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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The set of pseudo-gradient vector fields with the Smale property and a given set of trajectories is open (Proposition 3.4.3 in Audin-Damian)

In Audin Damian's proof of the Invariance of Morse Homology from vector field and Morse function, p.71, there's a proposition, where the Smale property is that the stable and unstable manifolds of the ...
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Transversality Condition and the Proof of Smale Theorem (Audin-Damian), Lemma 2.2.8 (Part 2)

In Audin and Damian, p.44-45, there seems to be a claim that one can prove transversality without directly showing the tangent spaces span the tangent space of the ambient manifold. In particular, in ...
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1answer
54 views

Milnor's Lectures on h-cobordism theorem: Lemma 6.2

In the book, Lemma 6.2 (stated below) talks about a corollary of the Thom's isomorphism theorem and Tubular neighbourhood theorem. The proof of the lemma is not provided by the author. And the ...
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Proof of Smale Theorem (Audin-Damian), Lemma 2.2.8

In Audin and Damian, p.43, there is a proof of the following lemma, relating pseudo-gradient vector fields adapted to $f$ on $V$, namely $X$ and approximation $X'$. Here, $\alpha_j$ a critical value ...
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1answer
31 views

Audin and Damian's notion of $C^1$ proximity

I am reading Audin and Damian's book on Morse Theory and Floer Homology. In pages 40-41 they state the Smale Theorem, whose statement uses a notion of $C^1$-proximity for vector fields. I believe the ...
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240 views

Derivative of differential of time-dependent flow of a vector field

I am reading the book "Morse Theory and Floer Homology" by Audin and Damian. And I am kind of stuck in understanding the last part of a proof in such book. Currently we are trying to prove ...
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2answers
54 views

Periodic solutions are Lyapunov Stable

Consider the ODE $$ \left\{ \begin{align*} \dot{x}=&y\\ \dot{y}=&-x^2-bx-c. \end{align*}\right. $$ Under the assumption that $b^2-4c>0$, we find the equilibria $P_1=\left(\frac{-b+\sqrt{b^2-...
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Question regarding the necessity of Cech cohomology in the proof of a paper.

I have a specific question regarding the understanding of a section of a paper by Wan. The link to the paper is https://core.ac.uk/download/pdf/82686957.pdf. I had the following question. Question: In ...
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49 views

Density and dimensionality of zeros in inverse square force fields of randomly distributed + and - charges in (at least) 1, 2 and 3 dimensions?

@mlk's answer to Density and dimensionality of zeros in inverse square force fields of randomly distributed sources in (at least) 1, 2 and 3 dimensions? is pleasing and straightforward, switching to ...
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690 views

Density and dimensionality of zeros in inverse square force fields of randomly distributed sources in (at least) 1, 2 and 3 dimensions?

Background: In this answer to Are there places in the Universe without gravity? in Astronomy SE I did a quick finite 2D calculation for 20 random sources to see if there was at least one zero, and ...
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Evanescent cycle of Lefschetz degenerescence.

I read in Voisin's book the following result which I cannot figure out. She said in her book that "it is easy to see..." so I think it is indeed easy to see but I don't see it... Let $B$ be ...
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112 views

Beyond calculus of variations

Calculus is a key component in analysis that has taken on many forms and generalisations. In the following table, we begin with the common notions of the optimisation of functions in one variable, to ...
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1answer
73 views

Elementary Morse Cobordism of Diffeomorphic Boundary Components

Let $(M,V,V')$ be a smooth manifold triads. I would like to find a Morse cobordism which is elementary, i.e. there exists Morse function $f:M\to[0,1]$ such that $f^{-1}(0)=V, f^{-1}(1)=V'$ and of only ...
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38 views

Morse function on torus and real projective plane.

I was considering following problem. Let us have a torus $T^2$ and real projective plane $\mathbb{R}P^2$. Let $f: T^2 \to \mathbb{R}$ and $g: \mathbb{R}P^2 \to \mathbb{R}$ be a Morse functions. Proof ...
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129 views

Invariance of Morse Homology, confusion in proof of book “Morse Theory and Floer Homology”

I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem. Basically we want to prove that the Morse ...
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1answer
68 views

Proof of Theorem 2.1.11 of book “Morse Theory and Floer Homology” by Audin and Damian

I am reading the book "Morse theory and Floer Homology " by Audin and Damian and I am stuck understanding the proof of this theorem. (Sorry I dont know the exact name of it that is why I ...
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42 views

Second-order derivative of functions $f \colon M \to \mathbb{R}$ where $M $ is a manifold

I am starting to read a book in Morse theory and Floer Homology. In the first few pages of morse theory I found a definition of second order derivative for functions $f \colon M \to \mathbb{R}$. ...
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Structure of a set of critical points of a Morse function given some topological information of a manifold

Let $M$ be a smooth , compact, connected manifold with a boundary. Let $f:M\to \mathbb{R}$ be a Morse function. Suppose that $\partial M = f^{-1}(y_1) \cup f^{-1}(y_2)$. What can we say about ...
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71 views

Recommendations for Cerf theory?

