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30
votes
3answers
2k 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 ...
30
votes
0answers
919 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 ...
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 ...
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 ...
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 ...
19
votes
2answers
991 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 ...
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 ...
18
votes
1answer
794 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 ...
17
votes
2answers
369 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 ...
17
votes
2answers
736 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 ...
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 ...
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 ...
15
votes
1answer
864 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 ...
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 ...
14
votes
1answer
872 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 ...
12
votes
1answer
430 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 ...
12
votes
1answer
626 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 $f^{−1}[-\epsilon,\epsilon]$ ...
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 ...
10
votes
2answers
301 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.
10
votes
2answers
590 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 ...
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 ...
9
votes
2answers
533 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 ...
8
votes
2answers
691 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 ...
8
votes
0answers
216 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 $X$: $X = \bigsqcup_{w\in W} C_w$ ...
8
votes
0answers
583 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 ...
7
votes
5answers
626 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 ...
7
votes
2answers
280 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 ...
7
votes
2answers
458 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 ...
7
votes
1answer
311 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 ...
7
votes
0answers
232 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. ...
7
votes
0answers
176 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 : ...
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?
6
votes
4answers
519 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 ...
6
votes
2answers
560 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 ...
6
votes
1answer
649 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 ...
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 ...
6
votes
1answer
434 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 ...
6
votes
3answers
233 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 ...
6
votes
1answer
197 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 ...
6
votes
1answer
839 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 $E$ ...
5
votes
4answers
448 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 ...
5
votes
2answers
228 views

Using Discrete Morse Theory to represent hom classes

In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical ...
5
votes
2answers
442 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 ...
5
votes
1answer
776 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 ...
5
votes
1answer
197 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 $G$ ...
5
votes
0answers
107 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 ...
4
votes
1answer
254 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 ...
4
votes
1answer
212 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 ...
4
votes
1answer
483 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
467 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 ...