For questions about or involving the fundamental group.

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1answer
21 views

What is a primitive element in a Fuchsian group?

I am reading some introductory texts about Selberg's trace formula for hyperbolic surfaces and I have encountered the concept of "primitive element" $\gamma$ in a hyperbolic Fuchsian group $\Gamma \...
1
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1answer
42 views

Can't see why particular homotopy is continuous

I'm checking the group laws for the fundamental group of $(X,x_{0})$: in particular I'm trying to show that $\gamma \simeq \gamma \cdot e$ , where $\gamma$ is a loop based at $x_{0}$, $e$ is the ...
0
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1answer
50 views

A relation between the index of a fundamental group and the covering map

My professor just enunciated this statement: $|\pi_1(X,x_0):\pi_1(\tilde X,\tilde x_0)|=\#P^{-1}(x_0)$ where $P$ is the covering $P:\tilde X\rightarrow X$, such that $P(\tilde x_0)=x_0$. I've ...
0
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1answer
69 views

Cayley graph of the fundamental group of the 2-torus

Does anybody knows some description or some good picture on the internet for the Cayley graph of the fundamental group of the 2-torus, when this is constructed by connected sum of two torus. I know, ...
3
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2answers
70 views

Is there any 'nice' space with fundamental group $\mathbb{Z}_3$?

I'm trying to build up intuition for the fundamental group, as it occurs in physics. In the simplest examples, the fundamental group is trivial, or $\mathbb{Z}$, or $\mathbb{Z}^n$. We can also get $\...
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0answers
70 views

How can we assume the first homology group of the complement is zero when constructing a Casson handle?

I am currently working through Scorpan's Wild World of 4-Manifolds specifically the section on Casson Handles. On page 78, he says if $D$ is the core of the handle after $n$ stages we may assume $\...
5
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1answer
77 views

Why does Van Kampen Theorem fail for the Hawaiian earring space?

The Hawaiian earring space has notoriously complicated fundamental group, and is essentially not as simple as the wedge sum of countably many circles whose fundamental group is straightforwardly given ...
0
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0answers
37 views

Calculating The Fundamental Group of the Hopf Fibration

I want to calculate the fundamental group of the Hopf fibration (or rather, the fundamental group of the total space of the fibre bundle that is the hopf fibration). I know that $S^3$ is simply ...
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0answers
11 views

Fundamental group of a topological group [duplicate]

Given $(G,\cdot)$ a topological group with identity $e$, is it always true that $\pi_1(G,e)$ is abelian? For what it's worth: I have already shown that if $f,g\in\Omega(X,e)$, then $[f \times g]=[f\...
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0answers
19 views

Equivalent definitions for simple connectivity

For a path connected space $X$, it is simply connected iff any two paths sharing endpoints are homotopically equivalent. Here simple connectivity means it has trivial fundamental group. I'm doing ...
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0answers
38 views

The complement $\mathbb{R^3}-A$ of a single circle $A$ deformation retracts onto the wedge of a Circle and a sphere. 2$

Here is my intuitive idea. We can pull all the points lying outside a sphere of some fixed radius containing the circle onto the sphere without disturbing the points inside the sphere. Now we have a ...
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0answers
40 views

Calculate the fundamental group and homology $S^2\cup T$

I am currently working through an old qualifying exam problem: Calculate the fundamental group and homology groups of the space $X$ obtained from the union $S^2\cup T$ where $S^2$ is the 2-sphere and ...
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1answer
48 views

Presentation of $\pi_1$ of compact orientable surface by induction?

I need to prove by induction $\pi_1(\Sigma_g)= \left\langle a_1,b_1,\dots ,a_g,b_g\mid \prod_i [a_i,b_i] \right\rangle$. For genus 1 this holds since $\pi_1(T^2)\cong \mathbb Z\times \mathbb Z$. For ...
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0answers
63 views

Why is the fundamental group of the plane with two holes non-abelian?

I know $\pi_1(\mathbb{R}^2\setminus\{x,y\}) = \mathbb{Z}\ast\mathbb{Z} = \langle a,b\rangle$, but it's non-abelian-ness isn't obvious to me. Specifically, I draw a box and two points to represent $x$...
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0answers
45 views

Fundamental group of cylinder

I calculated the fundamental group of the cylinder, $C$, using the following method: triangulate $C$ find max contractable subspace realise generators on remaining 1-simplices I found the ...
3
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1answer
91 views

“Representability” of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$?