Are there any good references for Cerf theory, written in English? The main references I see are usually Cerf's own papers in French. In particular, I am looking for a source developing the ...
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1answer
35 views

Relative homology of sublevels is the homology of the attached $k$-cell

Let $M$ be a smooth manifold of dimension $N$, $f: M \to \mathbb R$ smooth and let $p$ be a nondegenerate critical point, $f(p) = c$. Suppose that it is the only nondegenerate critical point in $f^{-1}...
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51 views

The topology of a $k$-dimensional ball in $\mathbb R^N$ and the definition of attaching a $k$-cell

Let $B_k$ be the $k$-dimensional ball in $\mathbb R^N$: $$ B_k = \left\{x \in \mathbb R^N \ : \ \sum_{i = 1}^k x_i^2 < 1, \ x_{k + 1} = \ldots = x_N = 0 \right\}. $$ In the context of Morse Theory, ...
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43 views

In the proof of Morse's Lemma, why can it happen that $h_{ij} \neq h_{ji}$?

The following is the first step in the proof of Morse's Lemma: Without loss of generality, we can assume that $\overline x$ is the origin in $\mathbb R^N$ (in the sense that the chart takes $\...
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1answer
33 views

$f: M \to \mathbb R$, smooth, $\varphi_t$ 1 parameter group of diffeomorphisms, then $df(\varphi_t(x))/dt = 1$

I will first introduce some preliminaries and notation. The question is highlighted below. $M$ smooth manifold of dimension $N$, $f:M \to \mathbb R$ smooth. Fact: a given vector field $\underline v$ ...
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If a two variable smooth function has two global minima, will it necessarily have a third critical point?

Assume that $f:\mathbb{R}^2\to\mathbb{R}$ a $C^{\infty}$ function that has exactly two minimum global points. Is it true that $f$ has always another critical point? A standard visualization trick is ...
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138 views

An alternative proof of the Morse Lemma using Taylor's theorem

I am trying to write out an alternative proof of the Morse Lemma suggested but not completely written in Elementary Classical Analysis by Marsden and Hoffman. This is my attempt. Without loss of ...
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1answer
80 views

Was the Lie bracket once called the Poisson bracket? (Milnor's Morse Theory)

Multiple times in Milnor's Morse Theory, he refers to the "Poisson bracket" (or, once, an obvious typo: "poison bracket") of two vector fields as $$[X,Y](f)=X(Yf)-Y(Xf).$$ (See, e....
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39 views

Definition of first fundamental form

I'm having some trouble understanding the first fundamental form of a manifold. Here are the definitions I'm working with (from Milnor's Morse Theory): Let $M$ be a $k$-manifold differentiably ...
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1answer
50 views

Proof of the Extended Reeb's Theorem for dimensions less than 7

Reeb's Theorem states that every compact smooth manifold which admits a Morse funtion with exactly two critical points is homeomorphic to n-sphere. I have heard an extension of this theorem for ...
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1answer
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Morse Theory proof of Fundamental Theorem of Algebra

Suppose that p(z) is a nonconstant polynomial with no roots. The complex plane with additional point ∞ is homeomorphic to the 2-sphere. At each z in the plane, let the vector at z be 1/p(z), which ...
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54 views

Relation between Morse number and Heegaard number

Given a closed 3-manifold $M$, a self-indexing Morse function $f:M\to[0,3]$ produces a Heegaard splitting by : $$M=f^{-1}([0,3/2])\cup f^{-1}([3/2,3]).$$ (This question discussed a bit on the topic.) ...
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1answer
49 views

Beginner's question about homotopy type in Milnor's Morse Theory

In the introductory section of Milnor's Morse Theory, he gives an example of the torus. The set of points of the torus which have height less than $a$ for some $p<a<q$ is just a solid disk, i.e....
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25 views

Geometric intuition of the Morse lemma

Here is the statement of the Morse lemma: For a smooth manifold $M$, $f:M\to \mathbb{R}$ a smooth function and $p\in M$ a nondegenerate critical point of $f$, we have a choice of local coordinates ...
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Eigenvalues for Hessian of Morse function

Let $f\in C^2(\Omega, \mathbb{R})$, $\Omega \subset\mathrm{R}^2$. If $f$ is a Morse function (each critical point is non-degenarate, i.e. $\det$ Hess$f(\zeta)\neq0,$ for any $\zeta$ s.t. $\nabla f(\...
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109 views

Genericity of an induced projection map

Let $X,Y$ be smooth manifolds, $S'$ a submanifold of $Y$, and $f:\mathbb{R}\times X\to Y$ a smooth function. Generically, we have that $f$ is transverse to $S'$, which implies that $S:=f^{-1}(S')$ is ...
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1answer
107 views

Question on Reeb's theorem.

