# Tagged Questions

In the area of mathematics known as differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. (Def: http://en.m.wikipedia.org/wiki/Morse_theory)

70 views

### Relation between tangent spaces of (un)stable manifolds in Morse theory

After asking this question about signs in the Morse complex, I realised that my confusion is really about how tangent spaces to different (un)stable manifolds are related. So suppose we have a Morse ...
57 views

### Explicit verification of signs in Morse complex

I'm trying to check by hand that the signs in the Morse complex, defined via choices of orientations on the unstable manifolds, lead to $\partial^2=0$. The books I've looked in seem to say either ...
21 views

83 views

### Doubt in definition of symmetric continuous function and norm in Kupka's paper

In this article "Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds" of I. Kupka has the following passage: For $H=l^{2}$ "Let $H^{*}$ be the dual of $H$. A base ...
145 views

### Potential proof for the Slice-Ribbon conjecture (may be wrong).

Let $f:(D^2,S^1)\to(D^4,S^3)$ be a smooth embedding (so called a slice disk), and we set $M:=f(D^2)$. Then, is the restriction map $C^{\infty}(D^4)\to C^{\infty}(M)$ open map with relative to $C^2$ ...
44 views

29 views

### Finding a domain of an integral curve of a vector field

Studying Morse theory, I am stuck on some problem. Let $M$ be a compact smooth manifold, and $f$ is a smooth real-valued function on $M$. Choose a Riemannian metric $g$ on $M$, let $X$ be the vector ...
100 views

### Self-indexing Morse function on a torus which is a height function

A Morse function $f: \Bbb T^2\to [0,2]$ is called self-indexing if $f^{-1}(n)$ is the set of critical points of index $n$. It is relatively easy to see that on any compact manifold, any Morse function ...
38 views

### Construct a smooth function on $T^2$ that has exactly three critical points

By the results in Morse theory, a smooth function on $T^2$ has at least three critical points, and at least one of them is degenerate. I'm asked to construct a smooth function that has exactly three ...
106 views

### Morse functions and connected sum

My question is closely related to this post but it is slightly different. Let $M_1$ and $M_2$ be two smooth closed $n$-manifolds such that there is a Morse function $f_i:M_i\rightarrow \mathbb R$ for ...
86 views

### Hints for an exercise on Morse theory

Exercise: Let $M$ be a $3$-dimensional smooth manifold with boundary $\partial M$ which is a surface of genus $g$. Moreover let $f:M\longrightarrow [0,1]$ be a Morse function with the following ...
39 views

### Does the Morse homology depend on the orientation?

Before asking my question I need to define some objects. I will follow the book "M. Audin, M.Damian - Morse theory and Floer homology", but the terminology is quite standard: Let $M$ be a smooth ...
52 views

### Problem in Morse Theory: critical points on the exotic sphere

I've taken a course on Morse theory a couple of years ago, but I have no idea about how to solve the following problem. Could you give some hints? Problem: Let $M$ be a smooth manifold homeomorphic ...
31 views

### Extending Morse-Smale pair from submanifolds?

The following proposition is extracted from Audin & Damian's Morse Theory and Floer Homology, Proposition 4.6.3: Let $(f,X)$ be a Morse-Smale pair on $V$ (a submanifold of $W$). Then there ...
44 views

### Equivalency of h-cobordisms

I'm reading lectures on the h-cobordism theorem by Milnor and I have a little problem understanding some basic points. I can't understand this theorem , not the theorem , not the proof. I appreciate ...
39 views

### Schwarz's Morse Homology

I have been struggling to understand the proof of proposition 2.9 of the Morse Homology book by Matthias Schwarz. I do understand the mathematical steps in the proof but I don't really see why we are ...
115 views

### Boundary of a manifold is a submanifold?

I was reading in the book Morse Theory and Floer Homology by Audin and Damian (translated in english) that the boundary of a manifold is not always a submanifold. I cannot see why that is true. Any ...
166 views

### Index of zero of a gradient vector field at a critical point

Let $M$ be a Riemannian manifold with a Morse function $f: M \to \mathbb{R}$. The zeroes of the gradient vector field of $f$ are the critical points of $f$. How do you show that a critical point ...
66 views

### Does the Morse-Bott index of a critical point depend on the choice of metric?

By the Morse lemma there exists a coordinate chart $(x_1,...,x_n)$ in the neighbourhood of a critical point $p$ of a Morse function $f : M^n \to \mathbb{R}$ such that \begin{equation*} f(x) = f(p) - \...
27 views

### Heegaard splitting via a Morse function - twisted union or not?

Let $M$ be a smooth, closed, connected, oriented 3-manifold and let $f: M \rightarrow \mathbb{R}$ be a self-indexing Morse function. Since $\frac{3}{2}$ is a regular value of $f$ it follows from Morse ...
135 views

### Proof of the last part of the Reeb theorem

I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. Suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both non-degenerate)...
113 views

