morse functions defined on a smooth compact manifold $M$, and where $t$ is in $[0,1]$ (So for all value of $t$, $...
7
votes
1answer
211 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 ...
6
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1answer
141 views

Understanding Morse homology with the example of a circle

I am studying Morse Homology using the book "Lectures on Morse Homology" by Augustin Banyaga and David Hurtubise and I am having a hard time to understand how to count flow lines with sign. Now I am ...
6
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1answer
674 views

Looking for a good book on Morse-Bott functions.

I am looking for a book to study for the first time Morse-Bott functions. Does anyone know one that is easy to follow and detailed? If there is one connecting this subject with symplectic geometry, it ...
6
votes
1answer
467 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 ...
6
votes
4answers
395 views

Prove no existing a smooth function satisfying … related to Morse Theory

i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ ...
6
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0answers
96 views

Existence of covers on manifolds with certain properties

I'm trying to prove the existence of Morse functions on differentiable manifolds, by adapting the proof found on Matsumoto's textsbook, which works for compact manifolds, to the non-compact case. I ...
5
votes
2answers
974 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 ...
5
votes
2answers
194 views

“Simple to state, but difficult to solve” problems which require analyzing topology of simplicial complexes?

In a User's Guide to Discrete Morse Theory, Robin Forman writes: A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial ...
5
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1answer
765 views

Manifold allowing function with two critical points is sphere

The only closed manifolds which allow a function with two (maybe degenerate) critical points are spheres. In dimension 2 it is quite easy to prove, but what is about higher dimensions?
5
votes
2answers
419 views

Milnor's proof that a smooth manifold has the homotopy type of a CW complex

I have some questions about the proof of Theorem 3.5 of Milnor’s “Morse Theory”: At the end of the proof of this theorem, Milnor addresses the case when $f$ has infinitely many critical points: ...
5
votes
3answers
150 views

About a Morse function on the euclidean $n$-sphere.

If you are in a hurry feel free to jump directly to the emphasized parts. Setup. Let $A$ be a real invertible symmetric square matrix of size $n+1$ whose eigenvalues are: $$\lambda_0>\lambda_1>...
5
votes
2answers
106 views

Existence of a proper Morse function

Given a manifold $M$, I know there exists a proper function $f: M \to \mathbb R$ (using the usual partition of unity argument) and a Morse function $g: M \to \mathbb R$ (genericity of Morse functions)....
5
votes
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 ...
5
votes
2answers
305 views

Is there a function like this?

Let $A=[0,1]$ and $C=\{0\}\cup\{\frac{1}{n},\ n\in\mathbb{N}\}$. i) Is there a function $f:A\rightarrow\mathbb{R}$ such that $f\in C^{r}(A)$, $r\geq 2$ and the set of critical "Values" of $f$ is $C$? ...
5
votes
1answer
66 views

What is the $S^1$-equivariant cup product on $S^2$?

Consider the sphere $S^2 = \mathbb{CP}^1$ with the $S^1 = \{ \tau \in \mathbb{C} \mid |\tau| = 1 \}$ action given by $$ \tau \cdot [z_1, z_2] = [\tau ^ k \cdot z_1, z_2] $$ The corresponding $S^1$-...
5
votes
1answer
234 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 ...
5
votes
1answer
132 views

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 ...
5
votes
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 ...
5
votes
1answer
450 views

Prove directly Morse lemma for real line $\mathbb{R}$

The exercise 9 section 7 chapter 1 in Guillemin & Pollak state the next Prove directly Morse lemma for real line $\mathbb{R}$.(Hint:Use this elementary calculus lemma: for any function on $\...
5
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0answers
68 views

What is the most surprising / interesting application of the inverse function theorem you have ever seen?

The inverse function theorem states that: Suppose that $f: U \subset \mathbb{R}^m \rightarrow \mathbb{R}^m$ is a $C^k$-function and that there exists $a \in U$ such that $f'(a): \mathbb{R}^m \...
5
votes
0answers
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 ...
5
votes
1answer
252 views

Making a gradient-like vector field a gradient vector field via choosing a Riemannian metric.

Let $\xi$ be a vector field on manifold $M^n$ which is a gradient-like vector field for a some Morse function $f$. Prove that there exists a Riemannian metric on $M$ such that $\xi$ is a gradient ...
4
votes
1answer
105 views

Poincare Duality in Morse Homology

I'm reading M. Audin and M, Damian's Morse Homology book right now and i have some issue regarding its section about Poincare Duality (Section 4.3, page 83). In this version of "Poincare Duality", it ...
4
votes
2answers
310 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 ...
4
votes
1answer
98 views

The Euler characteristic of a cubic Fermat surface

Let $F$ denote the cubic Fermat surface in $\mathbf{P}^3$ (everything is over the complex numbers): $$ F = \{ X^3 + Y^3 + Z^3 + W^3 = 0\}\subseteq \mathbf{P}^3.$$ I wish to compute the Euler ...
4
votes
1answer
131 views

Continuity of retraction on $r : \text{Int }\Omega^c \to B$

I'm reading Milnor's Morse Theory and I have difficulty verifying some claim (which is easy according to Milnor) on page $88$, section $\S 16$ in the book. Here's the setup for my question. In the ...
4
votes
2answers
108 views

Existence of only finitely many geodesics

The following question arose from Theorem 16.3 (p.90) in the book Morse theory from John Milnor. We are dealing with a complete Riemannian manifold $M$, an $a \in \mathbb{R_{>0}}$ and two points $...
4
votes
1answer
178 views

Manifolds with diffeomorphic boundaries

Suppose $X$, $Y$ are two $2n$-dimensional manifolds with handles of index $0$ and $n$. In particular, $X$ and $Y$ have boundary. Suppose that $X$, $Y$ have isomorphic homology and intersection form ...
4
votes
1answer
463 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 ...
4
votes
2answers
747 views

Preimage of singular points of smooth map between manifolds

Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ where $V$ is a (finite dim, real) vector space (of potentially very large dimension) and $SU(n)$ is the special unitary Lie group, what ...

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