47 votes
Accepted

Arithmetic Morse theory?

One usually considers the analogue of Morse theory in algebraic geometry to be the theory of vanishing cycles and Lefschetz pencils. Because of the nature of algebraic functions, Morse theory must be ...
  • 119k
19 votes
Accepted

CW complex of iterated loop spaces

By a result of Milnor, the space of maps from a finite CW complex to any CW complex is homotopy equivalent to a CW complex. This gives a general reason why spaces like $\Omega^k M$ have a CW structure....
18 votes
Accepted

Morse number of the Poincaré homology sphere

It's 6: you need one critical point of index 0 and index 3, and two of index 1 and index 2. It's at least 6, since the only 3-manifolds with lower Morse number are the 3-sphere (2) and lens spaces (4)...
  • 8,914
17 votes
Accepted

Intuition behind the Morse inequalities?

If you understand intuition behind the fact that the Euler characteristic is the alternating sum of the betti numbers, then I think you can grasp the Morse inequalities. A Morse function gives rise to ...
  • 62.2k
16 votes
Accepted

Easiest example where pseudo-isotopy fails to be the same as isotopy?

In high dimensions ($\geq 5$) the most basic examples arise on manifolds with nonempty boundary, where one requires that diffeomorphisms restrict to the identity on the boundary. The simplest case is ...
15 votes
Accepted

What are good Morse Theory lecture notes and books?

If you are looking for the classical approach to Morse theory, I feel nothing beats Milnor's book on the subject: Milnor, J. Morse theory. Annals of Mathematics Studies, No. 51 Princeton ...
  • 6,745
14 votes

Unstable manifolds of a Morse function give a CW complex

(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps. I will retract this for now. I do recall being told this, but I am not aware at this point in time where the gaps ...
  • 17.7k
12 votes
Accepted

Artin vanishing for Stein manifolds and restriction maps

The pairs (U,X) are called Runge pairs. The homology version of your statement is proved in the paper of Andreotti and Narasimhan Annals of Math vol 76 no 3 (1962) 499-509 using Morse Theory.The title ...
11 votes

Why not develop a Hamiltonian-based Morse theory?

To any continuous dynamical system $\Phi_t$ on a reasonable space $X$ (say a compact metric space) there is an associated Morse like theory, namely the Conley index theory, C. Conley: Isolated ...
11 votes

Can a Morse function define a unique structure on a closed manifold?

In looking for counter-examples it helps to notice that $M$ and $N$ have CW structures with the same numbers of $i$ cells, and hence they have the same Euler characteristic. It turns out that there ...
  • 33.5k
11 votes

Next steps for a Morse theory enthusiast?

A recent breakthrough result which uses Morse theory in a substantial manner is Watanabe's disproof of Smale conjecture in dimension 4. In it, he provides a method to compute Kontsevich's ...
10 votes

Witten's proof of Morse inequalities, question on eigenvalues?

Why not go to Witten's actual paper? He explains it well, and here is a sketch: The operator $\Delta_s$ depends on a Morse function $f:M\to\mathbb{R}$, and it takes on the form $$\Delta_s = \Delta+ s^...
  • 16.1k
10 votes

Height function on 2-torus with only 3 critical points

Warning: this is not an immersion (it has twelve Whitney-umbrella-like pinch points) Here is a relatively simple explicit realization: the $z$ coordinate for the parametric surface$$(x,y,z)=(\sin(2u),...
10 votes
Accepted

Normal form of functions $(x^2+y^2)^n+$ higher terms

The expression $(x^2+y^2)^2 + x^5 + y^5$ cannot be written in the form $(z^2+w^2)^2$ for any smooth functions $z$ and $w$ of $x$ and $y$. (Just look at the Taylor series expansion.) Similarly, $n>...
10 votes
Accepted

Sard's theorem and Cantor set

It is not hard to construct a smooth function $f$ on $\mathbb R$ such that $f \ge 0$ with $f(x) = 0$ if and only if $x$ is in the Cantor set $E$. If $F$ is an antiderivative of $f$, the critical ...
9 votes
Accepted

Is a Morse function always the height function of some embedding?

