1
vote
0answers
32 views

Geometric definition of the stable commutator length

In his book, D.Calegari proves the equivalence of the algebraic and geometric definitions of stable commutator length (Proposition 2.10, p. 15). I actually have some difficulties in understanding the ...
3
votes
1answer
67 views

simple closed curve is nullhomologous iff is separable

A simple closed curve $\gamma$ in an orientable genus $g$ surface $M$ is nullhomologous if and only if $M \setminus \gamma$ consists of two connected components, one of which is a surface $N$ with ...
0
votes
1answer
31 views

Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
1
vote
2answers
117 views

Representation of nullhomologous loop on compact surface as a product of commutators.

Why this sentence is true?: Assume that $M$ is compact surface and $f: S^1 \to M$ is nullhomologous and without selfintersections. Letting $g$ be the genus and $b$ the number of boundary components ...
3
votes
0answers
32 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
-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 ...
3
votes
1answer
61 views

For compact surface $M$ and loop $f$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such that $f \notin \ker(\phi)$

Why is this sentence true? For every not nullhomologous loop $f$ without selfintersections on orientable compact surface $M$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such ...
3
votes
1answer
42 views

Does a pseudo-Anosov homeomorphism of a punctured surface possess infinitely many periodic points?

In A Primer on Mapping Class Groups by Farb and Margalit theorem 14.19 implies that every pseudo-Anosov homeomorphism $f:S \rightarrow S$ on a compact surface $S$ possesses infinitely many periodic ...
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
49 views

the fundamental group of punctured surface

Let $S_{g,m}$ be a surface of genus $g$ with $m$ punctured, we know the fundamental group of $S_{g,0}$ is $$ \pi_1(S_{g,0}) = \left\langle a_1, b_1, \dots, a_g, b_g {~\large\mid~} [a_1, b_1] \dots ...
3
votes
1answer
93 views

Homology subgroups generated by non-intersecting cycles

Suppose I have a closed genus $g$ surface. I can pick a canonical homology basis for the surface by picking $g$ "A-cycles" $a_1,\ldots,a_g$, and then $g$ "B-cycles" $b_1,\ldots,b_g$, represented by ...
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 ...
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 ...
1
vote
1answer
124 views

Covering of orientable surface (Hatcher)

The following is an exercise from Hatcher, Algebraic Topology, that I'm struggling with (exercise 2.2.23): Show that if the closed orientable surface $M_g$ of genus $g$ is a covering space of $M_h$, ...
1
vote
2answers
231 views

Homology of orientable surface of genus $g$

I came across the problem of computing the homology groups of the closed orientable surface of genus $g$. Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems ...
2
votes
0answers
30 views

Difference between diffeomorphisms fixing a point or a whole neighborhood.

Let $S_g$ be a closed orientable surface of genus $g$ and $S_{g}^1$ a closed orientable surface with one boundary component. Let $p$ be in $S_g$ and let's note $\mathrm{Diff}_+(S_g,p)$ the set of ...
3
votes
2answers
59 views

Are there odd-sheeted coverings of non-orientable surfaces by orientable surfaces?

For any non-orientable surface (compact,connected) $X$ with genus $h$, we have a $2n$-sheeted cover of $X$ by an orientable surface $Y$ first by covering $X$ by $\Sigma_{h-1}$ (a double cover) and ...
2
votes
0answers
64 views

Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
2
votes
0answers
95 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
1
vote
2answers
49 views

Is there a Covering Map $\Sigma_3^1\to \Sigma_2^1$

Let $S_{g,n}^b$ be a genus $g$ surface with $b$ boundary components and $n$ punctures. I'm having some trouble with these past qualifying exam questions: Is there a covering map $p\colon ...
1
vote
1answer
143 views

Difference between free homotopy and isotopy. Numer of non-isotopic curves.

I realize that whenever I think of two simple closed curves in a surface being isotopic I actually think of them as being freely homotopic (intuitively). I am really confused now. So I have the ...
7
votes
1answer
110 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic)

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
0
votes
1answer
121 views

About zeros of vector fields in compact surfaces

I'm studying compact surfaces and in particular the relationship between zeros of vector fields defined on them and Euler characteristic of the surface herself. Let be $S$ a compact (smooth) surface ...
1
vote
1answer
94 views

Hatcher's Problem 3.2.18

How can I prove that: For the closed orientable surface $M$ of genus $g \geq 1$, show that for each nonzero $a \in H^1(M; \mathbb Z)$ there exists $b \in H^1(M; \mathbb Z)$ with $ab \neq 0$.
1
vote
1answer
74 views

Identify the Euler characteristic of the edge word $ abc^{-1}b^{-1}da^{-1}d^{-1} c $

