Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

7
votes
2answers
195 views

Compute fundamental group _visually_ by the polygonal representation of the space

Some time ago, before I learnt about covering spaces and Seifert-Van Kampen theorem, I tried to compute visually the fundamental group of some spaces. For example I figure out by myself that the ...
1
vote
1answer
26 views

References for injectivity/surjectivity $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$

I have read (Theorem 29.3, line 7, pg 144) the following statement: $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$ is surjective for $i \le 2(n-k)+1$ and injective for $i \le 2(n-k)$. and ...
0
votes
1answer
30 views

Proof by induction of path composition

Let $w, \alpha_n: I=[0,1] \rightarrow \mathbb{S}^1$, $w(s)=e^{2\pi i s}$, $\alpha_n(s)=e^{2 \pi i n s}$. Let $[u] \in \pi_1(\mathbb{S}^1,1)$ be the path homotopy class of the path $u$, an element of ...
0
votes
2answers
89 views

Can we always view loops as maps from $S^1\to X$?

I am working on exercises from Hatcher, algebraic topology and in a certain exercise (1.1.5), we identify a loop, which is a path $\gamma:I\to X$ with a map $f:S^1\to X$, which we can do since $\gamma(...
4
votes
2answers
116 views

Find finite path connected topological space with $π_1 (X, x_0 ) \cong \Bbb Z/2\Bbb Z$

Find a finite non-empty topological space $X$ that satisfies: $X$ is path connected Its fundamental group $π_1 (X, x_0 ) \cong \Bbb Z/2\Bbb Z$ for $x_0 \in X$ Is the connected pair $\{\{0\},\{1\},\{...
1
vote
1answer
33 views

Fundametal group and continuous extension to $\Bbb R^n$

Let $X ⊂ \Bbb R^n$ be a non empty subset with $n>0$ and let $x_0 ∈ X$. Let $Y$ be a non empty topological space and $g : X → Y$ a continuous map. Suppose $g$ has a continuous extension defined on ...
0
votes
2answers
84 views

What can we say about two topological spaces with the same fundamental group?

Let's consider two topological spaces. If they are homeomorphic, or homotopic equivalent, they have isomorphic fundamental groups, but the converse is not true. My question is: is there a (non trivial)...
0
votes
1answer
52 views

Hatcher Lemma 1.19

Let $f$ be some loop about $x_0$. From what I understand, we want to show that $\varphi_{0*}([f]) = \beta_h\varphi_{1*}([f])$ or $[\varphi_{0} f] = [h \ast (\varphi_{1}f) \ast \overline{h}]$ From what ...
0
votes
1answer
26 views

First fundamental group and connectedness of $X:=\mathbb{R^4}\setminus \{\pi_1 \cup \pi_2 \}$

On $\mathbb{R^4}$ consider $\pi_1 := \{x_1=x_2=0\}$ and $\pi_2 :=\{x_3=x_4=0\}$. Let $X:=\mathbb{R^4}\setminus \{\pi_1 \cup \pi_2 \}$ . Show that $X$ is arc-connected and find $\pi_1 \left(X\right)$...
2
votes
1answer
92 views

Loops as maps from $S^1\to X$, Hatcher 1.15 [duplicate]

I am working on Hatcher's problem 1.1.5. Show that for a space $X$, the following three conditions are equivalent. $\textit{a)}$ Every map $S^1\to X$ is homotopic to a constant map. $\textit{b})$ ...
0
votes
1answer
67 views

If $f_0=\omega_m$ and $f_1=\omega_n$, why is it not automatic that $\tilde f_0 = \tilde \omega_m$ and $\tilde f_1 = \tilde \omega_n$?

In the second paragraph of Hatcher's proof, fourth sentence, it says The uniqueness part of (a) implies $\tilde f_0 = \tilde \omega_m$ and $\tilde f_1 = \tilde \omega_n$. How does the uniqueness part ...
0
votes
1answer
25 views

Finding an open simply connected subset in a punctured open simply connected set

Let $X$ be an open , simply connected (path connected with trivial first homotopy group) subset of $\mathbb R^2$. Let $0\in X$. Is it true that for every $p,q\in X\setminus \{0\}$, there is an open ,...
1
vote
1answer
66 views

Let $X$ be a connected CW complex and $G$ a group such that every $\pi_1(X)\to G$ is trivial. Show that every $X\to K(G, 1)$ is nullhomotopic.

