# Tagged Questions

For questions about or involving the fundamental group.

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### 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 ...
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### 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 ...
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### 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(\...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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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 $[f \times g]=[f\... 1answer 25 views ### Equivalent definitions for simple connectivity For a path connected space$X$, it is simply connected iff any two paths sharing endpoints are homotopically equivalent. Here simple connectivity means it has trivial fundamental group. I'm doing ... 0answers 39 views ### The complement$\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 ...
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### 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 ...
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### 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 ...
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### 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 $x$...
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### 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 ...
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### “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 ...
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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...