For questions about or involving the fundamental group.

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1answer
22 views

Fundamental Confusion Regarding the Fundamental Group

Consider the following question. Let $X$ be a topological space and $x_0$ be any point in $X$. Let $\gamma\in \pi_1(X,x_0)$ be a non-constant loop about the base point $x_0$. Then can it happen ...
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0answers
19 views

How to calculate the fundamental group of $S^3$ without two linked cirles

I need to find: the fundamental group of the space obtained by cutting out the three-dimensional $S^3$ sphere of two circles, once linked with each other. Can you help me? I have no idea about it, i ...
-1
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0answers
29 views

Example of noncommutative fundamental group [closed]

I am looking for a space such that its fundamental group $\pi_1$ is noncommutative.
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0answers
29 views

Fundamental Group of an n-dimensional manifold with finite k punctures

Problem: Show that $\pi_1(M\setminus${k points}$) = \pi_1(M)$, where $M$ is an n-dimensional manifold ($n\ge 3)$ and k is a positive integer. In class, we went over the proof for the above ...
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0answers
21 views

Proving the continuity of a homotopy

Let $X$ be a path connected space such that for any continuous map $f:S^{1}\rightarrow X$ there exists $F:D^{2}\rightarrow X$ continuous such that $F|S^{1}=f$. I have to prove that $X$ is simply ...
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1answer
46 views

maps between spheres, torus and projective plane [closed]

How to solve these questions by direct and valid argument? Various methods are wanted. Thanks.
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2answers
29 views

Fundamental group of surface of genus $g$

Suppose we have a compact surface of genus $g$. How to calculate its fundamental group ?
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1answer
38 views

Fundamental group of $\mathbb{P}^n(\mathbb{C})$

We know that $$\Pi_1(\mathbb{P}^n(\mathbb{R})) \cong \mathbb{Z}_2 $$ for $n \geq 2 $. Is there a similar statement for $\Pi_1(\mathbb{P}^n(\mathbb{C}))$ ?
2
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1answer
42 views

Not all finitely-presented groups are fundamental groups of closed 3-manifolds

It is a well-known result that, for any finitely-presented group $G$ and any integer $n \geq 4$, there exists a closed $n$-manifold whose fundamental group is isomorphic to $G$ (a sketch of proof can ...
4
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2answers
95 views

What categorical limits and colimits does $\pi_1$ preserve?

$\pi_1$ is a functor from the category of pointed topological spaces to the category of groups. It's clear that this functor preserves products, since $\pi_1(X\times Y)=\pi_1(X)\times\pi_1(Y)$, and a ...
1
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1answer
51 views

Explicitly building Top. space with trivial homology group and non trivial fundamental group, w/o CW-complexes

I know this is a pretty famous question here, but I was asked to show explicitly such space, during a bachelor lecture, without using any CW-complex result. I started working using some covering ...
11
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2answers
230 views

Functoriality of fundamental group

I'm trying to prove the following statement: If $\mathcal{C}_1$ is the category of path connected topological spaces and $\mathcal{C}_2$ is the category of groups, then the mapping $\mathcal{C}_1 \ni ...
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1answer
57 views

Fundamental group of a genus-2 surface

I want to calculate the fundamental group of a genus-2 surface, i.e. a double torus. Using Van-Kampen I obtain ( with the notation generators- relations) $$\Pi_1(X,p) = < \alpha, \beta, \alpha_1, ...
2
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1answer
37 views

Finding the fundamental group of the torus

I'm trying to prove that the fundamental group of the torus is $\Bbb Z \times \Bbb Z$, and I've been checking different questions on this site, however in many of the questions people use the ...
2
votes
1answer
53 views

Fundamental group of $GL(n, \mathbb{C}) $

I want to prove that the fundamental group of $GL(n, \mathbb{C}) $ is infinite. I don't know how to proceed, any hint ?
0
votes
1answer
34 views

$\mathbb{R}P^2$ and its fundamental group by identification of edges of unity square

Suppose we identify edges of the the unity square $[0, 1] \times [0, 1]$, as in the picture: http://de.wikipedia.org/wiki/Datei:ProjectivePlaneAsSquare.svg Now to compute the fundamental group, for ...
2
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1answer
90 views

Showing that Möbius band can't retract onto the boundary circle

Trying to prove that Möbius band can't retract onto the boundary circle, I got an idea that I must show the below thing. If $\alpha\in\pi_1 (\partial M)$ is a generator, its image $i^*(\alpha) \in ...
2
votes
1answer
40 views

Given a group $G$, the existence of a space such that $\pi_1(X)\simeq G$.

