# 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.

227
questions

0
votes

1
answer

71
views

### A question about Marino–Prodi perturbation

In this paper N. Ghoussoub,
the author claims the following version of Marino–Prodi perturbation, that is :
Let $H$ a Hilbert space.
Let $f\in C^2(H, \mathbb{R}),$ $K$ is a compact subset of $K_c$ (...

2
votes

0
answers

131
views

### Exponential map of cotangent bundle and Morse theory on based loop space

Consider $M$ to be a compact manifold and $q_0,q_1\in M$. Let $L_t$ be a lagrangian and $\mathcal{E}_L$ the lagrangian action functional on the based loop space $\Omega(M,q_0,q_1)$ defined as $\...

1
vote

1
answer

98
views

### Is an $L^p$-sphere in Sobolev space $H_2^{s}(\Omega)$ a Hilbert manifold?

For a bounded smooth domain $\Omega$, let $H_2^{s}(\Omega)$ be the usual Sobolev space on $\Omega$.
Define $A:=\{f\in H_2^{s}(\Omega)| \lVert f\rVert_{L^p(\Omega)}=1\}$ where $2<p<2_{s}^*$.
...

4
votes

0
answers

376
views

### What are your common strategies/remedies when your new theory/idea stuck in most cases?

Sorry if this is not a suitable post for MO.
Sometimes after reading the origin of a theory/idea in differential topology I put myself in the shoes of that mathematician and ask myself, Did you do the ...

1
vote

0
answers

45
views

### Proper Morse function on open set

Let $M$ be a compact submanifold with boundary of $\mathbb{R}^n$ of dimension $n$. Let $f:M \to \mathbb{R}$ be a Morse function. Then $f$ is proper. Let $N:=M-bd(M)$. How can I get a proper Morse ...

4
votes

1
answer

269
views

### Building a manifold from a CW complex inductively

Given a finite dimensional finite $CW$ complex $X$ of dimension $d$, I want to build a compact manifold $M$ (with least dimension possible) with boundary with the property that,
$M$ has the same ...

1
vote

1
answer

155
views

### What is the definition of a height function for a subsurface?

What is the definition of a height function for a subsurface? Is a height function exactly a Morse function?
In the paper:
"On the Teichmüller tower of mapping class
groups
By Allen Hatcher at ...

1
vote

0
answers

48
views

### Morse function on the Sphere base on functions on the disc

Lets consider a Morse function $f:\mathbb{S}^2\rightarrow \mathbb{R}$ such that it has two maximal points, one minumun and one saddle at $c$. Notice that $f^{-1}(-\infty,c)$ is topologically a disk.
...

4
votes

2
answers

262
views

### Dismissing pseudoholomorphic curves in embedded contact homology

In the papers
The periodic Floer homology of a Dehn twist,
Rounding corners of polygons and the embedded contact homology of $T^3$,
and Combinatorial embedded contact homology for toric contact ...

3
votes

0
answers

142
views

### Homological stability of Chow varieties

Given a connected component $C$ of the degree $d$ Chow variety of $r$ cycles on $C_{d,r}(X)$ ($X$ is smooth projective variety over $\mathbb{C}$), let $C'$ be another connected component of $C_{d',r}(...

0
votes

0
answers

29
views

### Smoothness of an extension of the length function of gradient trajectories

Let $M$ be a closed manifold, $(f,g)$ a Morse-Smale pair, and $\varphi:\mathbb{R}\times M\to M$ the flow generated by $-\mathrm{grad}_gf$. Let $c_0,c_1$ and $c_2$ be critical points of $f$ with $l_i:=...

3
votes

0
answers

137
views

### Some clarifications on the PSS isomorphism in Hamiltonian Floer cohomology

I'm looking for some help in understanding the PSS isomorphism map in the context of Hamiltonian Floer cohomology and Morse cohomology with universal Novikov coefficients $\Lambda_{\omega}$ (à la ...

1
vote

0
answers

171
views

### Existence of Morse function on suspension

Let $X$ be a smooth simply connected compact manifold of dimension $n$ with boundary. Let $Y$ be a smooth compact manifold of dimension $n+1$ without boundary such that $H_{i+1}(Y)=H_{i}(X)$(reduced ...

1
vote

0
answers

167
views

### Questions related to Morse theory

I asked this question long back but could not get a satisfying answer. Starting with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded it inside $R^{2n}$. Now take a tubular neighborhood ...

