For questions about or involving the fundamental group.

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23 views

Homology and fundamental group of closed ball minus a “T”

I wish to solve the following problem: Let $D = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 1\}$. Let $Y = \{(x, 0, 1/2) \in \mathbb{R}^3 \mid x \in \mathbb{R}\} \cup \{(0, 0, z) \in ...
2
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0answers
59 views

Homotopic maps between connected spaces inducing the same homomorphism between the fundamental groups

This is Problem 7-9 in Lee's Introduction to Topological Manifolds: Suppose $X$ and $Y$ are connected topological spaces, and the fundamental group of $Y$ is abelian. Show that if $F,G: X ...
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2answers
42 views

A trivial fundamental group

I am reading fundamental groups from Munkres book. As stated in the definition, I understand a fundamental group relative to a base point $x_0$ includes all the loops based at point $x_0$. Later ...
2
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0answers
25 views

When does “every closed path is homotopic to a point” imply the space is path connected?

In the middle of talking about primitives and the Cauchy integral theorem, my Complex Analysis teacher came up with this sentence: This reasoning can be done in any simply connected set, because ...
9
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2answers
350 views

Does the definition of the fundamental group implicitly assume the Axiom of Choice?

Okay, I'm a little foggy around the axiom of choice, so help me out here. The standard way the fundamental group of a connected space $X$ is defined is as follows. You consider the set of all loops ...
0
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0answers
57 views

How do I visualize this quotient space?

If $V = [0,1] \times [0,1] \subset \mathbb{R}^2$. We define the equivalence relation $\sim$ on $V$ as follows: every element $(x,y) \in V$ is equivalent with itself and besides that the three ...
1
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1answer
30 views

van Kampen theorem for fundamental groupoid of $X$ relative to $A$

Let $X$ be a manifold with submanifold $A \subseteq X$. Let $\Pi_{1}(X,A)$ denote the homotopy classes of paths with endpoints lying in $A$. This is a Lie groupoid with set of objects $A$. For ...
5
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1answer
57 views

Is simply connectedness preserved after deleting a high codimension set

Suppose $X$ is a complex manifold of complex dimension $n$, $Z$ is a subvariety of complex codimension at least $2$. Suppose $\pi_1(X)=0$, do we have $\pi_1(X-Z)=0$? Do we have $\pi_1(X-Z)=\pi_1(X)$ ...
2
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0answers
65 views

Fundamental Group of a Surface [closed]

I came across the term "fundamental group of a surface" while reading a paper, and I'm not sure what it it all about. As well, what is understood by the generators of the fundamental group of a ...
1
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1answer
54 views

Fundamental group of a modified annulus

Let $A\subseteq C$ be the annulus given by $A=\left\{z|1\geq|z|\geq\frac12\right\}$. Define an equivalence relation on $A$ as follows: two different points $z, w$ are equivalent if $|z| = |w| = 1$ ...
0
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1answer
44 views

Calculate the first homology group of $P^2\#T$, that is $H_1(P^2\#T)$

I've already found that the fundamental group of the connected sum $P^2\#T$, by the labelling scheme $aabcb^{-1}c^{-1}$, to be $F_3/<aabcb^{-1}c^{-1}>$. How would I find the first homology ...
0
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2answers
66 views

Homotopy/fundamental group question: Why group axioms fail when defined on paths?

Neither Munkres nor Lee in their textbooks explicitly show why (fundamental) group properties like associativity fail when defined at the level of paths but work fine for homotopy classes of paths. ...
0
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0answers
56 views

Calculate the fundamental group of $S^1/\mathbb Z_n$

Calculate the fundamental group of $S^1/\mathbb Z_n$ ,where $\mathbb Z_n$ acts naturally on $S^1$ by rotations of $2\pi /n$ The origin of this problem is the following unclear solution of another ...
2
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1answer
89 views

A map $h:S^1\to X$ Induces a Trivial Homomorphism of Fundamental Groups Iff it is Nullhomotopic.

I recently started reading Algebraic Topology from Part II of Munkres' book Topology(Second Edition). A part of Lemma 55.3 in the book proves the following: Let $h:S^1\to X$ be a continuous ...
0
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2answers
77 views

Lists of the first fundamental group of spaces. [closed]

Here are some list to start with $$\begin{array}{c|c|c|} \hline Space(S)& \pi_1(S) \\ \hline \mathbb{R}^2&0 \\ \hline \mathbb{S}^1& \mathbb{Z} \\ \hline 1-Torus& ...
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1answer
74 views

The winding number (topological definition) is well-defined

can someone please tell me if my proof of the lemma below is legit. At first I like to give a definition so you know what the lemma is about. Definition: Let $g:[0,1] \to S^{1}$ be a closed path in ...
2
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0answers
83 views

Fundamental group of the Klein bottle

Proof that fundamental group of the Klein bottle have the next two different presentations: a) $Gen(a,b \,; baba^{-1}=1)$, b) $Gen(a,b \,; a^{2}b^{2}=1)$. I have the proof of (a) and I can prove that ...
8
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2answers
192 views

Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?

