For questions about or involving the fundamental group.

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-1
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0answers
90 views

Why every fundamental group isn't a trivial fundamental group?

I don't understand exactly the definition of the Fundamental Group. Munkres defined Fundamental Group in your book 'Topology' like "Let $X$ be a space; let $x_0$ be a point of $X$. A path in $X$ that ...
0
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1answer
17 views

Why is semi-locally simply connected defined this way?

I would like to know why we define a space $X$ to be semi-locally simply connected if $\forall p\in X \exists U\ni p: i(\pi_1(U))=0\subset \pi_1(X)$ (SLSC), where $i$ is induced by ...
1
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0answers
15 views

Inverse limit defining etale fundamental group.

Let $(S,s)$ be a connected scheme with geometric point $s$. In many places, I can find the etale fundamental group being defined as $$\varprojlim_{X \to S} \text{Aut}_S(X)$$ Where $X \to S$ ranges ...
0
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0answers
21 views

Calculating fundamental groups using retracts

I want to know if my calculations are correct. I am trying to find the fundamental group of the figure $X_n$ in $\mathbb{R}^2$ obtained by drawing $n$ lines between two points (the lines only meet at ...
0
votes
2answers
48 views

Compute explicitly a fundamental group

I want to compute the $\pi_1(X)$ where $$X=\mathbb{R}^2-(([-1,1]\times \{0\})\cup (\{0\}\times [-1,1]))$$ my only tools at the moment are the basic definitions and the fundamental group of a circle, I ...
0
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1answer
33 views

If an open set is evenly covered by $p$, then its open subset is also evenly covered by $p$.

If $U$ is an open set evenly covered by $p: E\to B$ and $W$ is an open set contained in $U$, then $W$ is also evenly covered by $p$. I'm trying to prove this statement, but have a difficulty. So ...
1
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1answer
59 views

Computation of the fundamental group of $\mathbb{C}P^n$ using induction on $n$.

Let $n\geqslant 2$, I am asked to prove the following: Proposition. $\pi_1(\mathbb{C}P^n,\cdot)$ is isomorphic to $\pi_1(\mathbb{C}P^{n-1},\cdot)$. Proof. First, let us introduce some notation: ...
1
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0answers
33 views

What is the fundamental group of a cone?

I am reading an article on orbifolds and it describes the cone as the quotient of unit 2-dim. disc by a finite cyclic group of rotations. But how is it's fundamental group finite cyclic?
0
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1answer
38 views

exercize about the foundamental group of $\mathbb{P}^n(\mathbb{R})$

Let $p$ be a point in $\mathbb{P}^n(\mathbb{R})$ and $\Sigma$ the set containing all the projective lines passing through $p$. Given $s\in \Sigma$ we can define a continous closed path (let's say ...
1
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0answers
29 views

On the existence of a continuous section of $\zeta\in\mathbb{S}^1\mapsto(\zeta^m,\zeta^n)\in\mathbb{S}^1\times\mathbb{S}^1$.

Let $(m,n)\in\mathbb{Z}^2$ and let define the following map: $$f:\left\{\begin{array}{ccc} \mathbb{S}^1&\rightarrow&\mathbb{S}^1\times\mathbb{S}^1\\ \zeta&\mapsto&(\zeta^m,\zeta^n) ...
0
votes
0answers
14 views

Permuted action of the ramified covering

Let $f:E\rightarrow S^2$ be a ramified covering of degree n, and let $t_1,t_2,..t_m$ be all its points of ramifications. Pick a point $t\in S^2$ distinct from all $t_i$ and connect it with the points ...
4
votes
1answer
52 views

Glueing manifolds with boundaries and Seifert-Van Kampen theorem

I've seen many times the following application of the SVK theorem: Let $M$ and $N$ two smooth $n$-manifolds ($n\ge 3$) with boundary and suppose that they have the same boundary $B$. Now, after ...
1
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2answers
56 views

Fundamental group of a manifold minus a submanifold

Let $X$ be a smooth $n$-manifold, with $n\ge 3$, such that $\pi_1(X)=\left<a_1,\ldots,a_m\right>$ (free group over $m$ elements) and suppose that there is an embedding: $$S^1\times ...
2
votes
1answer
35 views

Fundamental Group of Solid Octagon with Labelling Scheme

I am studying for the qualifying exams I am taking last week and am struggling with the following problem: Here's what I've tried. Say we call the left side $X$ and the right side $X'$. Pick ...
3
votes
1answer
105 views

The fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed

Determine whether the fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed is trivial, infinite cyclic, or isomorphic to the figure eight space. I found this answer: ...
-1
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1answer
61 views

Counterexamples on homotopy equivalence and infinite product [closed]

Let $(X,a),(Y,b)$ be pointed spaces and $f:(X,a)\rightarrow (Y,b)$ be a continuous function. If the natural homomorphism $f_*:\pi_1(X,a)\rightarrow \pi_1(Y,b)$ is a group isomorphism, is $f$ ...
1
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0answers
44 views

Theta-space is a deformation retraction of the doubly-punctured plane, how to find equations.

