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.

Filter by
Sorted by
Tagged with
4
votes
1answer
114 views

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 ...
1
vote
1answer
51 views

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 ...
2
votes
0answers
56 views

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 ...
0
votes
0answers
28 views

Local maximum and critical points smooth manifolds surjectivity

A local maximum of a smooth function $f:M\rightarrow \mathbb{R}$ is a critical point of $f$. Attempt: Let $p\in M$ be a local maximum of $f$. Let $X_p\in T_pM$. Let $c:(-\epsilon,\epsilon)\rightarrow ...
0
votes
1answer
32 views

All zeros of the gradient of a morse function on a riemannian manifold are non-degenerate

Let $f$ be a Morse function. Define the gradient vector field on a Riemannian manifold by $df_p(w)=\langle \text{grad } f|_p, w\rangle \; \text{ for all } p\in M, w\in T_pM.$ A zero $p$ of $\mathrm{...
0
votes
0answers
23 views

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 ...
0
votes
0answers
32 views

A Problem about Hermitian Form

$X$ is a linear space over $\mathbb C$,$q$ is a nondegenerate $(X^q=0)$ Hermitian form on $X$. $V$ is a subspace of $X$,$V^q= \{x \in X|q(x,y)=0, \forall y \in V \}$,$dim X/(V+V^q) $ is finite, $(V \...
1
vote
0answers
27 views

Morse Homology and The Space of Broken Trajectories

I am reading Morse Homology from the book by Audin and Damian. I have read the proof about $\overline{\mathcal{L}(a,b)}$ (space of broken trajectories connecting two critical points) being a manifold ...
0
votes
1answer
24 views

Unique index 0 critical point if all critical points are of index 0?

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a Morse function whose all critical points are of index 0. Is it true that $f$ has a unique critical point if it has at least one critical point? Since $\mathbb{R}...
4
votes
1answer
65 views

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....
0
votes
0answers
25 views

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 ...
1
vote
0answers
35 views

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. ...
0
votes
0answers
27 views

Generalization of Myers theorem

I need a hint to this problem, or anything that gives a pointer in the right direction would be helpful. The question I need to prove is: Let $M$ be a compact $n$-manifold such that Ric$(U,U) \geq (n-...
4
votes
1answer
98 views

The Euler characteristic of a cubic Fermat surface

Let $F$ denote the cubic Fermat surface in $\mathbf{P}^3$ (everything is over the complex numbers): $$ F = \{ X^3 + Y^3 + Z^3 + W^3 = 0\}\subseteq \mathbf{P}^3.$$ I wish to compute the Euler ...
0
votes
0answers
17 views

Compactification theorem in Morse Homology

I'm reading "Morse Theory and Floer Homology" by Audin-Damian and got stuck in the proof that the Morse differential indeed induces a homology (i.e. $\partial^2=0$). This is basically the ...
1
vote
0answers
103 views

Classification of compact surfaces

I would like to review the classification of compact surfaces (orientable or not) and without boundary (first). I know that they are classified topologically by Euler characteristic (and therefore by ...
1
vote
0answers
37 views

What are some applications of the Morse inequalities?

I'm currently learning some Morse theory, and I've just learned about the Morse inequalities. They appear to be very powerful - being able to deduce so much topological information from a single real ...
1
vote
1answer
28 views

Composition of Morse function with diffeomorphism isotopic to identity is a morse function

Suppose I have a Morse function $f$ on a compact smooth manifold $M$, potentially with boundary, and that $h$ is an automorphism of $M$ isotopic to the identity automorphism. Then is $f\circ h$ a ...
0
votes
0answers
26 views

Morse theory prerequisites

I'm wondering what the official and formal prerequisites for introduction to Morse theory by Yukio Matsumoto are. It seems really interesting in my opinion, so I'm wondering.
1
vote
1answer
18 views

A regular Morse map $f:C_K\to S^1$

A circle-valued Morse map $f:C_K \to S^1$ on the complement of a knot $K$ is said to be regular if there is a $C^\infty$ trivialisation $\Phi : N(K)\to K\times B^2(0,\epsilon)$ of a tubular ...
2
votes
0answers
36 views

Thom's transversality theorem on non-compact manifolds

I am studying at the moment Thom's transversality theorem for compact and smooth manifolds, which goes like this: Let $M$ be a compact, smooth manifold. Then the set of Morse functions is a dense and ...
1
vote
0answers
67 views

Details Related to One-variable Calculus in a proof of Morse Lemma.

