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1answer
100 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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0answers
21 views

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$(x_1,y_1)\in\mathbb{R}^2$, ...
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4answers
243 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, $x_1,x_2\in\mathbb{R}$ ...
3
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1answer
44 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 ...
3
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0answers
44 views

Equivariant version of Morse theory [closed]

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 MathOverflow.
3
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1answer
54 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
81 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 ...
3
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0answers
38 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
45 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
46 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?
2
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1answer
41 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
26 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 ...
1
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1answer
31 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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0answers
45 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
34 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
128 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
62 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
39 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
57 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 ...
1
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1answer
22 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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0answers
27 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 ...
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1answer
67 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
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1answer
85 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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0answers
121 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
45 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
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1answer
82 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
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2answers
51 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
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1answer
65 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
33 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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0answers
27 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 ...
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0answers
29 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
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0answers
44 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
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1answer
125 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
79 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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0answers
64 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
64 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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0answers
41 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 ...
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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 ...
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1answer
147 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
38 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
41 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 ...
4
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2answers
75 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
46 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 ...
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1answer
95 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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0answers
46 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
91 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 ...
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1answer
69 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, ...
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1answer
463 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
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1answer
124 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 ...
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1answer
85 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 ...