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4
votes
1answer
159 views

cell decomposition of real homogeneous hypersurfaces

Let $f(x_1,\ldots,x_n)\in\mathbf{R}[x_1,\ldots,x_n]$ be a homogeneous polynomial and consider the real hypersurface $H=\{(x_1,\ldots,x_n)\in\mathbf{R}^n:f(x_1,\ldots,x_n)=1\}$. Assume to simplify ...
8
votes
1answer
428 views

Is there a holomorphic Morse-Bott lemma?

It asks for a generalization of the question in the post Normal form for a holomorphic Morse function Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...
10
votes
2answers
242 views

Equivariant version of Morse theory

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 StackExchange.
3
votes
1answer
77 views

A reference for an equivariant Morse Lemma

Does anybody knows a reference for the following statement? Let $S^1$ acts on $\mathbb{C}^n$ in the usual (diagonal) way and $f:\mathbb{C}^n\to\mathbb{R}$ a smooth $S^1$-invariant function defined in ...
1
vote
0answers
64 views

restriction to the boundary in Morse theory

Given a compact manifold with boundary $(M,\partial M)$. There is a natural pullback map $$ H^*(M) \to H^*(\partial M) $$ I'm wondering if there is a reference that: 1) constructs this map in ...
2
votes
1answer
370 views

Contractibility of a configuration space

For a topological space $X$ and a positive integer $k\in \mathbb{N}_{>0}$ let $F_k(X):= \{ (x_1,\ldots,x_k)\in X^k |x_i\neq x_j \text{ for } i\neq j \}$ be its $k$-configuration space. Let ...
7
votes
0answers
221 views

diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...
1
vote
2answers
191 views

Question on Morse inequalities

I want to understand why: From K.C Chang's book "Infinite Dimensional Morse Theory and Multiple Solution Problems": if i have then $(4.1)$ is formal : it means that EDIT1: $(4.1)$ tel us that ...
1
vote
2answers
64 views

Stable/Unstable Manifold Theorem for a Morse-Bott function

Good night, anyone know of any reference where I can find the proof of the Stable/Unstable Manifold Theorem for a Morse-Bott function. I'm interested in the dimensions of the stable and unstable ...
4
votes
1answer
470 views

Morse theory and adiabatic limits

Let's start with a product manifold $M\times N$, with a product Riemannian metric $g_M\oplus g_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and ...
15
votes
2answers
329 views

Has Witten's perturbation on de Rham complex been studied on other elliptic complexes?

In his famous work, Supersymmetry and Morse theory, Witten perturbs de Rham complex by perturbing the exterior derivative $$d_h=e^{-ht}de^{ht}.$$ And he proves Morse inequality using some spectral ...
30
votes
0answers
889 views

Two-convexity ⇒ Lefschetz?

Assume that $\Omega$ is an open simply connected set in $\mathbb R^n$ (two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$. Is it true that any component of ...
6
votes
3answers
200 views

Is the 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 ...
3
votes
2answers
468 views

Self-indexing Morse functions on non-compact manifolds

Hi, given a compact manifold M we can always alter a given Morse function f to a self-indexing one (i.e., one where every critical point c has $f(c) = \operatorname{index}(c)$) - a proof of this may ...
2
votes
1answer
357 views

Question about a Lefschetz hyperplane type theorem

Let $X$ be a simply connected projective manifold of dimension $n$ over $\mathbb{C}$ and $D = \cup D_i$ be a divisor with normal crossings such that its all components $D_i$ are smooth and ample. I ...
7
votes
2answers
272 views

Existence of Morse functions on simply connected manifolds

Is it true that any simply connected closed manifold possesses a Morse functions that does not have critical points of index one? If the dimension is at least 5, this is a consequence of the results ...
3
votes
1answer
167 views

Measuring almost-critical values of smooth functions.