I'm reading M. Audin and M. Damian's book $\textit{Morse Theory and Floer Homology}$ and i'm having an issue on a proposition that is "Morse Lemma" here page 13. This proof about Morse Lemma is a ...
5
votes
2answers
104 views

Existence of a proper Morse function

Given a manifold $M$, I know there exists a proper function $f: M \to \mathbb R$ (using the usual partition of unity argument) and a Morse function $g: M \to \mathbb R$ (genericity of Morse functions)....
4
votes
1answer
100 views

Poincare Duality in Morse Homology

I'm reading M. Audin and M, Damian's Morse Homology book right now and i have some issue regarding its section about Poincare Duality (Section 4.3, page 83). In this version of "Poincare Duality", it ...
1
vote
0answers
32 views

Isomorphism $\ker(B)/Im(A)\cong \ker(A^t)/Im(B^t)$ for chain of linear maps.

Consider chain of linear maps between finitely dimensional vector spaces $E$, $F$ and $G$ over $\mathbb{Z}/2$: $E\xrightarrow{A}F\xrightarrow{B}G$ then we take transpose $A^t$ and $B^t$ and consider ...
1
vote
0answers
26 views

Generalization of Morse Lemma that appears in 'singular points of complex hypersurfaces'

In the book 'singular points of complex hypersurfaces', lemma 2.12, Milnor claims the following version of the Morse Lemma: $Let 0\in M\subset \mathbb{R}^{m}$ be a smooth manifold, and $r:\mathbb{R}^...
0
votes
0answers
46 views

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 ...
0
votes
0answers
91 views

Understanding the statement “almost all functions are morse”

Following up on my last question, some folks were kind enough to direct me to this theorem from Guillemin and Pollack, Differential Topology, Chapter 1.7 If $U \subseteq \mathbb{R}^n$ open, $f:U \...
0
votes
0answers
7 views

Number of eigenvalues in a hessian across a closed region

Suppose $F:\mathbb{R}^N\rightarrow \mathbb{R}^1$ is in $C^2$. Denote the Hessian of $F(x)$ at point $x$ by $H(x)$. Let $S$ be a closed subset of $\mathbb{R}^N$ and $k_1,k_2$ be positive real values....
1
vote
1answer
48 views

Why are embedded spheres removed in the connected sum but not in the handle attachment of (smooth) manifolds?

So i am currently studying differential manifolds and morse-theory. When i came across the connected sum, i learned that we glue two manifolds $M_1$ and $M_2$ along the boundaries of removed disks ...
3
votes
0answers
80 views

“A typical function is morse”

On the wikipedia page for Morse theory it states the following A smooth real-valued function on a manifold M is a Morse function if it has no degenerate critical points. A basic result of Morse ...
0
votes
1answer
37 views

Question about the directional derivative in Matsumoto's Introduction to Morse Theory

I'm currently reading Matsumoto's Introduction to Morse Theory and on p.58 he states Given a tangent vector, we can differentiate a function in its direction. Let uns expain this using the ...
0
votes
0answers
48 views

Lemma 2.2. Lemma of Morse - Milnor's Morse Theory, application of inverse function theorem.

I'm learning a very tiny bit of Morse theory, from this book. (Lemma of Morse) : let $p$ be a non degenerate critical point for $f$. Then there's a local coordinate system $(y^1, \ldots, y^n)$ in a ...
2
votes
2answers
73 views

Does the “easy” part of the Morse lemma follow from Ehresmann's fibration theorem?

Corollary 2.3 of these notes on Morse theory seem to suggest the "easy" part of the Morse lemma is a corollary of Ehresmann's fibration theorem. That is, if $f:X\to \mathbb R$ is a proper smooth map ...
2
votes
1answer
70 views

Where can I find this Morse Theory result?

