Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

2
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1answer
179 views

Find a covering space of $T$ corresponding to the subgroup of $\mathbb{Z}\times\mathbb{Z}$ generated by the element $m\times 0$

I know that $p: \mathbb{R}\to \mathbb{S}^1, t\mapsto(\cos2\pi t, \sin2\pi t)$ is a covering space and so $p\times p:\mathbb{R}\times\mathbb{R}\to \mathbb{S}^1\times \mathbb{S}^1$ is also, and since $\...
0
votes
1answer
110 views

Fundamental group of a triangle with edges identified

To construct the fundamental group of a triangle with edges identified, say ab, bc and ca, I considered the interior $U$ which is contractible and $V$ a thin neibourhood of the perimeter which can be ...
0
votes
1answer
136 views

Fundamental groups (isomorphism)

Regarding the proof of the theorem that says the fundamental group of a product space is isomorphic to the product the fundamental groups of the two spaces. A more detailed proof that I manage to find ...
0
votes
1answer
54 views

Fundamental group of $\mathbb{C}^2\setminus \Delta?$

Let $\Delta=\{(z,z): z\in \mathbb{C}\}$. What will be the fundamental group of $\mathbb{C}^2\setminus \Delta?$ I am unable to proceed at all. Please give me a hint.
0
votes
1answer
72 views

Show that if $n>1$, every continuous map $f:\mathbb{S}^n\to \mathbb{S}^1$ is nulhomotopic

Show that if $n>1$, every continuous map $f:\mathbb{S}^n\to \mathbb{S}^1$ is nulhomotopic, and show that every continuous map $f: P^2\to \mathbb{S}^1$ is nulhomotopic. I have already demonstrated ...
1
vote
1answer
85 views

Spaces with fundamental group isomorphic to a free group with $(n-1)$ generators

I'm trying to solve the following question: Let $X=A\cup B$ be an open cover of $X$ where $A,B$ are simply connected and $A\cap B$ consists of $n \ge 2$ path connected components. I want to show that ...
1
vote
1answer
160 views

Fundamental group of infinite row of tori [duplicate]

Consider a family $ \lbrace T_i \rbrace_{i\in\mathbb{Z}} $ of tori indexed on the integers such that every torus is attached to the preceding one and to the subsequent by a single point. How can we ...
0
votes
1answer
33 views

On the quotient space $X^n/S_n$, for a Hausdorff, contractible, locally path connected topological space $X$

Let $X$ be a topological space. For $n \ge 2$, the group $S_n$ acts naturally on $X^n$ as $\sigma . (x_1,...,x_n)=(\sigma(x_1),...,\sigma (x_n)), \forall (x_1,...,x_n) \in X^n$ . So we can consider ...
1
vote
1answer
159 views

When does conservative imply faithful for functors?

I was thinking about homotopy theory and classical questions such as "do homotopy groups characterize the homotopy type of a space ?", and came up with this argument : $hTop$ (the category of ...
1
vote
1answer
28 views

For a vector subspace $W$ of $\mathbb R^n$, where $n \ge 4$ and $\dim W\le n-3$, is $\mathbb R^n\setminus W $ simply connected?

Let $W$ be a vector subspace of $\mathbb R^n$, where $n \ge 4$ and $\dim W\le n-3$. Then is $\mathbb R^n\setminus W $ simply connected ? I can only see that it is path connected. Please help.
0
votes
1answer
45 views

Simple connected ness of $\mathbb R^n$ minus a compact line segment , for $n \ge 3$

Let $n \ge 3$ and let $L$ be a compact line segment i.e. of the form $[a,b]$ in $\mathbb R^n$. Then is $\mathbb R^n \setminus L$ simply connected ? I can only see it is path connected. Please help....
4
votes
0answers
123 views

For $n \ge 3$, is $\mathbb R^n \setminus A$ simply connected for any countable set $A$?