In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic ...
0
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1answer
45 views

Reference for: simple closed curves generate the fundamental group

In a 2-complex $X$, it is "obvious" that the simple closed curves through the $0$-cell $v$ generate the fundamental group $\pi_1(X, v)$. (By a "simple closed curve" I mean a loop which does not self-...
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2answers
37 views

Is this the projective plane or the Klein bottle? (Fundamental polygon)

I am trying to identify the topological type of this fundamental polygon and I think it is the projective plane or the Klein bottle If we treat the top green and red arrows a single arrow then we ...
0
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1answer
39 views

Link between $\mathbb{Z}$ and the fundamental group's of 'common' topological spaces

I have noticed that some of the most common topological space have fundamental groups related to $\mathbb{Z}$, as we can see below: Why is this the case? Is it is because they are all realted ...
0
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1answer
28 views

Fundamental group of simple graphs

Find the fundamental groups of graphs A, B and C as shown: They look simple but I am unsure what their fundamental groups would be. I was thinking that $A$ and $B$ are generated by just one ...
1
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1answer
25 views

principal bundle morphism preserves fundamental group

Two related questions. What is the morphism for principal bundles? Does it "preserve" fundamental groups? Fibre bundle morphisms usually preserve "the structure on the fibre". I am not sure how to ...
0
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1answer
29 views

Triangulation of double-holed torus to calculate fundamental group

Show that the fundamental group of the double-holed torus is given by: $\pi_1=<a, b, c, d | aba^{-1}b^{-1}=cdc^{-1}d^{-1}>$ I have ended up with the following identification diagram ...
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1answer
47 views

The fundamental group of a point is $1$

Show that the fundamental group of the point space $p$ is given as $\pi(p, w_0)=1$ where $w_0$ is the base point This is probably somewhat trivial, but I am looking for a proof. I am familiar with ...
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6answers
176 views

Examples of $\pi_1 (X) = \mathbb{Z}$ [closed]

I want to know some examples of topological spaces whose fundamental group is isomorphic to set of integers. First, of course i know $\mathbb{S}^1$, and its deformation retract, $\mathbb{R}^2 - \{ 0,...
2
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1answer
60 views

Fundamental group of the n-holed torus

I am trying to determine the fundamental group of the n-holed torus I know that the fundamental group of the torus is $\mathbb{Z} \times \mathbb{Z}$ The n-holed torus deformation retracts onto n ...
1
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1answer
34 views

Fundamental group of the complement of $S^1 \cup Z \subset \mathbb{R^3}$

I want to calculate the fundamental group of the complement of this space: This is the complement of $S^1 \cup Z \subset \mathbb{R^3}$ where $S^1$ is the unit circle in the $xy$-plane and $Z$ is ...
0
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0answers
18 views

Are Fuchsian groups without elliptic and parabolic elements at most countable? [duplicate]

Let $G \subset PSL(2, \Bbb R)$ be a discrete subgroup without elliptic or parabolic elements. Does it follow that it is at most countable? Subgroups as above have the property that the quotients of ...
0
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1answer
40 views

Understanding the Fundamental Group

Let us have a space $X$. Define the fundamental group $\pi_{1}(X,x_0)$ for some point $x_0 \in X$. If I understood it well, this group contains the path-homotopy classes that are consisting of paths ...
2
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1answer
24 views

Given that $H^1(X)=0$ on a connected space, show that all maps to $X\to S^1$ are null homotopic

Let $X$ be a path-connected, locally path-connected topological space, with $H^1(X)=0$. I would like to show that any map $f:X\to S^1$ is null homotopic, but I haven't really made any progress. ...
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1answer
27 views

Fundamental group of cylinder -triangulation method

Is this correct? Can we conclude that the fundamental group is trivial since there are no remaining generators on 1-simplices?
0
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1answer
20 views

Fundamental group Klein Bottle triangulation

I have been trying to find the FG of the Klein bottle, and I was wondering if someone could verify that this process is correct. After triangulating it, I then found a maximal tree (shown in yellow) ...
0
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1answer
39 views

Fundamental group of the sphere via triangulation

I know that the fundamental group of the sphere is zero, i.e. $\pi(S^2)=0$ I want to show this by triangulation, i.e: Triangulate the sphere Draw maximal tree Draw maximal contractable subspace ...
0
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1answer
41 views

Fundamental group of a tree?

