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.
396
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Handle body $H_g$ of genus $g$ as a CW complex
Often is the torus (as a $2$-dimensional manifold) given as an example of a CW-complex (see here, for example, for the genus $g$ manifold).
However, one can ask how to decompose the $3$-manifold (...
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Examples of degenerate (Floer) Hamiltonian orbits?
Say I have a periodic hamiltonian $H: S^1 \times M \longrightarrow M$ defined on a symplectic manifold $M$.
Then, a $1$-periodic Hamiltonian orbit of $H$ is the same thing as a fixed point of the ...
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What is the image of the Homology group of the boundary of a space under the map induced by inclusion?
Suppose $M$ is a compact manifold of dimension $n+1$ and $\delta M$ its boundary. What can we say about the image of $H_n(\delta M)$ under the inclusion $H_n(\delta M)\to H_n(M)$?
I was originally ...
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Orbit Space of Moduli Space in Morse Homology
If I have a group $G$ acting on a topological space $X$, and this action is free (no fixed points), what can I say about the orbit space $X/G$?
I am aware that freeness plus proper discontinuity leads ...
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Is there any version of (discrete) Morse Theory for subcomplexs?
I am now working on the (discrete) Morse theory and its application about simplices and cell complexes. However, I cannot find any statement about the property on closed subcomplexes / submanifolds of ...
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Placement of critical points of Morse function on the sphere
On a similar note to my previous question, I am still thinking about functions on the sphere.
Assume I have a $C^2$ function on the sphere $ f : S^{d-1} \mapsto \mathbb{R} $ which is Morse (i.e. all ...
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Can a real-valued function on the sphere have exactly 2 critical points which are not antipodal?
I came across this question while working on something quite different:
Can we build a $C^2$ function on the sphere of radius one, i.e. $f : S^{d-1} \mapsto \mathbb{R}, d \geq 3 $ such that it has ...
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Real analytic periodic function $f$ such that $\nabla f(x)=0 \Rightarrow \nabla^2f(x)=0$.
Is there any non-constant real analytic periodic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\{x\in\mathbb{R}^n\mid\nabla f(x)=0 \}\subset\{x\in\mathbb{R}^n\mid\nabla^2 f(x)=0 \}? $$
...
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Saddle point of a Morse function on SO(3)
Let $A\in\mathbb{R}^{3\times 3}$ be a symmetric, positive definite matrix with distinct eigenvalues and $R \in SO(3)$.
Is the function $f_A(R)=trace(A(I−R))$ has a saddle point? How to prove or ...
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Showing $f(z)=\frac{|z^m+1|^2}{(|z|^2+1)^m}$ is a morse function on a Riemann sphere.
Let $\hat {\mathbb{C}}$ be the Riemann sphere. The function
$$f(z)=\frac{|z^m+1|^2}{(|z|^2+1)^m}$$ gives a smooth function on $\hat {\mathbb{C}}$. Show that if $m\ge 3$, then $f$ is a morse function.
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Exercise 12 Chapter 1 - Computational topology for DA by Krishna Dey & Wang
I am solving problem 12 (chapter 1) in the book Computational topology for DA by Krishna Dey & Wang.
I have a question about notation. It is asked to prove that $f^{(-\infty,0]}\cap S^{2}$ and $f^{...
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Critical points a Morse function on SO(3)
Let $f_A: SO(3) \to \mathbb{R}$ and is given by $R\mapsto \frac{1}{2}|| A - R ||_F$, where $A \in \mathbb{R}^{3\times 3}$, and the norm is the Frobenius norm. As indicated in [1], the gradient of $f_A$...
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Local Modification of Morse Functions
If $f_0 , f_1$ are two Morse functions defined on some smooth manifold $M$ with a common critical point $p\in M$, the same value $f_0 (p)=f_1 (p)$ and with the same stable discs $W^s_{f_1} (p)=W^s_{...
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Diffeomorphism of level sets of functions depending on a parameter
Let $H : \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function with arguments $(x, \alpha)$, where $\alpha$ is a parameter. Fix a constant $c$ and suppose the set $S_{\alpha_0} :=...
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Structural stability of Morse-Smale diffeomorphisms
I'm trying to wrap my head around why Morse-Smale diffeomorphisms are structurally stable, using the real line as a toy example. Say $f$ is a Morse-Smale diffeomorphism. Proofs for more generalized ...
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On the definition of the second-order derivative of a Morse function
In Morse Theory and Floer Homology by Michele Audin and Mihai Damian (Morse theory and Floer Homology) they give the following definition to the second-order derivative at a critical point x.
$$(d^2f)...
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Morse-Bott inequalities in $\mathbb{CP}^n$
I want to prove the following statement (which I heavily believe is true):
Let $g:\mathbb{CP}^n\to \mathbb{R}$ be a non-constant Morse-Bott function and denote by $\text{Icrit}(g)$ the set of ...
