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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$L_2$ norm for matrix of funtion

In Audin's book Morse theory and Floer homology, they claimed Proposition 6.1.5. Let $H$ be a function on $\mathbb{R}^{2n}$, so that $X_H$ is a vector field on $\mathbb{R}^{2n}$. If $|dX_H|_{L_2}...
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the definition of differential of a Hamiltonian vector field

In Audin's book Morse theory and Floer homology, they claim: Proposition 6.1.5. Let $H$ be a function on $\mathbb {R^{2n}}$, so that $X_H$ is the Hamiltionian field on $\mathbb {R^{2n}}$. If $|...
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Question about critical point [closed]

Assume $M$ is a manifold, $f$ is a smooth function on it, the differential $df$ is a map $df:M\rightarrow T^*M$, Assume $x$ is a critical point, then it is on the zero section, moreover, prove $x$ is ...
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Question in Milnor's Morse Theory

In Milnor‘s book 'Morse Theory' p12 and p13, the proof of Theorem 3.1, which is Let $f$ be a smooth real valued function on a manifold $M$. Let $a < b$ and suppose that the set $f^{-1}[a,b]$, ...
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apply theorem of differentiation through an integral sign

When I came across the Morse Lemma, which is on page 5 of 'Morse Theory' by Milnor, I found one little calculus thing in its proof, in lemma 2.1 regarding the function $g_i$, which I fail to ...
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Consequence of the Graph Transform in Morse Homology

I have been reading some notes on Morse Theory and at some point the following claim is made: Suppose we have $f$ a Morse function and $M$ a compact smooth manifold, with $x,y\in Crit(f)$ such that $m(...
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Morse theory of a global function $f$ on a hypersurface $g=0$

Let $f$ and $g$ be two smooth functions on $\mathbb R^N$. Suppose $f$ is a Morse function on the manifold $M=\{g=0\}$. To study the Morse theory, we need to find the critical points of $f$ on $M$. ...
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Milnor, Lectures on h-cobordism theorem: Lemma 5.9

In lectures on the h-cobordism theorem, Milnor writes of an isotopy $h_t:\mathbb{R}^n\to\mathbb{R}^n$, with $\mathbb{R}^n=\mathbb{R}^a\oplus\mathbb{R}^b$ in his notation, the following lemma: The ...
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Milnor, Lectures on h-cobordism theorem, Handle Cancellation Theorem, Theorem 5.4, Assertion 6

In the Lectures, Theorem 5.4 (handle cancellation), Assertion 6, specifically page 57, Milnor is proving we can assume the two critical points of consecutive index can be assumed to be in the same ...
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How to show that $(x_1,\dots,x_{n+1})\in\mathbf S^n\mapsto x_{n+1}$ is a Morse function?

How to show that $(x_1,\dots,x_{n+1})\in\mathbf S^n\mapsto x_{n+1}$ is a Morse function? According to my course definition, I have to demonstrate two things: $f$ is $\mathcal C^2$, all critical ...
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Finding a specific sequence in Morse Homology

Consider $M$ to be a compact manifold and $f$ a morse function on $M$ so that we can do Morse Homology. Let $q_1,q_0$ be two critical points of $f$ such that $m(q_0)=m(q_1)+1$, where $m$ denotes the ...
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Condition for the exponential of a point to be in a specific unstable manifold

Consider $M$ to be a compact manifold and $f$ a morse function on $M$. Consider the unstable manifolds $W^{u}(x)$ defined using the flow of the negative gradient vector field of $f$. Let $c$ be a ...
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Subharmonic functions, their critical points and values

Let $f \colon \mathbb{C} \to \mathbb{R}_{+} \cup \left\{ 0 \right\}$ be a $C^{\infty}$ subharmonic function. Be given a compact domain $K \subset \mathbb{C}$, we let $D_{K}(f)$ be the set of critical ...
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Homotopy equivalence onto special fiber

The following proposition appears in Peters and Steenbrink's book on Mixed Hodge Structures. Proposition([Peters--Steenbrink, Proposition C.11]) If $f\colon X\to\Delta$ is proper and smooth over $\...
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Time derivative of Pushforward equality

In Audin and Damian's "Morse Theory and Floer Homology", In Prop. 5.4.5, there is a statement about the time derivative of a pushforward that I am having trouble understanding. In the last ...
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Morse index and conjugate points

Let $M$ be a compact smooth manifold and consider $L:S^1\times TM\rightarrow \mathbb{R}$ a smooth $1$-periodic Lagrangian. If we assume that $d_{vv} L(t,q,v)\geq l_0I$ for some $l_0>0$ then one can ...
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Morse Homology in $H^{1}(S^1,M)$

Let $M$ be a compact manifold, and consider the space $H^1(S^1,M)$ of loops that are of sobolev class $W^{1,2}$. Under suitable conditions on a lagrangian $L$, i.e., non-degenerancy, quadratic at ...
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Some questions about critical point

I’m reading Morse theory written by J.Milnor and find some questions: Let $c_1<c_2<\dots$ be critical values of $f \colon M\to\Bbb{R}$. If we assume $M^a=\{x\in M:f(x)\le a\}$ is compact for ...
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cancellation theorem in h-cobordism

I'm reading milnor's book h-cobordism, in beginning of the section cancellation theorem, milnor give an example that composition of two elementary cobordism with index $0$ and $1$ may be a product ...
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How to check that a tangent vector field is outward pointing at the boundary?

