The morse-theory tag has no wiki summary.

**10**

votes

**2**answers

437 views

### Witten's proof of Morse inequalities, question on eigenvalues?

See here. I present Theorem 6 and Corollary 7 as follows.
Theorem 6. For large $s$, $\Delta_s$ and $H$ have the same number of eigenvalues in the interval $[0, s^{-2/5}]$.
Corollary 7. $\dim ...

**9**

votes

**1**answer

365 views

### Intuition behind the Morse inequalities?

Forgive me if this is sort of a vague question, but can someone supply me with their intuition behind the Morse inequalities?

**2**

votes

**1**answer

188 views

### Construction of appropriate Morse functions

I am interested in the properties of connectedness of level sets of Morse functions. Let $M$ a compact smooth $n$-manifold, and $1\leq k<n$. Is it possible to construct $k$ Morse functions ...

**6**

votes

**0**answers

152 views

### Smooth morse theory of Riemannian distance functions

Let $(M,g)$ be a Riemannian manifold, and $p\in M$. As $R>0$ increases, the topology of the ball $B(p,R)$ changes, but the changes happen only at a Lebesgue measure zero set of $R$. For instance, ...

**1**

vote

**0**answers

37 views

### Parametric Sard-Smale theorem - when is the generic set open?

I am learning Morse/Floer Theory and in my work on my master's thesis I want to apply the parametric Sard-Smale theorem. I.e. I consider a Banach bundle $\mathcal{L} \to \mathcal{M}\times \mathcal{G}$ ...

**1**

vote

**0**answers

41 views

### Perturbation of a Fredholm sections which preserves compactness of 0-set

I am learning Morse-Bott-Floer theory and found the following cool paper
http://de.arxiv.org/abs/1310.5080
by P. Albers and D Hein. In order to prove a cup-length estimate on the number of critical ...

**2**

votes

**0**answers

66 views

### Is there any topological information encoded by the zero locus of a complex Hessian?

On a Kahler manifold one can always find a function $K$ so that (up to some constant) the Kahler form $\omega$ can be written as $\omega = \partial \bar{\partial}K$ (sometimes, under conditions that I ...

**2**

votes

**1**answer

171 views

### Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine

This question was not answered on math.stackexchange.
Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a ...

**3**

votes

**0**answers

43 views

### Morse index in variational calculus

Let $f(t,x,v)$ be a smooth function on $\mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^n$, $\mathcal{C}_{x,y}$ be the set of smooth paths $c:[0,1]\to \mathbb{R}^n$ with $c(0)=x,c(1)=y$, $E(c)$ be a ...

**1**

vote

**0**answers

167 views

### How does a level set look like when the minimum point of a function degenerate?

I apologize in advance if this question is well-known. I would like to know an explicit example of a compact, connected manifold $M$ and a smooth function $f: M \to\mathbb{R}$ which satisfy the ...

**10**

votes

**1**answer

303 views

### Morse number of the Poincaré homology sphere

What is the Morse number of the Poincaré homology sphere? What about the stable Morse number?

**7**

votes

**2**answers

556 views

### Why not develop a Hamiltonian-based Morse theory?

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory ...

**0**

votes

**0**answers

60 views

### On Properties Of Lusternik-Schnirelmann Category

I have this part of proof from "Analysis and Topology in Nonlinear Differential Equations" book page 292:
I don't see how we find that $cat(\Omega)\leq cat( N_{\varepsilon}\cap ...

**1**

vote

**0**answers

71 views

### Global topological equivalence of Morse functions

Two Morse functions $f$ and $g$ are called topologicaly equivalent if there are diffeomorphism $h$ of the source and orientational-preserving diffeomorphism $k$ of the target such that $f=k\circ ...

**8**

votes

**1**answer

600 views

### What is the analogue of a Lefschetz Thimble for Morse-Bott critical components (sets of non-isolated critical points)?

Small pre-face: I did an applied math PhD in the UK, but the problem I ended up studying has important ramifications in pure math, specifically to do with the Gauss-Manin connection in the presence of ...

**5**

votes

**1**answer

123 views

### Does there always exist a sequence of handle moves between handle decompositions that does not increase index? (+ ref. request)

Reference request: Firstly, I'm looking for a proof of the following well-known result about handle decompositions:
($\ast$) Given two handle decompositions of a smooth $n$-manifold $M$, there ...

**1**

vote

**0**answers

105 views

### 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:
...

**6**

votes

**2**answers

268 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 ...

**3**

votes

**2**answers

235 views

### 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 ...

**3**

votes

**2**answers

354 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 ...

**5**

votes

**0**answers

119 views

### 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 ...

**11**

votes

**2**answers

412 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.

**3**

votes

**1**answer

110 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 ...

**1**

vote

**0**answers

84 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 ...

**6**

votes

**1**answer

201 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 ...

**7**

votes

**0**answers

241 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. ...

**1**

vote

**3**answers

141 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 ...

**1**

vote

**2**answers

212 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 ...

**10**

votes

**2**answers

632 views

### 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 ...

**17**

votes

**2**answers

417 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 ...

**5**

votes

**3**answers

320 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 ...

**8**

votes

**2**answers

310 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 ...

**2**

votes

**1**answer

374 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 ...

**2**

votes

**0**answers

154 views

### 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 ...

**0**

votes

**0**answers

101 views

### 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 ?
...

**1**

vote

**2**answers

270 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?

**2**

votes

**1**answer

403 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 ...

**3**

votes

**1**answer

361 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 ...

**1**

vote

**0**answers

129 views

### 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 ...

**7**

votes

**1**answer

341 views

### 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 ...

**3**

votes

**0**answers

158 views

### 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 ...

**1**

vote

**1**answer

221 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 ...

**6**

votes

**1**answer

470 views

### 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 ...

**7**

votes

**2**answers

328 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 ...

**8**

votes

**2**answers

759 views

### 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 ...

**4**

votes

**1**answer

188 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 ...

**6**

votes

**4**answers

555 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 ...

**8**

votes

**0**answers

231 views

### 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 $X$:
$X = \bigsqcup_{w\in W} C_w$ ...

**11**

votes

**2**answers

582 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 ...

**2**

votes

**1**answer

145 views

### 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 ...