# Questions tagged [morse-theory]

In differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold.

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### A question about Marino–Prodi perturbation

In this paper N. Ghoussoub, the author claims the following version of Marino–Prodi perturbation, that is : Let $H$ a Hilbert space. Let $f\in C^2(H, \mathbb{R}),$ $K$ is a compact subset of $K_c$ (...
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### Invariance of morse homology, doubt in proof in 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. Link to the statement of the theorem ...
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### Upper bound on index of geodesics in terms of length

Let $(M, g)$ be a compact Riemannian manifold. Let $i(-)$ be the index of a geodesic and let $l(-)$ be the length. Is there an inequality of the form $i(\gamma) \leq C l(\gamma)$ for some $C>0$ ...
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### Next steps for a Morse theory enthusiast?

I don't know if this question is really appropriate for MO, but here goes: I quite like Morse theory and would like to know what further directions I can go in, but as a complete non-expert, I'm ...
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### Is a homotopy sphere with maximum Morse perfection actually diffeomorphic to a standard sphere?

The Morse perfection of a closed differentiable manifold $\Sigma^n$ is defined to be the largest integer $k$, such that there exists a smooth mapping $$p:S^k\times\Sigma^n\rightarrow\mathbb{R}$$ where ...
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### Index and length of closed geodesics

Consider the round metric on $S^n$. The geodesics are (multiples of) great circles, and one can verify that this metric is of Morse-Bott type. The Morse indices of the n-covered great circles are (if ...
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### A generalization to Bott‘s theorem (from Milnor’s “Morse theory”)

This is Theorem 22.1 of Milnor‘s Morse theory: Let $M$ be a complete Riemannian manifold, let $p,q\in M$ be so that the space $\Omega’$ of minimal geodesics joining $p$ to $q$ is a topological ...
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### CW-structure induced by Morse function on Riemannian manifold [duplicate]

I have heard a statement in the following direction. Given a compact Riemannian manifold $M$ with a Morse function on it possibly satisfying some extra assumptions. Then this data induces a CW-...
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### Connections between spectral geometry and critical point/Morse theory

I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...
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### Sard's theorem and Cantor set

Sard's famous theorem asserts that Theorem. The set of critical values of a smooth function from a manifold to another has Lebesgue measure $0$. I am asking for the curiosity that is it possible to ...
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### Finite-dimensional argument for Morse-Smale pairs?

Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $M$ (i.e. pairs $(f,g)$ where $g$ is a metric on $M$, $f$ is a Morse function on $M$, and the stable/...