Identify the Euler characteristic of the edge word $ abc^{-1}b^{-1}da^{-1}d^{-1} c $. The Euler characteristic is $$ X=V-E+F$$ where $V$, $E$ and $F$ are the vertices, edges and faces ...
1
vote
1answer
56 views

Uniqueness of Seifert graphs

If we make the bands and disks of a Seifert surface really small and really thin the surface collapses to a graph. It is called a Seifert graph. If it is not a directed and weighted graph, can we ...
2
votes
1answer
59 views

Graphs from Seifert surfaces

Given a Seifert surface if we make the disks and bands infinitely small and thin it becomes a graph where the disks are vertices and the bands are edges. Can we say that following theorem, For ...
1
vote
2answers
124 views

Uniqueness of Seifert surfaces of knots

I know the theorem that Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. So, can we also say that for every unique Seifert surface there is an unique knot and vice ...
4
votes
1answer
166 views

Representation of (co)homology classes of $3$-manifolds by embedded surfaces

Let $M$ be a closed oriented $3$-manifold. Theorems in algebraic topology allow us to identify $$H_2(M) \ \cong \ H^1(M) \ \cong \ \langle M,S^1\rangle$$ where (co)homology is meant with integer ...
3
votes
1answer
223 views

Classification of fundamental groups of non-orientable surfaces

I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$. I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
12
votes
1answer
477 views

A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
0
votes
1answer
201 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 ...
6
votes
0answers
132 views

Normal subgroups of the fundamental group of a non-orientable surface.

Let $N^2_g$ be a non-orientable closed genus $g\geq 2$ surface. Is there a way to explicitly list the normal subgroups of $\pi_1(N^2_g)$ in terms of generators and relations? I am interested in ...
1
vote
2answers
539 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
2answers
120 views

What's the K-group of a surface?

What's the K-group of a surface? I also want to know how to calculate such group and if there is a explicit characterization of the generators.
4
votes
0answers
206 views

Morse theory and homology of an algebraic surface (example)

Let $T_n$ denote the $n$-th Chebyshev polynomial and define $f_n(x,y,z)\!:=\!T_n(x)\!+\!T_n(y)\!+\!T_n(z)$ and $$Z_n:=\mathcal{Z}(f_n) \subseteq \mathbb{R}^3,$$ the Bachoff-Chmutov surface, where in ...
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 ...
3
votes
1answer
295 views

Rank of first homology group for surface with punctures?

I feel like this question will be a head-slapper once I figure out the answer, but for the moment I'm having trouble! Let $M$ be a compact, connected, orientable 2-manifold of genus $g$ with $b$ ...
3
votes
1answer
159 views

triangulation of pair of pants

How can we triangulate a pair of pants in a simple way? I am looking for some triangulation where I can compute the Euler characterstic easily (which is -1 for a pair of pants).
3
votes
2answers
85 views

Group of automorphisms of an orientable surface

If we consider the group of automorphisms of an orientable surface, then the subgroup that contains the orientation-preserving automorphisms will be of index two. Why is that? Any explanation will ...
7
votes
1answer
149 views

Why can all surfaces with boundary be realized in $\mathbb{R}^3$?

I'm having trouble comprehending an informal proof of the fact that all compact surfaces with boundary can be realized in $\mathbb{R}^3$. I'm trying to find a proof of it on the internet, but I can't ...
2
votes
2answers
199 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!!
2
votes
0answers
41 views

ruling out non Pseudo-anosov automorphisms

We are given a fibration $S\to M\to S^1$ where S is a compact hyperbolic surface, M a 3-manifold and $S^1$ the circle. Topologically speaking, it is clear that M has to be the mapping torus ...
2
votes
1answer
160 views

Supposedly “trivial” implication that injective surfaces are incompressible

My question is about a passage in Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev. Let $F$ be a surface in some $3$-manifold $M$. $F$ is called incompressible if for every ...
8
votes
2answers
500 views

What are all topological spaces obtained by gluing the edges of a triangle?

I am currently learning about polygonal presentations of surfaces. In the notation I'm using (following Lee's "Topological Manifolds"), $\langle a, b \ |\ aba^{-1}b^{-1}\rangle$ is a presentation of ...
6
votes
1answer
179 views

How to correct a wrong proof about the Birman exact sequence?

I've given a proof of the exactness of the Birman exact sequence of groups: $$1\to\pi_1(S_{g,r}^s)\to MCG(S_{g,r}^{s+1})\overset{\lambda}{\to} MCG(S_{g,r}^s)\to 1$$ making use of classifying spaces ...
4
votes
1answer
121 views

Article or book explaining rigorously facts about the mapping class group

I would like to know more about relationships between the mapping class group of an orientable surface with negative Euler's characteristic and moduli spaces. In particular, I would like to have a ...