Find the fundamental group of the space $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$. $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$ where $T$ is a graph $T$ is the graph made of $3$ edges with a ...
0
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0answers
35 views

I don't understand what a Fundamental group is

I have been staring at the definition for days, drew diagrams but I don't understand as to what its elements are The fundamental group $\pi_1(X,x)$ at a base point $x$ is a set of rel $\{0,1\}$ ...
3
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2answers
61 views

Fundamental group of $\mathbb{R}^3$ minus trefoil knot

Let $ \ T \subset \mathbb{R}^3 \ $ be the trefoil knot. A picture is given below. I need a hint on how to calculate the fundamental group of $ \ X = \mathbb{R}^3 \setminus T \ $ using Seifert-van ...
1
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1answer
31 views

Computing fundamental group of the mapping torus $\pi_1(T_{z\mapsto z^r}(S^1))$

I am trying to compute the fundamental group of the mapping torus of $f(z) = z^r$ for $r\in \mathbb{R}$ on the domain $S^1$. So, the space is $S^1\times I$ with $(z, 0)$ identified with $(z^r, 1)$. ...
0
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1answer
23 views

Problem computing the fundamental group of the double torus.

To compute this I want to use the Seifert-van Kampen Theorem, then I choose $U=\text{Left Torus}/{point}$ and $V=\text{Right Torus}/{point}$ so then the intersection $U \cap V$ is a cylinder with out ...
1
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1answer
61 views

Computing the fundamental group

I want to compute the fundamental group of the double Torus using the Seifert-van Kampen theorem so then I choose $U=\text{double Torus} / \{\text{point} =x_1\}$ and $V=D$ the disc. The thing is ...
0
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1answer
50 views

Unique homomorphisms between fundamental groups of topological spaces

Let $u:A\rightarrow B$ be a continuous map of topological spaces, $a\in A$, $b=u(a)$. How do I prove that there exists a unique group homomorphism $$u':\pi_1(A,a)\rightarrow\pi_1(B,b)$$ ...
5
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2answers
41 views

Show that $\pi(X, x) = [e_x]$ if $X$ is a finite topological space with the discrete topology.

Show that $\pi(X, x) = [e_x]$ if $X$ is a finite topological space with the discrete topology. I want to show this by showing that there does not exist any path $f$, $\forall x, y \in X$. Assume for ...
6
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2answers
57 views

Is the Fundamental Group of space with contractible universal cover torsion free?

Some classmates and I were working on the following question - is the fundamental group of the Klein Bottle $K$ torsion free? We have the following presentation: $$\pi_1(K) = \langle a,b: aba = b \...
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1answer
40 views

CW structures of identification spaces, fundamental groups and universal covers

I am working on this exercise concerning the relation between some identification spaces and their CW structure and I am a bit confused. First, the question is the following: For each identification ...
3
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1answer
80 views

Relations between center (fundamental group) and (co)root and weight lattices for Lie groups

I would like to find some explanation or reference for the following facts, provided they are correct, and clarify some of the assumptions. Denote by $G$ a (perhaps semisimple compact connected) Lie ...
2
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1answer
37 views

Infinite wedge sum of circles and a space homotopy equivalent

Given a space $G = \bigcup_{n=1}^{\infty} A_n$ where $ A_n $ is a circle $ C[ (n,0),n] \in R^{2}$ I'd like to show that its fundamental group is an infinately generated free group. So let's say $a_i $ ...
0
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0answers
40 views

Fundamental group and universal cover for this quotient space

For a non-zero integer $p$ define the topological space $L_p$ by: \begin{equation} L_p=\mathbb{D}^2\sqcup\mathbb{S}^1\big{/}z\sim{z^p} \end{equation} Check that $L_1\cong\mathbb{D}^2$, and more ...
5
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0answers
30 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces $...
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0answers
21 views

Image of map of étale fundamental groups

Let $X$ be a connected (locally Noetherian) scheme and $\phi: Y \rightarrow X$ be a finite etale Galois cover of degree $d$, i.e. $Y$ is connected and there are $d$ $X$-invariant automorphisms of $Y$. ...
1
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1answer
28 views

IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$?

IF $S$ is a torus and $p,q$ are two different points of $S$, how can I calculate $\pi (S \setminus\{p\})?$ and $\pi (S \setminus \{p,q\})$? $\pi(X)$ denotes the fundamental group of $X$.
4
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0answers
55 views

group actions of fundamental groups on homotopy groups

Let $\pi_n(\mathbb{R}P^n)$ be the $n$-th homotopy group of the $n$-dimensional projective space. Then by the long exact sequence of homotopy groups associated to the fibration $S^n\to \mathbb{R}P^n\to ...
0
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0answers
57 views

Computing fundamental group of the complement to three infinite straight lines, and of complement to $S^1 \cup {Z} $

Question 1: Find the Fundamental group of the complement to three infinite straight lines that have no common points in $\mathbb{R^3}$ Question 2 Compute the fundamental group of the complement of $...