That theta space is given by $S^1\cup(0\times[-1,1]) \subset\mathbb{R}^2$ it is said that this space is a deformation retract of the doubly punctured plane, here is the explanation I found: The ...
2
votes
1answer
64 views

The fundamental group of $B^2\times S^1$

In one exercise we are supposed to find the fundamental group of $B^2\times S^1$. It is given that the fundamental group is $\mathbb{Z}$, because we can show that $S^1$ is a deformation retract of ...
1
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1answer
69 views

Fundamental Group of Torus with Axis

I am studying for my comprehensive exams and have come across the following question, which I have been struggling with: Let $T$ be the torus given by rotating the circle ...
6
votes
2answers
203 views

What would the fundamental group of disjoint union look like?

Fundamental group of a wedge sum is a free product of fundamental groups. Hence, $\pi_1$ maps a coproduct of topological spaces with base points to a coproduct of groups. Since disjoint ...
1
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1answer
40 views

Can the exercise be solved this alternative way?(homomorphisms of fundamental groups)

I have this exercise: Let A be a subspace of $\mathbb{R}^n$; let $h \colon (A,a_0)\rightarrow (Y,y_0)$. Show that if h is extendabe to a continuous map of $\mathbb{R}^n$ into Y, then $h_*$ is ...
0
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1answer
75 views

Fundamental group of sphere with 2 handles with 2 mobius bands.

Is it real to calculate?? We have a sphere with with 2 handles and with 2 glued mobius bands (red on picture). So, i think we need to use Van Kampen Theorem 2 times. 1) $ X = X_1 \cup X_2 $ , ...
2
votes
1answer
41 views

If $Y$ is path-connected, then there is only one homotopy class of maps $[0,1] \to Y$

I have this exercise: If Y is path-connected, show that there is only one homotopy-class of continuous functions from $[0,1]$ to Y. My attempt: What I need to show is that if I have two ...
9
votes
3answers
131 views

Homotopy equivalent spaces have homotopy equivalent universal covers

A problem in section 1.3 of Hatcher's Algebraic Topology is Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of the path-connected, locally path-connected spaces $X$ and $Y$. ...
0
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1answer
48 views

How is this connection between the groups made?

I'll give some background before my question, you may skip that if you like: Lets say that you have the unit circle $S^1$. It can be proved that the map $p: \mathbb{R} \rightarrow S^1$, given by ...
4
votes
1answer
29 views

Deck transformations of cover of double mapping cylinder

On page 66 of Hatcher's Algebraic Topology, he discusses the universal cover of a space $X$ which is a cylinder with its edges glued to a circle by maps $z \mapsto z^m$ and $z \mapsto z^n$. He ...
2
votes
2answers
48 views

Why do we need surjectivity of f to get injectivity of $f_*$?

If we have two topological spaces X,Y and a continuous function between f, and $f(a)=b$. We get a function between the fundamental groups: $f_*: \pi(X,a)\rightarrow\pi(Y,b)$, which is given by ...
0
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0answers
43 views

Topology information of the object

I would like to know how to determine fundamental group, topological and homotopical equivalence of this object. OBJECT Can we know additional information from visual representation of this object?
2
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0answers
46 views

Generators of commutator subgroup of fundamental group of genus-2 surface

Recall the fundamental group of a genus-2 surface: $$ \pi_1(\Sigma_2) = < a_1, b_1, a_2, b_2 \mid [a_1, b_1][a_2, b_2] = 1 > $$ By which I mean a free group of four variables, divided out by ...
1
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0answers
22 views

Covering space of a topological group has same isotropy subgroup for each element in the fiber

A problem in May's Algebraic Topology book (on page 33) is Suppose $p:H \to G$ is a covering map of topological groups, and let $K < H$ be its kernel. Show that $k \mapsto (g \mapsto kg)$ ...
4
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1answer
70 views

Why is $\pi_1(F_g)^{ab} = \Bbb Z \langle a_1, b_1, \ldots , a_g, b_g \rangle$?

I am told that for a surface with genus $g$, call it $F_g$, the abelianization of $\pi_1(F_g) = \langle a_1, b_1, \ldots , a_g, b_g \mid [a_1, b_1] \cdots [a_g,b_g] = e \rangle$ is $\pi_1(F_g)^{ab} = ...
2
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1answer
33 views

Fundamental Group using Van Kampen

Let $ U = \mathbb{S}^{1} \times \mathbb{S}^{1} - interior\ of\ a\ disc$ and $V=\mathbb{K} - interior\ of\ a\ disc$, where $\mathbb{K}$ is the Klein bottle. Let $f : \partial U \to \partial V$ be a ...
4
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1answer
58 views

Exercise 1.2.19 of Hatcher's Algebraic Topology

I've been trying to prove exercise $1.2.19$ of Hatcher's algebraic topology: Show that the subspace of $\mathbb{R}^3$ that is the union of the spheres of radius $\frac{1}{n}$ and center ...
2
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1answer
36 views

