# 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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### 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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### 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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### 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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### 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 ...
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 ... 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 ...