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14 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 ...
0
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1answer
57 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
3
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0answers
79 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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0answers
29 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 ...
3
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0answers
18 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 ...
1
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1answer
43 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 ...
2
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1answer
43 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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0answers
19 views

Cutting a surface at critical levels produces cylinders

Let $F$ a closed surface with isolated critical points and a homeomorphism $g: F \rightarrow F$ that maps critical levels to critical levels. Let us cut the surface $F$ by the critical levels. Do we ...
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0answers
34 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 ...
2
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1answer
44 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 $f$ ...
4
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1answer
78 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?
2
votes
1answer
27 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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0answers
68 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: ...
0
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1answer
31 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
5
votes
0answers
201 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 ...
1
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1answer
41 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 ...
0
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0answers
58 views

Open problems in variational analysis/PDEs

I wasn't sure whether this question was more appropriate for StackExchange or Overflow, but in any case I would really appreciate it if any active researchers in the field responded. I'm a PhD ...
1
vote
1answer
38 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 ...
5
votes
2answers
127 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 ...
7
votes
2answers
109 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 $t$, ...
3
votes
0answers
53 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 ...
1
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1answer
111 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
22 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$(x_1,y_1)\in\mathbb{R}^2$, ...
6
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4answers
276 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
votes
1answer
75 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
votes
1answer
109 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 ...
1
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2answers
131 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
votes
0answers
46 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
votes
1answer
59 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
votes
1answer
47 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
votes
1answer
57 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
vote
2answers
34 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 ...
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1answer
45 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
59 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
37 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
163 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
68 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
votes
1answer
43 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 $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
3
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0answers
72 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
vote
1answer
33 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
votes
0answers
29 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
73 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 ...
2
votes
1answer
213 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
votes
1answer
170 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
73 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 ...
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1answer
111 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
60 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
86 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
37 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
30 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 ...