# 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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### 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 ...
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, $x_1,x_2\in\mathbb{R}$ ...
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 ...
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 ...
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 ...
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 ...
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-...
42 views

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 ...
54 views

### Getting rid of square root via integration

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