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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Height function on the sphere is Morse

I'm trying to prove that the height function on the sphere $\mathbb{S}^2$ is a Morse function. Somehow from my onw calculations, I conclude that it's not Morse. Using the embedding with spherical ...
58 views

How to prove a function is Morse

In general, if I have a function $g:X \rightarrow R$ defined on a manifold $X$, how can I show it is a Morse function? Is the best way to just compute $g \circ \phi$, where $\phi$ is a parametrization ...
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Proving height function is Morse

We have a differentiable manifold $X$ embedded in $\mathbb{R}^{n+1}$ and let $h: X \to \mathbb{R}$ as $h(x_{1},...,x_{n+1})=x_{n+1}$. We need to prove that $h$ is a Morse function. I don't know how to ...
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Morse theory graphic

How can I make the images that appear in Morse theory?. Type Beginner's question about homotopy type in Milnor's Morse Theory, i try in geogebra but i cannot.
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Birth-death : Always more than 1 bifurcation?

Say I have a (smooth) function $f : \mathbb{R}^n \to \mathbb{R}$, and a critical point $x$ (ie, $f'(x) = 0$). I call this point degenerate if $\det \text{Hess}_x f = 0$ (so, equivalently, if the ...
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Stationary phase approximation, compact support needed

I am trying to understand the stationary phase approximation as described on Wikipedia https://en.wikipedia.org/wiki/Stationary_phase_approximation. As necessary condition, they mention a compact ...
1 vote
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Topology of blow up of a cone

Let $X$ be a real algebraic surface with a unique $A_1$ point at $p \in X$ (that is, the germ of $X$ at $p$ is isomorphic to the germ of $\{x^2+y^2=z^2\}$ at the origin). Let $Y \to X$ be the blow up ...
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Reference request - "cones with multiple components" near higher order critical point

I am looking for information and relevant terminology for a phenomenon related to the geometry of sublevel sets of functions. I believe that the possibly related fields are Morse theory and algebraic ...
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Morse flow: cancelling handle pairs away from deformation retract

Given a smooth manifold (not closed, maybe with boundary) $M$ in $R^n$, take a section with a hyperplane $H$ of some dimension $d$. Assume that $M$ has $M\cap H$ as deformation retract. For example, a ...
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1 vote
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Why no locally trivial stratification in $C^1$ sense of Whitney's four sheets example

I am reading this paper https://www.math.ias.edu/~goresky/pdf/MatherBio.pdf from Mark Goresky. He talked about an example on page 5 saying that The first example is an algebraic subset of Euclidean ...
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CW complex structure on manifolds with certain dimensional cells

I started reading Morse theory and I learnt one of the fundamental theorem which says that any smooth closed manifold admits a CW complex structure. I had a following question: Given a smooth, closed ...
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1 vote
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Nice Morse chart with respect to a Riemannian metric

Let $(M,g)$ be a smooth Riemannian manifold and let $f: M \to \mathbb{R}$ be a Morse function (i.e. critical points are nondegenerate). The Morse lemma says that around every critical point $p$ of $f$ ...
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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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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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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 ...
112 views

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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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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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 ...