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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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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0answers
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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2answers
78 views

Hessian of Morse function on $S^{n}$ mistake

I am trying to get that $f(x_{0},...,x_{n+1})=x_{n+1}$ has $index_{(0,...,0,1)}=n$ Can you find my mistake or post a partial solution? My attempt I evaluate df using inverse of stereographic proj. (...
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1answer
357 views

Lefschetz Hyperplane Theorem: reference request

I've just begun working on my bachelor thesis on the "Lefschetz Theorem on Hyperplane Sections" (see for example http://en.wikipedia.org/wiki/Lefschetz_hyperplane_theorem). The goal of the thesis is ...
2
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1answer
247 views

Proof of Morse inequalities?

Can you think of a proof of Morse inequalities without using the Morse cohom. $\cong$ sing.cohom or Witten's approach? http://en.wikipedia.org/wiki/Morse_theory#The_Morse_inequalities Any references ...
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48 views

Morse cohomol. $\cong$ De Rham cohomol.

Are you aware of any short proofs for that fact (references)? And also for the fact that Morse cohomol. isomorphic to Singular cohomol. Also, a silly question: In the following thesis, he proves ...
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99 views

Morse-Smale Complex, boundary on the number of segments by the number of critical points.

I am looking for a known upper bound on the number of monotone regions of a Morse function by the number of its critical points in the interior of the manifold and on its boundary. Here I try to ...
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1answer
248 views

Implicit function theorem and derivative (proof of splitting lemma)

I have this theorem with a part of the proof: $\quad$ Let $V$ be a Hilbert space, $U$ an open neighborhood of $u\in V$, and let $\varphi\in C^2(U,\mathbf R)$. Define implicity the linear operator $...
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1answer
204 views

Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...
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94 views

Question on Theorem 5.1 from K.C Chang's book

I have this theorem and in the first part of the proof , I don't understand why $d_z f(\theta_1+\theta_2)=\theta_1$ ? Theorem $5.1$. Suppose that $U$ is a neighborhood of $\theta$ in a Hilbert ...
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1answer
339 views

About existence of Morse functions

Let's consider 4-manifold $M$, $\partial M = \partial M_1 + \partial M_2 = S^1 \times S^2 + \mathbb{RP}^3$. Is it true that there exist a Morse function $$f\colon M^4 \to [0,1],\quad f^{-1}(0) = \...
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1answer
508 views

Reconstructing a manifold from critical points

I am teaching theoretical calculus this semester, and on the last discussion section we were discussing critical points of functions. I explained the idea of Morse theory, and a student of mine asked ...
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1answer
307 views

Is 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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1answer
608 views

Morse index and degeneracy

The function is as follows: $$f(x,y,z) = e^x(xy-y^2-z^2)$$ I have found the critical points to be $(0,0,0)$ and $(-2,1,0)$. The question asks to determine the morse index of the points and the ...
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2answers
512 views

Why is the moduli space of gradient flow lines $\widehat{\mathcal M}(p,q) = \mathcal M(p,q) / \mathbb R$ a smooth manifold?

Let $(M,g)$ be a compact smooth orientable riemannian manifold, and let $f: M \to \mathbb R$ be a Morse function. All functions here are assumed to be smooth. We will be considering the integral ...
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0answers
137 views

Uniqueness of the “asymptotic limit” of a sequence of gradient flow lines

Let $(M,g)$ be a compact smooth orientable riemannian manifold, and let $f: M \to \mathbb R$ be a Morse function. All functions here are assumed to be smooth. We will be considering the integral ...
2
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1answer
235 views

Morse lemma question

Consider the statement of the Morse lemma: Let $b$ be a non-degenerate critical point of $f:M \to \mathbb R$. Then there exists a chart $(x_1, ..., x_n)$ in a neighborhood $U$ of $b$ such that $\...
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196 views

how to obtain a generalized Morse function out of a fiber bundle?

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth ...
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3answers
134 views

What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?

While for $n=2$ it is pretty clear that the contours of a non-constant $f:\mathbb R^n\to\mathbb R$ are either extrema (and therefore points) or (the union of) 1-dimensional isolines, for $n=3$ I am ...
2
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1answer
481 views

Parametrization of level sets of a smooth function

Let $H:\mathbb{R}^2\rightarrow\mathbb{R}$ be given by $H(q,p)=p^2/2+3q^2/2$ (single-well potential). This function has a critical point at $(0,0)$. Define $T:\mathbb{R}^+\rightarrow \mathbb{R}$ by, $...
27
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1answer
3k views

When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
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1answer
1k views

Background for reading Milnor's Morse Theory book

I wish to study the book 'Morse Theory' by J. Milnor, but I am not sure whether I have the necessary prerequisites. I know basic point set topology, real analysis (limits, continuity, differentiation, ...
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1answer
233 views

Fundamental theorem of Morse theory for $\Omega(S^n )$

Using the Fundamental theorem of Morse Theory we can prove that $\Omega(S^n)$ is homotopically equivalent to a CW complex with one cell each in dimensions $o,n-1,2(n-1), \cdots$ and so on. But how can ...
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48 views

Finite approximation of path space.

Let $M$ be a connected Riemannian manifold, $\Omega(M)=\Omega$ the path space and $E: \Omega \to \mathbb{R}$ the energy function. We can define $\Omega^c:=E^{-1}([0,c])$ and $\Omega(t_0, \dots, t_k)$ ...
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1answer
109 views

Path space of $S^n$

Suppose that $p,q$ are two non conjugate points on $S^n$ ($p \ne q,-p$). Then there are infinite geodesics $\gamma_0, \gamma_1, \cdots$ from $p$ to $q$. Let $\gamma_0$ denote the short great circle ...
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0answers
264 views

Non degenerate critical points.

