For questions about or involving the fundamental group.

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Quotient Groups and Covering Spaces in Painting Hanging

Consider the $1$-out-of-$n$ painting hanging problem: Given $n$ nails in a wall, how can we hang a painting such that upon removal of any nail, it falls. This has a nice interpretation as a problem in ...
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1answer
23 views

Why is this proof that the group operation on $\pi_1(X,x_0)$ is well defined?

Let $g_1,g'_1, g_2, g'_2$ be loops on a topological space $X$ at $x_0 \in X$. Suppose that $[g_1]=[g'_1]$ and $[g_2]=[g'_2]$. Then let a map $F: [0,1]\times [0,1] \to X$ be defined as $$ F = \Big\{ ...
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The Seifert-Van Kampen theorem as a push-out

My question concerns the proof of the Seifert-Van Kampen theorem. The version of such a statement that interests me is the following. Let $X$ be a topological space, and $U, V\subseteq X$ two ...
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39 views

Fundamental groups of the projective plane

I'm practicing for my topology final. Munkres' Topology, Theorem 74.4 states that: Let $X$ denote the $m$-fold projective plane, that is the space obtained by taking the connected sum of the ...
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Punctured plane

What does one point compactification of singly, doubly, triply punctured plane $\mathbb{R}^2$ look like? What would their fundamental groups look like? I'm trying to visualize but can't seem to draw ...
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1answer
39 views

Fundamental group of Topologists sine curve

How can one prove that the fundamental group of the topologists sine curve is trivial? I haven't been able to make any progress on this. A hint in the right direction preferred over a complete ...
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0answers
38 views

Path structure of an odd-looking higher inductive type.

Consider the (putative) higher inductive type $\mathtt{PNest}$, with the constructors: $\mathtt{base} : \mathtt{PNest}$ $\mathtt{nest} : \mathtt{base} = \mathtt{base} \to \mathtt{base} = ...
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2answers
64 views

Hatcher problem 1.2.12

In this problem a modified Klein bottle (say $X$) is taken in account which is seen as embedded space in $\mathbb R^3$ (giving subspace topology on the usual self intersecting figure of Klein bottle ...
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86 views

Need help on how to compute the fundamental group of a space.

I'm studying for an oral qualifying exam and going through various past exams I find on the interwebs, including this mock exam from the University of Bath. One of the questions seems like it should ...
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0answers
77 views

Mobius band does not retract to boundary circle - specific part

(I'm trying to ask this in a way such that it isn't a duplicate question) The proofs that I've seen for the fact that there is no retraction from the Mobius band to its boundary circle usually say ...
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36 views

Proving $S^{4}/G$ is simply connected where $G$ is not a free group action

Consider the sphere $S^{4}$ as a subset of $\mathbb{R}^{5}$ and consider the action of the group $G$ of homeomorphisms generated by $(x_1, x_2, x_3, x_4, x_5) \rightarrow (-x_2, x_1, -x_4, x_3, x_5)$ ...
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1answer
40 views

Surjection of the fundamental group of a manifold onto a free group induces a map onto a wedge of circles, why?

Why does a surjective map $\pi_1(M)\twoheadrightarrow F$ of the fundamental group of a manifold $M$ onto a free group $F$ over $n$ generators induce a continuous map $M\twoheadrightarrow\bigvee^n S^1$ ...
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1answer
45 views

Fundamental Group is free on infinite generators.

This is question 16 of section 1.2 in Hatcher's Algebraic Topology. I have to show that the fundamental group of the space $X$ is free on an infinite number of generators. So here is my approach. ...
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2answers
76 views

Fundamental groupoid

Let $(X,x_0)$ be a pointed topological space. The homotopy groups $\pi_n(X,x_0)=Hom((S^n,s_0),(X,x_0))$ are groups because $S^n$ is a cogroup object in the pointed homotopy category. Removing the ...
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1answer
58 views

Fundamental group Pi1(SU(n)) and Pi2(SU(n))

I need to find the fundamental group $\pi_1(SU(n))$ and $\pi_2(SU(n))$ for all $n$. I don't have any idea.
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1answer
46 views

Fundamental group of composition of function

Let Y be a simply connected space, and let f : X → Y and g : Y → Z be continuous functions. What is π1(g ◦ f )? Prove your answer. I think the fundamental group of the composition would be the ...
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2answers
78 views

Fundamental group is a homotopy invariant

I am a newbie to topology and am not able to understand how to attack this problem: Any hints would be appreciated Assuming that: $$ f \sim g \Rightarrow \pi_1(f) = \pi_1(g). $$ Prove that the ...
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1answer
52 views

How can I show $G_0$ and $G_1$ are conjugate subgroups?

Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the induced homomorphism from the fundamental group of $E$ to the fundamental group of $B$. Let $e_o$ and $e_1$ be points in E ...
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1answer
72 views

on Cayley diagrams

is the picture the Cayley Graph of the group $\langle a,b,c\mid a^2, b^2,c^2\rangle$ ? What would it be for $\langle a,b,c\mid a^2b^2c^2\rangle$?
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2answers
117 views

Fundamental group a collection of six 2-spheres touching pairwise in a cyclic order, and each touching a common plane.

Find the fundamental group of the space comprising a collection of six 2-spheres touching pairwise in a cyclic order, and each touching a common plane. Touching means having one point in common. I ...
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A space having exactly three coverings up to equivalence

Q: Give an example of a topological space having exactly 3 coverings up to equivalence (including a covering by the space itself). Proof: There is a theorem that says that given a topological ...
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32 views

Algebraic fundamental group vs arithmetic fundamental group

I'm trying to read about anabelian geometry and obviously the things to start with is the algebraic (etale) fundamental group. Every now and again I encounter authors talking about the arithmetic ...
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54 views

Explicit expression for the topological invariant of O(n)

I learned that the fundamental group of $O(n)$ is $\Bbb{Z}/2\Bbb{Z}$ (for $n>2$). What is the explicit expression for its topological invariant? To be specific: Given a smooth path ...
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1answer
29 views

Based covering maps for a bouquet of two circles

For each of the following subgroups of $$ \left \langle x,y \right \rangle = \pi_{1}(S^{1}\vee S^{1}) $$ construct a based covering map $$ \ p:(\tilde{X},\tilde{b})\rightarrow (S^{1}\vee S^{1},b) ...
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2answers
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wedge product of projective planes

if we have the wedge product of the real projective plane $P^2$ V $P^2$ Then how would i use Seifert Van Kampens theorem to compute the fundamental group $\pi_1$($P^2$ V $P^2$ ) ? i'm some what ...
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1answer
26 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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31 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 ...
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34 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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26 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
74 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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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
42 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 $\Pi_1(\mathbb{P}^n(\mathbb{C}))$ ?
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1answer
59 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 ...
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2answers
107 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 ...
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1answer
58 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 ...
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2answers
261 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
106 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, ...
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1answer
42 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 ...
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1answer
58 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 ?
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1answer
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$\mathbb{R}P^2$ 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 ...
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1answer
135 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 ...
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1answer
41 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 ...
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1answer
71 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 ...
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1answer
44 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 ...
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1answer
81 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
58 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, ...
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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 ...
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1answer
59 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 ...
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3answers
78 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 ...
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1answer
89 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 ...