Question 2 in Chapter 1.B in Hatcher's Algebraic Topology: Let $X$ be a connected CW complex and $G$ a group such that every homomorphism $\pi_1(X)\to G$ is trivial. Show that every map $X\to K(G, ...
0
votes
2answers
39 views

Surjectivity of sending loop from fundamental group to the endpoint of its lift?

This question refers to the following map defined on a fundamental group of some topological space $X$ covered by a covering map $p:\bar{X}\rightarrow X$: $$r: \pi_{1}(X, x_{0}) \rightarrow p^{-1}(x_{...
6
votes
1answer
109 views

Fundamental group of uncountable space with cofinite topology

So it is easy to show that such a space is path connected (assuming CH, injective maps $f:I \rightarrow X$ are continuous) but I'm not sure how to start computing the fundamental group. Will it depend ...
0
votes
0answers
45 views

Homotopy groups of $\mathbb R^n-\mathbb Z^n (n>1)$?

How to compute the homotopy groups of $\mathbb R^n-\mathbb Z^n (n>1)$? The fundamental group may be an easy exercise using Van Kampen theorem, but how about higher ones?
9
votes
3answers
484 views

How is going around the circle once in each direction homotopic to a point?

Two paths are homotopic if one can be continuously deformed to the other, right? So I've been told that the fundamental group of the circle is isomorphic to the integers, since you can't deform e.g., ...
0
votes
2answers
54 views

find two topological spaces $Y_1$, $Y_2$ such that $\pi (Y_1)=\pi (Y_2)= (0)$ but $\pi (Y_1 \cup Y_2)\ne (0)$

I everybody, I have to solve this exercise. I have to find two topological spaces $Y_1$, $Y_2$ such that $\pi (Y_1)=\pi (Y_2)= (0)$ but $\pi (Y_1 \cup Y_2)\ne (0)$. Can you help me?
4
votes
1answer
184 views

Hatcher Exercise 1.2.8

I am trying to prove the following excersise (1.2.8) from Hatcher's Algebraic Topology: Given 2 tori $S^1\times S^1$ and identifying $S^1\times \{x_0\}$} compute the fundamental group. My approach is ...
0
votes
1answer
80 views

The fundamental group of the Lattice - (R x Z) U (Z x R)

I am trying to show that the identity map $ id:S_h \vee S_v \rightarrow S_h \vee S_v$ does not lift to L = $(\mathbb{R} \otimes \mathbb{Z}) \cup (\mathbb{Z} \otimes \mathbb{R}) $ via the covering ...
0
votes
0answers
43 views

Does the center of $\pi_1(Y)$ act trivial on $[X,Y]_\star$?

Let $X$ and $Y$ be based (and well-pointed) and connected. We have an action of $\pi_1(Y)$ on the set $[X,Y]_\star$ of based homotopy classes of based maps. The quotient is just the set $[X,Y]$ of ...
1
vote
0answers
30 views

Degree of universal cover of simple Lie group

I have seen a statement that if $\mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $\mathfrak{g}$. Equivalently, the simply connected group with Lie ...
3
votes
1answer
83 views

Fundamental group of the $m$-fold suspension of a finite discrete space

Recall that the suspension of a topological space $X$ is the space $SX$ resulting by identifying $X\times\{0\}$ and $X\times\{1\}$ to single points of the "cylinder" $X\times[0,1]$. Now let $X_m$ be a ...
0
votes
1answer
53 views

covering spaces and equivalency of these three propositions [closed]

This question is really important for me since the answer will give me the a way for solving similar proofs at algebraic topology lessons,so i need your help.. I need to prove these following ...
1
vote
1answer
45 views

The fundamental group of $(S^1\times S^1)/(S^1\times \{x\})$

What is the fundamental group of $(S^1\times S^1)/((S^1\times\{x\})$ where $x$ is a point in $S^1$? My guess is that $S^1$ is a deformation retract of $(S^1\times S^1)/(S^1\times \{x\})$. Thus $\pi_1(...
2
votes
0answers
38 views

Can nonisomorphic groupoids have homotopy equivalent classifying spaces?

