Questions tagged [morse-theory]
In differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold.
254
questions
3
votes
1
answer
289
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 ...
- 1,073
3
votes
2
answers
196
views
When is the Morse equivalence local?
Let $f:X \to \mathbb{R}$ be a Morse function on some compact submanifold $X \subset \mathbb{R}^n$, and assume that $p \in X$ is not a critical point of $f$. For some $\epsilon > 0$ let $D_\epsilon(...
- 15.5k
3
votes
1
answer
320
views
Geometric Morse theory ( and its complex analogy)
In the literature are there some concept of geometric version of Morse or Picard Lefschets theory? That is the comparison of level sets as Riemannian submanifold not merely as ...
- 312
3
votes
1
answer
581
views
the free loop space fibration is a locally trivial fiber bundle - reference?
Let $Q$ be a compact Riemannian manifold. Then $\Lambda Q\rightarrow Q,$ $\gamma\mapsto \gamma(0)$ can be shown to be a locally trivial fiber bundle of Hilbert manifolds. Here, $\Lambda Q$ denotes the ...
- 2,875
3
votes
1
answer
220
views
Handle attachment information from Morse function and triangulation
First, allow me to setup the relevant information. It is well known that a Morse function $f:M\to\mathbb{R}$ induces a handle decomposition of $M$.
For simplicity, let's restrict for now to the ...
- 159
3
votes
1
answer
136
views
Perturbation of vector fields in Morse Homology
Recently I have been reading on Morse Homology. Suppose we have a compact manifold $M$ and a smooth function $f:M \rightarrow \mathbb{R}$ and a Morse vector field $X$ such that we can do Morse ...
- 781
3
votes
1
answer
163
views
Morse functions inducing Heegaard diagrams
Let $(\Sigma, \alpha, \beta)$ be a Heegaard diagram for a 3-manifold $M$, corresponding to a Heegaard splitting $M = H_1 \cup_\Sigma H_2$. There may be many self-indexing Morse functions $f: M \to \...
- 31
3
votes
1
answer
236
views
Finite-dimensional argument for Morse-Smale pairs?
Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $M$ (i.e. pairs $(f,g)$ where $g$ is a metric on $M$, $f$ is a Morse function on $M$, and the stable/...
- 1,175
3
votes
1
answer
454
views
Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine
This question was not answered on math.stackexchange.
Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a ...
- 31
3
votes
1
answer
305
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 $\...
- 15.5k
3
votes
1
answer
195
views
bounding the probability that a polynomial is near 0
Given a polynomial $p(x_1,\ldots, x_k)$ in $k$ variables with maximum degree $n$, and $x_1,\ldots, x_k \in [0,1]$. Suppose $\max_{x \in [0,1]^k} p(x) = 1$, can we get an upper bound on the probability ...
- 4,354
3
votes
1
answer
377
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 $f_{z\...
- 1,171
3
votes
0
answers
33
views
Integral equivariant formality for Hamiltonian T-actions
What is the simplest example of a compact symplectic manifold $M$ with Hamiltonian $T$-action for which the integral $T$-equivariant cohomology is not formal
i.e. $$H_T^*(M,\mathbb{Z}) \not \cong H^*(...
- 459
3
votes
0
answers
174
views
Morse theory for isotopies of codimension 1 submanifolds and generalized bigon moves
I seem to have managed to convince myself of the validity of a certain result. If it does indeed hold (modulo non-catastrophic adjustments) I could really use a reference. Otherwise, I would also be ...
- 141
3
votes
0
answers
170
views
Exponential map of cotangent bundle and Morse theory on based loop space
Consider $M$ to be a compact manifold and $q_0,q_1\in M$. Let $L_t$ be a lagrangian and $\mathcal{E}_L$ the lagrangian action functional on the based loop space $\Omega(M,q_0,q_1)$ defined as $\...
- 781
3
votes
0
answers
152
views
Homological stability of Chow varieties
Given a connected component $C$ of the degree $d$ Chow variety of $r$ cycles on $C_{d,r}(X)$ ($X$ is smooth projective variety over $\mathbb{C}$), let $C'$ be another connected component of $C_{d',r}(...
- 5,851
3
votes
0
answers
121
views
Upper bound on index of geodesics in terms of length
Let $(M, g)$ be a compact Riemannian manifold. Let $i(-)$ be the index of a geodesic and let $l(-)$ be the length. Is there an inequality of the form $i(\gamma) \leq C l(\gamma)$ for some $C>0$ ...