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Morse function "induced" by Heegard splitting
Suppose I have Heegard splitting $M^3 = S_g \bigcup_f S_g$, where $S_g$ are solib bodies of genus g. I know how to construct Morse function on $S_g$ , so my question is:How can i fing a Morse function ...
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Poincaré-Hopf theorem and Morse theory on three-dimensional torus
I am asking this question to clarify a comment that appears below Eq.(32) of this paper, which applies Morse theory to classify van Hove singularities in energy bands of crystalline solids. These ...
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Morse functions with minimal number critical points
Is it true that $RP^n$ has a Morse function with n critical points,and dont have Morse function with n-1 critical points
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Construction of Morse function on $RP^1$
I wanna construct Morse function on $RP^1$ with one critical point,how can i do it.
P.S. i wanna construct such function on every $RP^n$,but at first wanna understand it on the simplest example.
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Issue with proof of theorem 2.1.11 in Audin-Damian
The theorem in question is as follows:
Let $V$ be a closed smooth manifold and $f: V\to \mathbb{R}$ a Morse function. Let $a$ be a critical point of $f$ with index $k$ and $\alpha=f(a)$. Suppose that ...
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Attaching a handle at a local maximum of Morse function
I am studying Morse theory and I am bit confused about what happens to a manifold when going through a critical value of a Morse function.
Here is the setting: $M$ is a compact manifold and $f : M \to ...
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How to show there is an $S^4$ included in a simplicial complex?
A $\mathbb{Z}_2$-space is a pair $(T, \nu)$, where $T$ is a topological space and $\nu: T \rightarrow T$, called the $\mathbb{Z}_2$-action, is a
homeomorphism such that $\nu \circ \nu= id_{T}$ . If $(...
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Signature property of the Maslov (Conley-Zhender) index
In the proof of the signature property of Maslov index, one uses the fact that if $S$ is a symmetric matrix whose norm is less than $2\pi$, then the matrix $\exp(J_0S)$ does not admit $1$ as ...
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a manifold on which every smooth function has at least two local minima
I am wondering whether there exists such a compact and connected manifold such that every smooth function on it has at least two local minima.
Because it is compact, so there is at least one local ...
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Does a self-indexing Morse function gives information about the Heegaard genus?
This question comes up when I am working on an exercise finding the Heegaard genus of the 3-torus $S^1\times S^1\times S^1$.
By definition of Heegaard genus, it is the minimal possible genus of the ...
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How to find a discrete Morse function on this simplicial complex
I have a graph called partition graph. This graph gives rise to a simplicial complex called box complex $B_{edge}$. Since this simplicial complex is too big and studying the topological features of it ...
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What is operator calculus? Article by Palais.
I'm reading a paper titled Morse theory on Hilbert Manifolds by R. Palais. And in the demonstration of the Morse Lemma (pg 307), he use something called operator calculus, for example he take the ...
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There exists a Morse function on $M$ taking different values at different critical points
Let $M$ be a compact smooth manifold (without boundary). Prove: there exists a Morse function $f$ on $M$, such that $f$ takes different values at different critical points.
This is an exercise on the ...
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Problem regarding a theorem from Milnor's Morse Theory book .
$\mathbf {The \ Problem \ is}:$ The doubt is from Milnor's ''Morse Theory'',theorem $3.2$, pg $14(attached).$
In pg $18$, in showing that $F(q)\leq F(p) < c-\epsilon$ as $\frac{\partial F}{\partial ...
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two surgeries on a collection of circles.
in this paper page 4 we have :
There are five essentially distinct possibilities, corresponding to the
various ways of performing two surgeries on a collection of circles so that the
resulting 2-...
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How to show Hessian is independent of co-ordinate system at critical point?
A smooth manifold $M$ is a topological manifold with compatible smooth atlas (in the following all manifolds are assumed to be $n$-dimensional, smooth, oriented, closed, and without boundary.)
A ...
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Why are these two figures homotopy equivalent?
In basically every introduction to Morse theory, we see the example of a height function on a torus. Morse theory then predicts that a torus minus a disk is homotopy equivalent to a cylinder with a 1-...
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Finding a Morse chart for $f$ where $\nabla f$ points "outward"
Let $(M,g)$ be a Riemannian manifold, and let $f$ be a smooth function whose Hessian is conformal to the metric, i.e. $\text{Hess} f=\lambda g$ for some smooth $\lambda:M\to \mathbb{R}$. Suppose that $...
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Is there a good set of lectures or notes on topology for studying characteristic classes?
I was watching the lectures of Dr. Tadashi Tokieda on "topology and geometry", which was pretty amazing that I finished all the the lectures in 3 days. What particularly helpful was the ...
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$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 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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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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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 ...
user1001232
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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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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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