For questions about or involving the fundamental group.

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2
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1answer
46 views

Equivalent presentation for the fundamental group of the projective plane

We know that $\langle a,b;(ab)^2=1\rangle$ and $\langle z;z^2\rangle$ are presentations of the fundamental group of the projective plane. Therefore, one is obtained from the other via Tietze ...
4
votes
1answer
120 views

A question on Hawaiian earring

Consider the Hawaiian earring. Suppose $f_n$'s are the loops representing the circles of radius $1/n$ centered at $(1/n, 0)$ for $n=1,2\cdots$. Suppose the following holds in the fundamental group ...
1
vote
1answer
55 views

On the associative property of a binary operation of the fundamental group.

I was reading about the proof of associativity property of the operation on the fundamental group here. The book gives the following diagram then it says the reader should supply the elementary ...
0
votes
2answers
35 views

Show the two fundamental groups representation of the Klein bottle are the same

Show the two different representation of the fundamental groups of the Klein bottle $G=<a,b:aba^{-1} b>$ and $H=<c,d:c^2 d^{-2}>$. As we calculate in different and gett one result as ...
2
votes
0answers
37 views

Fundamental group of infinitely many glued copies of a space

Let $C=\{p\in \mathbb{R^3}: \left\|p\right\|_{\infty}\le 1\}$ be a cube in $\mathbb{R}^3$. Let $X$ be a compact, path-connected topological space in $\mathbb{R}^3$ such that $X\subset C$, ...
2
votes
1answer
30 views

How to use Seifert–van Kampen to find $\pi_1( RP^n ∨ RP^n)$

Let $X=\{(x,y) \in RP^n\times RP^n;x=x_0, or, y=x_0\}$ where $x_0$ is some fixed point of $RP^n$. In other words, $X$ is two copies of $RP^n$ with one point $x_0$ in common. find $\pi_1( X,x_0)$? I ...
0
votes
2answers
70 views

Explain why the following statement is false

Let $f:S^1 \to S^1$ be given by $f(z)=z^2$, where $z=x+iy, x^2+y^2=1$. Then there is a unique lift $\bar f: S^1 \to \mathbb{R}$ with the properties that (i) $\bar f(1)=0$ and ...
2
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0answers
21 views

Étale Fundamental group for $\mathbb{A}^n$ (prime to $p$-part)

I apologize if this is silly - let $k$ be a separably closed field, I wish to calculate $\pi_1(\mathbb{A}^n)$ completed away from the characteristic :($Hom(\hat{\pi_1(\mathbb{A}^n)}, G)$ classifies ...
3
votes
3answers
86 views

Show the projective space $RP^2$ minus a point is homotopy equivalent to the unit circle $S^1$

Show the projective space $RP^2$ minus a point is homotopy equivalent to the unit circle $S^1$. I can image how to do by the graph,as think of $RP^2$ as the unit disk with opposite boundary points ...
0
votes
0answers
36 views

Seeking guide for project.

I have to submit a project within 2 months for 4th semester(M.Sc). I wish to do it on knot theory, although I know little about it. My plan is to make it an introduction to the subject and to ...
2
votes
1answer
62 views

Associativity of concatenation of closed curves from $I$ to some topological spaces $X$

I'm looking for some example of closed curve such that $f*(g*h)=(f*g)*h,$ in some topological space $X$. I tried to use $X$ like the Sierpinski space, but I can't find such closed curve.
1
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1answer
44 views

Exercise 2 in Hatcher, section 1.2: the union of convex sets is simply connected

Here is my problem : I have convex subsets $X_1, \dots, X_n$ of $\mathbb R^m$ such that $X_i \cap X_j \cap X_k$ is never empty. I have to show that $\bigcup\limits_{i=1}^nX_i$ is simply ...
1
vote
1answer
59 views

$\pi_1([S^2-\{(0,0,1),(0,0,-1)\}]/x\sim -x)$ [closed]

I'm interested in calculating $\pi_1(X)=\pi_1([S^2-\{(0,0,1),(0,0,-1)\}]/x\sim -x)$ Moreover, I'm interested in $Y=[S^2\cap \{x\in\mathbb R^3:\:x_1 x_2 x_3>0\}]/x\sim -x$. Appreciate any kind ...
0
votes
1answer
29 views

isomorphic fundamental groups of quotient space

I have a set $X\subset \mathbb R^n$ an equivalence relation $x∼-x$. Say $Y\cong X$ i.e they are homeomorphic. I would like to conclude $\pi_1(X/∼)=\pi_1(Y/∼)$. Is that true? It does look reasonable. ...
2
votes
2answers
41 views

calculate $\pi_1(\mathbb D-\{(0,0)\})$

I'm interested in calculating $\pi_1(\mathbb D-\{(0,0)\})$. My guess would be that $\mathbb D-\{(0,0)\}$ is homotpic to $S^1$ and so the fundamental group would be $\mathbb Z$. Am I right? How would ...
3
votes
0answers
124 views

Generators of the braid group

Let $C$ be the plane curve in $\mathbb{C}^2$ defined as $\{x²-y^3=0 \}$. The fundamental group of $\mathbb{C}^2 \backslash C$ is the same of the trefoil knot : $\langle a_0,a_1 \; : \; a_0a_1a_0= ...
5
votes
1answer
78 views