We know that two discrete groups having the same classifying space up to homotopy are isomorphic. One can just take fundamental groups and conclude. The situation with topological groups is subtler. ...
2
votes
1answer
45 views

Fundamental group via deck transformations considering rotation on sphere

Let $Z_m$ act on $S^1$ by multiplication with $e^{2\pi ki/m}$ for $k \in Z_m$. Let $X = S^1 / Z_m$ be the orbit space of this action. Then we have a universal cover $q:S^1 \rightarrow X$ given by the ...
2
votes
0answers
68 views

The small étale topos of a scheme is equivalent to the category of finite $\pi_1(X,x)$-sets for every scheme $X$ and every geometric point $x$

Recall Milne, Etale cohomology, Theorem I.5.3: Let $x$ be a geometric point of a connected scheme $X$. The functor $Hom(x,-):FEt/X\to Sets$ ($FEt/X$ being the category of $X$-schemes finite and ...
1
vote
1answer
82 views

Homotopy of continuous map from a space with finite fundamental group

Problem source: 2c on the UMD January, 2018 topology qualifying exam, seen here https://www-math.umd.edu/images/pdfs/quals/Topology/Topology-January-2018.pdf I have an argument for it, but I am not at ...
1
vote
3answers
100 views

How the two non null-homotopic equivalence classes generate the null-homotopic loop on the torus

I am new in Alebraic Topology. Given the torus, we say that the fundamental group of the torus is generated by two loops (or more exactly two equivalent classes of loops). One writes $\pi=\mathbb{Z}\...
2
votes
1answer
126 views

Fundamental Group Contains Infinite Cyclic Subgroup

Let $X \subseteq \Bbb{R}^2$ s.t. $\Bbb{R}^2$ such that $S^1 \subseteq X \subseteq \Bbb{R}^2 - \{(0,0)\}$. Prove that contains $\pi_1(X,(0,1))$ an infinite cyclic subgroup. Here's what I did. Let $j : ...
2
votes
2answers
63 views

Geometric presentation of fundamental group of a surface

Let $S = S_g$ be a closed surface. An author of a paper writes: We say $\langle a_1, b_1, \cdots, a_{2g}, b_{2g} \ | \ R \rangle$ is a geometric presentation of the fundamental group $\pi_1(S)$ ...
1
vote
1answer
51 views

Fundamental groups of the configuration spaces of all triangles and right triangles

This is a question from a past comprehensive exam: Consider triangles in the plane, with vertices given by non-colinear points as usual. The space $T$ of all plane triangles can be given a natural ...
1
vote
2answers
55 views

Fundamental group of two linked spheres

I want to calculate the fundamental group of this space: $$X=\{(x,y,z)\in\mathbb{R}^3:(x^2+y^2+z^2-100)[(x-10)^2+y^2+z^2-1]=0\}.$$ I’m not good with this kind of exercises: I know the Seifert-van ...
2
votes
0answers
95 views

Is $\left<a,b,c|aba^{-1}b^{-1},aca^{-1}c^{-1}\right>$ the fundamental group of two tori joined by a circle?

If we connect $S^{1}_a\times S^1_b$ to $S^1_c\times S^1_d$ by identifying $S^1_a\times x_0$ with $S^{1}_c\times x_1$ to form $T'$ is $\pi_1(T')=\left<a,b,c|aba^{-1}b^{-1},aca^{-1}c^{-1}\right>$. ...
5
votes
0answers
69 views

First fundamental group $\pi_1(X)$ of $\mathbb{R^3} \setminus \{$a circumference $\cup$ a line tangent to a point on the circumference$\}$

I have to find the fundamental group $\pi_1(X)$ of $X=\mathbb{R^3} \setminus \{$a line $\cup$ a circumference$\}$. Line passing through the centre of the circle, we have: $\pi_1(X) \cong\left(\mathbb{...
2
votes
1answer
55 views

What is wrong with this application of van Kampen's Theorem?