I'm reading a paper (https://www.jstor.org/stable/pdf/25151781.pdf) which gives a definition of a outward-pointing tangent vector field (see Assumption 1 below). Poincare-Hopf Theorem. Let $M \subset \...
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Morse function induced on fibered product

Let $A,B$ and $C$ be three smooth manifolds. Suppose that $F:A\to C$ and $G:B\to C$ are smooth and transverse functions, making the fibered product $$S=A\underset{F,C,G}{\times}B=\{(x,y)\in A\times B\,...
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A Morse function on $\mathbb RP^2$ whose all critical points are non-degenerate. [duplicate]

$\mathbf {The \ Problem \ is}:$ Find a Morse function on $\mathbb RP^2$ with one non-degenerate critical point . It may have other critical points . $\mathbf {My \ approach}:$ If we take a homogenous ...
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3 votes
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Signed Morse homology of $\mathbb{R}P^2$

Here's my understanding of computing the signs in Morse homology (following the book by Audin and Damian). Let $f$ be a Morse function on a manifold $M$ with a negative pseudo-gradient $X$. Let $W^s(p)...
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what is level curves? what is non-critical level curves?

in this paper we have non-critical level curves. "On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps at Besançon " We have :...
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2 votes
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What is the condition on the hypersurface, such that all height functions are Morse functions?

When we choose the morse function of a hypersurface of affine space. We always choose a height function. But when must a height function be a morse function?I mean, can we add some curvature ...
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7 votes
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128 views

Morse functions invariant under diffeomorphisms

Let $f:M \to \mathbb{R}$ be a Morse function of a compact manifold $M$. Assume $\sigma:M \to M$ is a diffeomorphism such that $f$ is invariant under $\sigma$, i.e. $f(\sigma x)=f(x)$ for all $x \in M$....
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Morse Theory for sublevel sets with strict inequality

Let $M$ be a compact manifold and $f:M \to \mathbb{R}$ a Morse function. For a real number $s$, define $$M^{\leq s}:=\{x \in M \ : \ f(x) \leq s\}.$$ In what follows, let $s<t$ be real numbers such ...
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Can a function who is linear in each variable separately have a degenerate critical point that is not a saddle point?

Given a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ that is linear in each variable $x_i$ separately but not in the vector $x$, e.g, $f(x)=2x_1x_4x_6-7x_1x_2x_3+15x_3x_5x_6$. This family of ...
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Length of a broken geodesic

Suppose $M$ is a Riemannian manifold, with injectivity radius larger than $\delta$. Define $M^n_\delta$ to be the set of points $(x_1,...,x_n)\in M$ with $d(x_i,x_{i+1})\le \delta$. Hence, each pair ...
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On exercise $1.7.15$ of Guillemin-Pollack

$\mathbf {The \ Problem \ is}:$ If $X$ is an embedded submanifold in $\mathbb R^N$ ,show there exists a linear map $L : \mathbb R^N \to \mathbb R$ such that $L|_X$ is a Morse map . $\mathbf {My \ ...
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3 votes
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Milnor: Morse Theory Theorem 3.1 why do we need $f^{-1}[a,b]$ to be compact?

In Milnor's Morse Theory the Theorem 3.1 is given as follows: Let f be a smooth real valued function on a manifold $M$. Let $a < b$ and suppose that the set $f^{−1}[a,b]$, consisting of all $p \in ...
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4 votes
2 answers
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Analogous of Poincaré Duality for relative homology and relative cohomology

I am studying Morse Theory on finite dimensional and compact manifolds using homology groups and relative homology groups on $\mathbb{Z}$. I want to show that this theory could be developed using De ...
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"Negative definite" in the definition of Morse index

Wikipedia and other sources say that the index of a non-degenerate critical point $p$ of a manifold $M$ is "the dimension of the largest subspace of $T_p\left(M\right)$ on which the Hessian is ...
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The flux of a the negative gradient flow of a Morse-Bott function on a compact manifold converges to a critical value?

Let $(M, g)$ be a compact Riemannian manifold and $f: M \rightarrow \mathbb{R}$ be a Morse-Bott function, i.e. the set a critical points of $f$, $Crit(f)$, has connected components which are smooth ...
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2 votes
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Soft/Hard question on the relation of homology of finite covering spaces and the base (for singular homology, Morse homology and ECH)

I would like to know how much is to be expected from the relation of the homology of a $n$-sheeted covering manifold $M$ and its base $N$ (let's say $\pi:M\mapsto N, deg \space\pi=n$). I'm interested ...
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5 votes
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Intuitive explanation of singular homology

Let $J$ be a $\mathcal{C}^1$-functional over a inner product space. The local behavior of $J$ near an isolated critical point $u$ is described by the sequence of critical groups $$C_q(J, u):= H_q(J^c\...
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1 vote
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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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2 votes
1 answer
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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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3 votes
1 answer
100 views

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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1 vote
1 answer
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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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8 votes
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309 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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4 votes
2 answers
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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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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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27 votes
1 answer
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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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2 votes
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175 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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2 votes
1 answer
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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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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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