This Prove that $\mathbb R ^n $ without a finite number of points is simply connected for $n\geq 3$ shows that $\mathbb R^n$ minus a finite set is simply connected for $n \ge 3$. I know that $\mathbb ...
0
votes
2answers
343 views

CW complex with fundamental group $\Bbb Z/n$

I'm just learning about CW complexes. I came across this answer: "One way of constructing a connected $2$-dimensional CW complex with fundamental group $\Bbb Z/n$ (for some integer $n\geq 2)$ is to ...
1
vote
1answer
74 views

Fundamental group S^2/~

Let $X = \mathbb{S}^2/\sim$ where $(cos(\theta), sin(\theta), 0 ) \sim (cos(\theta + \pi), sin(\theta + \pi), 0 )$ for all $\theta \in [0, 2\pi]$ Calculate fundamental group of $X$ I try use ...
1
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2answers
58 views

Let $\mathbb{D}^3=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2\leq 1\}$. Define $X=\mathbb{D}^3/\sim$ with the relation $(x,y,z)\sim(-x,-y,-z)$

Let $\mathbb{D}^3=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2\leq 1\}$. Define $X=\mathbb{D}^3/\sim$ with the relation $(x,y,z)\sim(-x,-y,-z)$ for $x,y,z$ such that $x^2+y^2+z^2=1$. Calculate the ...
0
votes
1answer
88 views

Let $A$ be a finely generated Abelian group. Prove that there is a topological space $X$ with a basic point $x_0$ such that $\pi_1(X,x_0)\cong A$. [duplicate]

Let $A$ be a finely generated Abelian group. Prove that there is a topological space $X$ with a basic point $x_0$ such that $\pi_1(X,x_0)\cong A$. I'm trying to solve this problem but I do not know ...
0
votes
1answer
29 views

Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$.

Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$. I think that this space is the free product $\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$, but I do not know how to ...
2
votes
1answer
53 views

Let $X$ be the space obtained from $\mathbb{R}^3$ by removing the axes $x,y$ and $z$. Calculate the fundamental group of $X$.

Let $X$ be the space obtained from $\mathbb{R}^3$ by removing the axes $x,y$ and $z$. Calculate the fundamental group of $X$. I am trying to use the Van Kampen theorem but I do not know how to apply ...
2
votes
2answers
88 views

Fundamental group of intersection trivial

Let $U,V$ be open sets in $\mathbb{R}^n$ such that $U,V,U\cap V$ are path connected, and $U\cup V=\mathbb{R}^n$. Let $x_0\in U\cap V$. Show that if $\pi_1(U,x_0)\ncong\{1\}$ then $\pi_1(U\cap V,x_0)\...
1
vote
3answers
48 views

Possible groups for fundamental group of $\mathbb{R}$

I want to know if there exist a topology on $\mathbb{R}$ such that $\pi_1(\mathbb{R})\ne (0)$? If not, we conclude that the fundamental group is a property of the underground set, not the topology it ...
3
votes
0answers
264 views

Quotient of universal cover by fundamental group

The circle $S^1$ has fundamental group $\pi_1(S^1)=\mathbb{Z}$, universal cover $\mathbb{R}$, and satisfies $S^1=\mathbb{R}/\pi_1(S^1)$. Similarly, $\mathbb{R}P^2$ has fundamental group $\pi_1(\...
3
votes
1answer
74 views

Induced homomorphism of a dominating map

Let $A$ be a topological space homotopy dominated by a space $X$; i.e. there exist continuous maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ so that $g\circ f\simeq id_A$. Assume that $\...
1
vote
0answers
52 views

Combinatorial method for fundamental group

I recall learning a "combinatorial" method for computing fundamental group of simplicial complexes (not graphs) involving maximal tree. It is something like this: a group is generated by generators $...
2
votes
0answers
73 views

Find a topological space whose fundamental group is $D_4$ [duplicate]

$D_4$ here indicates the dihedral group of order 8. Does there exist any trivial example of topological spaces such that it has $D_4$ as it's fundamental group.
0
votes
1answer
188 views

computing the fundamental group of a wedge sum

I'm trying to solve the problem $18 (a$ from the chapter $1$ of Hatcher's Algebraic topology. it is: Prove that the wedge sum $S^1 \vee S^2$ has fundamental group $\mathbb{Z}$. The next chapter is ...
0
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0answers
133 views

Induced homomorphism on $S^1$ non-trivial

Let $f:S^1\rightarrow S^1$ be a map such that $f(-x)=-f(x)$ for all $x\in S^1$. Show that the induced homomorphism $f_\ast:\pi_1(S^1,x_0)\rightarrow\pi_1(S^1,f(x_0))$ is non-trivial. What I know: 1. ...
1
vote
1answer
94 views

Prove that $\pi_1(X\times Y, (x_0,y_0))$ is isomorphic to $\pi_1(X,x_0)\times \pi_1(Y,y_0)$.