### Manifold in Milnors Morse Theory

While reading "Morse Theory" by Milnor, I noticed that certain arguments would not work, if the considered manifolds have nonempty boundary. Example: Proof of 3.5 I could not find the definition ...
I am learning Morse homology and I have been thinking about the following observation. One way of doing, say finite-dimensional, Morse theory is by fixing a Morse function $f\in C^{\infty}(M)$, where $... 0answers 111 views ### Application of Morse Inequalities I am an undergraduate student interested in morse theory. I understand, that the morse inequalities provide an lower bound for the number of critical points morse functions on a manifold can take. One ... 2answers 192 views ### How does Morse theory on non-compact manifolds differ from compact manifolds? What is the Morse homology of a non-compact manifold? When is it, as in the compact case, isomorphic to singular homology of the underlying manifold? What other constructions can be identified with ... 2answers 186 views ### Preimage of singular points of smooth map between manifolds Given a smooth ($C^{\infty}$) map$\phi: V \rightarrow SU(n)$where$V$is a (finite dim, real) vector space (of potentially very large dimension) and$SU(n)$is the special unitary Lie group, what ... 1answer 113 views ### Morse lemma implies nondegenerate critical points are isolated It might be a stupid question. I don't quite know how we get the conclusion " Non-degenerate critical points are isolated" from Morse lemma. I know around each non-degenerate critical point we have a ... 1answer 94 views ### References on cancellation of critical points I'm not sure if I am using the terms correctly. Suppose you have, for example, a Morse function$f : S^2 \rightarrow \mathbb{R}$with 2 critical points of index 0, 1 critical point of index 1, and 1 ... 1answer 132 views ### Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine. Let$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 smooth function. The theorem of Sard gives us that$$f+\... 1answer 54 views ### Understanding Result on Non-Degenerate Critical Points I read a result in a collected works of Steven Smale and one result leapt out at me which I'm clearly not understanding. Stated: Theorem 1.1 (a): Suppose$J: M \to \mathbb{R}$is a$C^2$... 0answers 118 views ### Proof of Reeb's theorem without using Morse Lemma I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. That is, suppose we have an$n$-manifold$M$together with a smooth function$f$with exactly two critical points (both non-... 0answers 50 views ### Why gradient-like dynamical systems are special case of Morse-Smale systems? I'm studying Morse Theory and my question is exactly as stated in the above title. I can't see how a gradient-like dynamical system could be considered as a Morse-Smale system? Thanks in advance for ... 2answers 151 views ### Is Gauss curvature a Morse function? Given a Gauss map$\nu: M \rightarrow S^k$of a orientable, compact manifold, we define the shape operator$S_p = -d \nu: T_p M \rightarrow T_{\nu(p)} S^k$to be the negative differential. Define the ... 0answers 183 views ### How level sets look like when a critical point degenerate? I'd like to know an explicit example of a compact, connected manifold$M$and a smooth function$f\colon M \to \mathbb{R}$which satisfy the following properties: We denote by$m$the minimal value ... 0answers 70 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 ... 1answer 78 views ### Question About Local minimum I have this definition of a local minimum: We say that$u$is a local minimum of$f$is there exist a neighborhood$V$of$u$such that for all$v\in Vf(v)\geq f(u).$So we say that$u$is not a ... 0answers 42 views ### The space of minimal geodesics on$SU(2m)$In the proof of Bott periodicity for the unitary group in Milnor's Morse theory (Lemma 23.1, page 128), it is asserted that the space of minimal geodesics from$I$to$-I$in the special unitary group ... 1answer 80 views ### The main theorem of discrete Morse theory. I don't understand this part of the proof on page 16 of the following paper. http://www.maths.ed.ac.uk/~aar/papers/forman5.pdf 0answers 111 views ### Making a gradient-like vector field a gradient vector field via choosing a Riemannian metric. Let$\xi$be a vector field on manifold$M^n$which is a gradient-like vector field for a some Morse function$f$. Prove that there exists a Riemannian metric on$M$such that$\xi$is a gradient ... 0answers 46 views ### Sheaf-theoretic approach to Morse functions? It is known that one can define a smooth structure on a manifold using a sheaf-theoretic formulation via defining the algebra of the (a fortiori) smooth functions on it (which satisfies the usual ... 0answers 52 views ### Limits of trajectory of gradient flow in Hilbert space I have been studying about gradient flow in Hilbert space of a Morse function$f$. Specifically, let$X$be a Hilbert space and$f : X\to \mathbb R$be$C^3$function. The gradient flow here is ... 1answer 124 views ### On the Euler characteristic in Morse Theory Let$f:M\to \Bbb{R}$be a Morse function, where$M$is a$k$-manifold. The index$i_{f,p}$is defined to be the number of negative eigenvalues of the Hessian$H_f$at the critical point$p$. For ... 1answer 185 views ### Prove directly Morse lemma for real line$\mathbb{R}$The exercise 9 section 7 chapter 1 in Guillemin & Pollak state the next Prove directly Morse lemma for real line$\mathbb{R}$.(Hint:Use this elementary calculus lemma: for any function on$\...
We have defined: a Morse function on $X$ is a smooth function $f:X\rightarrow\mathbb R$ with only non-degenerate critical values. I tried to get some intuition about this, and found the section Basic ...