- 927
3
votes
0
answers
124
views
A generalization to Bott‘s theorem (from Milnor’s “Morse theory”)
This is Theorem 22.1 of Milnor‘s Morse theory:
Let $M$ be a complete Riemannian manifold, let $p,q\in M$ be so that the space $\Omega’$ of minimal geodesics joining $p$ to $q$ is a topological ...
- 1,610
3
votes
0
answers
247
views
What mathematical background to i need in order to understand proofs of the h-cobordism theorem?
I am about to finish my undergraduate studies and I really enjoyed the topology and differential-geometry classes. I'd love to continue studying differentialtopology and i considered doing some ...
- 173
3
votes
0
answers
232
views
Existence of connections in a vector bundle whose parallel transport preserves a function on a total space
Let $p:E \to M$ be a vector bundle over a smooth manifold $M$, $M\times 0$ be the image of its zero section of $p$, $\mathcal{X}(M)$ be the space of vector fields on $M$, and $\Gamma(E)$ be the space ...
3
votes
0
answers
100
views
What's a completely computational/syntactical model for handle decompositions of manifolds?
Simplicial sets, CW complexes
Simplicial sets can be described completely algebraically, by specifying a family of sets, and maps between them satisfying certain relations. This description can be ...
- 5,535
3
votes
0
answers
93
views
Is there a finite dimensional subcomplex of the Morse-Bott complex that computes the cohomology of a manifold?
In the paper Morse-Bott theory and equivariant cohomology, Austin and Braam built the Morse-Bott (geometric) complex which computes the de Rham cohomology of the manifold $M$ (this paper can be found ...
- 219
3
votes
0
answers
90
views
Decompositions from torus actions and compactness of (sub-)level sets
Let $T=\mathbb{C}^{*}$ act on a smooth complex quasi-projective variety $X$. Assume that the limit point $\lim_{t\to 0}t\cdot x$ exists for every $x\in X$.
From the induced $U(1)$-action and its (...
- 22.7k
3
votes
0
answers
153
views
Morse-Bott functions without critical manifolds of index 1 and n-1
I am now reading the article of M.F.Atiyah "Convexity and commuting hamiltonians" and I can't understand lemma 2.1. which says that if $\varphi \colon M \to \mathbb R$ is a Morse-Bott function without ...
- 2,235
3
votes
0
answers
481
views
Number of critical points of a smooth function
Are there any examples of closed manifolds with the property that the minimal number of critical points (possibly degenerate) of a smooth function on this manifold is strictly bigger than the stable ...
- 31
3
votes
0
answers
82
views
If $M$ is a globally symmetric space, do we need it to be compact to prove that all the critical submanifolds in $\Lambda M$ are nondegenerate?
In "The free loop space of globally symmetric spaces" by Ziller, he proves the following theorem:
Theorem 2. For a globally symmetric space the critical submanifolds in $\Lambda M$ are all ...
- 401
3
votes
0
answers
170
views
Morse index in variational calculus
Let $f(t,x,v)$ be a smooth function on $\mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^n$, $\mathcal{C}_{x,y}$ be the set of smooth paths $c:[0,1]\to \mathbb{R}^n$ with $c(0)=x,c(1)=y$, $E(c)$ be a ...
- 2,194
3
votes
0
answers
671
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 $g$ ...
- 1,197
3
votes
0
answers
416
views
Is the Hessian almost everywhere nondegenerate?
Let $M$ be a complete Riemannian manifold. For a fixed point $p$ in $M$, the Riemannian distance to $p$ is denoted by $d_p$. Fix a strongly convex geodesic ball $B(o,R)$ in $M$ and some disjoint ...
- 265
2
votes
4
answers
1k
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 ...
- 1,171
2
votes
3
answers
2k
views
Index of a Morse function via the Hessian tensor
For a smooth function $f:M\to \mathbb{R}$ one usually defines the degeneracy and index of a critical point $p\in M$ in terms of the eigenvalues of the Hessian matrix $(\partial^2 f/\partial ...
- 2,233
2
votes
1
answer
617
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 \...
- 3,282
2
votes
1
answer
147
views
Is an $L^p$-sphere in Sobolev space $H_2^{s}(\Omega)$ a Hilbert manifold?
For a bounded smooth domain $\Omega$, let $H_2^{s}(\Omega)$ be the usual Sobolev space on $\Omega$.
