The morse-theory tag has no wiki summary.

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### Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here:
...

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### Is there a holomorphic Morse-Bott lemma?

It asks for a generalization of the question in the post
Normal form for a holomorphic Morse function
Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...

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253 views

### Gradient of the energy functional in $H^{1,2}$-norm

I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy ...

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509 views

### Induced maps in Morse Homology

Let $M,N$ be closed manifolds. Given a differentiable map $f:M\rightarrow N$, I am interested in computing $f_k:H_k(M)\rightarrow H_k(N)$, in Morse Homology. This problems seems difficult, and the ...

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225 views

### Using Discrete Morse Theory to represent hom classes

In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical ...

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### Morse theory Vs degree theory

I asked this question on http://math.stackexchange.com but no unswers!
I have this paragraph from K.C. Chang Infinite dimensional Morse theory
In comparison with degree theory, which has proved ...

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286 views

### Is a Morse function always the height function of some embedding? [closed]

Pictures in introductory texts to Morse theory are often drawn as to interpret a Morse function as a height function. Typically, an embedding of a torus into $\mathbb{R}^3$ is drawn, and the Morse ...

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196 views

### cell decomposition of real homogeneous hypersurfaces

Let $f(x_1,\ldots,x_n)\in\mathbf{R}[x_1,\ldots,x_n]$ be a homogeneous polynomial
and consider the real hypersurface $H=\{(x_1,\ldots,x_n)\in\mathbf{R}^n:f(x_1,\ldots,x_n)=1\}$. Assume to simplify ...

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### Parametrized cancelations in stable Morse theory

Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...

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278 views

### Equivariant version of Morse theory

Is there some variant on Morse or Morse-Bott theory yielding equivariant (co)homology instead of singular homology?
Any reference/idea would be greatly appreciated.
Crossposted on StackExchange.

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83 views

### A reference for an equivariant Morse Lemma

Does anybody knows a reference for the following statement?
Let $S^1$ acts on $\mathbb{C}^n$ in the usual (diagonal) way and $f:\mathbb{C}^n\to\mathbb{R}$ a smooth $S^1$-invariant function defined in ...

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69 views

### restriction to the boundary in Morse theory

Given a compact manifold with boundary $(M,\partial M)$. There is a natural pullback map
$$ H^*(M) \to H^*(\partial M) $$
I'm wondering if there is a reference that:
1) constructs this map in ...

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**1**answer

375 views

### Contractibility of a configuration space

For a topological space $X$ and a positive integer $k\in \mathbb{N}_{>0}$ let $F_k(X):= \{ (x_1,\ldots,x_k)\in X^k |x_i\neq x_j \text{ for } i\neq j \}$ be its $k$-configuration space.
Let ...

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229 views

### diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...

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196 views

### Question on Morse inequalities

I want to understand why: From K.C Chang's book "Infinite Dimensional
Morse Theory and Multiple Solution Problems":
if i have
then $(4.1)$ is formal : it means that
EDIT1: $(4.1)$ tel us that ...

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74 views

### Stable/Unstable Manifold Theorem for a Morse-Bott function

Good night, anyone know of any reference where I can find the proof of the Stable/Unstable Manifold Theorem for a Morse-Bott function. I'm interested in the dimensions of the stable and unstable ...

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477 views

### Morse theory and adiabatic limits

Let's start with a product manifold $M\times N$, with a product Riemannian metric $g_M\oplus g_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and ...

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351 views

### Has Witten's perturbation on de Rham complex been studied on other elliptic complexes?

In his famous work, Supersymmetry and Morse theory, Witten perturbs de Rham complex by perturbing the exterior derivative
$$d_h=e^{-ht}de^{ht}.$$
And he proves Morse inequality using some spectral ...

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### Two-convexity â‡’ Lefschetz?

Assume that
$\Omega$ is an open simply connected set in $\mathbb R^n$
(two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$.
Is it true that any component of ...

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215 views

### Is the set of focal points of a submanifold on a normal geodesic discrete?

Let $M$ be a complete riemannian manifold, $L$ a smooth submanifold of $M$ and $\gamma$ a geodesic with $\gamma'(0)$ normal to $L$. A focal point of $L$ is a critical value of the normal exponential ...

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489 views

### Self-indexing Morse functions on non-compact manifolds

Hi,
given a compact manifold M we can always alter a given Morse function f to a self-indexing one (i.e., one where every critical point c has $f(c) = \operatorname{index}(c)$) - a proof of this may ...

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360 views

### Question about a Lefschetz hyperplane type theorem

Let $X$ be a simply connected projective manifold of dimension $n$ over $\mathbb{C}$ and $D = \cup D_i$ be a divisor with normal crossings such that its all components $D_i$ are smooth and ample.
I ...

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277 views

### Existence of Morse functions on simply connected manifolds

Is it true that any simply connected closed manifold possesses a Morse functions that does not have critical points of index one?
If the dimension is at least 5, this is a consequence of the results ...

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172 views

### Measuring almost-critical values of smooth functions.

Consider a compact sub-manifold $X \subset \mathbb{R}^n$ of Euclidean space and let $f:X \to \mathbb{R}$ be any smooth function. Recall that $x \in X$ is a critical point of $f$ if the gradient ...

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### Microlocal geometry - A theorem of Verdier

(1) In "Geometrie Microlocale", Verdier states the following theorem.
Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$.
Then for $\ell$ a linear form on $E$, we have a ...

