# Tagged Questions

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### Is there a cayley graph for the Klein bottle?

When studying algebraic topology we learned about the fundamental group of the $2$-torus $T^2$ which is isomorphic to $$\langle a, b \mid aba^{-1}b^{-1} \rangle$$ (the free abelian group on two ...
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### Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
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### Fundamental group of the Poincaré Homology Sphere

I'm working on the Poincaré Homology Sphere $P_3$ and would like to compute it's Homology $H_1$ and fundamental group. I would like to identify it's fundamental group with the binary icosahedral group ...
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### Appropriateness of using an interval in $\Bbb{R}$ as the parameter to a continuous path in space $X$.

The fundamental group $\pi_1(X,x)$ is usually defined using continuous paths $p : [0,1] \to X$. But... (1) Can you use other spaces besides a closed interval of $\Bbb{R}$ in the usual topo? (2) How ...
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### $G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint ...
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### How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
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### Mapping Class Group of Simply Connected Spaces

I was wondering the following: If we take $M$ to be some orientable, simply-connected $n$-manifold. What can be said about $\pi_0(Homeo(M))$? We know that $\pi_1(M)=0$ and I know that the group is ...
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### Free action of finite direct product

Let's consider free action of finite abelian group $G = G_1 \oplus G_2$ on a manifold $X$. Is it true that $X/G$ is diffeomorphic to $(X/G_1)/G_2$?
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### Basis for singular chains group

The singular chain group $S_p(X)$ is defined as the free abelian group generated by continuous functions $T \in C( \Delta_p , X)$. What I understand this means is that we define $T' \in S_p(X)$ as ...
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### if we set topology on a group like that, is it important?

Let $G$ be a group and $\omega$ be set of all subgroup of $G$. Since $\omega$ is closed under intersection, it is trivial to check that $\omega$ satisfies to conditions to be a base. Thus,Let $T$ be ...
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### Fundamental group of Poincaré sphere

Do the two presentations below, $$G=\langle d,v \mid dv^2d=vdv, dv^3d=v^2 \rangle$$ and $$\langle r,s,t \mid r^2=s^3=t^5=rst \rangle = \langle s,t \mid (st)^2=s^3=t^5 \rangle,$$ define the same group? ...
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### Set of generators of the commutator subgroup of a surface group

Good morning, I am having a hard time trying to describe the commutator subgroup of a surface group. Namely, if $S$ is a compact orientable surface and $G$ its fundamental subgroup, let's recall that ...
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### Different profinite topologies on a group?

I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...
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### Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
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### Abelian group with cyclic subgroup and cyclic quotient is generated by two elements

I have a number of questions that I think are related. I'm studying Algebraic Topology by Hatcher. I have essentially the same question as here. When talking about homology groups, the book says that ...
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### relation between the group O(3) and SU(2)

Base on relations between groups $O(3)$, $SO(3)$ and $SU(2)$: a) $O(3)=SO(3)\otimes \{1,-1\}$ b) $SO(3)\simeq SU(2)/Z_2$ Can I say $\{1,-1\}$, i.e. $Z_2$, also the center of the group $O(3)$? If ...
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### Free product of the trivial group with another group

I'm new to the idea of a free product.. Basically I was wondering if G is an arbitrary group and 1 is the trivial group then is $1\star G \cong G$. If not.. what whould it look like?
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### HNN extensions as fundamental groups

I have heard that the Seifert–van Kampen theorem allows us to view HNN extensions as fundamental groups of suitably constructed spaces. I can understand the analogous statement for amalgamated free ...
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### Meaning of Fundamental group of a graph

I am a computer science student working in graph algorithms. I am unable to understand what the fundamental group of a graph means. I have some intuition regarding the fundamental group of a ...
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### Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
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### Group of continuous automorphisms of cylindrical plane is a lie group

How do I prove this theorem? Theorem: The group of continuous automorphisms of a cylindrical plane is a lie group. In this context, cylindrical means the Laguerre plane. I found a paper it ...
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### Topology of a 3D wired Mandala?

There is a so called 3D-wired Mandala, based upon $2$ large circles each flowered symmetrically on its circumference by two sets of each $8$ half-circles. The circles are interconnected together by ...
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### On the quotient group $\pi_{1}(K)/N$ for the Klein bottle $K$

I know that the Klein bottle $K$ is obtained from the unit square by making identifications on the boundary with the appropriate directional arrows. Usually, what is done is that we identify the point ...
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### Zero exponent sum w.r.t group words in knot group's presentation

I am reading, "Plane Curves Associated to Character Varieties of 3-Manifolds" by Cooper, Culler, Gillet, Long, and Shalen and on page 28 ( http://www.math.uic.edu/~culler/papers/PlaneCurves/curves.pdf ...
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### Is there a nontrivial topological group that's isomorphic to its fundamental group?