Let $K$ be a compact subset of the Euclidean space $\mathbb{R}^n$; let $U$ be a neighborhood of $K$ and let $f:U \rightarrow \mathbb{R}$ be a smooth function such that all critical points of $f$ in $K$...
3
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1answer
176 views

How does $\operatorname{Ric} \ge 0$ guarentee the Busemann function is regular in the splitting theorem?

Cheeger-Gromoll's famous splitting theorem says If $(M,g)$ contains a line and $\operatorname{Ric} \ge 0$. Then $(M,g)$ is isometric to a product. I want to know how does $\operatorname{Ric} \ge 0$...
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156 views

Question on Morse lemma

I have this: (Page 421, heading Asymptotically quadratic functionals) Remark 2.2. (a) If $N$ is any neighbourhood of $x_0$, then the excision property of homology theory implies $$C_k(f,x_0) ...
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0answers
46 views

Critical group and morse Lemma

I have this theorem with a part of it's prove I have two questions: 1) what is the spectral decomposition of A ? 2)How to see that $B_{\varepsilon}\cap f_0 = \lbrace x\in H,||x||\leq \varepsilon , ...
3
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2answers
170 views

Morse Theory and critical groups

Please I have a question: What is the relation between Morse theory and critical point theory ? I studied the Morse inequalities and critical groups, but i can not not find or at least i do not ...
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2answers
82 views

Question about “THE MORSE INEQUALITIES”in Milnor's book

in this paragraph what is $H_{*}$ ? Please help me Thank you .
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1answer
159 views

Question on Morse theory

What is the difference between the theory of Morse study in the book of Milnor: "Morse theory "and that studied in the book" ciritical point theory and Hamiltonian systems " Please Thank you
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1answer
86 views

question about Morse theory in Hilbert space

This is sade to be the Morse theory in Hilbert space ,and i want to know the definition (or where i can find it ) of : The qth singular relative homology groupe The qth critical group Please;
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1answer
65 views

Question on the demonstration of Morse theorem

We have theorem of Morse and this is the proof i dont understand this : "$(c_i)$ has no cluster point since each $M^a=f^{-1}]-\infty,a]$ is compact " Thank you.
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2answers
1k views

Question about index of critical points.

I don't really understand what index of a critical point is and I am trying to do a very simple example. I was wondering if someone could help me figure out what the index of the critical point $(0,0)...
3
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1answer
584 views

Question about theorem 3.2 from Morse theory by Milnor

The demonstration of the theorem 3.2 in the book Morse theory by Milnor THEOREM $\mathbf{3.2.}$ Let $f:M\to\bf R$ be a smooth function, and let $p$ be a non-degenerate critical point with index $\...
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1answer
325 views

An other question about Theorem 3.1 from Morse theory by Milnor

In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that: for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla f\right\rangle=\...
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1answer
82 views

Retraction by deformation

in Theorem 3.1 in the book Morse theory by Milnor , in the end of the proof they say that : $r_t$ is a deformation retract ? how to prove this ? please thank you
3
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1answer
332 views

Lemme 2.4 in Morse theory by Milnor

This is lemma 2.4 from "Morse theory" by Milnor ,with the prove I have some questions about this prove : 1) why $\displaystyle\frac{dc}{dt}(f)=\lim_{h\rightarrow 0} \frac{fc(t+h)-fc(t)}{h}$ and ...
2
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1answer
169 views

Question on Theorem 3.1 from Morse theory by Milnor

In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that: for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla f\right\rangle=\...
8
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1answer
499 views

How to check whether a vector field is Morse-Smale?

Setup and notation: Let $f:M\to \mathbb{R}$ be a Morse-function on the compact $m$-dimensional manifold $M$ and let $X$ be a gradient-like vector field for the function $f$. Denote the unstable ...
8
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1answer
743 views

Prove Poincare duality theorem with Morse theory.

First let us consider a smooth n-manifold. And find a Morse function $f$. Now let's consider $-f$. A singular point of $f$ with index $k$ is a singular point of -f with index n-k. Thus we have a ...
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0answers
539 views

Proof of the Morse Lemma in dimension 1

This is the Morse lemma in dimension1 : Let $M$ be a smooth $1$-manifold and $f: M \longrightarrow \Bbb R$ be a smooth function. Suppose $p$ is a non-degenerate critical point of $f$. Then ...
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1answer
91 views

Lemma2.1 (in dimension 1)in the book of Morse theory by Milnor

i have this lemma : Let $f$ be a $C^{\infty}$ function in a convex neighborhood $V$ of $0$ in $\mathbb{R}$ , with $f(0)=0$ then $f(x)= x g(x)$. for suitable $C^\infty$ function $g$ defined in $...
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1answer
194 views

Lemma of Morse in dimension 1

I want to write the Morse lemma which is in dimension $n$ : Let $p$ be a non-degenerate critical point for $f$. Then there is a local coordinate system $(y^1,...,y^n)$ in a neighborhood $U$ ...
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1answer
1k views

Index of Morse function

What is the definition of the index of a Morse function in dimension one?
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1answer
231 views

Non-degenerate smooth functions on a manifold

I am trying to prepare a presentation on "the use of differential geometry in the theory of critical points", but only the case where there is a single variable. (only in dimension 1), and i ask ...
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1answer
1k views

Gradient-like vector fields

Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function. My goal is to better understand gradient-like vector fields for $f$. Question: Do any two ...

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