Intuition for an abelian fundamental group

Any topological group has an abelian fundamental group by the Eckmann-Hilton argument. Is there some intuition behind the fundamental group being abelian that would enable one to predict this ...
0
votes
2answers
78 views

Fundamental group via Van Kampen

I want to compute the fundamental group of the set C defined below : $A_{1}:=[0,1]²,A_2:=[-1,0]\times[0,1],C=\partial A_1 \cup \partial A_2$. I have to use the Van Kampen theorem and so I know that ...
3
votes
3answers
127 views

Proving fundamental group is commutative if its space is a group

I am working on this proof problem on fundamental group: A fundamental group $\pi_1 (X, x_0)$ is commutative if its space $X$ is group. Here are what I know of: (1) The proof should begin, I ...
2
votes
1answer
44 views

Classification of All Maps $T^2 \to S^4$ up to homotopy

From my study of physics I have arrived at the question of how to classify all maps $\mathbb{T}^2\to S^4$ where $\mathbb{T}^2$ is the two-torus. The classification should be up to homotopies. The ...
1
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2answers
38 views

Homotopy class of a loop in the $2$-skeleton of a simplicial complex

Suppose I have a loop $\sigma : [0, 1] \rightarrow X$ in a path-connected finite simplicial complex $X$. I know that $\sigma$ can be homotoped so that it lives in $\text{sk}^2(X)$, but is this ...
1
vote
1answer
45 views

Computing fundamental groups of products

Let $X$ be a connected graph and $S^{1}$ the usual circle and consider the product $X \times S^{1}$. How would one compute the fundamental group $\pi_{1}(X \times S^{1})$ in this case? I know that one ...
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2answers
49 views

Topological group closed path

A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$, $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let $f,h$ be closed paths ...
1
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1answer
26 views

homotopic closed paths in topological group

A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$ $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let f,h be closed paths in ...
0
votes
1answer
21 views

Reference for a particular case of the classification theorem of covering spaces

Let $X$ be a connected topological space (maybe some other hypothesis should be imposed on $X$). Then I'd like a reference for the following result: The sets: $$A=\{\text{equivalence classes ...
0
votes
1answer
34 views

Retraction and fundamental groups

If we know $\pi_1(X),\pi_1(Y)$, what we can say about existence of retraction $X$ onto $Y$ and vice versa. I think if $\pi_1(X)$ is 'smaller' than $\pi_1(Y)$ there is no retraction $X$ onto $Y$. For ...
2
votes
1answer
16 views

map $GL^+(n)\rightarrow GL^+(n+1)$ homotopy equivalence

Let $GL^+(n)$ be the $n\times n$ real matrices with positiv determinant, and let $i\colon GL^+(n)\mapsto GL^+(n+1)$, $i(A)=\begin{pmatrix} 1 & 0 \\ 0 & A \end{pmatrix}$. Is $i$ a homotopy ...
7
votes
2answers
71 views

What's the “easiest” closed 3-manifold with a nonabelian fundamental group?

I'm looking for some easy compact, oriented 3-manifolds without boundary that have a nonabelian fundamental group. It needn't be perfect. "Easy" means that it has an easy Heegard diagram, say, one ...
0
votes
1answer
44 views

Difference between $\pi(X,x)=0$ and $\pi(X,x)=\{1\}$

Are $\pi(X,x)=0$ and $\pi$$(X,x)=${1} denoted the same thing, the trivial group? Both notation come from a same book: A first course in algebraic topology by Kosniowski. Note that $\pi(X,x)$ is the ...
0
votes
0answers
74 views

Fundamental group of a Coke can

I have to compute the fundamental groups of these sets, but I have no idea of how they can retract!! 1) A closed Coke can (cylinder) with the pull-tab lifted (the pull-tab is a circle). 2) Like the ...
3
votes
1answer
50 views

First examples for topology of non-Hausdorff spaces

I have absolutely no intuition about non-Hausdorff spaces. I would like to understand the topology of non-Hausdorff spaces (in particular spaces obtained by "bad" group actions). As a first example, ...
8
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0answers
134 views

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 ...
1
vote
1answer
27 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\{ ...
3
votes
0answers
92 views

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 ...
3
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0answers
50 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 ...
2
votes
0answers
64 views

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 ...
1
vote
1answer
62 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 ...
5
votes
0answers
57 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} = ...
0
votes
2answers
105 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 ...
7
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0answers
121 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 ...
2
votes
0answers
94 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 ...
2
votes
0answers
38 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 $(x_1, x_2, x_3, x_4, x_5) \rightarrow (-x_2, x_1, -x_4, x_3, x_5)$ ...
1
vote
1answer
54 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 $M\twoheadrightarrow\bigvee^n S^1$ ...
0
votes
1answer
58 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. ...
4
votes
2answers
90 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 ...
1
vote
1answer
63 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.
0
votes
1answer
47 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 ...
0
votes
2answers
83 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 ...
0
votes
1answer
53 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 ...
6
votes
1answer
85 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 $\langle a,b,c\mid a^2b^2c^2\rangle$?