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Results tagged with gt.geometric-topology
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user 25510
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
32
votes
How would a topologist explain "every Riemann surface of genus $g$ is hyperelliptic if and o...
A 19th century topologist would explain this by dimension count. By Riemann-Hurwitz, a surface of genus $g$ covering the sphere
with $2$ sheets has $2g+2$ ramification points which gives $2g-1$ free c …
- 91.8k
27
votes
Is every rational realized as the Euler characteristic of some manifold or orbifold?
The answer for connected 2-dimensional orbifolds is no. Euler characteristic is
$$\chi(O)=\chi(M)-\sum\left(1-\frac{1}{q}\right)-\frac{1}{2}\sum\left(1-\frac{1}{p}\right),$$
where $p,q\geq 2$ are inte …
- 91.8k
17
votes
What are the shapes of rational functions?
There is a characterization of Schwarzian derivatives of rational maps:
section 3 in the text:
http://www.math.purdue.edu/~eremenko/dvi/schwarz3.pdf
There is something similar also in arXiv:math/0512 …
- 91.8k
15
votes
Accepted
Do all combinatorially distinct fundamental polygons correspond to surfaces?
Yes. 2. Yes. (I suppose that the surfaces are "the same" if they are homeomorphic).
For 1, it is sufficient to check the definition of surface: that every point has a neigborhood homeomorphic to t …
- 91.8k
15
votes
Accepted
How many metrics of constant curvature exist on a Riemannian surface?
First on terminology. "Riemannian surface" is a surface already equipped with a Riemannian metric. So the question "how many metrics of constant curvature exist on a Riemannian surface" makes sense on …
- 91.8k
12
votes
Accepted
Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{...
On the first question (the universal cover of the complement of a lattice). The missing points are in the image, so it is not the map that "behaves" but the inverse map.
The inverse map behaves in a v …
- 91.8k
12
votes
Accepted
The behaviour of holomorphic mapping of curves
You are asking too many questions, some of them are very difficult.
Here are some answers.
Image of a Jordan curve under a rational function. Take a circle for $\gamma$.
Every continuous function o …
- 91.8k
10
votes
Does every orientable surface embed in $\mathbb{R}^{3}$
Here is a reference:
MR0304649 (46 #3781)
Rüedy, Reto A.
Embeddings of open Riemann surfaces.
Comment. Math. Helv. 46 (1971), 214–225.
He talks about Riemann surfaces, but every orientable topological …
- 91.8k
7
votes
Which polygons have *simple* periodic billiard paths?
Consider the simple polygonal billiard path itself. It is a polygon and it is convex,
because all interior angles are less than $\pi$. Now start with an arbitrary convex polygon.
It is a billiard path …
- 91.8k
7
votes
Are there some other notions of "curvature" which measure how space curves?
There are many things that are called "curvature". For example, Menger's curvature which
plays an important role in Anaysis. Search "Menger curvature" anywhere in Mathscinet, to find out what is this …
6
votes
Accepted
Nonexistence of sphere with one conical point [reference request]
The proof is very simple. Let $f$ be the developing map (take an isometry of some small disk on your surface to a region in the plane with constant curvature metric, and then perform analytic continu …
- 91.8k
6
votes
Accepted
Is a simple closed curve always a free boundary arc?
The answer is positive. For any Jordan curve, there is a homeomorphism of the Riemann sphere, which sends this curve
to a circle. Your statement immediately follows.
One proof of this statement involv …
- 91.8k
5
votes
Are there any books/articles that apply abstract coordinate free differential geometry to ba...
There is a calculus textbook which does this:
Bamberg and Sternberg A course in mathematics for students in physics, vol. II,
Chap. 22, "Thermodynamics".
- 91.8k
4
votes
A convex curve inside the unit circle
This has nothing to do with circles (or polygons). If one convex curve is inside another,
then the length of the inner curve is smaller.
- 91.8k
4
votes
Accepted
Construction of self-covering map of any surface
First of all, the Riemann-Hurwitz formula with $\chi<0$ implies that
for every self-covering $d=1$ so it is an automorphism. The only punctured surfaces with $\chi\geq 0$ are torus, sphere, and sphere …
- 91.8k