The formal answer is yes. Moreover, the function does not have to be Morse, just any smooth function. Indeed let $f\colon M\to \mathbb{R}$ be any smooth function. Let us fix an imbedding $i\colon M\...
  • 19.3k
9 votes

Is a Morse function always the height function of some embedding?

This is trivially true: take an embedding $g:M\to \mathbb R^{n-1}$ and consider $(g,f):M\to \mathbb R^{n-1}\times \mathbb R\to \mathbb R$.
  • 24.5k
9 votes
Accepted

Kähler manifold with even-only singular cohomology

Assuming that by `cohomology' you mean integer coefficients, there is a general result saying what you want for simply connected manifolds of dimension $5$ or more, without the Kähler condition. ...
9 votes

Unstable manifolds of a Morse function give a CW complex

A generic perturbation of the metric makes the flow Morse Smale, (stable and unstable manifolds meet transversally). In this situation, the unstable manifold do form a CW-complex. The unstable ...
  • 51.9k
9 votes

Smooth Morse function from Forman's discrete Morse function

You can do the next best think. To a Forman-Morse function $f$ one can associate a flow on the manifold whose stationary points are precisely the barycenters of the faces of your simplicial ...
8 votes

Regular CW complex arising from a Morse decomposition

If the Morse function $f$ is perfect, then, for any choice of metric $g$, the attaching maps cannot be homeomorphism. Indeed if the Morse function was perfect, then the boundary operator of the ...
8 votes

Generalizations of the handle trading techniques

Yes; see Smale, On the structure of manifolds (Amer. J. Math. 84 1962 387–399) where it's shown that in high dimensions, you can eliminate handles under various connectivity assumptions. The h-...
8 votes
Accepted

Realizing Morse functions on $S^2$ as height functions

Any Morse function on $S^2$ may be realized by an embedding $S^2\hookrightarrow \mathbb{R}^3 \to \mathbb{R}$. For a Morse function $F:S^2\to \mathbb{R}$, take the equivalence relation with equivalence ...
  • 62.2k
8 votes

The handlebody decomposition of S^1 bundles over surfaces?

There is a standard way to get a Heegaard splitting that works more generally for 3-manifolds fibering over $S^1$. Take two copies of the fiber surface, and "tube" them together on either side. The ...
  • 62.2k
7 votes
Accepted

Using Discrete Morse Theory to represent hom classes

In section 11 of this paper I show that a discrete Morse function on a simplicial complex leads to a dynamical description of Forman's theory. More precisely there is a canonical flow associated ...
7 votes

Height function on 2-torus with only 3 critical points

I would recommend to look at the paper (here is a free original in russian) Elena Kudryavtseva, Realization of smooth functions on surfaces as height functions. (Russian) Mat. Sb. 190 (1999), no. 3, ...
7 votes

Can the constant rank theorem for smooth manifolds be generalized to nonconstant rank?

Your motivating question has a negative answer: Consider the inclusion $\iota:S^m\to\mathbb{R}^{m+1}$, which has rank at most $m$. If there were a smooth extension $f:D^{m+1}\to\mathbb{R}^{m+1}$ of ...
7 votes

What are good Morse Theory lecture notes and books?

These lecture notes were actually mainly devoted to the Morse Complex in the infinite dimensional setting; but they were thought to be suitable for finite dimensional manifolds as well (btw, you don't ...
  • 51.9k
7 votes
Accepted

Morse index in PDEs

In finite dimensional Morse theory, you study a function $f:M\to\mathbb{R}$ and look for it's critical points, i.e., where $(df)_p = 0$. Then, Morse theory says that a count of these critical points ...
  • 6,073

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