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1k views

### Conjugate points in Lie groups with left-invariant metrics

For any Lie group $G$ there exist many left-invariant Riemannian metrics, namely, one just takes any inner product on the tangent space at the identity $T_eG$ and then left translate it to the other ...

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### Is there a Morse theory proof of the Bruhat decomposition?

Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup_{w \in W} BwB$ of cells given by double ...

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### complex Morse function on a four-manifold

If we have a complex Morse function on a complex four-manifold, $f: X\to \mathbb{C} $, can we tell from the function how the genus of inverse images $f^{-1}(z)$ (for regular values) may change? under ...

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### Application of Morse theory to second order systems

Hello
I'm looking for some applications of Morse theory to second order differential system,( or boundary value problems )
Someone can help me with a pdf or a book which has these applications ?
...

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245 views

### What does it mean that homotopy is generic?

Due to Cerf, there's exist a certain homotopy between two Morse functions. It is said that this homotopy is "generic". What is a precise definition of the property to be "generic" in this case?

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### Yang-Mills and Chern-Simons functionals as Morse functions

Can the Yang-Mills or Chern-Simons action functionals be considered as [possibly perfect] Morse functions? I assume we would be in an equivariant scenario due to considering the configuration spaces ...

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319 views

### Is the space of gradient-like vector fields contractible?

Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function.
Question: Is the space $GVect(M,f)$ of all gradient-like vector-fields for $f$ contractible ...

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### Detecting Non-Transversality

Suppose $f \colon \mathbb{R}^n \to \mathbb{R}$ is Morse and has finitely many critical points. Is there an algorithm for determining whether there exists a saddle-saddle connection (an orbit of grad ...

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### A description of cellular boundary maps in terms of a Morse function

I'm writing a paper on classical Morse Theory and I'm interested in applying Morse functions to the computation of homology groups of a compact manifold $M$. The standard way in which this is done is ...

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### Perturbation of Morse functions at critical points leaving stable manifolds invariant

Let $f$ be a given Morse-Smale function $f$ on $\mathbb R^n$ with finite many critical points and sufficient growth at $\infty$ like $\langle x, f(x)\rangle \geq |x|$ (cp. MO120858).
Is there a way ...

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218 views

### Integration by parts wrt. a Morse function on its basin of attraction

Let $f:\mathbb R^n\to\mathbb R$ be a Morse function with uniform nondegenerate Hessian at critical points, i.e. for some $\delta>0$
$$
\forall x \in \{\nabla f=0\}\;\forall \xi\in\mathbb R^n: \quad ...

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### When is the determinant a Morse function?

This might be ridiculously obvious, but...
For each $n \in \mathbb{N}$, let $M_n$ denote the manifold of $n \times n$ matrices with real entries. It is well known that the $n$-dimensional determinant ...

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211 views

### Morse lemma with least amount of regularity.

I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ...

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### Notes for Bott's 1963 lectures on Morse theory

Would anybody happen to know where I could obtain a scanned version of
Lectures on Morse theory - [revised and expanded version of notes of lectures delivered at Professor R. Bott's topology seminar ...

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175 views

### Concerning strata in $C^\infty(M)$

The Morse functions are dense in $C^\infty(M)$, and you can ask if a 1-parameter family of smooth functions between two given Morse functions will be a homotopy through Morse functions. Well, Cerf ...

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### Intersection of plus/minus cells in Bialynicki-Birula decomposition

Let $X$ be a projective variety endowed with an algebraic $\mathbb{C}^*$-action. Assume the fixed point set $W$ is discrete. Then we have two cell decompositions of $X$:
$X = \bigsqcup_{w\in W} C_w$ ...

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527 views

### Discrete Morse function from smooth one

Suppose I have a smooth manifold $M$ and a smooth Morse function $f$ on $M$. Is there a standard way to replace $M$ with a cell complex and $f$ with a discrete Morse function such the resulting ...

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### Invariance group of Morse charts

Suppose I have a smooth function $\varphi$ that vanishes at $p$ and has a positive definite Hessian at that point (suppose that we are on a smooth manifold of dimension $M$). Then the Morse lemma ...

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### Any algebraic substitute for Morse theory (and homology) in arbitrary characteristic?

As far as I know, Morse theory yields much information on the topology of smooth manifolds; in particular, it can be used to prove Artin's vanishing (that the singular cohomology of smooth complex ...

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### Is there a discrete Cerf theory?

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the ...

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### Operations via Morse Theory

I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)homology ...

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### 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 $G$ ...

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296 views

### Special Morse function on a Riemann surface

Let $f: S \to \Bbb R$ be a Morse function on a Riemann surface. Let $x_0$ be a saddle point of $f$. Since $x_0$ is a critical point of $f$, it makes sense to talk about the bilinear forms ...

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250 views

### Classifying smooth embeddings which yield Morse functions

Let $\mu:M \to \mathbb{R}$ be a fixed surjective smooth function on a smooth manifold $M$. Let $N$ be a smooth compact manifold that embeds smoothly into $M$ via $\iota:N \to M$.
What conditions ...

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### Looking for the perfect morse

A Morse function $f(x)\colon \mathbb{R}^n\to \mathbb{R}$ is a smooth function s.t. all singular points are non-degenerate. A theorem of Sard implies that for any smooth $f(x)$ and almost all $a\in ...