Tagged Questions

3
votes
1answer
36 views

Cutting a sphere along a curve

i have a question. I think that is not difficult, but i can't find a solution.7 I want to show the following: Cutting a sphere along a curve always results in two discs. Therefore i want to use the ...
3
votes
1answer
59 views

If radial projection is bijective then is it a homeomorphism?

Suppose $S$ is a regular surface in $\mathbb{R}^3 $ and $0\not\in S$. Now consider the radial projection $f: S\to\mathbb{S}^2$ given by $$f(x)=\frac{x}{||x||} \hspace{5mm}\mbox{ for all $x\in S$}$$ ...
3
votes
1answer
70 views

Projective lines which can be viewed in some sense as surfaces

The complex projective line can be viewed as the $2$-sphere. I'd appreciate some examples of other projective lines (over any field or even ring) that can be viewed in some sense as a surface. I am ...
1
vote
0answers
44 views

Virtual knot diagrams on surfaces with genus?

To the best of my limited understanding, a virtual knot diagram may be thought of as the projection of an embedding of $\mathbb{S}^1$ in a 2-manifold with genus onto $\mathbb{R}^2$. That is to say it ...
0
votes
1answer
130 views

$S-\{p\}$ admits a bouquet of circles as deformation retract.

Let $S$ be a closed compact surface, $p\in S$ and $X=S-\{p\}$. Show that X admits a bouquet of circles as deformation retract. How many circles? I'm starting to study algebraic topology and I can't ...
0
votes
2answers
171 views

Euler characteristic of a surface

It is known that a closed orientable surface of genus $g$ has Euler characteristic $2-2g$. According to this, the open disc being of genus $0$ should have Euler characteristic $2$, but this ...
3
votes
0answers
59 views

Do we really need to use the Jordan-Schönflies Theorem to prove that every surface can be triangulated?

I have read that most proofs of the triangulability of surfaces require the use of the Jordan-Schönflies Theorem. However, is such high-tech machinery really needed? The problem is that 3-manifolds ...
1
vote
0answers
35 views

Sign-preservation of continuous map in a small neighborhood

So I was reading a small book on surfaces called "Mostly Surfaces" which is available for free in the internet: http://www.math.brown.edu/~res/Papers/surfacebook.pdf In page 32, the author decides ...
1
vote
1answer
68 views

How do I find the number of vertices on a planar diagram of a surface?

Cheers, I have a question which I just do not seem to see the answer for: I am proving the classification theorem for compact surfaces and use planar diagrams as representation of the surfaces. I ...
1
vote
1answer
99 views

Pinched torus generalization

The pinched torus is homeomorphic to a sphere with two (different) points identified.           What is the name and topological structure of the ...
2
votes
1answer
264 views

Parametric Equations for a $2$-torus

I know that for a torus (with one hole) the parametric equations describing it are $x= (c + a\cos v)\cos u, y= (c + a\cos v)\sin u, z= a\sin v$, where $c$ is the radius from the center of the hole to ...
2
votes
1answer
94 views

The classification of surfaces

Can we completely classify the simply-connected surfaces (with or without boundary) in $\mathbb R^3$ up to homeomorphism?
2
votes
2answers
126 views

Replacing two cross-caps by a handle

For a non-orientable surface, we can replace a handle by two cross-caps. Can we do the opposite i.e replace any two cross-caps by a handle? Any help is appreciated!!
13
votes
3answers
889 views

Cutting a Möbius strip down the middle

Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right ...
5
votes
2answers
116 views

Prove that any shape 1 unit area can be placed on a tiled surface

Given a surface of equal square tiles where each tile side is 1 unit long. Prove that a single area A, of any shape, but just less than 1 unit square in area can be placed on the surface without ...