# 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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### 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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198 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 ...

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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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197 views

### 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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303 views

### 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 (...

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### Normal form of functions $(x^2+y^2)^n+$ higher terms

By Morse lemma for any $C^{\infty}$ function $f$ on $\mathbb R^2$ with Taylor series $(0,0)$ starting with $x^2+y^2$ one can find local $C^{\infty}$ coordinates $(x',y')$ such that locally $f(x',y')=...

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### First order decidability of limit of gradient flow?

Let $f: \mathbb{R}^n\to\mathbb{R}$ be a polynomial function, and let $p$ be a critical point. Consider the ascending manifold $A_p$ consisting of all points whose limit under the gradient flow of $f$ ...

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### Cylindrical Decomposition vs Morse decomposition

Suppose I have a polynomial Morse function $f: \mathbb{R}^n \to \mathbb{R}$. Consider the ideal $I(\nabla f)$ generated by the partial derivatives $\partial_i f$, and assume that the real zero-set of ...

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### Constructing ($\infty, 1)$-category from Morse theory on a manifold

It is well known that a topological space is more or less the same as an $\infty$-groupoid. I'm wondering if there is an analogous construction which starts with a manifold endowed with Morse theory (...

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289 views

### Geometric Morse theory ( and its complex analogy)

In the literature are there some concept of geometric version of Morse or Picard Lefschets theory? That is the comparison of level sets as Riemannian submanifold not merely as ...

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### Could an inverse of (weak) Morse inequality exists in some special case?

Could an inverse Morse inequality hold in some sense? More precisely I wish the following result to be true:
Problem
$M$ is a smooth simply connected compact manifold, $dim(M)=n$, $f$ is a morse ...

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### Is there a finite dimensional subcomplex of the Morse-Bott complex that computes the cohomology of a manifold?

In the paper Morse-Bott theory and equivariant cohomology, Austin and Braam built the Morse-Bott (geometric) complex which computes the de Rham cohomology of the manifold $M$ (this paper can be found ...

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### What functions have the same persistence diagrams?

The panels in the figure below show, from left to right:
a piecewise affine function with support equal to a bounded interval and an indication of its superlevel filtration;
the corresponding ...

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### Change of Morse complex when changing the metric

Suppose $(M,g)$ is a closed Riemannian manifold and $f$ is a Morse function whose critical point are isolated points (not Morse-Bott). If the Morse-Smale condition is satisfied, then we can define a ...

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### On unstable manifold and incidence number of Novikov complex

Novikov complex is an extension of Morse theory to (closed) Morse 1-form $\omega$, which is not necessarily exact. Suppose for simplicity, $\omega$ is in the integer cohomology class and the universal ...

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### Dense set of functions on manifold with no local optima

Given a smooth manifold $M$ and another $S$, consider a smooth function $\psi: S \times M \rightarrow \mathbb{R}$, and use this to define $\psi_s:M\rightarrow \mathbb{R}$ by $\phi_s(p):= \psi(s,p)$.
...

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148 views

### Bott Periodicity: Morse and K-Theory [duplicate]

Is there an easy way to see the equivalence of the two statements of Bott periodicity via K-Theory and the original Morse Theory proof?
So, $$BU \times \mathbb{Z} \simeq \Omega^2BU$$ and
$$K(X)\...

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### Homotopy type of spaces of functions with few critical points

Given a closed manifold $M$ and an integer $k\geq 0$, let $G_k(M)$ denote the space of smooth functions $f:M\to\mathbb R$ with at most $k$ critical points.
To what extend has the topology of the ...

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### Null-homotopies in the space of framed functions on a surface

Let $M$ be a smooth manifold. Morse functions on $M$ are smooth functions $M \to \mathbb{R}$ with only very nice singularities.
Fact: The space of Morse functions on $M$ is not, in general, ...

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### Decompositions from torus actions and compactness of (sub-)level sets

Let $T=\mathbb{C}^{*}$ act on a smooth complex quasi-projective variety $X$. Assume that the limit point $\lim_{t\to 0}t\cdot x$ exists for every $x\in X$.
From the induced $U(1)$-action and its (...

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### Relation between Morse Theory and integration against Euler Characteristic

I'm studying Robert Ghrist papers on integration against Euler Characteristic. I am particularly interested in the relation with Morse Theory. I am trying to understand the proof of Theorem 25.1 (page ...