13
votes

1
answer

914
views

### Artin vanishing for Stein manifolds and restriction maps

In the setting of complex Stein manifolds $X$ of complex dimension $d$, the theorem of Andreotti--Frankel implies the vanishing of the singular cohomology group $H^i(X,\mathbb Z)=0$ for $i>d$. With ...

2
votes

0
answers

67
views

### Existence of a proper Morse function

I started with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded (local embedding) it inside $R^{2n}$. Now take a regular neighbourhood $U$ of $X$ in $R^{2n}$ which has the same homotopy ...

1
vote

1
answer

80
views

### Morse approximation with bounded number of critical points

Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...

10
votes

0
answers

138
views

### A minimal $\mathbb Z/2$-invariant Morse function on $U(n)$

Consider the group $U(2n)$ of unitary matrices. This has two standard and important homomorphic involutions. The most famous one $A \mapsto \overline A$ is complex conjugation, with fixed-set $O(2n)$, ...

5
votes

0
answers

231
views

### Existence of an affine variety with homotopy type of suspension of another affine variety

Let $X$ be an affine variety. My question is does there exist another affine variety with the homotopy type of the suspension of $X$?

8
votes

1
answer

143
views

### Analog of Cerf theory in PL

Is there an analog of Cerf theory in PL?
More specifically, given two handle decompositions of a PL (relative) cobordism $W$, is it always possible to go from one handle decomposition to the other via ...

8
votes

2
answers

621
views

### Bialynicki-Birula decomposition for real analytic varieties

Let $X$ be a smooth complex algebraic variety endowed with a $\mathbb{C}^*$ action. We assume also to have an antiholomorphic involution $\sigma$ over $X$ such that it anticommutes with the action ...

1
vote

0
answers

159
views

### Circle-valued Morse function and minimal genus

I think the following two statements are true, and most likely are in the literature. If so, could someone point me to some references? If not, counterexamples?
Let $Y$ be a closed oriented connected ...

2
votes

0
answers

147
views

### Genericity of an induced projection map

I am cross-posting a question asked on Math Stackexchange that has not been answered, in which I am still interested in.
Let $X,Y$ be smooth manifolds, $S'$ a submanifold of $Y$, and $f:\mathbb{R}\...

5
votes

1
answer

379
views

### Invariance of morse homology, doubt in proof in book "Morse Theory and Floer homology"

I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem.
Link to the statement of the theorem
...

2
votes

0
answers

107
views

### Existence of a certain Morse function

Let $\mathbb{B}^4=\{(x_1, x_2, x_3, x_4): x_1^2+x_2^2+x_3^2+x_4^2\leq 1\}$ be the closed unit 4-ball. Consider first the Morse function
$$
f(x_1,x_2,x_3,x_4)=\frac{1}{2}\left(x_1^2+x_2^2-x_3^2-x_4^2\...

1
vote

0
answers

105
views

### Relating the Morse index with the Maslov index

In the following paper https://arxiv.org/pdf/math/0408280.pdf there is created an isomorphism between the Floer Homology of an hamiltonian functional $H$ in the cotangent bundle and the the Morse ...

7
votes

1
answer

291
views

### How do the strong/weak Morse inequalities depend on the ring we are working over?

The Morse inequalities relate the number of critial points to global invariants of the manifold $M$.
The weak version states the following:
$$
\# \operatorname{Crit}_k f \ge \dim HM_k(M, \mathbb ...

3
votes

0
answers

111
views

### Upper bound on index of geodesics in terms of length

Let $(M, g)$ be a compact Riemannian manifold. Let $i(-)$ be the index of a geodesic and let $l(-)$ be the length. Is there an inequality of the form $i(\gamma) \leq C l(\gamma)$ for some $C>0$ ...

22
votes

3
answers

1k
views

### Next steps for a Morse theory enthusiast?

I don't know if this question is really appropriate for MO, but here goes: I quite like Morse theory and would like to know what further directions I can go in, but as a complete non-expert, I'm ...

6
votes

0
answers

104
views

### Equivariant Morse theory for non-compact Lie groups

Let $G$ be a Lie group acting properly on a smooth manifold $M$. The (non-equivariant) definition of a Morse function does not carry over to equivariant functions $M \rightarrow \mathbb{R}$ (where $\...

