48
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 ...
- 137k
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....
- 10.7k
17
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 ...
- 19.4k
16
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 ...
- 7,373
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 ...
- 18.6k
14
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 ...
- 53.6k
13
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 ...
- 4,111
11
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),...
- 17.5k
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 ...
- 35k
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
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>...
- 106k
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. ...
- 19.2k
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 ...
- 56.6k
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 ...
- 34.1k
9
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 ...
- 7,087
8
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 ...
- 106k
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-...
- 19.2k
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 ...
- 34.1k
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 ...
- 66.8k
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 ...
- 66.8k
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
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 ...
- 56.6k
6
votes
Accepted
Generalizations of the handle trading techniques
You might find the paper by C.T.C Wall: Geometrical connectivity I, J. London Math. Soc. 3 (1971), p. 597-604, interesting.
What Wall proves, entirely by handle trading, is that if $W:M_0 \to M_1$ is ...
- 20.6k
6
votes
Is the bordism from disjoint union to connected sum universal for connected manifolds?
I think the idea in Thomas Rot's answer (now deleted) can be modified to give a counterexample when $M_1$, $M_2$ and $N$ are connected.
Namely, assume $M_1$, $M_2$ and $N$ are all null-cobordant, and ...
- 35k
6
votes
Accepted
Constructing ($\infty, 1)$-category from Morse theory on a manifold
This problem has a long history going back at least to the paper of Cohen, Jones and Segal (available here: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0883-04.pdf).
In terms of your ...
- 18.6k
6
votes
Accepted
First order decidability of limit of gradient flow?
Overview: The boundaries of the basins of attraction are lower dimensional stable manifolds. In two dimensions, they are the arcs flowing from repelling fixed points to saddle points. I expect that ...
- 151k
6
votes
Unstable manifolds of a Morse function give a CW complex
Let us show that there exists a metric for which stable and unstable manifolds of a given Morse function are transverse.
By Kupka-Smale theorem, a Morse function $f$ on a manifold with a Riemannian ...
- 2,582
6
votes
Accepted
Index and length of closed geodesics
This follows from Bonnet-Myers. For a metric near the round metric (in the $C^\infty$ topology), the sectional curvature will be pinched below by $k > 0$, where $k\thickapprox 1$. Hence a segment ...
- 66.8k
6
votes
Accepted
Invariance of morse homology, doubt in proof in book "Morse Theory and Floer homology"
Oh, I happen to know the guy who wrote that PDF. As to your questions.
Yes, that's the idea. In $V \times A$, $F = f_0$ so the critical points are in one-to-one correspondence with those of $f_0$. ...
- 508
6
votes
Accepted
Bialynicki-Birula decomposition for real analytic varieties
No, consider the following $\mathbb{C}^*$-action on $\mathbb{CP}^2$ : $$z.[z_0:z_1:z_{2}] = [z_0:z.z_1:z^2.z_2] ,$$ along with the antiholomorphic map $\sigma ([z_0:z_1:z_2]) = [\bar{z_{0}}:\bar{z_{1}...
- 6,933
Only top scored, non community-wiki answers of a minimum length are eligible