# 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.

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### 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
...

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### 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\...

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### 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 ...

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### Parametric general position theorem for foliations

The situation is the following: let $M$ be a manifold endowed with a smooth foliation $\mathcal{F}$ of codimension one (suppose orientable, transversely orientable) and let $F_t : S \rightarrow M$ be ...

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### 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 ...

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### 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 $C>0$ ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 ...

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### 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 ...

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### $1$-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 \...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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 ...

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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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### 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(...

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### 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-...

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### 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, ...

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### 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 ...

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### 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/...

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### 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 $\...

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### 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 ...

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

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### 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 ...

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### 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 ...

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### Topological invariants of a certain “stratified” manifold, with pieces of different “dimensions”

Disclaimer: I don't fully understand what I'm talking about in the question below. I'm still trying to figure out the right question to ask. Quotations and question marks in brackets mean that I'm not ...

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### Visualizing a normalized hyper-elliptic curve as a honey-flow on a multi-doughnut

Trying to visualize in 3D the normalized model $y^2=f(x)$ of a hyper-elliptic curve AND the degree two ramified cover $(X,Y)\mapsto X$, $\mathbb{C}\times \mathbb{C}\to \mathbb{C}$. Is this projection ...

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### The handlebody decomposition of S^1 bundles over surfaces?

What is the most natural handlebody decomposition of $F_g \times S^1$, if $F_g$ is an orientable closed surface of genus $g$?

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### In a manifold, $\angle xpy>\frac{\pi}{2}$, for $q$ on $px$ or $py$, $B_q(r)$ homeomorphic to $B_p(r)$?

Let M be an n-dimensional Riemannian manifold without boundary, with sectional curvature $\geqslant -1$. For a point $p\in M$, suppose there exist $l, \delta>0$, $x,y \in M$ with $d(p,x),d(p,y)>...

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### Realizing Morse functions on $S^2$ as height functions

Let $h: \mathbb{R}^3 \to \mathbb{R}$ be the usual height function (i.e. $h(x,y,z) = z$). One way that Morse functions on $S^2$ are often described is by picking an embedding $i: S^2 \to \mathbb{R}^3$ ...

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### The norm squared of a moment map

I am studying the paper by E. Lerman:
https://arxiv.org/abs/math/0410568
Let $(M,\sigma)$ be a connected symplectic manifold with a Hamiltonian action of a compact Lie group $G$, so that there exist ...

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### Splitting of chains of loop space

Evaluation at base point induces a splitting of homology of free loop space $LM$ of a compact manifold $M$, i.e. $H_*(LM)\cong H_*(M) \oplus H_*(LM, M)$. Can such splitting be realised on cellular ...

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### $A_{\infty}$ multiplications on Morse cochain complex

Can the higher order $A_{\infty}$ multiplications defined by Fukaya be made trivial(by perturbing gradient trees) when Morse cochain complex is isomorphic to Morse cohomology, in which case the cup ...

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### What mathematical background to i need in order to understand proofs of the h-cobordism theorem?

I am about to finish my undergraduate studies and I really enjoyed the topology and differential-geometry classes. I'd love to continue studying differentialtopology and i considered doing some ...

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### 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 ...

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### Generic properties of Jacobians of smooth functions

Let $f = (f_1, \dotso, f_n):\mathbb{R}^n \to \mathbb{R}^n$ be a smooth map and let $J$ be its Jacobian (determinant of the matrix with $ij$-th entry $\partial_i f_j$). We introduce the zero sets of $J$...

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### Elementary questions about Morse-Bott functions

Let $M$ be a manifold, $F$ be a Morse-Bott function, $c$ be a critical level, and $M_c$ be the corresponding critical submanifold. Let us assume that $M_c$ is connected, and the index of $M_c$ is ...

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### Morse theory for pairs of submanifolds of complementary dimension

If you have a closed monotone symplectic manifold $M$, then to any pair of closed monotone Lagrangian submanifolds $L_1$, $L_2$ you can associate (modulo some bubbling assumptions) a $\mathbb{Z}_N$-...

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### Critical points and high homotopy groups

Is there any known or interesting relation between critical points (possibly degenerate, or maybe only nondegenerate) of a function on a manifold and generators/relations of high homotopy groups? I ...

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### CW complex of iterated loop spaces

In Milnor's book Morse Theory, it is proved that the loop space $\Omega S^n$ of the n sphere has the homotopy type of a CW complex with one cell each in the dimensions 0, n-1, 2n-2, 3n-3, ... Or more ...

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### Deformation sublevel sets of functions which preserve boundary

I'm interested in proving the following fact, which seems to naturally arise from gradient flow deformations, but appears to be a bit tricky.
Consider a smooth family
$$f_s : M \to \mathbb{R}, \quad ...

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### Generalized Sard's lemma

Let $f: X \to \mathbb{R}$ be a $C^{1,1}$ (that is $C^1$ with Lipschitz differential) function on a manifold $X$. Suppose that $f$ is smooth at all points of a subset $C \subset \text{Crit}f$ of ...

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### Existence of connections in a vector bundle whose parallel transport preserves a function on a total space

Let $p:E \to M$ be a vector bundle over a smooth manifold $M$, $M\times 0$ be the image of its zero section of $p$, $\mathcal{X}(M)$ be the space of vector fields on $M$, and $\Gamma(E)$ be the space ...

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### Pullback of Morse form satisfies Palais Smale

Let $(\alpha,g)$ be a Morse-Smale pair on a closed smooth manifold $M$, i.e. $\alpha$ is a Morse form and $g$ a Riemannian metric on $M$ such that stable and unstable manifolds of the gradient vector ...

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### What's a completely computational/syntactical model for handle decompositions of manifolds?

Simplicial sets, CW complexes
Simplicial sets can be described completely algebraically, by specifying a family of sets, and maps between them satisfying certain relations. This description can be ...

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### What are good Morse Theory lecture notes and books?

Searching on the net I couldnt find any recent lecture/course notes on Morse Theory. I found an old set of notes (http://www.math.toronto.edu/mgualt/Morse%20Theory/mfp.pdf) by Mike Hutchings and these ...

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### Cap product for (co)homology from handle decompositions/Kirby diagrams

Since handle decompositions and Morse functions are intimately related, I'm imagining that a given explicit handle decomposition allows for an explicit description of the cellular complex and thus of (...