For questions about or involving the fundamental group.

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1answer
25 views

How do I compute the fundamental group of this space?

Let $X$ denote the real projective plane. How do I compute the fundamental group of the connected sum $X\#X$? I'd like to use Van Kampen's theorem, but I have trouble visualizing what this space ...
2
votes
2answers
66 views

Manifolds with a finite but not trivial fundamental group

I came across this nice result: Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial: $$H^1_{dR}(M)=0.$$ However, I don't ...
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3answers
47 views

Requirement “closed under finite intersection” in Van-Kampen-Theorem

Given a topological space $X$, a point $x_0 \in X$ and an open cover $\mathcal{U}$ of $X$ of path-connected subsets containing $x_0$ which is closed under finite intersections, we have by Van-Kampen ...
2
votes
1answer
63 views

Does trivial fundamental group imply contractible?

Let $X$ be a path-connected topological space with a trivial fundamental group: $$\pi_1(X,x_0)=\{e\}.$$ Does $X$ have to be homotopic to a point? I know that the converse is true: a ...
3
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2answers
52 views

Help finding the fundamental group of $S^2 \cup \{xyz=0\}$

let $X=S^2 \cup \{xyz=0\}\subset\mathbb{R}^3$ be the union of the unit sphere with the 3 coordinate planes. I'd like to find the fundamental group of $X$. These are my ideas: I think the first thing ...
4
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0answers
57 views

Showing that a space is not homeomorphic to $\mathbb{R}^4$.

I have a space $X$ which is defined to be the quotient of $[0,1)\times T^3$ ($T^3$ is the 3-torus) by the relation $(0,x)\sim (0,y)$ for all $x,y \in T^3$. It is a kind of cone over $T^3$, except that ...
3
votes
1answer
71 views

Integral homology of real Grassmannian $G(2,4)$

I would like to compute $\pi_1$ and the integral homology groups of the real Grassmannian $G(2,4)$. (This is a question on an old qualifying exam.) The hint for the computation of $\pi_1$ is to put a ...
4
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1answer
49 views

Is an open subset of a compact surface with connected boundary completely determined by its fundamental group?

Is an open, connected subset of a compact surface with connected boundary determined (up to homeomorphism) by its fundamental group? If we weaken the hypotheses, I can see how this can fail: A ...
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0answers
27 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
66 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
43 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
35 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
votes
2answers
368 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
59 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
vote
1answer
31 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
votes
1answer
64 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
68 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
vote
1answer
60 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
votes
1answer
45 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
58 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
95 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
78 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& ...
1
vote
1answer
75 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
91 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
votes
2answers
210 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
votes
1answer
45 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
71 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
87 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
votes
1answer
38 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
38 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
221 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
votes
2answers
109 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 ...
1
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1answer
46 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
69 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
votes
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
votes
1answer
63 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
votes
0answers
52 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
votes
1answer
61 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
165 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
78 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
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
votes
0answers
47 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
45 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
81 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
votes
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
163 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
vote
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 ...