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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27
votes
1answer
3k views

When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
6
votes
4answers
395 views

Prove no existing a smooth function satisfying … related to Morse Theory

i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ ...
0
votes
3answers
134 views

What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?

While for $n=2$ it is pretty clear that the contours of a non-constant $f:\mathbb R^n\to\mathbb R$ are either extrema (and therefore points) or (the union of) 1-dimensional isolines, for $n=3$ I am ...
5
votes
2answers
974 views

How does Morse theory on non-compact manifolds differ from compact manifolds?

What is the Morse homology of a non-compact manifold? When is it, as in the compact case, isomorphic to singular homology of the underlying manifold? What other constructions can be identified with ...
8
votes
1answer
184 views

Smooth classification of vector bundles

I am trying to understand the smooth classification of $n$-disk bundles over $S^n$. As vector bundles, these are classified by $\pi_{n-1}(SO(n))$ via the clutching construction but I am interested in ...
4
votes
1answer
521 views

Can a proper Morse function $\mathbb{R}\to\mathbb{R}$ have infinitely many critical points?

Depending on interpretation, there may be an assumption missing from Exercise 6.1.4(a) in Liviu I. Nicolaescu's Invitation to Morse Theory: Suppose $f : \mathbb{R} → \mathbb{R}$ is a proper Morse ...
14
votes
0answers
553 views

Proof of Reeb's theorem without using Morse Lemma

I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. That is, suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both non-...
2
votes
0answers
42 views

Discrete Morse Function for $n$-simplex

I am trying to find a "useful" discrete Morse function for the $n$-simplex. According to (https://www.emis.de/journals/SLC/wpapers/s48forman.pdf page 12), a possible discrete Morse function is $f(\...
2
votes
1answer
324 views

An other question about Theorem 3.1 from Morse theory by Milnor

In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that: for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla f\right\rangle=\...
5
votes
1answer
136 views

Explicit verification of signs in Morse complex

I'm trying to check by hand that the signs in the Morse complex, defined via choices of orientations on the unstable manifolds, lead to $\partial^2=0$. The books I've looked in seem to say either ...
0
votes
1answer
54 views

Getting rid of square root via integration

How do we prove, for positive $D$, this result? $$ e^{-2\sqrt D} \sqrt{\pi} = \int_0^\infty s^{-1/2} e^{-(s+D/s)} ds $$
4
votes
0answers
130 views

gradient flows on Hilbert manifolds

I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded. To be more precise, a ...
4
votes
1answer
168 views

Relation between tangent spaces of (un)stable manifolds in Morse theory

After asking this question about signs in the Morse complex, I realised that my confusion is really about how tangent spaces to different (un)stable manifolds are related. So suppose we have a Morse ...
3
votes
0answers
538 views

Proof of the Morse Lemma in dimension 1

This is the Morse lemma in dimension1 : Let $M$ be a smooth $1$-manifold and $f: M \longrightarrow \Bbb R$ be a smooth function. Suppose $p$ is a non-degenerate critical point of $f$. Then ...
3
votes
1answer
233 views

Fundamental theorem of Morse theory for $\Omega(S^n )$

Using the Fundamental theorem of Morse Theory we can prove that $\Omega(S^n)$ is homotopically equivalent to a CW complex with one cell each in dimensions $o,n-1,2(n-1), \cdots$ and so on. But how can ...
3
votes
0answers
130 views

Classification of highly-connected manifolds

In http://www.maths.ed.ac.uk/~aar/papers/n-1con2n.pdf, Wall studied $(2n, n)$-handlebodies, ie. 2n-manifolds that have a handle presentation with only 0 and n handles. Given a presentation, he ...
3
votes
0answers
82 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 ...
2
votes
1answer
276 views

The connection of Morse function

Suppose $M$ and $N$ are two manifolds, $f$ is a Morse function on $M$, $g$ is a Morse function on $N$, can you find a new manifold $P$ as the connection of $M$ and $N$ and a Morse function $h$ on the $...
2
votes
1answer
712 views

On the Euler characteristic in Morse Theory

Let $f:M\to \Bbb{R}$ be a Morse function, where $M$ is a $k$-manifold. The index $i_{f,p}$ is defined to be the number of negative eigenvalues of the Hessian $H_f$ at the critical point $p$. For ...
2
votes
1answer
109 views

Path space of $S^n$

Suppose that $p,q$ are two non conjugate points on $S^n$ ($p \ne q,-p$). Then there are infinite geodesics $\gamma_0, \gamma_1, \cdots$ from $p$ to $q$. Let $\gamma_0$ denote the short great circle ...
1
vote
1answer
194 views

Lemma of Morse in dimension 1

I want to write the Morse lemma which is in dimension $n$ : Let $p$ be a non-degenerate critical point for $f$. Then there is a local coordinate system $(y^1,...,y^n)$ in a neighborhood $U$ ...
1
vote
0answers
54 views

Critical points in a simple case of manifold with boundary.

Let $X$ be a smooth, connected, compact manifold and $Y=X\times [a,b]$. Let $f:Y\to\mathbb{R}$ be a Morse function on $Y$ such that $f|_{X\times \{a\}}=a$ and $f|_{X\times \{b\}}=b$. What can we say ...
1
vote
0answers
38 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
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 $f: U \to \mathbb R$ ...