I'm having trouble understanding Corollary 1.28 of Hatcher which proves that for every group $G$ there is a space $X_G$ such that $\pi_1(X_G)=G$. Since $G$ is the quotient of a free group $F$, we ...
1
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1answer
64 views

Every Group is a Fundamental Group

I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and ...
3
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1answer
41 views

Map from $\mathbb{R}\mathbb{P}^{2}$ to Klein bottle homotopic to constant map

I'm currently struggling with the following exercise: Show that any continuous map $$f: \mathbb{R}\mathbb{P}^{2} \to K$$ where $K$ is the Klein bottle, is homotopic to a constant map. I know that ...
4
votes
1answer
77 views

Fundamental group of the complement of an eight shape

What is $\pi_1(\mathbb{R}^3 \setminus (S^1 \vee S^1))$? I would say that the space deformation retracts to a sphere $S^2$ surrounding the missing eight with two sticks stuck in the two loops of the ...
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2answers
50 views

How do we prove that the fundamental group is a group?

My understanding of the fundamental group is that it's the set of all loops starting and ending at a point $x_0$ in a space $X$, along with the operation of composition. For it to be a group, ...
0
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0answers
31 views

lifting loops on surface with abelian fundamental group for decrease their self-intersection number

I found a proof of following statement (which is available here ), but I'm not sure if we must assume that fundamental group of surface $M$ is non-abelian: Lemma: Let $M$ be a compact orientable ...
0
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1answer
40 views

Classical presentation of fundamental group of surface with boundary

It is well known fact about fundamental group of orientable compact surface: Letting $g$ be the genus and $b$ the number of boundary components of surface $M$. There is a generating set ...
2
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3answers
71 views

Hatcher, p.35, $\mathbb{R^{n}} - \{x\}$ is homeomorphic to $S^{n-1} \times \mathbb{R}$

Could some help me understand how $\mathbb{R^{n}} - \{x\}$ is homeomorphic to $S^{n-1} \times \mathbb{R}$. Also, if the one point set is replaced by any finite set, how does the argument work. This ...
3
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1answer
79 views

The nature of isomorphism between fundamental groups with different base points

New to algebraic topology. Munkres (Topology, 2 ed.) in the last paragraph on page 332 says that "If $X$ is path-connected, all the groups $\pi_1(X,x)$ are isomorphic, so it is tempting to try to ...
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1answer
44 views

Does the Wirtinger presentation extend to compliments of graphs and links?

In a previous question I asked about a specific fundamental group problem, which was resolved via SVK but I was also interested in whether or not the Wirtinger presentation was valid in some way. In ...
2
votes
3answers
134 views

Fundamental group of complement of circle with a line passing through it.

Let $X= S^1 \cup y$-axis in the $xy$-plane. Compute the fundamental group of $\mathbb{R}^3 - X$ Edit: To clarify, $S^1 =\{(x,y)|x^2 + y^2 = 1\}$ is the unit circle in the $xy$-plane, so that the ...
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2answers
75 views

what is the fundamental group of a torus with $k$ points removed

I was about to use the Seifert-van Kampen to compute the fundamental group of a torus of genus 2. In the process, I need to know the fundamental group of a torus (of genus one) with a hole removed. is ...
3
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0answers
47 views

Is there fundamental group “with coefficients”

Constructing fundamental group in usual way, one uses $I=[0,1] \in \mathbb R$, so uses real numbers, and gets a group classifying coverings for nice spaces: locally connectible and so on, in fact ...
2
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1answer
44 views

Fundamental group of the quotient of a cylinder by a rotation at either end

The following is a past qual problem: let $X = S^1 \times [0,1]$ be the cylinder, and define an equivalence relation on $X$ by $(z,1) \sim (iz,1)$ and $(w,0) \sim (e^{i\pi /7} w, 0)$. Compute ...
6
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1answer
117 views

An algebraic topology proof of a result from analysis

A colleague of mine recently brought up the following result from real analysis: Theorem: If $f:\mathbb{R}^2\to\mathbb{R}^2$ is continuous and $|f(x)-f(y)|\geq |x-y|$ for all $x,y$, then $f$ is onto. ...
4
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1answer
82 views

Fundamental group of quotient of $S^1 \times [0,1]$

I have a past qual question here: Let $X = S^1 \times [0,1] /{\sim}$, where $(z,0) \sim (z^4,1)$ for $z \in S^1 = \{ z \in \mathbb{C} \colon \| z \| = 1 \}$. Compute $\pi_1(X)$. I've been trying to ...
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0answers
40 views

How to describe a homomorphism from a fundamental group to a finite group?