Consider a compact sub-manifold $X \subset \mathbb{R}^n$ of Euclidean space and let $f:X \to \mathbb{R}$ be any smooth function. Recall that $x \in X$ is a critical point of $f$ if the gradient ...
26
votes
0answers
2k views

Microlocal geometry - A theorem of Verdier

(1) In "Geometrie Microlocale", Verdier states the following theorem. Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$. Then for $\ell$ a linear form on $E$, we have a ...
16
votes
2answers
1k views

Conjugate points in Lie groups with left-invariant metrics

For any Lie group $G$ there exist many left-invariant Riemannian metrics, namely, one just takes any inner product on the tangent space at the identity $T_eG$ and then left translate it to the other ...
24
votes
5answers
2k views

Is there a Morse theory proof of the Bruhat decomposition?

Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup_{w \in W} BwB$ of cells given by double ...
2
votes
0answers
143 views

complex Morse function on a four-manifold

If we have a complex Morse function on a complex four-manifold, $f: X\to \mathbb{C} $, can we tell from the function how the genus of inverse images $f^{-1}(z)$ (for regular values) may change? under ...
0
votes
0answers
92 views

Application of Morse theory to second order systems

Hello I'm looking for some applications of Morse theory to second order differential system,( or boundary value problems ) Someone can help me with a pdf or a book which has these applications ? ...
1
vote
2answers
241 views

What does it mean that homotopy is generic?

Due to Cerf, there's exist a certain homotopy between two Morse functions. It is said that this homotopy is "generic". What is a precise definition of the property to be "generic" in this case?
4
votes
2answers
506 views

Discrete Morse theory and existence of minimal complex

A minimal complex is a CW complex whose only cells are the homology cells. Is there some sort of criterion on CW complexes about existence of minimal complexes? Actually I am working on a problem ...
11
votes
5answers
1k views

Yang-Mills and Chern-Simons functionals as Morse functions

Can the Yang-Mills or Chern-Simons action functionals be considered as [possibly perfect] Morse functions? I assume we would be in an equivariant scenario due to considering the configuration spaces ...
3
votes
1answer
313 views

Is the space of gradient-like vector fields contractible?

Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function. Question: Is the space $GVect(M,f)$ of all gradient-like vector-fields for $f$ contractible ...
1
vote
0answers
120 views

Detecting Non-Transversality

Suppose $f \colon \mathbb{R}^n \to \mathbb{R}$ is Morse and has finitely many critical points. Is there an algorithm for determining whether there exists a saddle-saddle connection (an orbit of grad ...
7
votes
1answer
298 views

A description of cellular boundary maps in terms of a Morse function

I'm writing a paper on classical Morse Theory and I'm interested in applying Morse functions to the computation of homology groups of a compact manifold $M$. The standard way in which this is done is ...
3
votes
0answers
139 views

Perturbation of Morse functions at critical points leaving stable manifolds invariant

Let $f$ be a given Morse-Smale function $f$ on $\mathbb R^n$ with finite many critical points and sufficient growth at $\infty$ like $\langle x, f(x)\rangle \geq |x|$ (cp. MO120858). Is there a way ...
1
vote
1answer
212 views

Integration by parts wrt. a Morse function on its basin of attraction

Let $f:\mathbb R^n\to\mathbb R$ be a Morse function with uniform nondegenerate Hessian at critical points, i.e. for some $\delta>0$ $$ \forall x \in \{\nabla f=0\}\;\forall \xi\in\mathbb R^n: \quad ...
6
votes
1answer
407 views

When is the determinant a Morse function?

This might be ridiculously obvious, but... For each $n \in \mathbb{N}$, let $M_n$ denote the manifold of $n \times n$ matrices with real entries. It is well known that the $n$-dimensional determinant ...
4
votes
1answer
208 views

Morse lemma with least amount of regularity.

