# 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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### 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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### 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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### 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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### 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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### Thom's gradient conjecture and analyticity

Suppose we have an analytic function $f: U \to {\mathbb R}$, where $U\subset {\mathbb R}^n$ is an open subset and $0 \in U$ is a critical point of $f$. Thom conjectured that if a trajectory $x(t)$ of ...
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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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### 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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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 ... 0answers 82 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 ... 2answers 324 views ### Regular CW complex arising from a Morse decomposition Suppose (M,g) is a Riemannian manifold equipped with a Morse function f: M \rightarrow \mathbb R. It's been shown that f gives rise to a CW decomposition homeomorphic to M under the generic ... 1answer 1k views ### How difficult is Morse theory on stacks? The title is a little tongue-in-cheek, since I have a very particular question, but I don't know how to condense it into a pithy title. If you have suggestions, let me know. Suppose I have a Lie ... 3answers 2k views ### Height function on 2-torus with only 3 critical points It is well-known that a Morse function on T^2 has at least 4 critical points, but also that there exist functions f\colon T^2\to\mathbb R with only 3 critical points (the least possible number ... 0answers 221 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 ... 0answers 107 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 ... 0answers 88 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 ... 0answers 153 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 (... 1answer 357 views ### Equivariant handle decompositions Suppose I have some smooth closed high-dimensional manifold M acted on smoothly by a finite group G. By a metric averaging procedure, we can equip M with a smooth Riemannian metric so that G ... 1answer 259 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 ... 1answer 153 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')=... 0answers 91 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 ... 1answer 365 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 (... 1answer 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 ... 1answer 109 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 ... 0answers 83 views ### 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 ... 1answer 140 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 ... 1answer 84 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 ... 0answers 72 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 ... 0answers 50 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). ... 0answers 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)\...
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 ...
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 (...