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2
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1answer
32 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 ...
1
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2answers
22 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 ...
0
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0answers
19 views

Morse Ffunction's definition

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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0answers
27 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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0answers
30 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 ...
2
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0answers
28 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!!
1
vote
1answer
47 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 ...
0
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0answers
32 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 $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
2
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0answers
50 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 ...
0
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0answers
10 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, ...
0
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0answers
25 views

Question about the Morse index of $p_0$ where $f''(p_0)=id-T'(p_0)$

I have this perturbation $f'(x)=x-T(x)$ where $T$ is compact, i have that $p_0$ is non degenerate and i want to see if it's Morse index (i.e. the suprimum of the dimensions of subspaces where ...
0
votes
1answer
63 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 ...
1
vote
1answer
52 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 ...
3
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0answers
75 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
37 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 ...
1
vote
1answer
61 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 ...
2
votes
2answers
45 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 ...
1
vote
1answer
47 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 ...
1
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0answers
19 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 ...
0
votes
0answers
24 views

Derivative and Morse lemma

can someone explain me this writing ? this is from K.C Chang book's "Infinite Dimensional Morse Theory and Multiple Solution Problems" i don't understand how to find exactly (5.9) , what it means ...
0
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0answers
26 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 ...
3
votes
0answers
37 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 ...
1
vote
1answer
98 views

Implicit function theorem and derivative (proof of splitting lemma)

I have this theorem with a part of the proof: And I have this question: why $\nabla\hat{\varphi}(w)=(I-Q) \varphi(w+g(w))$ ? Please help me. Thank you.
3
votes
0answers
65 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 ...
1
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0answers
59 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 ...
2
votes
1answer
59 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) = ...
0
votes
0answers
37 views

Question about saddle point

i have this paper http://www.sysmath.com/jweb_xtkxyfzx/CN/article/downloadArticleFile.do?attachType=PDF&id=10691 and i dont understand how to prove in page 3 that $\overline{c}$ is a critical ...
0
votes
0answers
6 views

Question on critical groups

if i have that $C_n(\phi,u)=\delta_{n,k}F, n=0,1,....$ where $k$ is the morse index of $u$, $F$ is a vector field and $\delta$ is the Kronecker delta is this implise that $$\dim ...
17
votes
1answer
121 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 ...
1
vote
1answer
28 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 ...
0
votes
1answer
32 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 ...
3
votes
1answer
57 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 ...
2
votes
0answers
40 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 ...
1
vote
1answer
82 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 ...
3
votes
0answers
41 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 ...
0
votes
3answers
90 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
votes
1answer
64 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, ...
15
votes
1answer
366 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 ...
2
votes
0answers
83 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 ...
1
vote
1answer
74 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 ...
1
vote
0answers
26 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 $\Omega(t_0, \dots, t_k)$ ...
2
votes
1answer
73 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 ...
1
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0answers
71 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 ...
0
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0answers
26 views

Critical groups

I have a small question : What is the relation between the critical groups and the change in topology near a critical point ? ,how we can see this changement ? Please help me Thank you .
3
votes
1answer
72 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 ...
2
votes
0answers
78 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 ...
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0answers
21 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 ...
2
votes
1answer
90 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 ...
0
votes
2answers
48 views

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

in this paragraph what is $H_{*}$ ? Please help me Thank you .
1
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1answer
61 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