# 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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### Continuity of retraction on $r : \text{Int }\Omega^c \to B$

I'm reading Milnor's Morse Theory and I have difficulty verifying some claim (which is easy according to Milnor) on page $88$, section $\S 16$ in the book. Here's the setup for my question. In the ...
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### A linear algebra related detail in a proof of Index Theorem

Here is a clipping from Milnor's Morse Theory. Since this question about linear algebra, I will present my question below so that no prior knowledge of the materials in the book required to answer ...
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### What information is contained in the 'list of cells' of CW-complexes

In Milnor's book on Morse theory there are many statements of the form '[...] has the homotopy type of a CW-complex with these kind of cells'. For example, Theorem 17.3 (fundamental theorem of Morse ...
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### Existence of Morse function and Immersion

I have started with differential topology and I try to solve exercises in the book Differential Topology by Victor Guillemin, Alan Pollack. There are 2 exercises in chapter Sard and Morse Theorem I do ...
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### Understanding Hessian on Manifold (without Riemannian Geometry)

I've been going through notes on Morse theory and Handlebody theory and I've been having some trouble with the definition of the Hessian provided. The notes are on pages 3-4 here http://people.math....
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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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### Proof that Morse complex is a complex using coherent orientation

I'm reading the book Morse Homology by M. Schwarz, which aims to develop Morse homology in strict analogy with Floer homology. For orientation matters, the book follows the paper A. Floer and H. ...
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### Coordinates functions and Morse function.

I'm trying to resolve the following problem: Let $X$ be a submanifold of $\mathbb{R}^N$ of dimension $k$. Show that there exists $l : \mathbb{R}^N \rightarrow \mathbb{R}$ linear such that its ...
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### Homology calculation using Morse theory

I am currently reading Morse Theory from the book written by Audin and Damian. And faced the Problem. Let $V$ be a compact connected manifold of dimension $n$ without boundary and let $D$ be a disk ...
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### Is having critical points a necessary condition for a Morse function?

The definition of a Morse function requires that its critical points are all degenerate and no two of them share the same function value. Now, I'm wondering whether or not the criticality condition is ...