I'm reading through chapter 1 of Curve and Surface Reconstruction: Algorithm and Mathematical Analysis. There's lemma 1.1 where in the proof apparently a result of Morse Theory is used, and I'd like ...
0
votes
1answer
33 views

Question regarding description of a $\lambda$-handle in Matsumoto's “ An Introduction to Morse Theory”

On Page 76 in Y. Matsumoto's "An Introduction to Morse Theory" he introduces an $m$-dimensional $\lambda$-handle including the following figure: he then says The lightly shaded area corresponds ...
1
vote
0answers
21 views

Singular point for Morse functions

I am working in the following setting. $\newcommand{\R}{\mathbb{R}} \newcommand{\Vreg}{{V^{reg}}}$ I have two functions $F \colon \R^{n+1}\to \R^{n}$ and $L \colon \R^{n+1} \to \R$, such that $F$ is ...
5
votes
2answers
194 views

“Simple to state, but difficult to solve” problems which require analyzing topology of simplicial complexes?

In a User's Guide to Discrete Morse Theory, Robin Forman writes: A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial ...
1
vote
1answer
60 views

Homotopy type of the loop space of a compact Lie group

The following theorem is proved in Milnor's famous book "Morse theory". Theorem 21.7 (Bott). Let $G$ be a compact, simply connected Lie group. Then the loop space of $G$ has the homotopy type of a CW-...
1
vote
0answers
57 views

Show that is a Morse function.

Let $f:M\to\mathbb{R}$ a Morse function, and $\pi:\tilde{M}\to M$ a covering map, show that $f\circ \pi:\tilde{M}\to \mathbb{R}$ is also a Morse function. Let $\tilde{f}=f\circ \pi$, then $$d\tilde{f}...
0
votes
0answers
29 views

Zeroth dimensional relative Morse Homology vanishes

Currently I am reading Morse Homology. And I faced a problem, which is easy to show in case of Singular Homology. But I can not do it for Morse Homology. So the problem is, for any submanifold $V$ of ...
3
votes
1answer
80 views

Simple proof for bound on the sum of Betti numbers for piecewise linear complex

Let a piecewise linear $d$-dimensional complex $M \subset \mathbb{R}^m, m \geq d$ be obtained by attaching $K$ convex $d$-dimensional polytopes $\{ C_i \}_{i=1}^K$ (cells), along their facets without ...
2
votes
1answer
32 views

Proving the existence of non-minimum critical points

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be an analytic function such that $$\lim_{|\vec{x}|\rightarrow \infty}f(\vec{x}) = \infty$$ Let $M$ be the set of points at which $f$ achieves its minimum ...
0
votes
0answers
107 views

Can a smooth function with compact sublevel sets only admit local minimizers?

I asked a similar question here but did not specify all the details of the function I encountered. As that question has been answered, I ask the clarified version here. Suppose $f: U \to \mathbb R$ ...
5
votes
0answers
65 views

What is the most surprising / interesting application of the inverse function theorem you have ever seen?

The inverse function theorem states that: Suppose that $f: U \subset \mathbb{R}^m \rightarrow \mathbb{R}^m$ is a $C^k$-function and that there exists $a \in U$ such that $f'(a): \mathbb{R}^m \...
2
votes
1answer
58 views

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 ...
0
votes
0answers
18 views

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 ...
6
votes
1answer
136 views

Understanding Morse homology with the example of a circle

I am studying Morse Homology using the book "Lectures on Morse Homology" by Augustin Banyaga and David Hurtubise and I am having a hard time to understand how to count flow lines with sign. Now I am ...
2
votes
0answers
72 views

Morse Homology of some function on torus

Consider the torus $\Bbb T^2=\Bbb R^2/\Bbb Z^2$. And define a Morse function $f:\Bbb T^2\to \Bbb R$ by $f([(x,y)])=\cos(2\pi x)+\cos(2\pi y)$. Now the critical points are minimum $\big[\big(\frac{1}{2}...
0
votes
0answers
42 views

Finding the entropy

Let $M$ be a compact manifold and $F:M\to \mathbb{R}$ Morse function. Let $\phi_t$ be a flow generated by $F$ in the following way: $\frac{d\phi _t}{dt}=-\nabla F(\phi _t)$. Let $f=\phi _1$. Find $h(f)...

1
2 3 4 5 6