Let $S$ be a connected locally noetherian scheme with $s$ a geometric point of $S$. I read something like this: to give a surjective continuous homomorphism from $\pi_1(S, \bar{s})$ to a finite group ...
2
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1answer
33 views

Fixed point equivalence [duplicate]

Let $\Bbb D^n$ be n-dimensional ball and $S^{n-1}$ the $n-1$ dimensional sphere realized as boundary of $\Bbb D^n$. Prove that following are equivalent. There is no retraction $\Bbb D^n\to S^{n-1}$ ...
3
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1answer
68 views

For compact surface $M$ and loop $f$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such that $f \notin \ker(\phi)$

Why is this sentence true? For every not nullhomologous loop $f$ without selfintersections on orientable compact surface $M$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such ...
4
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1answer
95 views

Fundamental Group of a Hexagon with Edge Identifications

What's the easiest way to compute this thing's fundamental group? I've been playing with it for a little while, and I'm getting $\mathbb{Z}+\mathbb{Z}$. After making the ID's I think the 1-cells ...
5
votes
3answers
175 views

Do freely homotopic maps induce the same homomorphism on fundamental groups?

Let $f,g\colon X\to Y$ be two continuous maps that are freely homotopic, such that there is some $x_0\in X$ with $f(x_0)=g(x_0)$. Is it true that the induced homomorphisms $f_*,g_*\colon ...
4
votes
2answers
90 views

Funky Fundamental Group Question

Let $D$ be a closed disk (w/ boundary $C$) and let $D_a$, $D_b$ be two disjoint closed disks in the interior of $D$ (w/ boundaries $C_a$ and $C_b$, resp.) . Now remove the interiors of $D_a$ and ...
1
vote
1answer
63 views

the fundamental group of punctured surface

Let $S_{g,m}$ be a surface of genus $g$ with $m$ punctured, we know the fundamental group of $S_{g,0}$ is $$ \pi_1(S_{g,0}) = \left\langle a_1, b_1, \dots, a_g, b_g {~\large\mid~} [a_1, b_1] \dots ...
0
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1answer
55 views
4
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1answer
101 views

Homology and Fundamental group of $\mathbb{R}^4\setminus S^1$

I was attempting to help someone with this problem and realized I could not solve it myself. Let $S^1=\{(x,y,0,0):x^2+y^2=1\}$ be the unit circle in $\mathbb{R}^4$ and consider $M=\mathbb{R}^4 ...
9
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1answer
133 views

Is the homotopy type of an aspherical space determined by its fundamental group?

Question: Let $X$ and $Y$ be path-connected spaces that admit a contractible universal cover, with $\pi_1(X) \cong \pi_1(Y)$. Is $X$ homotopy equivalent to $Y$? Comments: $X$ and $Y$ are both ...
2
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1answer
101 views

Fundamental group of a connected graph

Is this a legit way to prove that a fundamental group of a connected graph $\Gamma$ is a free group? Without using quotient and homotopy extension property from Hatcher's "Algebraic Topology": Take ...
2
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0answers
22 views

Dual of etale fundamental group, field functoriality

Suppose $k$ is an arbitrary field and $X/k$ is a finite-type and integral scheme (hence connected). Let $K$ be the fraction field of $X$. Via the canonical morphism $$\phi: \rm{Spec}(K)\rightarrow ...
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vote
1answer
52 views

Determing if the fundamental group of the following is isomorphic to either the trivial, infinite cyclic, figure eight fundamental groups

Hello there i am having trouble to determine isomorphisms of the following fundamental groups: 1) the torus $T$ with a removed point. 2) $\mathbb{R}^3$ with nonnegative axes 3) $S^1 \cup ...
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6answers
7k views

Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down

A friend of mine told me that it's possible to hang a picture on the wall from a string using two nails in such a way that removing either of the two nails will make both the string and picture fall ...
3
votes
2answers
80 views

Showing that $\pi(G/H, 1) = H$ under a condition

Problem: Let $G$ be a simply connected (i.e., $\pi(G)=1$) topological group, and let $H$ be a discrete normal subgroup. Prove that $\pi(G/H,1) = H$. I know that since $H$ is a discrete subgroup ...
4
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0answers
31 views

Isomorphism of the etale fundamental group

Given a birational proper morphism $f\colon X \rightarrow Y$ of complex algebraic varieties. It is always true that $f^* \colon \pi^{et}_1 (X)\rightarrow \pi^{et}_1(Y)$ is an isomorphism? I think ...
2
votes
2answers
47 views

Higher Homotopy Group of a Product of Spaces

I want to show that for two toplogical spaces $ X_1,X_2$ and for $x_1\in X_1 , x_2 \in X_2$ we have an isomorphism between $\pi_n (X_1 \times X_2 , (x_1,x_2)) $ and $ \pi_n (X_1, x_1) \times \pi_n ...