For questions about or involving the fundamental group.

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2
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0answers
41 views

What can be said about a space with solvable fundamental group?

Let $X$ be a topological space. Suppose $\pi_1(X)$ is solvable, can we say something about $X$ ? This question is probably broad, however I am interested in knowing if there is anything at all that ...
2
votes
2answers
70 views

Fundamental group contains $\mathbb{R}$ or $\mathbb{Q}$

Is there any topological space whose fundamental group contains $\mathbb{Q}$ or $\mathbb{R}$? In case of (singular) homology or cohomology, we can change its coefficients to any abelian groups (with ...
4
votes
1answer
44 views

Definition of second topological $K$-group of a Banach algebra

The question is a about the definition of the second topological $K$-group of a Banach algebra $A$. I was reading a text of Alain Valette (Prop. 3.3.7) where he proves that $$ K_1(SA) \cong \pi_1(\...
1
vote
2answers
61 views

How to show that $\pi_1(\mathbb{RP}^2) = \mathbb{Z}_2$?

How to show that $\pi_1(\mathbb{RP}^2) = \mathbb{Z}_2$? In general is well known that $\pi_1(\mathbb{RP}^n) = \mathbb{Z}_2, ~ n \ge 2.$ But how to show this assertion? I have a few knowledge about ...
5
votes
0answers
49 views

Fundamental group of the plane minus a Cantor set

If $C⊆ℝ$ is the Cantor set, what is the rank of the (necessarily free) fundamental group $π_1(ℝ^2 - C×\{0\})$? Since the complement of the Cantor set is open, and an open set in $ℝ$ is always a union ...
1
vote
0answers
45 views

Fundamental group of hole-punched torus with boundary identification

I'm trying to find the fundamental group of $Y$ obtained from the torus by removing a small disk and identifying the boundary with the torus meridian. Here's my idea. The torus has the polygon ...
1
vote
1answer
31 views

Fundamental group of $S^{1}$ unioned with its two diameters

Is my solution correct? Call $X$ the riflescope space (I made this name up). I let $p$ be the point of intersection of the two diameters, and $q$ be the right point of intersection of the horizontal ...
2
votes
1answer
34 views

Surjective map between fundamental group of surfaces

Let $f\colon S_m\rightarrow S_n$ be a continuous map of degree $\pm1$. Then the induced morphism $f_\bullet \colon \pi_1(S_m) \rightarrow \pi_1(S_n)$ is onto. How can I prove this? I know that the ...
0
votes
0answers
28 views

Homotopy on a cylinder

Given a cylinder $C := \mathbb{R} \times S^1$, the fundamental group is $\pi_1 \cong \mathbb{Z}$. My basic question is: Why? I completely fail to see what the set of non-homotopic loops on the ...
3
votes
1answer
53 views

Fundamental group of the unit tangent bundle on the genus 2 torus?

I'm interested in the 3-dimensional model geometries; specifically $\widetilde{SL}(2,\mathbb{R})$. I'm looking for a good (see, easily visualizable) example of a compact manifold formed as a quotient ...
0
votes
1answer
39 views

Which is the fundamental group of $\mathbb{R}^2-\{(0,0),(0,1)\}?$

Which is the fundamental group of $\mathbb{R}^2-\{(0,0),(0,1)\}?$ Making a picture of this space with two closed curves with point basis on $(-1,0)$ and both disjoint each one involving $(0,0)$ and $(...
0
votes
1answer
40 views

Nested sequence of compact connected sets

Suppose that $K_1 \supset K_2 \supset K_3 \supset \dots $ is a nested sequence of compact connected subsets of $S^2$ such that $\pi_1(K_j)\simeq \mathbb{Z}$ for all $j$. Prove or provide a ...
1
vote
1answer
37 views

Construct a space with given fundamental group

I am trying to find out how to construct a space with the following fundamental group: $ \pi_{1}(X)= \langle a,b, c \mid b^{2}ac, c^{-1}a^{2} \rangle$ What is the main strategy for solving this kind ...
2
votes
1answer
50 views

Does a group action lifted to the universal cover commute with the fundamental group action?

Question: Let $\varphi \colon G \to \text{Homeo}(X)$ be a group action on a topological space $X$ with basepoint $x_0$ and universal covering $\pi \colon \widetilde{X} \to X$. Then the subgroup of ...
2
votes
2answers
28 views

Fundamental group of a compact space with compact universal covering space

I have this problem for Riemannian manifold, but think that it is just a topological problem. I know that this is probably a silly question, but it is since a while that I don't study general topology ...
0
votes
1answer
35 views

Computing Fundamental Group of $S^1$

I have a hard time in understanding this from books and papers I found online. So anyone please suggest me some good reference or preferably an online lecture for computation of fundamental group of $...
6
votes
1answer
142 views

Does fundamental group distinguish between any two non homeomorphic topological space?

