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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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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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278 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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262 views

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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1answer
337 views

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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25 views

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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65 views

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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25 views

critical points relevant to the lowest order non-perturbative correction

I am interested in the Hyperasymptotics of multidimensional integrals of the form $$\mathcal{I}(\lambda) = \int_{\mathbb{R}^n} dz_1 \wedge dz_2 \wedge \dotsi \wedge dz_n \, g(z_1,\dotsi,z_n) \, e^{\...
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79 views

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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177 views

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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84 views

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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3answers
692 views

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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129 views

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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1answer
149 views

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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1answer
249 views

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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About Hessians of functions

Say one is given a twice differentiable (at least) function $f : \mathbb{R}^n \rightarrow \mathbb{R}$. What are some conditions when one can put a lowerbound on the smallest eigenvalue of its ...
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86 views

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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1answer
293 views

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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1answer
271 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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1answer
101 views

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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1answer
122 views

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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1answer
80 views

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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70 views

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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48 views

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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145 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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211 views

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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141 views

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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0answers
76 views

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 (...
9
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1answer
296 views

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 ...
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84 views

Wrinkling smooth functions

I am interested in applying a result from the work by Eliashberg and Mishachev on wrinkling. Namely, in their first paper on wrinkling, they prove Theorem 1.6 B (Theorem 1.6 A is a non-parameterized ...
2
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130 views

A gradient trajectory connecting boundary components in an annulus

In a course of writing a paper I realised that I need a lemma below, and found a half page proof. I have not seen this lemma before and wonder if maybe someone here knows this lemma or can prove it in ...
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0answers
129 views

Understanding the proof of Morse Lemma using homotopy method

On pages no. 52-56 of "Lectures on Morse Homology" by Augustin Banyaga and David Hurtubise presented a proof of Morse Lemma using homotopy method AKA Palais proof using Moser "path method". There are ...
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1answer
790 views

Can the constant rank theorem for smooth manifolds be generalized to nonconstant rank?

The constant rank theorem says that if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest ...
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1answer
245 views

“Explicit” perturbations of Morse-Bott functions

There are explicit perturbations of Morse-Bott functions $f:X\to\mathbb{R}$ used in the literature (ex: Austin-Braam, Banyaga-Hurtubise, Bourgeois) to help solve various problems (ex: building Morse ...
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1answer
2k views

Arithmetic Morse theory?

Is there any analogue of Morse theory in Number theory? Naive idea arising in my head is that defining a Morse function on scheme and find etale cohomology using that function. Since I'm not an expert ...
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3answers
385 views

Is the bordism from disjoint union to connected sum universal for connected manifolds?

Let $M_1$ and $M_2$ be two oriented, connected, closed $n$-manifolds. It is known that the disjoint union $M_1 \sqcup M_2$ and the connected sum $M_1 \# M_2$ are cobordant, via a bordism $\Sigma_{M_1, ...
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0answers
133 views

Existence of a perfect discrete Morse function

Let $X$ denote a regular cell structure on a closed (orientable) $n$-manifold (If it helps, the cells are polytopal and the attaching maps are affine). Recall that a discrete Morse function on this ...
2
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1answer
240 views

Bijection of critical points on two manifolds

Suppose that $f$ and $g$ are two smooth functions defined on $R^n$. Assume that $(a-\epsilon, b+\epsilon)$ contains no critical point of $g$. Then $g^{-1}[a,b]$ it homomorphic to $g^{-1}(a)\times [a,b]...
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111 views

Morse-Bott functions without critical manifolds of index 1 and n-1

I am now reading the article of M.F.Atiyah "Convexity and commuting hamiltonians" and I can't understand lemma 2.1. which says that if $\varphi \colon M \to \mathbb R$ is a Morse-Bott function without ...
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1answer
204 views

Derivative of smooth function change sign infinitely on [0,1]? [closed]

Can the derivative $f^\prime$ of a smooth function $f\in C^\infty[0,1]$ change sign infinitely many times (or $f$ have infinitely many isolated critical points)? If yes, how about an analytic function ...
11
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1answer
329 views

Easiest example where pseudo-isotopy fails to be the same as isotopy?

This question concerns diffeomorphism of manifolds. Let $f: M \to M$ be a self-diffeomorphism. We will say that it is isotopic to the identity if there is a continuous one-parameter family of ...
4
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1answer
169 views

Finding a critical point on a product of two balls under some boundary conditions

Let $\mathbb {\overline B}^n\subset \mathbb R^n$ be a closed unit ball in and let $\mathbb B^k\subset \mathbb R^k$ be a open unit ball. Suppose $F$ is a smooth function on $\mathbb {\overline B}^n\...
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0answers
229 views

Is there a derived geometric interpretation of morse functions?

Given a smooth affine scheme $X = \mathbb{V}(g)$ over a field of characteristic 0, let $f:X \to \mathbb{A}^1$ be a morphism of schemes. Then, the critical locus is given by $\pi_*(dg \cap df)$ for $\...
6
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2answers
663 views

Can a Morse function define a unique structure on a closed manifold?

I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have ...
7
votes
2answers
394 views

Generalizations of the handle trading techniques

As Theorem 8.1 in "Lectures on the h-cobordism theorem (written by J.Milnor)" show, we can choose a handle decomposition of cobordism (satisfying some connectivity and dimensional assumptions) with no ...
3
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0answers
370 views

Number of critical points of a smooth function

Are there any examples of closed manifolds with the property that the minimal number of critical points (possibly degenerate) of a smooth function on this manifold is strictly bigger than the stable ...
3
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1answer
131 views

Morse function on slicing disk complement determines ribbon?

It is well-known that given a ribbon knot and the corresponding slicing disk in the 4-ball, the distance function (maybe squared) to the origin defines a Morse function in the complement of the ...
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22 views

Sufficient conditions for a conormal vector to be regular for an orbit stratification

Let a complex reductive group $G$ act on a $\mathbb{C}^{n}$ with finitely many orbits. Let $\mathcal{S}$ be the stratification of $\mathbb{C}^{n}$ according to these orbits. Let $(x,\xi) \in T_S^{*}\...