I am curious if there is a decent "bare hands" proof that the fundamental group of $S^1$ is $\mathbb Z$ that does not invoke covering space theory. One must show two claims. First, that $f(t)=e^{2\pi ...
2
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1answer
42 views

Find a presentation for the fundamental group of $P^2\#T$

I have to find a presentation for the fundamental group of $ P^2\# T $. Which I believe to be identified by the following labeling scheme: $(a_1b_1a_1b_1)(a_2b_2a_2^{-1}b_2^{-1})$. $P^2$ is the ...
2
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2answers
65 views

A space $X$ with $S^{2n+1}$ as universal covering space must be orientable.

Problem If $X$ has $S^{2n+1}$ as universal covering space, then show that $X$ must be orientable. My idea: By contradiction, suppose $X$ is non-oreintable. Then we consider the orientation covering ...
3
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1answer
85 views

The fundamental group of $S^n$

I want to prove that $\pi_1(S^n,x_0)$ is trivial if $2\leq n,$ BUT using universal covering. So let $p:\tilde S^n \rightarrow S^n$ the universal covering. Define $f:D^n\rightarrow S^n$ such that ...
0
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1answer
37 views

Why is $Z*Z/({xyx^{-1}y} )= 1$?

I was reading notes on the Fundamental Group and came across the statement $Z*Z/({xyx^{-1}y}) = 1$. Why is this true? If a quotient group is trivial doesn't it mean that the two factors are ...
0
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1answer
36 views

Fundamental groups in path-connected space

I'm studying Fundamental groups and today I saw the follow theorem: Theorem: Let be $X$ a topological space path-connected and $x,y\in X$. Then, the application $\psi:\pi_1(X,x)\to \pi_1(X,y)$ is a ...
6
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4answers
218 views

What is an “inner isomorphism” between different groups?

It is well known that if $X$ is a path-connected topological space containing points $x$ and $y$, then the fundamental groups $\pi_1(X,x)$ and $\pi_1(X,y)$ are isomorphic. Wikipedia makes the further ...
3
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2answers
106 views

intuition on the fundamental group of $S^1$

I am familiar with the proof that the fundamental group of the unit circle $S^1$ is $\mathbb Z$, yet I couldn't develop intuition for why it is true. For example, why would I fail if I try to find ...
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1answer
44 views

Fundamental group of a quotient of $S^2 \times I$.

Let $X=S^2 \times [0,1]$ and let Y be the quotient space obtained from X identifying each point $x\in S^2 \times \{1\}$ with its antipodal in $S^2 \times \{0\}$. How can I calculate $\pi_1 (Y)$? All ...
0
votes
0answers
68 views

Fundamental group of projective plane with handles

I was told that the fundamental group of the projective plane with g handles is isomorphic to $\langle c_1, \ldots, c_{2g+1} | c_1^2 \cdot \ldots \cdot c_{2g+1}^2\rangle$. How can I show it? I can ...
0
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0answers
12 views

Etale Fundamental group of an algebraic group

I want to calculate the algebraic fundamental group of a an algebraic group over a riemann surface over $\mathbb C$ (or a smooth algebraic projective curve). Let me state the first case where ...
2
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1answer
58 views

Approximating a CW-complex with a trivial fundamental group

Let $X$ be CW-complex and $\pi_1(X) = 0$. I want to prove that there is $Y \simeq X$ with only one $0$-cell and without $1$-cells. Geometrically it's obvious, but I have no idea for rigorous proof. I ...
5
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1answer
62 views

Bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_n/\pi_1$

I would like to prove that there is a bijection between the free homotopy classes $[S^{n},X]$ and the orbit space $\pi_{n}(X,x_{0}) / \pi_{1}(X,x_{0})$ where the action of $\pi_{1}(X,x_{0})$ over ...
0
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0answers
49 views

Fundamental group of the mapping cone of a loop

You can read in Wikipedia the following: Given a space X and a loop $\alpha\colon S^1 \to X$ representing an element of the fundamental group of $X$, we can form the mapping cone $C_α$. The effect of ...
4
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2answers
113 views

Finding the Fundamental Groups of Some Modular Spaces

I'm looking to compute the fundamental group of a couple of different quotients of the $n$-torus. The first of these I'm interested is the space $\mathbb{T}^n/S_n$ where the symmetric group $S_n$ ...
2
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1answer
60 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
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1answer
160 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
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1answer
77 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
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2answers
60 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
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0answers
43 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
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1answer
42 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
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2answers
80 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
23 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 ...
4
votes
3answers
146 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 ...
1
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0answers
40 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
74 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.
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1answer
67 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
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1answer
60 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
35 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
44 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
137 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
89 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
92 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 ...