# 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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### Morse theory for vector-valued functions

Let $f:\mathbb{R}^{m+k}\mapsto\mathbb{R}^k$ be a smooth function. I have seen quite a few books for Morse theory for $f$ when $k=1$. Is there a generalization to $k\geq2$? When $k=1$, we can define a ...
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### Reference request: complete, rigorous proof of compactification of moduli spaces of flow lines in Morse homology?

The result I'm looking for can be stated as follows (taken from Hutchings' notes): Here the moduli spaces are referring to the spaces of flow lines of the negative gradient flow induced by the Morse ...
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### Unstable manifolds of a Morse function give a CW complex

A coauthor of mine and I want to use the following innocent looking statement in a forthcoming paper: Statement. Let $M^{2n}$ be a compact manifold and let $f$ be a Morse function with critical ...
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### Is the deformation along flow lines a simple homotopy equivalence?

Let $(M,g)$ be a compact, smooth $n$-manifold with boundary $\partial M$ and let $f: M \to [a,b]$ be a Morse function, whose critical points are interior and which satisfies $f^{-1}(b) = \partial M$. ...
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### What does one call a Morse function whose nondegenerate condition is relaxed?

In robotics, navigation functions are of utmost interest to plan a path from an initial location $q_0$ to a target location $q^t$. A function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is a navigation ...
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### Kähler manifold with even-only singular cohomology

Given a simply connected smooth projective variety (hence a compact Kähler manifold) with singular cohomology generated in even degrees, do we know that there is a Morse function on it such that all ...
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### Topological invariants of a certain “stratified” manifold, with pieces of different “dimensions”

Disclaimer: I don't fully understand what I'm talking about in the question below. I'm still trying to figure out the right question to ask. Quotations and question marks in brackets mean that I'm not ...
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### Visualizing a normalized hyper-elliptic curve as a honey-flow on a multi-doughnut

Trying to visualize in 3D the normalized model $y^2=f(x)$ of a hyper-elliptic curve AND the degree two ramified cover $(X,Y)\mapsto X$, $\mathbb{C}\times \mathbb{C}\to \mathbb{C}$. Is this projection ...
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### The handlebody decomposition of S^1 bundles over surfaces?

What is the most natural handlebody decomposition of $F_g \times S^1$, if $F_g$ is an orientable closed surface of genus $g$?