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21
votes
1answer
212 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 ...
4
votes
1answer
104 views

Bott-Samelson construction of a perfect Morse function on G/T

An undergraduate student of mine is interested in writing a "senior" thesis on the topology of Lie groups. Let $G$ be a simply connected compact Lie group and $T$ a maximal torus. One way of ...
3
votes
1answer
63 views

Convergence of local stable manifolds

This question is a kind of local version of a previous post (MO224171). Let $\ f_n \ $ be a sequence of Morse functions on $\mathbb{R}^d$, converging, in the $C^\infty$-topology, to a limit Morse ...
6
votes
1answer
94 views

Stable manifolds of a sequence of Morse functions

Let $\ f_n \ $ be a sequence of Morse functions on $\mathbb{R}^d$, adequately converging (in the $C^2$-topology, say) to a limit Morse function $\ f$: $$ f_n \to f \ .$$ At any critical point $\ p\ ...
4
votes
1answer
120 views

Is Hessian operator self-adjoint on infinite dimensional environment?

As we know the Hessian matrix is symmetric in a finite-dimensional environment. What about the Hessian operator $D^2F$ for a functional $F:H\rightarrow \mathbb{R}$, where $H$ is a Hilbert space and ...
8
votes
2answers
142 views

Banach manifold of paths with endpoints on submanifolds

Fix a Riemannian metric on a manifold $M$. Suppose that we fix two points $x,y \in M$. We start with the space $C^{\infty}_{\searrow}(x,y) = \left\{\gamma: \mathbb{R}\to ...
3
votes
0answers
61 views

If $M$ is a globally symmetric space, do we need it to be compact to prove that all the critical submanifolds in $\Lambda M$ are nondegenerate?

In "The free loop space of globally symmetric spaces" by Ziller, he proves the following theorem: Theorem 2. For a globally symmetric space the critical submanifolds in $\Lambda M$ are all ...
5
votes
2answers
227 views

Transversality in Morse theory, linear algebra version

I am working on a product in Morse-Bott homology which has led me to the following considerations and unanswered question. I would be very grateful if anyone could help. Suppose $H:\mathbb{R}^n \to ...
10
votes
2answers
556 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 ...
14
votes
1answer
442 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?
3
votes
1answer
205 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
0answers
185 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
0answers
57 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
0answers
52 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
0answers
71 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 ...
3
votes
1answer
216 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
0answers
54 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
0answers
177 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
1answer
342 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
2answers
591 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 ...
1
vote
0answers
75 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 ...
9
votes
1answer
633 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
1answer
132 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
0answers
114 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
2answers
292 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
2answers
248 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
2answers
379 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
0answers
127 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 ...
12
votes
2answers
489 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
1answer
132 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
0answers
90 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
1answer
205 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
0answers
244 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
3answers
166 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
2answers
213 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 ...
11
votes
2answers
659 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
2answers
442 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
3answers
355 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
2answers
327 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
1answer
384 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
0answers
159 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
0answers
102 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
2answers
278 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
1answer
413 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
1answer
384 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
0answers
133 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
1answer
352 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
0answers
164 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
1answer
229 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 ...
7
votes
2answers
626 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 ...