5
votes
0answers
104 views

Parametrized cancelations in stable Morse theory

Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...
6
votes
1answer
196 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 ...
7
votes
0answers
228 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. ...
6
votes
3answers
207 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
1answer
317 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 ...
6
votes
1answer
414 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
210 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 ...
3
votes
1answer
295 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 ...
1
vote
4answers
365 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 ...
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
2answers
432 views

Reference request: embedded Morse theory

For references of embedded Morse theory, or so-called relative morse theory of a pair, I have found R.W. Sharpe's "Total Absolute Curvature and Embedded Morse numbers" totally not helpful. For further ...
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 ...
6
votes
2answers
531 views

Morse Function on 3-Torus from Heegaard Splitting

It is easy to visualize some Morse functions on surfaces (such as the torus) via the height function, but this seemingly doesn't work for 3-manifolds. I am looking for an explicit one on the 3-torus ...
8
votes
0answers
551 views

Good introduction to Morse-Novikov theory?

Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued ...
14
votes
1answer
861 views

``Nice'' metrics for a Morse gradient field: counterexample request

Ralph Cohen (professor at Stanford) is teaching a class on algebraic topology and moduli spaces this quarter, beginning by reviewing his perspective of Morse theory. He defined "nice" metrics, proved ...
4
votes
1answer
474 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 ...
4
votes
2answers
464 views

Gluing in Morse homology for Hilbert manifolds

Is there any reference for gluing in the context of Morse homology on Hilbert manifolds? Gluing is pretty standard in Morse homology for finite-dimensional manifolds. Unfortunately, in the ...
2
votes
0answers
290 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 ...
0
votes
3answers
762 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 ...
3
votes
4answers
676 views

literature on geometrical viewpoint on calculus of variations for physics

What is a good reference for a geometrical viewpoint on the calculus of variations for physics, using differential forms etc. to derive Yang-Mills equations and other topics of the standard model? ...
6
votes
2answers
1k views

existence of Morse functions satisfying the Palais-Smale condition

Given an arbitrary Hilbert manifold, can on find a complete Riemannian metric and a Morse function satisfying the Palais-Smale condition?
5
votes
1answer
767 views

Morse-Bott homology for infinite-dimensional manifolds

Is there any work on Morse-Bott homology for infinite-dimensional manifolds (e.g. Hilbert manifolds). I am particularly interested in the case where we have a locally trivial fiber bundle and the ...
9
votes
6answers
2k views

CW-structures and Morse functions: a reference request

The following is probably well known, but I wasn't able to locate a reference in the literature. Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...
6
votes
2answers
1k views

Critical points on a fiber bundle

Consider a (smooth) bundle Eā†’_B_, and a (smooth) function f: E ā†’ R on the total space. Then it makes sense to talk about the derivatives of f along the fibers. Let C be the subspace of E consisting ...