I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ...
8
votes
2answers
649 views

Notes for Bott's 1963 lectures on Morse theory

Would anybody happen to know where I could obtain a scanned version of Lectures on Morse theory - [revised and expanded version of notes of lectures delivered at Professor R. Bott's topology seminar ...
6
votes
3answers
438 views

Induced maps in Morse Homology

Let $M,N$ be closed manifolds. Given a differentiable map $f:M\rightarrow N$, I am interested in computing $f_k:H_k(M)\rightarrow H_k(N)$, in Morse Homology. This problems seems difficult, and the ...
4
votes
1answer
173 views

Concerning strata in $C^\infty(M)$

The Morse functions are dense in $C^\infty(M)$, and you can ask if a 1-parameter family of smooth functions between two given Morse functions will be a homotopy through Morse functions. Well, Cerf ...
8
votes
0answers
206 views

Intersection of plus/minus cells in Bialynicki-Birula decomposition

Let $X$ be a projective variety endowed with an algebraic $\mathbb{C}^*$-action. Assume the fixed point set $W$ is discrete. Then we have two cell decompositions of $X$: $X = \bigsqcup_{w\in W} C_w$ ...
9
votes
2answers
519 views

Discrete Morse function from smooth one

Suppose I have a smooth manifold $M$ and a smooth Morse function $f$ on $M$. Is there a standard way to replace $M$ with a cell complex and $f$ with a discrete Morse function such the resulting ...
2
votes
1answer
133 views

Invariance group of Morse charts

Suppose I have a smooth function $\varphi$ that vanishes at $p$ and has a positive definite Hessian at that point (suppose that we are on a smooth manifold of dimension $M$). Then the Morse lemma ...
15
votes
1answer
822 views

Any algebraic substitute for Morse theory (and homology) in arbitrary characteristic?

As far as I know, Morse theory yields much information on the topology of smooth manifolds; in particular, it can be used to prove Artin's vanishing (that the singular cohomology of smooth complex ...
17
votes
2answers
717 views

Is there a discrete Cerf theory?

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the ...
30
votes
3answers
1k views

Operations via Morse Theory

I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)homology ...
5
votes
1answer
194 views

Equivariant handle decompositions

Suppose I have some smooth closed high-dimensional manifold $M$ acted on smoothly by a finite group $G$. By a metric averaging procedure, we can equip $M$ with a smooth Riemannian metric so that $G$ ...
3
votes
1answer
293 views

Special Morse function on a Riemann surface

Let $f: S \to \Bbb R$ be a Morse function on a Riemann surface. Let $x_0$ be a saddle point of $f$. Since $x_0$ is a critical point of $f$, it makes sense to talk about the bilinear forms ...
4
votes
1answer
240 views

Classifying smooth embeddings which yield Morse functions

Let $\mu:M \to \mathbb{R}$ be a fixed surjective smooth function on a smooth manifold $M$. Let $N$ be a smooth compact manifold that embeds smoothly into $M$ via $\iota:N \to M$. What conditions ...
2
votes
1answer
396 views

Looking for the perfect morse

A Morse function $f(x)\colon \mathbb{R}^n\to \mathbb{R}$ is a smooth function s.t. all singular points are non-degenerate. A theorem of Sard implies that for any smooth $f(x)$ and almost all $a\in ...
7
votes
0answers
171 views

Is there a general theory of Sard measures?

Let $f:M \to N$ be a smooth map of smooth (second countable) manifolds. The set $C_f \subset M$ of critical points of $f$ is defined to be the set of all $m \in M$ such that the differential $df : ...
1
vote
4answers
359 views

Perturbation of Morse function

The following question is probably classic in Morse theory, so a reference to an existing result should be sufficient. I don't know much about Morse theory and I am dealing with the following ...
6
votes
1answer
629 views

Alexander duality in terms of morse function?

As well known, we can prove Poincare duality in terms of morse theory. (By comparing two chain complexes obtained from two morse functions, $f\colon M\to \mathbb{R}$,$-f\colon M\to \mathbb{R}$ for ...
1
vote
0answers
257 views

Transversality and isolated degenerate critical points

Maybe some of the following statements are not precise. Please correct them. Let $M$ be a compact smooth manifold. Let $f: M \to {\mathbb R}$ be a Morse function. Then a generic Riemannian metric $g$ ...
3
votes
0answers
413 views

Topological version of two results in smooth Morse theory

Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post Morse theory in TOP and PL categories? for references). It is well known ...