Let $X$ and $Y$ be two topological spaces with basic points $x_0$ and $y_0$, respectively, and give $X\times Y$ with the product topology. Prove that $\pi_1(X\times Y, (x_0,y_0))$ is isomorphic to $\...
2
votes
1answer
24 views

Does $Ab(\pi_1(\Sigma_2)/N)=1$ imply $\pi_1(\Sigma_2)/N = 1$?

Let $\Sigma_2$ be a genus two surface. If $N$ is a normal subgroup of $\pi_1(\Sigma_2)$ such that the abelianization of the quotient $Ab(\pi_1(\Sigma_2)/N)$ is trivial, does that also imply that the ...
1
vote
1answer
173 views

Show that the fundamental group of wedge of two circles is non Abelian without actually computing it.

Given that there is a pointed space $(Y,y_0)$, such that it has non Abelian fundamental group, need to show that the fundamental group of the wedge of two circles is non Abelian. What I was thinking ...
0
votes
2answers
49 views

Show that for an abelian countable group $G$ there exists a compact path connected subspace $K ⊆ \Bbb R^4$ such that $H_1(K)$ isomorphic to $G$

" Given an abelian countable group G thus there exists a compact path connected subspace $K ⊆\Bbb R^4$ such that $H_1(K) ≅ G$ ", where $H_1$ is the first singular homology. Can I prove it using the ...
1
vote
1answer
58 views

Manifolds covered by finitely many charts

Suppose $M$ is a connected topological manifold, which has a finite open cover $\{U_i\}$, where each $U_i$ is homeomorphic to $\mathbb{R}^n$. Is it necessarily true that $\pi_1(M)$ is finitely ...
4
votes
0answers
435 views

Representation of fundamental group and flat bundle

My question is inspired by this discussion where the two notions of flatness for vector bundle are discussed. I would like to understand the one-to-one correspondence between flat bundles over a ...
11
votes
1answer
257 views

Is there a subset of $\mathbb R^2$ that has fundamental group $\mathbb Z^2$?

I know the fundamental group of the torus is $\mathbb Z^2$. I am wondering is it possible to find a subset of the plane such that the fundamental group of this subspace is $\mathbb Z^2$? I can't seem ...
1
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0answers
124 views

Fundamental group of a mapping cone

Suppose $X$ is path connected and $f: S^n \to X$. I am interested in determining what $\pi_1(C_f)$ is. My guess is that the fundamental group is $\pi_1(X)$ since $C_f = (S^n \times I) \sqcup X /{\...
2
votes
1answer
61 views

Showing that $\pi_1(X/G) \cong G$.

Suppose G is a discrete topological group acting freely on a simply connected topological space X. I am trying to show that $\pi_1(X/G) \cong G$. Here is my progress so far: I have shown that $X \to ...
1
vote
1answer
44 views

Is the fundamental group of the image a subgroup of the fundamental group of the domain?

I’m trying to learn Algebraic Topology, and there’s a very basic notion that I think must be true, but I can’t seem to prove or disprove it. Let $f:X\to X$ be a continuous map. Then the ...
0
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0answers
33 views

A set of generators of $\pi_1(X,x_0)$ where $X=U\cup V$ and $U\cap V$ path connected

Suppose $X$ is a topological space and $U,V$ are open subsets of $X$ such that $X=U\cup V$ such that $U\cap V$ is pathwise connected. We have the two inclusions $i:U\to X$ and $j:V\to X$ which induce ...
4
votes
0answers
263 views

Monodromy representation.