Meaning of torsion elements in fundamental groups

I've been thinking today about the fundamental group of the projective plane which is the cyclic group of two elements ($\pi_1(\mathbb{R}P^2)=\mathbb{Z}_2$). This means there is only one class of ...
1
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1answer
148 views

Trivial Fundamental Group Notation (1 or 0)

I read in some books that have the notation $\pi_1(X)=0$ to mean $X$ has trivial fundamental group. My question is why 0 and not 1, i.e. $\pi_1(X)=1$? Is this a case of additive notation versus ...
2
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1answer
37 views

Map from Torus to Projective Plane inducing an Epimorphism on Fundamental groups

Is it true or false that there exists a continuous map from the torus to the projective plane inducing an epimorphism on the fundamental groups? I am quite lost on this problem, what I know is ...
4
votes
1answer
50 views

Homeomorphism from $U\subset \mathbb{R}^2 \rightarrow U$ which permutes points.

I'm trying to solve the following problem: Let U be a connected open set, $U \subset \mathbb{R^2}$, and consider $p_1,p_2,...,p_n \in U$ and $q_1,q_2,...,q_n \in U$. Show that exists an ...
2
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1answer
36 views

Finitely generated group acting cocompactly on a Manifold of bounded geometry

Want a justification of the following fact. If G is a finitely generated group, then there exist a Riemannian manifold $M$ such that the action of $G$ is cocompact isometric properly discontinuous. ...
3
votes
2answers
53 views

What is the fundamental group of the torus with two segments attached?

I'm trying to calculate the fundamental group of the following space: I've been thinking that I should apply Seifert - Van Kampen theorem but I haven't been able to choose some nice open sets $U$ ...
0
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1answer
46 views

Why is the fundamental group of a topological group independent of the basepoint?

Let $G$ be a topological group. I want to show that, up to isomorphism, $\pi_1(G, g)$ is independent of the choice of base point $g \in G$. Here is what I have so far: Since $ G $ is a ...
0
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1answer
39 views

Fundamental group of $GL^{+}_2(\mathbb{R})$

I know that fundamental group of $GL^{+}_2(\mathbb{R})$ is isomorphic to $\mathbb{Z}$. It's written everywhere. I don't know how to proceed to prove it, any hint ? Thanking you.
1
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1answer
23 views

How to prove that $\phi:G\rightarrow \pi_1(X/G,p(x_0))$ is a homomorphism of groups?

Let $G$ be a discontinuous group (this means that it acts discontinuously with finite stabilizers) of homeomorphisms of a simply connected, locally compact metric space $X$. Let $p:X\rightarrow X/G$ ...
2
votes
1answer
41 views

Surface groups and subgroups of fundamental groups

The fundamental group of any closed surface is a surface group. Let $S_3$ be the orientable surface of genus 3. Is $\pi_1(S_3)$ isomorphic to an index-3 subgroup of any surface group? We have 1 ...
0
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1answer
33 views

Getting intuition for Munkres' 2 dimensional CW complex construction:

I've been working my way through Munkres's Topology, and came across the following question which I'm having some difficulty wrapping my head around. Prove the following: Theorem: If G is a ...
1
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1answer
70 views

Fundamental group of the n-fold torus is nonabelian for n > 1

I'm attempting to show that the $n$-fold torus (i.e. the connected sum of $n$ tori, $T_1 \# T_2 \# \ldots \# T_n$) has non-abelian fundamental group when $n > 1$. It has been suggested that I ...
0
votes
2answers
31 views

Let $(G, *)$ be a group and let $\{g,h\}$ be a subset of $G$. Prove that $(g*h)^{-1}=h^{-1}*g^{-1 }$.

Let $(G, *)$ be a group and let $\{g,h\}$ be a subset of $G$. Prove that $(g*h)^{-1}=h^{-1}*g^{-1}$. I know that I should show that $X*Y=Y*X=e$. But I don't know how to calculate it.
1
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1answer
86 views

Does having the same fundamental group imply that two spaces have the same homotopy type?

I have to prove that two different topological spaces $X,Y$ have the same homotopy type. I've been able to prove so far that $\pi_1(X)=\pi_1(Y)$ but I don't know if this is enough to say that $X$ and ...
0
votes
0answers
11 views

Find and sketch a lift for the covering $p:\mathbb{R}\to S^1$ given by $t \to e^{2\pi i t}$

Actually my question is is it the same as the path $f: I\to S^1$ beginning at $(1,0)$ given by $f(s)=cos\pi s, sin\pi s)$ lifted to $f'(s)=s/2$ beginning at $0$ and ending at $1/2$?
1
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1answer
26 views

What are the conjugacy class and homotopy class of fundamental group in graph theory?

In the Zeta functions of graphs : a stroll through the garden 's 99 page , there is the define of conjugacy class, but I can't understand it well. Are there any difference between conjugacy class ...
1
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0answers
47 views

How does the fundamental group of the base space act on its universal cover?

I have a guess: Given $p : \tilde{X} \rightarrow X$, and fixing $x_0 \in X$, then $\pi_1(X, x_0)$ acts on $p^{-1}(x_0)$ in an obvious way. (Monodromy) Is this action the action that gives $X$ as a ...