I am reading through the first couple of chapters of Milnor's Morse theory, and I've gotten to Reeb's sphere theorem (theorem 4.1), If $M$ is a compact manifold and $f$ is a differentiable function ...
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46 views

Reference request: Morse theory exercises

I'm currently learning Morse theory for my masters (with an eye towards K-theory). I'm working through Milnor's Morse Theory however it doesn't have any exercises. From a quick search myself I found ...
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84 views

Trivial tangles and radial Morse functions

A $n$-dimensional tangle is a collection $\mathcal{C}$ of properly embedded $n-2$-disks in a $n$-disk $D^n$. A tangle is said trivial if all the disks in $\mathcal{C}$ are simultaneously ambient ...
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98 views

A Morse Function with Minimum Number of Critical Points

Let $M$ be a closed manifold. If $f$ is a Morse function on $M$, then by Morse inequalities we know that $f$ must have at least $\sum_i\beta_i(M;\mathbb{Z}_2)$ critical points. When is it possible to ...
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62 views

Milnor - Morse Theory, proof of Lemma 16.1

I have a question reading the proof of Lemma 16.1 in Milnor's "Morse Theory", pp.88-89. https://www.maths.ed.ac.uk/~v1ranick/papers/milnmors.pdf This lemma is asserting that if $M$ is a ...
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Homotopical Effects of Length and Energy Functional on The Loop Space

Let $M$ be a compact Riemannian manifold and set $\mathcal{M}$ to be the space of $W^{1,2}$ closed curves on $M$. This is a Hilbert manifold and for positive constant $r$ define $\mathcal{M}_E^{r^2}$ ...
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1answer
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Existence of a coordinate system on an embedded submanifold in $\Bbb R^n$ satisfying a certain condition

Let $M$ be an embedded submanifold of dimension $k$ in $\Bbb R^n$, and let $u^1,\dots,u^k$ be coordinates for a region of $M\subset \Bbb R^n$. Then the inclusion map $M\hookrightarrow \Bbb R^n$ ...
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Question about the proof of Reeb's theorem in Milnor's Morse Theory

Theorem. (Reeb) If $M$ is a compact manifold and $f$ is a differentiable function on $M$ with only two critical points, both of which are nondegenerate, then $M$ is homeomorphic to a sphere. Proof) ...
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47 views

Existence of a certain smooth function $\Bbb R\to \Bbb R$ whose derivative takes values in $(-1,0]$

I am reading Morse Theory of Milnor, and in the proof of Theorem 3.2, Milnor says let $\mu:\Bbb R\to \Bbb R$ be a smooth function satisfying $\mu(0)>\epsilon$, $\mu(r)=0$ for $r\geq 2\epsilon$, and ...
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65 views

Question about the proof of Theorem 3.1 in Morse Theory – Milnor

The following theorem is Theorem 3.1 in Morse theory of Milnor. $M^a$ denotes the sublevel set $f^{-1}(-\infty,a]$. Theorem. Let $f$ be a smooth real-valued function on a manifold $M$. Let $a<b$ ...
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25 views

Morse-Bott Lemma in Hilbert Manifolds

I was wondering if there is any proof available for Morse-Bott lemma in the infinite dimensional case. In the finite dimensional case we have the following: Suppose $f\colon M \to \mathbb R$ is a ...
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87 views

Morse Smale diffeomorphisms of the sphere

Are there examples of diffeomorphisms (or homeomorphisms) $f:S^2\rightarrow S^2$ on the 2-sphere which have an odd number of critical points and thus a finite, odd-size set of non-wandering points? I ...
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Morse distance functions characterization in terms of normal exponential map

I'm reading the book "Classical and Modern Morse Theory with Applications" by Mercuri, Piccione and Tausk. We want to characterize the points $q\in \mathbb{R}^{n+p}$ such that the distance ...
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1answer
54 views

Lefschetz hyperplane theorem through Morse Theory in G-H p158

I am reading the Morse theoretic proof of the Lefschetz Hyperplane theorem in Griffiths-Harris and I am missing a transition. They claim that since the matrix $$\dfrac{1}{4}\left(\left(\dfrac{\partial^...
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1answer
125 views

Milnor - Morse Theory, proof of Morse's lemma

Lemma 2.2. Lemma of Morse - Milnor's Morse Theory, application of inverse function theorem. I have a question about the linked one. I was reading the book "Morse Theory" of Milnor, and I ...
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1answer
63 views

Understanding distance function on Riemannian Manifold

We have the next definition. This is from the book "Classical and Modern Morse Theory with Applications" by Mercuri, Piccione and Tausk. Let $f:M \to \mathbb{R}^{n+p}$ be an isometric ...

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