In the area of mathematics known as differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. (Def: http://en.m.wikipedia.org/wiki/Morse_theory)

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Jacobian of the diffeomorphism in Morse's lemma

Morse's Lemma states that if $0$ is a non-degenerate critical point of $f:\mathbb{R}^n\to \mathbb{R}$, then there exists a neighbourhood $U$ of $0$ and a diffeomorphism $\phi$ such that $F = \phi\circ ...
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Such a manifold is homeomorphic to a sphere

I recently read that if a compact differentiable manifold admits a real function with only two critical points, then it is homeomorphic to a sphere. If the function is Morse, this follows from ...
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Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible.

Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible. The problem is from the following material. It contends that the result is by standard Morse ...
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Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
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Is there $f: U \to \mathbb{R}^{n}$ injective such that…

Let $f: U \to \mathbb{R}^{n}$ $C^{1}$ injective where $U$ is a open in $\mathbb{R}^{n}$ (so $f$ is open by invariance domain theorem). a) Is there exist $f$ such that dim $ker(df_{x}) >$ dim ...
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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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Integral Curves of Gradient-like Vector Fields

If $X$ is a gradient-like vector field of a Morse function $f\colon M\to \mathbb{R}$, then the integral curve $c_p(t)$ starting at an arbitrary point $p$ approaches critical points as $t\to \pm ...
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Regular CW complex arising from a Morse decomposition

Suppose $(M,g)$ is a Riemannian manifold equipped with a Morse function $f: M \rightarrow \mathbb R$. It's been shown that $f$ gives rise to a CW decomposition homeomorphic to $M$ under the generic ...
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Use of implicit function theorem in showing that $f(x) \leq a$ is a submanifold with boundary

This question comes from a statement in John Milnor's "Morse Theory" on page 4. Let $f: M \to \mathbb{R}$ be a smooth function on a manifold $M$. Milnor claims that if $a$ is not a critical value of ...
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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 ...
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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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Schwarz's Morse Homology

I have been struggling to understand the proof of proposition 2.9 of the Morse Homology book by Matthias Schwarz. I do understand the mathematical steps in the proof but I don't really see why we are ...
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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 ...
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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 ...
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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 ...
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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 ...
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Application of Morse Inequalities

I am an undergraduate student interested in morse theory. I understand, that the morse inequalities provide an lower bound for the number of critical points morse functions on a manifold can take. One ...
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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 ...
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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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101 views

Morse lemma implies nondegenerate critical points are isolated

It might be a stupid question. I don't quite know how we get the conclusion " Non-degenerate critical points are isolated" from Morse lemma. I know around each non-degenerate critical point we have a ...
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References on cancellation of critical points

I'm not sure if I am using the terms correctly. Suppose you have, for example, a Morse function $f : S^2 \rightarrow \mathbb{R}$ with 2 critical points of index 0, 1 critical point of index 1, and 1 ...
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Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine.

Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a smooth function. The theorem of Sard gives us that ...
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Understanding Result on Non-Degenerate Critical Points

I read a result in a collected works of Steven Smale and one result leapt out at me which I'm clearly not understanding. Stated: Theorem 1.1 (a): Suppose $J: M \to \mathbb{R}$ is a $C^2$ ...
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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 ...
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Why gradient-like dynamical systems are special case of Morse-Smale systems?

I'm studying Morse Theory and my question is exactly as stated in the above title. I can't see how a gradient-like dynamical system could be considered as a Morse-Smale system? Thanks in advance for ...
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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 ...
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How level sets look like when a critical point degenerate?

I'd like to know an explicit example of a compact, connected manifold $M$ and a smooth function $f\colon M \to \mathbb{R}$ which satisfy the following properties: We denote by $m$ the minimal value ...
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Why not develop a Hamiltonian-based Morse theory?

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory ...
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Question About Local minimum

I have this definition of a local minimum: We say that $u$ is a local minimum of $f$ is there exist a neighborhood $V$ of $u$ such that for all $v\in V$ $f(v)\geq f(u).$ So we say that $u$ is not a ...
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The space of minimal geodesics on $SU(2m)$

In the proof of Bott periodicity for the unitary group in Milnor's Morse theory (Lemma 23.1, page 128), it is asserted that the space of minimal geodesics from $I$ to $-I$ in the special unitary group ...
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The main theorem of discrete Morse theory.

I don't understand this part of the proof on page 16 of the following paper. http://www.maths.ed.ac.uk/~aar/papers/forman5.pdf
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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 ...
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Sheaf-theoretic approach to Morse functions?

It is known that one can define a smooth structure on a manifold using a sheaf-theoretic formulation via defining the algebra of the (a fortiori) smooth functions on it (which satisfies the usual ...
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Limits of trajectory of gradient flow in Hilbert space

I have been studying about gradient flow in Hilbert space of a Morse function $f$. Specifically, let $X$ be a Hilbert space and $f : X\to \mathbb R$ be $C^3$ function. The gradient flow here is ...
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On the Euler characteristic in Morse Theory

Let $f:M\to \Bbb{R}$ be a Morse function, where $M$ is a $k$-manifold. The index $i_{f,p}$ is defined to be the number of negative eigenvalues of the Hessian $H_f$ at the critical point $p$. For ...
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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 ...
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Intuition about Morse functions

We have defined: a Morse function on $X$ is a smooth function $f:X\rightarrow\mathbb R$ with only non-degenerate critical values. I tried to get some intuition about this, and found the section Basic ...
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A question related to the topology of the level sets of a particular type of smooth functions $f:\mathbb{R}^2\to \mathbb{R}$.

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a smooth function without critical points; i.e. such that $\nabla f(x)\neq (0,0)$, for all $x\in\mathbb{R}^2$. Is it true or false that all the level curves of $f$ ...
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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?