I am new to fundamental group. I was reading Munkres and found that need of fundamental group was to distinguish between non-homeomorphic topological spaces. So my question is, does fundamental ...
3
votes
2answers
56 views

Is there a retraction of $S^1\vee S^2$ onto $S^1$?

I am currently working through a problem that requires me to prove that $\pi_1(S^1\vee S^2)\simeq\mathbb{Z}$ directly by showing that the homomorphism induced by the inclusion of $S^1$ on $\pi_1$ is a ...
1
vote
1answer
63 views

Hatcher's exercise 1.2.22 on the Wirtinger presentation

Here exercise 1.2.22 is recalled, but the asker seems to know how to solve it assuming the "geometry is valid". I however, do not know to use the van Kampen theorem in order to find the relations $...
-2
votes
0answers
19 views

Pictures and illustrations of attaching a 2-cell to a wedge of circles?

I am having a lot of trouble visualizing how to attach a disk "along a word". Where can I find some pictures and illustrations of this procedure?
1
vote
1answer
27 views

What is a primitive element in a Fuchsian group?

I am reading some introductory texts about Selberg's trace formula for hyperbolic surfaces and I have encountered the concept of "primitive element" $\gamma$ in a hyperbolic Fuchsian group $\Gamma \...
1
vote
1answer
44 views

Can't see why particular homotopy is continuous

I'm checking the group laws for the fundamental group of $(X,x_{0})$: in particular I'm trying to show that $\gamma \simeq \gamma \cdot e$ , where $\gamma$ is a loop based at $x_{0}$, $e$ is the ...
0
votes
1answer
54 views

A relation between the index of a fundamental group and the covering map

My professor just enunciated this statement: $|\pi_1(X,x_0):\pi_1(\tilde X,\tilde x_0)|=\#P^{-1}(x_0)$ where $P$ is the covering $P:\tilde X\rightarrow X$, such that $P(\tilde x_0)=x_0$. I've ...
0
votes
1answer
73 views

Cayley graph of the fundamental group of the 2-torus

Does anybody knows some description or some good picture on the internet for the Cayley graph of the fundamental group of the 2-torus, when this is constructed by connected sum of two torus. I know, ...
3
votes
2answers
81 views

Is there any 'nice' space with fundamental group $\mathbb{Z}_3$?

I'm trying to build up intuition for the fundamental group, as it occurs in physics. In the simplest examples, the fundamental group is trivial, or $\mathbb{Z}$, or $\mathbb{Z}^n$. We can also get $\...
2
votes
0answers
76 views

How can we assume the first homology group of the complement is zero when constructing a Casson handle?

I am currently working through Scorpan's Wild World of 4-Manifolds specifically the section on Casson Handles. On page 78, he says if $D$ is the core of the handle after $n$ stages we may assume $\...
5
votes
1answer
81 views

Why does Van Kampen Theorem fail for the Hawaiian earring space?

The Hawaiian earring space has notoriously complicated fundamental group, and is essentially not as simple as the wedge sum of countably many circles whose fundamental group is straightforwardly given ...
0
votes
0answers
41 views

Calculating The Fundamental Group of the Hopf Fibration

I want to calculate the fundamental group of the Hopf fibration (or rather, the fundamental group of the total space of the fibre bundle that is the hopf fibration). I know that $S^3$ is simply ...
1
vote
0answers
13 views

Fundamental group of a topological group [duplicate]

Given $(G,\cdot)$ a topological group with identity $e$, is it always true that $\pi_1(G,e)$ is abelian? For what it's worth: I have already shown that if $f,g\in\Omega(X,e)$, then $[f \times g]=[f\...
0
votes
1answer
25 views

Equivalent definitions for simple connectivity

For a path connected space $X$, it is simply connected iff any two paths sharing endpoints are homotopically equivalent. Here simple connectivity means it has trivial fundamental group. I'm doing ...
0
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0answers
39 views

The complement $\mathbb{R^3}-A$ of a single circle $A$ deformation retracts onto the wedge of a Circle and a sphere. 2$

Here is my intuitive idea. We can pull all the points lying outside a sphere of some fixed radius containing the circle onto the sphere without disturbing the points inside the sphere. Now we have a ...
0
votes
0answers
42 views

Calculate the fundamental group and homology $S^2\cup T$

I am currently working through an old qualifying exam problem: Calculate the fundamental group and homology groups of the space $X$ obtained from the union $S^2\cup T$ where $S^2$ is the 2-sphere and ...
1
vote
1answer
50 views

Presentation of $\pi_1$ of compact orientable surface by induction?