My understanding of van Kampen's Theorem (simplified to just two neighbourhoods): Let $X$ be a topological space and let $\{N_a, N_b\}$ be a cover of $X$ such that $N_a \cap N_b$ is path-connected (...
1
vote
0answers
46 views

The fundamental group of a topological space is isomorphic with its connected component fundamental group

can you help me with this problem of fundamental groups? suppose that $X$ is a topological space, let's fix a point on $X$ like $p\in{X}$. Thus I want to prove that the fundamental group of $X$ at $p$ ...
4
votes
1answer
86 views

Showing $\langle a,b\mid abab^{-1}\rangle$ and $ \langle c,d \mid c^2d^2\rangle$ are isomorphic.

I computed the fundamental group of the Klein bottle in two different ways and obtained two seemingly different answers: $$ \langle a,b \mid abab^{-1}\rangle $$ and $$ \langle c,d \mid c^2d^2\rangle. $...
1
vote
1answer
213 views

Fundamental group of $\mathbb{R}^3\setminus$ line+2 circles

I am trying to find the fundamental group of the space $X=\mathbb{R}^3\setminus S$ where $S$ is the union of the $z$-axis, and two circles of radius 1 on the $xy$-plane, centered at $(0,0,1)$ and $(0,...
1
vote
1answer
86 views

Lemma 81.1 in Munkres

In the proof of Munkres, which is given in the picture below, I don't understand why we need to show that $h,\ h(e_0)=e_1$ exists iff $[\alpha]\in N(H_0)$. So I tried to organize the proof in the way ...
3
votes
1answer
128 views

Classification of double coverings and $H^1(X ; \mathbb{Z}/2\mathbb{Z})$

It is a well known fact (see for example Hatcher's "Algebraic Topology", chapter $1$) that there is a bijection between the $n$-sheeted coverings of $X$ up to isomorphism of covering spaces and the ...
3
votes
1answer
61 views

What does the following manifold look like?

In hyperbolic geometry, we know that any complete hyperbolic surface is the quotient $\mathbb{H}^{2}/\Gamma$ where $\Gamma < \text{Isom}(\mathbb{H}^{2})$ a discrete subgroup. We know $\text{Isom}^...
1
vote
1answer
52 views

Cannonical group isomorpshim for fundamental group in a path connected space.

My question is related to this one which tells us that fundamental group $\pi_{1}(X,x_0)$ is abelian $\textit{iff}$ for every pair $\alpha$ and $\beta$ of paths from $x_0$ to $x_1$, we have the same ...
3
votes
1answer
103 views

Is there a general way to calculate the fundamental group of a quotient space?

Suppose $X$ is a path-connected topological space, and $A$ is a path-connected subset of $X$. My question is, is there a way to calculate the fundamental group of the quotient space $X / A$ in terms ...
2
votes
1answer
78 views

the fundamental group of $X$ is the symmetric group $S_3$, then whether it has a universal cover?

Question: Suppose that $X$ is a path-connected space with $\pi_1(X)=S_3$, which is the 3-symmetric group. I just wonder that whether $X$ has a universal cover. Try: Based on Hatcher, $X$ has a ...
0
votes
2answers
102 views

Find the fundamental group of $\Bbb C^2 \setminus \{(x,y):xy=0 \}$. [closed]

What is the fundamental group of $\Bbb C^2 \setminus \{(x,y):xy=0 \}$? How do I proceed? Please help me in this regard. Thank you very much.
2
votes
0answers
62 views

Finding fundamental groups of two spaces. [closed]

What are the fundamental groups of $\Bbb R^3 \setminus \{\rm { circle} \}$. $\Bbb R^3 \setminus \{\rm {two\ disjoint\ circles} \}$. Please help me in this regard. Thank you very much.
0
votes
0answers
35 views

Finding fundamental groups.

What are the fundamental groups of $\Bbb R^3 \setminus {\rm (two\ parallel\ lines)}$ and $\Bbb R^3 \setminus {\rm (two\ intersecting\ lines)}$. How do I compute those groups? Thank you very much.
0
votes
1answer
43 views

Group of covering Transformation corresponding to proper discontinuous action of a group on a connected space is the group itself.

Suppose $G$ acts properly discontinuously on a connected space $X$. Show that the group $G(X,p,X/G)$ of the covering transformation of $p:X \rightarrow X/G$ is $G$. I have tried in this manner- ...