In a scientific article on the net, i find without further details, the following paragraph : Let $ X \subset \mathbb{C} $ be un open connected subset, and $ x \in X $. A linear differential ...
2
votes
1answer
123 views

Calculating the fundamental group of $S^1$ with SvK

The groupoid version of van Kampen's theorem states that if $X$ is a topological space and $X = U\cup V$ where $U,V$ are open, then the fundamental groupoid $\tau_{\leq 1}(X)$ of $X$ is the limit of ...
1
vote
1answer
28 views

If $H \le \pi_1(X,x)$ is conjugate to $P_*(\pi_1(Y, y))$, then $H \cong P_*(\pi_1(Y, y'))$ for some $y' \in P^{-1}(x)$

Assuming $P: Y \to X$ is a covering map, I want to show that if $H \le \pi_1(X,x)$ is conjugate to $P_*(\pi_1(Y, y))$, then $H \cong P_*(\pi_1(Y, y'))$ for some $y' \in P^{-1}(x)$. I started off with ...
0
votes
1answer
32 views

Homotopic maps between pointed sets induce same group homomorphism

I am trying to show that if $f, g: (X,x) \to (Y,y)$ such that $f \cong g$, then the induced homomorphisms $f_{*}, g_{*}: \pi_1(X,x) \to \pi_1(Y,y)$ are equal. I proved that for any $h \in \pi_1(X,x)$,...
1
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0answers
50 views

$f$ has square root if and only if image of $f_*$ contained in $2 \mathbb Z$

Let $X$ be a Hausdorff space that is connected and locally path connected, and let $f : (X,x) \to (S^1,1)$ be a continuous map. Prove that $f$ has a square root (a continuous mapping $g : X \to S^1$ ...
4
votes
2answers
376 views

Help resolving this contradiction in descriptions of the fundamental groups of the figure eight and n-torus

I'm faced with a few contradictory statements: 1) This article on Wolfram MathWorld says that "$\mathbb{Z}*\mathbb{Z}=\lbrace(n,m)\rbrace$ is a free abelian group of rank 2." (I thought that this is ...
4
votes
1answer
246 views

Confusion about free homotopy, based homotopy and homotopy groups

Unfortunately, this becomes a very general post: I have some questions concerning the homotopy invariance of homotopy groups. I start from what should be clear: If $f,g:(X,x_0)\to (Y,y_0)$ are based ...
1
vote
1answer
112 views

Prove the fundamental group of A is trivial $A= \{(x,y,z)\mid z \geq 0 \} \backslash \{(x,y,z)\mid y=0, 0 \leq z \leq 1 \}$

As stated i need to prove that the fundamental of the upper half of $R^3$ minus the line $y=0$ and the line segment $ 0 \leq z \leq 1 $ has a trivial fundamental group.Im just started in these kind ...
2
votes
1answer
69 views

how to prove the fundamental group isomorphic? [closed]

Suppose $f:S^1 \rightarrow S^1$ is continuous and has no fixed point. How can we prove that $f_*:\pi_1(X,x_0)\rightarrow\pi_1(X,f(x_0))$ is a group isomorphism where $f_*([\sigma]) = [f \circ \...
0
votes
0answers
182 views

Why the only finite fundamental group of compact surfaces are those of $\mathbb{S}^2$ and $\mathbb{RP}^2$?

I know that the fundamental group of orientable surfaces is of the form $F(a_1,b_1,\ldots,a_n,b_n)/N(\prod a_j*b_j*a_j^{-1}*b_j^{-1})$ where $F$ is the free products with certain generators and $N$ is ...
3
votes
0answers
82 views

Fundamental group of S_3 without curve

Define $S^3$ as $$ S^3 = \left\lbrace (z_1, z_2) \mid |z_1|^2 + |z_2|^2 = 2 \right\rbrace $$ and curve $C$ as $$ C(t) = (e^{imt}, e^{int}) $$ for $m,n \in \mathbb{Z}$, such that gcd(m,n) = 1. The ...
0
votes
1answer
71 views

Universal coverings and fully faithful fiber functors?

Consider a continuous map $\alpha:A\to B$ with the homotopy lifting propery and the unique path lifting property. Consider the induced fiber functor $$F:\pi_1B\longrightarrow \mathsf{Set}$$ taking a ...
2
votes
1answer
116 views

$X \setminus point$ not path-connected implies $X$ simply connected

Per $X$ be a path-connected space. Does $X \setminus point$ not path-connected implies $X$ simply connected? Thinking about curves and lines seem to suggest the truth of the statement, but I think ...