6
votes
1answer
192 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
2answers
276 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 ...
9
votes
2answers
522 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 ...
5
votes
1answer
195 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 $G$ ...
1
vote
0answers
264 views

Transversality and isolated degenerate critical points

Maybe some of the following statements are not precise. Please correct them. Let $M$ be a compact smooth manifold. Let $f: M \to {\mathbb R}$ be a Morse function. Then a generic Riemannian metric $g$ ...
4
votes
0answers
421 views

Topological version of two results in smooth Morse theory

Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post Morse theory in TOP and PL categories? for references). It is well known ...
5
votes
2answers
429 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 ...
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 ...
6
votes
2answers
527 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 ...
17
votes
2answers
990 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 ...
14
votes
1answer
853 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 ...
13
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 ...
17
votes
1answer
768 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
724 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 ...
3
votes
1answer
463 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
447 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 ...
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 ...
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 ...