12
votes

0
answers

186
views

### Topologies of level sets of nearby functions

Suppose we have two smooth, real valued functions $\Phi$ and $\hat{\Phi}$ on a manifold $M$. Suppose $\Phi$ and $\Phi$ are close under some function space topology like $L^2$ or $L^\infty$. I am ...

4
votes

1
answer

133
views

### Existence of perfect Morse functions on Fermat surfaces $x^n+y^n+z^n+w^n=0$

It seems that whether a simply connected 4 manifold needs 1-handles and 3-handles is still an open question, see Existence of Morse functions on simply connected manifolds.
I am wondering if it is ...

2
votes

0
answers

108
views

### Variation of Morse Functions: a reference request

Suppose I have a manifold $X$ and a family of Morse functions $F_t:X \times \mathbb R \to \mathbb R$ on it where $t$ is the second parameter. So, if we fix $t$, we get a regular Morse function for ...

2
votes

0
answers

73
views

### $1$-parameter analytic functions are almost everywhere Morse

Let $I = [t_{0}, t_{1}]$ be a closed interval with $t_{0} < t_{1}$ and let $M$ be a compact real analytic $n$-dimensional manifold without boundary. Furthermore, let $f:I \times M \rightarrow \...

7
votes

0
answers

156
views

### Is a homotopy sphere with maximum Morse perfection actually diffeomorphic to a standard sphere?

The Morse perfection of a closed differentiable manifold $\Sigma^n$ is defined to be the largest integer $k$, such that there exists a smooth mapping
$$p:S^k\times\Sigma^n\rightarrow\mathbb{R}$$
where ...

5
votes

1
answer

269
views

### Index and length of closed geodesics

Consider the round metric on $S^n$. The geodesics are (multiples of) great circles, and one can verify that this metric is of Morse-Bott type. The Morse indices of the n-covered great circles are (if ...

3
votes

0
answers

109
views

### A generalization to Bott‘s theorem (from Milnor’s “Morse theory”)

This is Theorem 22.1 of Milnor‘s Morse theory:
Let $M$ be a complete Riemannian manifold, let $p,q\in M$ be so that the space $\Omega’$ of minimal geodesics joining $p$ to $q$ is a topological ...

9
votes

1
answer

230
views

### Smooth Morse function from Forman's discrete Morse function

Let $M$ be a smooth manifold and $K$ a triangulation of $M$, so $K$ is a regular CW-complex and in particular a simplicial complex. Assume that $M$ is compact so $K$ is finite. Let $f\colon K \to \...

3
votes

2
answers

204
views

### 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 ...

2
votes

0
answers

144
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 ...

3
votes

2
answers

184
views

### When is the Morse equivalence local?

Let $f:X \to \mathbb{R}$ be a Morse function on some compact submanifold $X \subset \mathbb{R}^n$, and assume that $p \in X$ is not a critical point of $f$. For some $\epsilon > 0$ let $D_\epsilon(...

2
votes

1
answer

147
views

### CW-structure induced by Morse function on Riemannian manifold [duplicate]

I have heard a statement in the following direction. Given a compact Riemannian manifold $M$ with a Morse function on it possibly satisfying some extra assumptions. Then this data induces a CW-...

9
votes

0
answers

302
views

### Connections between spectral geometry and critical point/Morse theory

I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...

6
votes

2
answers

431
views

### Sard's theorem and Cantor set

Sard's famous theorem asserts that
Theorem. The set of critical values of a smooth function from a manifold to another has Lebesgue measure $0$.
I am asking for the curiosity that is it possible to ...

3
votes

1
answer

160
views

### Finite-dimensional argument for Morse-Smale pairs?

Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $M$ (i.e. pairs $(f,g)$ where $g$ is a metric on $M$, $f$ is a Morse function on $M$, and the stable/...

2
votes

0
answers

110
views

### Multilinear Morse functions on the n-torus

Consider the $n$-dimensional Torus $T^n = \prod_{i=1}^n S^1$ as a subset of $\mathbb R^{2n} = \prod_{i=1}^n \mathbb R^2$ in the standard way.
Is it true that a generic $n$-multilinear functional on $\...

15
votes

3
answers

1k
views

### 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 ...

6
votes

0
answers

107
views

### 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 $f^{-1}(b) = \partial M$.
...

2
votes

0
answers

233
views

### 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 ...

5
votes

1
answer

217
views

### 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 ...