I need to prove by induction $\pi_1(\Sigma_g)= \left\langle a_1,b_1,\dots ,a_g,b_g\mid \prod_i [a_i,b_i] \right\rangle$. For genus 1 this holds since $\pi_1(T^2)\cong \mathbb Z\times \mathbb Z$. For ...
1
vote
0answers
66 views

Why is the fundamental group of the plane with two holes non-abelian?

I know $\pi_1(\mathbb{R}^2\setminus\{x,y\}) = \mathbb{Z}\ast\mathbb{Z} = \langle a,b\rangle$, but it's non-abelian-ness isn't obvious to me. Specifically, I draw a box and two points to represent $x$...
0
votes
0answers
46 views

Fundamental group of cylinder

I calculated the fundamental group of the cylinder, $C$, using the following method: triangulate $C$ find max contractable subspace realise generators on remaining 1-simplices I found the ...
3
votes
1answer
91 views

“Representability” of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$?

In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic ...
0
votes
1answer
45 views

Reference for: simple closed curves generate the fundamental group

In a 2-complex $X$, it is "obvious" that the simple closed curves through the $0$-cell $v$ generate the fundamental group $\pi_1(X, v)$. (By a "simple closed curve" I mean a loop which does not self-...
0
votes
2answers
40 views

Is this the projective plane or the Klein bottle? (Fundamental polygon)

I am trying to identify the topological type of this fundamental polygon and I think it is the projective plane or the Klein bottle If we treat the top green and red arrows a single arrow then we ...
0
votes
1answer
40 views

Link between $\mathbb{Z}$ and the fundamental group's of 'common' topological spaces

I have noticed that some of the most common topological space have fundamental groups related to $\mathbb{Z}$, as we can see below: Why is this the case? Is it is because they are all realted ...
0
votes
1answer
28 views

Fundamental group of simple graphs

Find the fundamental groups of graphs A, B and C as shown: They look simple but I am unsure what their fundamental groups would be. I was thinking that $A$ and $B$ are generated by just one ...
1
vote
1answer
26 views

principal bundle morphism preserves fundamental group

Two related questions. What is the morphism for principal bundles? Does it "preserve" fundamental groups? Fibre bundle morphisms usually preserve "the structure on the fibre". I am not sure how to ...
0
votes
1answer
32 views

Triangulation of double-holed torus to calculate fundamental group

Show that the fundamental group of the double-holed torus is given by: $\pi_1=<a, b, c, d | aba^{-1}b^{-1}=cdc^{-1}d^{-1}>$ I have ended up with the following identification diagram ...
1
vote
1answer
48 views

The fundamental group of a point is $1$

Show that the fundamental group of the point space $p$ is given as $\pi(p, w_0)=1$ where $w_0$ is the base point This is probably somewhat trivial, but I am looking for a proof. I am familiar with ...
7
votes
6answers
181 views

Examples of $\pi_1 (X) = \mathbb{Z}$ [closed]

I want to know some examples of topological spaces whose fundamental group is isomorphic to set of integers. First, of course i know $\mathbb{S}^1$, and its deformation retract, $\mathbb{R}^2 - \{ 0,...
2
votes
1answer
66 views

Fundamental group of the n-holed torus

I am trying to determine the fundamental group of the n-holed torus I know that the fundamental group of the torus is $\mathbb{Z} \times \mathbb{Z}$ The n-holed torus deformation retracts onto n ...
1
vote
1answer
36 views

Fundamental group of the complement of $S^1 \cup Z \subset \mathbb{R^3}$

I want to calculate the fundamental group of the complement of this space: This is the complement of $S^1 \cup Z \subset \mathbb{R^3}$ where $S^1$ is the unit circle in the $xy$-plane and $Z$ is ...
0
votes
0answers
18 views

Are Fuchsian groups without elliptic and parabolic elements at most countable? [duplicate]

Let $G \subset PSL(2, \Bbb R)$ be a discrete subgroup without elliptic or parabolic elements. Does it follow that it is at most countable? Subgroups as above have the property that the quotients of ...
0
votes
1answer
41 views

Understanding the Fundamental Group

Let us have a space $X$. Define the fundamental group $\pi_{1}(X,x_0)$ for some point $x_0 \in X$. If I understood it well, this group contains the path-homotopy classes that are consisting of paths ...
2
votes
1answer
26 views

Given that $H^1(X)=0$ on a connected space, show that all maps to $X\to S^1$ are null homotopic

Let $X$ be a path-connected, locally path-connected topological space, with $H^1(X)=0$. I would like to show that any map $f:X\to S^1$ is null homotopic, but I haven't really made any progress. ...
1
vote
1answer
30 views

Fundamental group of cylinder -triangulation method

Is this correct? Can we conclude that the fundamental group is trivial since there are no remaining generators on 1-simplices?