Questions tagged [morse-theory]

For questions about Morse theory, which is a branch of differential topology to analyze topology of manifolds by studying differentiable functions on them.

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243 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 $C^2$ function,...
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0answers
553 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-...
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120 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 ...
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2answers
318 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 ...
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0answers
489 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 ...
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150 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 ...
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1answer
115 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 $v\in V$ $f(v)\geq f(u).$ So we say that $u$ is not a ...
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159 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 ...
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1answer
117 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
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1answer
252 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 ...
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101 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 ...
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290 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 ...
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1answer
712 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 ...
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1answer
450 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 $\...
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189 views

Intuition about Morse functions

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 ...
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1answer
109 views

A question related to the topology of the level sets of a particular type of smooth functions $f:\mathbb{R}^2\to \mathbb{R}$.

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a smooth function without critical points; i.e. such that $\nabla f(x)\neq (0,0)$, for all $x\in\mathbb{R}^2$. Is it true or false that all the level curves of $f$ ...
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1answer
765 views

Manifold allowing function with two critical points is sphere

The only closed manifolds which allow a function with two (maybe degenerate) critical points are spheres. In dimension 2 it is quite easy to prove, but what is about higher dimensions?
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1answer
158 views

Morse's polynomial and Poincaré's polynomial equality

Suppose $M$ is a compact smooth manifold with Morse's polynomial $\mathcal{M}(t)$ and Poincaré's polynomial $\mathcal{P}(t)$ satisfying $\mathcal{M}(t)=\mathcal{P}(t)$ for any coefficient field $\...
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167 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: http://www.mtm.ufsc.br/~...
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1answer
147 views

Basic idea for finding critical point via Morse theory

Please what is the basic idea for finding critical point via Morse theory and critical groups? Thank you
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2answers
538 views

Morse theory Vs degree theory

I have this paragraph from K.C. Chang Infinite dimensional Morse theory In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in estimating ...
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1answer
179 views

Morse functions dense in a trigonometric polynomial space

Let $V$ be the vector space of trigonometric polynomials of degree $\le D$ on the flat torus $\mathbb T^n$. That is, $$V=\operatorname{Span}\left\{\cos (2\pi \lambda \cdot x), \sin (2\pi \lambda \cdot ...
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1answer
187 views

Proof of the h-cobordism theorem

I am currently learning Morse theory. Having read Milnors "Morse Theory" I am now studying his "Lectures on the h-cobordism theorem". I have also read parts of "Lectures on Morse Homology" by Banyaga ...
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2answers
672 views

Cup product in Morse cohomology

Dualizing the Morse complex, we obtain the Morse cohomology, which is isomorphic to the usual singular cohomology and thus admits a cup product. Does anybody know how this cup product would look like ...
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2answers
151 views

Family of Morse functions made constant

I'm looking for a proof of the following theorem: Let $f_t$ be a family of real-valued Morse functions defined on a smooth compact manifold $M$, and where $t$ is in $[0,1]$ (So for all value of $t$, $...
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110 views

Converting a space to a simplicial set

Given a compact manifold and a Morse function, we obtain a convenient cellular decomposition. But suppose I have a (finite dimensional, possibly singular) space $X$ that is not a manifold and I wish ...
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1answer
231 views

How an empty set is collapsed to a point?

In the original book of Conley Index Theory: Isolated Invariant Sets and the Morse Index chp3.3, p6, Charles Conley mentioned that ...
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a question related with morse theory [duplicate]

Show that there exists no smooth function $f:\mathbb{R}^2→\mathbb{R}$,such that $f(x,y)\geq 0$ for any $(x,y)\in\mathbb{R}^2$, with exactly two critical points$(x_1,y_1)\in\mathbb{R}^2$, $(x_2,y_2)\in\...
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4answers
395 views

Prove no existing a smooth function satisfying … related to Morse Theory

i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ ...
4
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1answer
521 views

Can a proper Morse function $\mathbb{R}\to\mathbb{R}$ have infinitely many critical points?

Depending on interpretation, there may be an assumption missing from Exercise 6.1.4(a) in Liviu I. Nicolaescu's Invitation to Morse Theory: Suppose $f : \mathbb{R} → \mathbb{R}$ is a proper Morse ...
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1answer
674 views

Looking for a good book on Morse-Bott functions.