Define $A:=\{f\in H_2^{s}(\Omega)| \lVert f\rVert_{L^p(\Omega)}=1\}$ where $2<p<2_{s}^*$.
...
- 631
2
votes
1
answer
116
views
Generalize the conception of 'stable' solution and 'stable outside a compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold
I want to talk about generalizing the conception of 'stable' solution and 'stable outside the compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold.
I'm reading $\Delta u +e^u=0$...
- 729
2
votes
1
answer
144
views
Morse approximation with bounded number of critical points
Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...
- 81
2
votes
1
answer
188
views
CW-structure induced by Morse function on Riemannian manifold [duplicate]
I have heard a statement in the following direction. Given a compact Riemannian manifold $M$ with a Morse function on it possibly satisfying some extra assumptions. Then this data induces a CW-...
- 21.1k
2
votes
1
answer
131
views
Could an inverse of (weak) Morse inequality exists in some special case?
Could an inverse Morse inequality hold in some sense? More precisely I wish the following result to be true:
Problem
$M$ is a smooth simply connected compact manifold, $dim(M)=n$, $f$ is a morse ...
- 697
2
votes
1
answer
252
views
Modification of Morse lemma with two functions
Standard Morse lemma states that if the singularity of a smooth function $f$ is non-degenerate, one can choose coordinates such that function has a "quadratic" form $f = \sum \limits_i x_i^2 - \sum \...
2
votes
1
answer
538
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 $f:M\...
- 281
2
votes
1
answer
222
views
Morse Theory on pseudo-Hermitian manifold
I wonder if Morse Theory on pseudo-Hermitian manifold is developed. For example, I wonder if the following statement on pseudo-Hermitian manifold, which is corresponding to the Riemannian case, is ...
- 834
2
votes
1
answer
192
views
How to obtain the local bound on the length of the Morse function?
This is a follow-up of the question
Is there a bound on the length of the longest Morse trajectory?.
Consider the following setup. Let $(M,g)$ be a (complete) Riemannian Hilbert manifold and $f$ a ...
- 2,875
2
votes
2
answers
498
views
How to use that the Hessian is negative definite in this proof
Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...
- 129
2
votes
1
answer
304
views
Bijection of critical points on two manifolds
Suppose that $f$ and $g$ are two smooth functions defined on $R^n$. Assume that $(a-\epsilon, b+\epsilon)$ contains no critical point of $g$. Then $g^{-1}[a,b]$ it homomorphic to $g^{-1}(a)\times [a,b]...
- 21
2
votes
1
answer
578
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 ...
- 1,331
2
votes
1
answer
112
views
Morse theory for compact sets bounded by hypersurfaces in euclidian space
I am having trouble understanding precisely how some part of Morse Theory works.
More precisely, take $X$ to be a compact set of $\mathbb{R}^d$ such that $\partial X$ (topological boundary) is a ...
- 93
2
votes
1
answer
91
views
In a manifold, $\angle xpy>\frac{\pi}{2}$, for $q$ on $px$ or $py$, $B_q(r)$ homeomorphic to $B_p(r)$?
Let M be an n-dimensional Riemannian manifold without boundary, with sectional curvature $\geqslant -1$. For a point $p\in M$, suppose there exist $l, \delta>0$, $x,y \in M$ with $d(p,x),d(p,y)>...
2
votes
0
answers
115
views
Reference for Morse-Bott vector fields
I'm looking for a reference for the following result:
Let $X$ be a vector field on a manifold $M$ and let $C$ be a submanifold of $M$ formed by critical points of $X$. Assume that the Hessian of $X$ ...
- 1,379
2
votes
0
answers
130
views
Is Morse theory local?
I am currently trying to use Morse theory in my work. Say I have $X$ smooth compact manifold and $f : X \to \mathbb{R}$ a smooth function. The classical result I know is : when $f$ has non-degenerate ...
- 93
2
votes
0
answers
80
views
Understanding dimension of gradient flow trees for product on Morse complex
I am trying to square my intuition with the facts and hope this question is not too vague.
I am reading this paper which describes the $A_\infty$-category of Morse functions on a manifold $M$ of ...
- 121
2
votes
0
answers
129
views
Transformation of Morse function in $\mathbb{CP}^n$ trough symplectomorphism
I have a question I have been thinking for a while and feel like it should be true but have no idea of how to prove it. First let me introduce a piece of notation.
Consider $(\mathbb{CP}^n,\omega)$ ...
- 181