For questions about or involving the fundamental group.

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1answer
16 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 ...
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1answer
17 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
25 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
46 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 ...
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6answers
153 views

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

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 - \{ ...
2
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1answer
52 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 ...
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1answer
32 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 ...
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0answers
13 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 ...
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1answer
39 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
23 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. ...
2
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1answer
21 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?
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1answer
16 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
36 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
39 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
33 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
votes
2answers
55 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
30 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
17 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
60 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
48 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
48 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
39 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
56 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
35 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
35 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 ...
4
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0answers
26 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
18 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
52 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
55 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 ...
0
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0answers
31 views

Is the complementary of a 2 codimensional set in a complex space simply connected?

It is well-known that if X is a complex manifold and $A \subset X$ an analytic set of codimension at least $2$ then $\pi_1(X)=\pi_1(X\setminus A)$. Can we generalize this equality when $X$ is a ...
0
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1answer
31 views

Fundamental Group Computation $\mathbb R^2 \setminus S^1$ .

I need to compute the fundamental group of $\mathbb R^2 \setminus S^1$ . First I notice this is a disconnected set, hence I will have to differentiate between two cases. If I choose the based point ...
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3answers
76 views

How does attaching a 1-cell to a path connected CW-complex affect the fundamental group?

If A is the path connected CW complex and X is the new CW complex made by attaching a single 1-cell, is it true that the fundamental group of X is the same as that of A? I've tried to justify this ...
2
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0answers
24 views

Relation of fundamental group $\pi_1(X,x_0)$ and properties of C(X)

As stated by commutative Gelfand Naimark theorem, every unital C* algebra is of the form C(X) for some compact Hausdorff space X. Moreover two such algebras are isomorphic iff the corresponding ...
6
votes
1answer
116 views

Is this a different proof of the fundamental group being abelian?

I have proved the fundamental group of a topological group is abelian. But I've found nowhere the similar proof as mine. Everywhere I looked up, it was done either exploiting categorical properties or ...
4
votes
0answers
33 views

How to find fundamental groups and covering spaces of $\mathbb{RP}^2\vee \mathbb{RP}^2$?

The following is an exercise I was assigned in homotopy theory. Defined $X = \mathbb{RP}^2\vee \mathbb{RP}^2$. a) Find $\pi_1(X)$. b) Find the universal cover of $X$. c) Find all of its connected ...
0
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0answers
19 views

relation of fundamental group for different base point.

I want to know the relation between fundamental group based at $x_0$, $\pi_1(X,x_0)$ and fundamental group based at $x_1$, $\pi_1(X,x_1)$, where $X$ is a topological space. and $x_0, x_1 \in X$. What ...
4
votes
2answers
57 views

Fundamental group of Mapping Torus $M_h$: how to prove that the action is really $h_*$?

Let us work with the following setting: let $h$ be an automorphism (assume base point preserving) of a genus $g$ surface ($g>0$) to himself. $h \colon (\Sigma^g,\ast) \to (\Sigma^g,\ast)$ and ...
1
vote
1answer
22 views

How to show the constant path is the identity element in fundamental group?

Let $X$ be a topological space and $q$ is a point in $X$. Denote the fundamental group of $X$ based at $q$ by $\pi _1(X,q)$. Then how should I verify that the constant path $c_q(s)\equiv q$ is the ...
5
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1answer
64 views

Minimal-dimension example of (open) subset of $\mathbb{R}^n$ with trivial first cohomology but nontrivial fundamental group

As a follow-up to this question, I was wondering what dimension provides the minimal counterexample to the claims: If $U\subseteq\mathbb{R}^n$ is an open connected set with trivial $H^1(U)$, then ...
3
votes
1answer
41 views

Classification of line bundles by group homomorphisms from the fundamental group to $\mathbb{Z}_2$

Let $X$ be a path-connected manifold. Then by the classification of vector bundles, the collection of all (real) line bundles on $X$ is $$ \text{Vect}^1(X)\cong ...
3
votes
1answer
50 views

Does inclusion give an injective homeomorphism $\pi[G \cap B(a, r),a] \to \pi[G, a]$?

Let $G$ be a open, path connected subset of the plane and $a\in G$. Set $G' = G\cap B(a,r)$, where $B(a,r)$ is the open ball of center $a$ and radius $r$. Let $\gamma$ be a closed loop in $G'$ with ...
1
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1answer
36 views

When does triviality of $f_*$ imply null-homotopy of $f$?

I've been considering problems of the following type: Given certain topological spaces $X$ and $Y$ and a continuous function $f:X\to Y$, prove that $f$ is null-homotopic. The cases studied so ...
2
votes
1answer
53 views

Simply connected means universal-covering spaces

A covering space $p:(Y,y_0) \to (X,x_0)$ is called universal when $Y$ is simply connected (say that we restrict ourselves to path connected spaces, and locally path connected). I heard that this ...
2
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1answer
93 views

Past paper question without answers; could anyone provide an answer?

I apologize in advance I haven't my own approach here, it doesn't mean I haven't tried at all, I've just realized my lack of knowledge and practice in the subject. It would be great if I can get an ...
2
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2answers
88 views
+50

Fundamental group - space of copies of circle $S_1$

For $n>1$ an integer, let $W_n$ be the space formed by taking $n$ copies of the circle $S_1$ and identifying all the $n$ base points to form a new base point, called $w_0$. What is $\pi_{1}$($W_n , ...
2
votes
2answers
50 views

A technical detail in the construction of a covering space: Is the projection $p$ continuous without assuming semilocal simple connectedness?

I am studying the construction of a covering space with prescribed characteristic subgroup. For simplicity, I will outline the case where the characteristic subgroup is trivial (i.e. we're dealing ...
0
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0answers
34 views

Fundamental group from triangulation

How do I calculate the fundamental group of surface if I have been given it in triangulated form. I have attached an example triangulation I think I may have to simplify it further to a tree and ...
4
votes
1answer
266 views

Nontrivial element in first homology of Hawaiian earring

I am stuck at the following exercise from Hatcher section 3.3: 14. Let $X$ be the shrinking wedge of circles in Example 1.25, the subspace of $\mathbb{R}^2$ consisting of the circles of radius ...