I am looking for a book to study for the first time Morse-Bott functions. Does anyone know one that is easy to follow and detailed? If there is one connecting this subject with symplectic geometry, it ...
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2answers
2k views

Introductory references for Morse theory

In view of my master's thesis, I have to learn the basics of Morse theory (defining the morse complex, showing that Morse homology is isomorphic to singular homology...) I have been told that Milnor'...
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213 views

The Morse complex of a manifold with boundary

For a smooth manifold with boundary $M$ and $\partial M = V_+ \cup V_-$ two disjoint sets of boundary components, one usually defines the Morse complex of $M$ using a Morse-Smale pair $(f,X)$ such ...
2
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1answer
167 views

Using a Morse function to find the number of points of each index

During my Algebraic Topology course, we began to talk a bit about Morse functions. I was a bit lost on the topic, and my notes are lacking, so coming across this problem, I'm not really sure what to ...
2
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1answer
96 views

Number of cells in a minimal cell structure for a non-simply connected manifold?

I have obtained a cell structure of a connected (but not simply connected) manifold using Morse theory. Is there any way for me to know whether this cell structure is minimal?
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1answer
329 views

Why is $[\widetilde{v},\widetilde{w}]_p(f)=0$ when $f$ has a critical point at $p$?

Let $M$ be a smooth manifold and $f$ a smooth function $M\to\mathbb{R}$. Let $p$ be a critical point of $f$. We define the Hessian of $f$ at $p$ to be the symmetric bilinear functional $f_{**}$ on $...
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2answers
66 views

Sphere construction from cells

http://en.wikipedia.org/wiki/Morse_theory In the link, there is this statement 'The number of critical points of index $\gamma$ of $f : M → \mathbb{R}$ is equal to the number of $γ$ cells in the CW ...
3
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1answer
5k views

Definition of a Morse function.

http://en.wikipedia.org/wiki/Morse_theory Suppose $M$ is a manifold. Morse function $f:M \rightarrow \mathbb{R}$ is defined as a function in which all its critical points are non-degenerate. In the ...
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230 views

Sequence of critical values has no cluster point (Milnor, Morse Theory)

The following Claim is used in the proof of Theorem 3.5 in John Milnor's "Morse Theory": Claim: Let $f: M \rightarrow \mathbb{R}$ be a differentiable function on a manifold $M$ with no degenerate ...
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69 views

Question about Morse index

in general the Morse index of a critical point $p$ is the suprimum of the dimensions of sub spaces where $f''(p)$ is negative definite but whene $f''(p)=I-T$ ($f''(p)$ is a compact perturbation of ...
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2answers
442 views

History of Morse theory.

How can I get good references which give many information about history of Morse theory? Now I am interested in how and who found that Hessian have a lot of data. Thank you for your helping!!
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1answer
110 views

Symplectic submanifolds in $\mathbb{R}^{4}$

Which symplectic submanifolds can be realized in $\mathbb{R}^{4}$? It easy to show that such submanifolds aren't compact. So, they are spheres with some handles and holes. Which relations between the ...
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1answer
47 views

Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ Whene $t=-...
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0answers
113 views

handle moves: proof

In several 4-manifold textbooks, when handle moves (creation, cancellation, sliding) are discussed, they are explained using very helpful drawings. However, I would like to know if there is a ...
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0answers
153 views

Proof that I can always get a height function that is Morse.

So a height function $h(x_{1},...,x_{m})=x_{k}$ for mfld $M^{m}\subset \mathbb{R}^m$. I proved that Morse functions are dense in $C^{\infty}(M,\mathbb{R})$. So I can approximate h by Morse functions, ...
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1answer
86 views

Question about Morse inequality

Helli , i have question i Morse inequality why $$\sum_{q\geq0} M_q(a,b) t^q =\sum_{q\geq 0}\beta_q(a,b)t^q+(1+t)Q(t),$$ where $Q(t)$ is a polynomial with nonnegative integer coefficients implise ...
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1answer
2k views

Normal Bundle of a Manifold

I was reading "Morse Theory" by J.Milnor and at page number 32 there is remark "It is not difficult that N is an n-dimensional manifold differentiably embedded in $\mathbb{R}^{2n}$ ( N is the total ...
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1answer
502 views

Application of Lefschetz duality to prove Lefschetz hyperplane theorem

I'm trying to understand the proof of the Lefschetz hyperplane theorem in Milnor's book "Morse Theory", page 41 but I can't understand his use of Lefschetz duality. At this point it has been proven ...
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830 views

Morse functions are dense in $C^{\infty}(M,\mathbb{R})$ questions.

Hi here is a proof inspired from the reference below. Feel free to get very technical with your comments so that at the end I understand it well. I am more concerned about the questions I added below,...
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1answer
679 views

(Morse) non-degenerate iff transverse to the zero section

So Morse $f:M\to \mathbb{R}$ has nondegenerate critical point p iff $df|_{p}\pitchfork 0$-section. Attempt nondegenarate p iff Hessian has full rank at p iff $Im(D|_{p}df)=T_{df(p)}^{*}M\...

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