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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172 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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106 views

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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113 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:\mathbb{R}...
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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 ...
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79 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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144 views

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
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 ...
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1answer
124 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
503 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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159 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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67 views

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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315 views

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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121 views

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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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 ...
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187 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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87 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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231 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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62 views

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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148 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) \...
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844 views

What is the definition of Index of a point or vector field?

The Poincare-Hopf Index Theorem states that: Theorem: The index of a vector field with finitely many zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold. ...
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1answer
168 views

Diffeomorphism from level sets onto spheres

I am just learning Morse theory. I apologize if the solution to the following question is well-known. For a Morse function $f(x):\mathbb{R}^n\mapsto\mathbb{R}$, if the level set $f^{-1}(c)$ only ...
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2answers
263 views

Is the tangent space to a critical submanifold a subspace of the kernel of the Hessian?

Trying to solve a question I have been faced with another question. Let $f:M\to\mathbb{R}$ be a smooth function and $b\in \mathbb{R}$ a critical value of it. Now the following relation is true? $$...
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1answer
552 views

Hessian of a function at the critical points

Let $f:M\to\mathbb{R}$ be a smooth function and $p\in M$ is a critical point of it. The Hessian of $f$ at a critical point $p$ is a symmetric bilinear form $\operatorname{Hess} f_p$ s.t. $\forall v,...
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359 views

Reference request for Morse theory with complex valued functions

I was just wondering if there is a version of Morse theory by considering maps from $f: M \to \mathbb{C}$ where $M$ is a complex manifold and $f^{\prime \prime }(z) \neq 0 $ whenever $f^{\prime}(z)=0$....
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286 views

Minimal number of Morse critical points

I am interested in the minimal number of critical points of a Morse function on a closed manifold or a proper Morse function on a manifold with boundary; I will call this the Morse function of the ...
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72 views

Resolving a double-point singularity in terms of Morse theory

Background: I want to understand the "homology realization problem", a.k.a. Steenrod's problem, in 4-dimensional case. The precise statement that I'm considering is: Theorem. Any 2-dimensional ...
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130 views

Classification of highly-connected manifolds

In http://www.maths.ed.ac.uk/~aar/papers/n-1con2n.pdf, Wall studied $(2n, n)$-handlebodies, ie. 2n-manifolds that have a handle presentation with only 0 and n handles. Given a presentation, he ...
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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>...
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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 ...
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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 ...
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1answer
249 views

Morse functions on 4-manifolds

What are examples of 4-manifolds that have more Morse critical points than topologically required? For example, $\pi_1(M), H_*(M; \mathbb{Z})$ have less than $k$ generators but any Morse function on $...
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65 views

Contracting paths of framed functions

Let $M$ be a smooth manifold. Morse functions on $M$ are smooth functions $M \to \mathbb{R}$ with only very nice singularities. Fact: The space of Morse functions on $M$ is not, in general, ...
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109 views

Attaching cells in the Morse Theory

Every critical point of a Morse function tells us to glue a cell of a dimension equal to the index of the critical point. Is there any information about how to glue this cell? For instance, does the ...
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1answer
64 views

Is this a trivial case of the Morse lemma?

Say we have a $C^{\infty}$ function $f: \mathbb{R} \to \mathbb{R}$ and a point $x_0 \in \mathbb{R}$ such that $f'(x_0) = 0$ but $f''(x_0) \not= 0$. Then does the Morse lemma guarantee us that there ...
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1answer
72 views

Construction of a Morse-Smale system

I am having difficulties understanding the construction of Morse-Smale systems. They start with $M$ compact and connected smooth manifold, then they say there exists an inmersion (or embeddement) $i: ...
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98 views

Reeb graph of the tilted torus

Reeb graph :https://en.wikipedia.org/wiki/Reeb_graph some pictures: http://www.math.brown.edu/~banchoff/Beyond3d/chapter3/section08.html The Reeb graph of an upright torus, which I'll call "a ...
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1answer
115 views

f_a is a morse function

How can I prove: Let $f:\mathbb{R}^k\longrightarrow{\mathbb{R}}$, and, for each $a\in{\mathbb{R}^k}$ define $$f_a(x)=f(x)+a_1x_1+...+a_kx_k.$$ For almost all $a\in{\mathbb{R}^k}$, $f_a$ is a Morse ...
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61 views

Milnor - Morse theory - theorem 7.2

I am reading the book "Morse theory" by J. Milnor. In the proof of theorem 7.2 an assertion is done: Assertion 2: The focal points of $(M, q)$ along any normal line $\ell$ occur in pairs with the ...
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1answer
107 views

Question about a proof of Morse Inequalities

I have some question so this passage. (1) Are the $a_i$ the critical values of the function $f$, meaning that $f(p_i)=a_i$, where the $p_i$ are the critical points of $f$? (2) Why can we assume ...
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66 views

Any composition of the Finsler-Minkowski norm giving a Morse function?

Given a Finsler-Minkowski norm $F$ on $R^n$, let $\mathfrak{F}=\{\Sigma_r\}_{r\geq0}$, where $\Sigma_r=\{y\in R^n: F(y)=r\}$, be a partition of $R^n$. There exists any (infinitely) smooth Morse ...
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1answer
116 views

critical points of different index with same critical value

Let $f: M \to R$ be a morse function from a closed n-manifold $M$ with a critical value $c$. Is it possible that there are two critical points $P_1$, $P_2$ with different index, but with the same ...
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1answer
187 views

Subadditivity of betti number of pairs of spaces

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 $R_\lambda(X,Y)$ to be the ...
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1answer
222 views

Understanding the proof of Morse Lemma using homotopy method

On pages no. 52-56 of "Lectures on Morse Homology" by Augustin Banyaga and David Hurtubise presented a proof of Morse Lemma using homotopy method AKA Palais proof using Moser "path method". There are ...
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2answers
697 views

Reference for Picard-Lefschetz theory

What are some good references (papers/lecture notes/books) on Picard-Lefschetz theory, Morse theory and complex manifolds? I am looking for material at a level as introductory as possible, as my ...
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1answer
944 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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93 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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