2
votes
1answer
19 views

Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
0
votes
0answers
28 views

Doubt about local flatness of low dimensional embeddings

I would like to know if it is possible to have a simple curve $\gamma $ on a surface $S$ such that $\gamma$ is compact and embedded (i.e. with respect to the topology induced from $S$ it is ...
-2
votes
1answer
79 views

Cw complex $\Sigma_g$

Consider the oriented connected compact surface $\Sigma_g$ of genus $g$ with its standard CW structure. How do I write down the attaching map for the single $2$-cell and how can it be proven that it ...
1
vote
1answer
33 views

Topology of level surfaces

I have a level surface of the form $f(x,y,z,w)=0$ and also $g(x,y,z)=0$. Here f and g are differentiable! I need to decide if they are compact or not. Is there any criteria, theorem or anything? ...
1
vote
1answer
35 views

Is $\textrm{im}(f)$ homeomorphic to the torus less the inner equator?

Consider the map $id_{S^1}\times f:S^1\times [0, 1]\longrightarrow S^1\times S^1$ where $f:[0, 1]\longrightarrow S^1$ is given by $$f(t)=(\cos(\pi t), \sin(\pi t)).$$ Is it true that $\textrm{im}(f)$ ...
1
vote
1answer
32 views

Name of the surface with two sides and three boundaries

Once i have seen a 3d visualization of a surface with the following characteristics: it had three circular borders. If you imagine the surface inscribed in the earth globe, one of the borders would ...
2
votes
1answer
37 views

Whats the general idea for finding the mapping class group of the twice punctured sphere?

Consider the twice punctured sphere. Now I know the mapping class group of the twice punctured sphere is $\mathbb{Z}_{2}$, the cyclic group of order 2. However, I know one applies the alexander trick ...
1
vote
0answers
16 views

Action of Homeomorphisms on Proper Arc system.

Let $S_{g,n}$ be a surface of genus $g$ and with $n$ punctures. By an essential arc we mean an embeded arc (end points are in punctures) which is: Homotopically non-trivial i.e. not homotopic to a ...
2
votes
1answer
53 views

Does the connected sum depend on direction of gluing?

The connected sum of two surfaces (2-manifolds) is defined by removing a disk from each and gluing the cut edges: (Image adapted from Wikipedia) Does the resultant surface (up to homeomorphism) ...
2
votes
1answer
96 views

Topological surface thought experiment

Imagine a two-dimensional version of you lives on some compact, connected surface (orientable or non-orientable). How would you figure out on which surface you are living? Are there experiments you ...
6
votes
2answers
143 views

Closed, orientable surface whose genus is very hard to find intuitively

I'm introducing the Classification Theorem on closed and orientable surfaces in a talk on (intuitive) topology, and to motivate it I'd like an example of an embedding of a surface in $\mathbb{R}^3$ ...
1
vote
0answers
42 views

Proof of Euler Characteristic for Sphere

Theorem 1. All cell decompositions of a sphere $S$ have Euler characteristic 2. This is well-known, but I had this idea for an intuitive proof: for any cell decomposition $\Gamma$ with $V$ ...
3
votes
0answers
51 views

homomorphism of fundamental groups induced by a map of two surfaces

I am trying to find another proof of the following theorem Theorem Let $X$ and $Y$ be two compact surfaces of genus greater than $2$. Then every homomorphism $π_1(X,x_0)→π_1(Y,y_0)$ is induced by a ...
0
votes
1answer
35 views

Surface with border is homotopy equivalent to bouquet of circles

Why is any compact surface with non-trivial boundary homotopy equivalent to bouquet of circles? It was mentined in "Course homotopy topology" by Fomenko, Fuchs while calculating homotopy groups of ...
0
votes
2answers
110 views

Is a closed compact 2-Manifold that is embedded in euclidean 3-space always orientable?

I am sorry if this is a trivial question but I am a little confused right now so please bear with me. Since non-orientable compact 2-manifolds without boundary cannot be embedded in three-dimensional ...
2
votes
1answer
114 views

Is conformal equivalence the same as topological equivalence?

Is it true that if I take two surfaces that are topologically equivalent, I can find a conformal mapping between them?
3
votes
1answer
115 views

Euler characteristic for non-convex polyhedra.

I read a little introductory book to topology. It basically said that for any two-dimensional manifold (well maybe just the closed ones, as I think about it) its topology can be unambiguously defined ...
1
vote
0answers
47 views

some question in the proof of classification of compact connected surface

Each compact connected $2$-manifold $S$ has a proper triangulation $K$, so we can order all $2$-simplices of $S$, $F_1,F_2,\ldots,F_{k-2}$ such that $F_i$ meets $F_{i-1}\cup F_{i-2} \cup \ldots \cup ...
3
votes
1answer
59 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
218 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
79 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
1answer
66 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
203 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 ...
1
vote
2answers
549 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
69 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
47 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
81 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
201 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
549 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
105 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
200 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
2k 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
123 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 ...