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15
votes
4answers
1k views

Morse theory in TOP and PL categories?

Apparently there are topological and piecewise linear versions of Morse theory. I would like to know of references that treat these topics. How is a Morse function defined for compact manifolds (with ...
1
vote
1answer
168 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 ...
4
votes
1answer
222 views

Is any Morse trajectory contained in a contractible open set?

Suppose $f$ is a Morse function on a Riemannian Hilbert manifold $M$. Let $p_{\pm}\in \text{Crit}(f)$ be given and fix some $u:R\rightarrow M$ which is an integral curve of $-\nabla f$ connecting ...
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
571 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
598 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 ...
2
votes
0answers
169 views

Generators of local homology groups of an isolated critical point

This is a basic Morse theory question: Let $M$ be a smooth manifold, and $f:M\to\mathbb{R}$ a smooth function with an isolated critical point $x$. Set $c:=f(x)$. The local homology of $f$ is the ...
12
votes
1answer
437 views

Nonisotopic homotopy equivalent Morse functions

One can cut a manifold up along the critical levels of a Morse function and deduce something about the topology. In particular the critical points (and the connecting gradient flowlines) define a ...
18
votes
2answers
1k views

The difference between a handle decomposition and a CW decomposition

Let $M$ be a compact finite-dimensional smooth manifold. I have a question about the relationship between the statements that a Morse function induces a handle decomposition for $M$, and that it ...
3
votes
1answer
160 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 ...
2
votes
1answer
265 views

Thom's gradient conjecture and analyticity

Suppose we have an analytic function $f: U \to {\mathbb R}$, where $U\subset {\mathbb R}^n$ an open subset, with $0\in U$ a critical point of $f$. Thom conjectured that if a trajectory $x(t)$ of ...
12
votes
1answer
633 views

Most general context for the Morse Lemmas

Among the foundational results in differential topology are the Morse lemmas: Suppose that $f\colon\, M\to \mathbb{R}$ is a smooth function on a closed manifold M, that $f^{−1}[-\epsilon,\epsilon]$ ...
14
votes
1answer
879 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
484 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 ...
14
votes
4answers
1k views

Searching for an unabridged proof of “The Basic Theorem of Morse Theory”

Steven Smale labels the following statement "The Basic Theorem of Morse Theory" in A Survey of some Recent Developments in Differential Topology: Let f be a $C^\infty$ function on a closed manifold ...
4
votes
2answers
469 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 ...
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 ...
6
votes
1answer
840 views

Euler characteristics and characteristic classes for real manifolds?

Consider an oriented manifold $X$. To calculate its Euler characteristic, one might integrate the Euler class. Now if $X$ were a complex manifold, and given as a section of some complex bundle $E$ ...
2
votes
0answers
294 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 ...
18
votes
1answer
802 views

How does the Framed Function Theorem simplify Cerf Theory?

A handle decomposition of a manifold $M$ is a useful structure to carry around. It is induced by a Morse function $f\colon\, M\to \mathbb{R}$. How are two handle decompositions of $M$ related? The ...
3
votes
1answer
433 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 ...
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 ...
17
votes
2answers
741 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
0answers
925 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 ...
0
votes
0answers
89 views

Can Morse trajectories break if their first derivative is uniformly bounded?

Consider a compact Riemannian manifold endowed with a Morse function f. Fix two critical points x and y of f. Whenever you have a sequence $u_k$ of Morse trajectories connecting x and y it might ...
3
votes
1answer
475 views

Transversality in Morse theory for the (perturbed) geodesic action functional

I am interested in Morse homology on the loop space of a given compact (Riemannian) manifold. A small perturbation renders the geodesic action ("energy") functional Morse. Now I am interested in the ...
7
votes
2answers
460 views

Stratification of smooth maps from R^n to R?

I'm interested in stratifications of smooth maps $\mathbb{R}^n\to\mathbb{R}$ (or more generally of any $n$-manifold $M^n\to\mathbb{R}$). The codimension 0 stratum should be Morse functions, and the ...
6
votes
1answer
654 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
1answer
257 views

Separability of the space of bounded continuous maps

Let $O$ be an open subset of the separable Hilbert space H and $k\geq0$ . Consider $C_b^k(O, Sym(H))$, the space of k-times continuously differentiable maps with values in the bounded symmetric ...
2
votes
1answer
165 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
votes
1answer
298 views

When a Morse function is also a Moment map…

Are there any interesting results, if a Morse function $M \rightarrow \mathbb{R}$ happens to be the moment map, for some $S^1$ action on $M$ (equipped with $\omega$) as well? Thank you very much.
5
votes
4answers
449 views

Is there a bound on the length of the longest Morse trajectory?

Given a compact Riemannian manifold (with a fixed metric) and a Morse function on it (also fixed). Is there a bound (depending on the metric and the Morse function) on the length of the Morse ...
19
votes
2answers
992 views

Graphs, K-theory and combinatorial balls: conjectures

The following conjectures from Kapranov and Saito's Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions aren't as well-known as they aught to be, so I'd like to state ...
0
votes
0answers
256 views

Gradient of the energy functional in $H^{1,2}$-norm

I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy ...
1
vote
0answers
142 views

Do higher dimensional maxima of a real valued multivariable function form a cell complex?

Suppose $f: R^n \rightarrow R$ is a positive real valued function. Let $\lambda_1, \ldots , \lambda_i$ be the first $i$ ordered eigenvalues of the Hessian $Hess(f)$. Let $v_1, \ldots, v_i$ be the ...
0
votes
3answers
805 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
702 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?
2
votes
2answers
772 views

Morse Theory on Non-smooth Manifolds

Let $X$ be a circle that with one corner (i.e. think of a triangle where we smooth out two of the vertices). Now let us consider the topological torus $M \cong \mathbb{T}^n$ which is the product of ...
21
votes
2answers
2k views

Level sets of Morse functions

Every compact two dimensional manifold admits a Morse function such that any its regular level set is at most two circles. I am interested in a generalization of that phenomenon. Does there exist a ...
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 ...
5
votes
1answer
780 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 ...
7
votes
5answers
628 views

Is there a Morse theory for sections of bundles or more generally for maps?

This question was prompted by my interpretation of a question by cosmologist Berian James. Background Some cosmologists have suggested using the cosmological dark matter density, which defines a ...
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 E→_B_, and a (smooth) function f: E → 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 ...