All I know is that the topological group has to be Abelian. I have no idea how to prove or disprove this statement. Thanks in advance.
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### Double cover of symplectic groups

What is the normal definition of double cover of Symplectic group? I couldn't find a simple and understandable definition
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### Topics of Group Theory Required to Understand Betti Numbers

I am studying Group Theory. I made sure I have a problem at hand, as part of the motivation for my study. I have chosen the problem as being able to understand as well as compute Betti Numbers where ...
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### Prove that any subgroup of $F_5$ of index 3 is isomorphic to $F_{13}$

Let $F_n$ denote the free group on $n$ elements. Prove that any subgroup of $F_5$ of index 3 is isomorphic to $F_{13}$. I noted that the wedge product of 13 copies of $S^1$ is a 3 fold covering ...
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### Is $\langle abab^{-1}\rangle$ a normal subgroup of $\langle a,b|\varnothing\rangle$?

Let $G=\langle a,b | \varnothing\rangle$ and let $H\leq G$ s.t $H=\langle abab^{-1}\rangle$. Is $H\triangleleft G$? I'm asking this question in order to understand the fundamental group of the Klein ...
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### Showing path connected matrices of a group $G$ is a normal subgroup

Let $G$ be a subgroup of $GL_n(\Bbb{R})$. Define $$H = \biggl\{ A \in G \ \biggl| \ \exists \ \varphi:[0,1] \to G \ \text{continuous such that} \ \varphi(0)=A , \ \varphi(1)=I\biggr\}$$ Show that $H$ ...
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### Consequence of injectivity of projections from covering spaces

We have the theorem which says that the induced homomorphism $p_* : \pi_1(\tilde X,\tilde x_0)\rightarrow \pi_1(X,x_0)$ is injective (hence a monomorphism). Here $\tilde X$ is a covering space of $X$. ...
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### Homomorphism/map in both direction implies isomorphism/homeomorphism or not?

I was working on a homework, and my first attempt get me to a deadend, but I was eventually able to solve it using a different method. But the fail attempt make me curious, and I wonder if it could ...
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### Infinite products of a (finite) group

So I'm having a little trouble understanding the concept of infinite (cartesian) products of a group -- specifically, my notes (and, of course, homework questions) have concepts of, say ...
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### E measurable with m(E) < $\infty$?

Suppose that $E$ is measurable with $m(E)$ $<$ $\infty$. ii) Show that $\displaystyle \ \ \int_E 2f\,\,\,$ $=$ $2$$\displaystyle \ \ \int_E f\,\,\,$ if $f$ is bounded and measurable. I told my ...
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### Sufficient condition for a direct limit of abelian groups to be infinitely generated

I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
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### free groups and bouquet of circles

For any free group $F$ generated by the set $S$, one can construct a graph specifically a bouquet of circles $X$ s.t $\pi_1(X)=F$. My question is: Does this mean free groups are isomorphic to a free ...
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### Groups acting on polytopes

I am currently reading the paper "Polytopal Resolutions for Finite Groups" [1] by Graham Ellis, James Harris and Emil Skoeldberg and have a question regarding an early remark of theirs. Their basic ...
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### Given a topological space $X$, does $H_1(X)=\mathbb{Z}$ imply $\pi(X)=\mathbb{Z}$?

Let $X$ be any topological space with the first homology group $H_1(X)=\mathbb{Z}$ . I claim $\pi_1(X)=\mathbb{Z}$. By Hurewicz, we know that $H_1(X)$ is the abelianization of $\pi_1(X)$. ...
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### All the compact covering spaces of torus.

I know the covering spaces of the of a torus $T^2$ are homeomorphic to $T^2,S^1\times\mathbb{R},\mathbb{R}^2$. I am interested in finding all of the covers with covering space $T^2$. The subgroups of ...
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### Representation theorems for groups

There are two baffling representation theorems for groups: Every group is isomorphic to the automorphism group of some graph. (see Frucht's theorem) Every group is isomorphic to the fundamental ...
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### Groups not arising from certain centralizers

There's a lot of fuss in certain subfields of algebraic topology about giving fancy interpretations to the rings coming from the cohomology of groups, where "cohomology" is allowed to be taken to be ...
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### Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?

Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
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### proof by nuke of the fact that fundamental group of topological group is abelian

"The fundamental group of a topological group is abelian". does this problem admit a proof by nuke. This is inspired by a a question in mathoverflow. The usual proof is by a Eckmann-Hilton ...
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### Prerequisites for Algebraic Topology

I'd like to self-study Munkres' Topology. I'm already comfortable with point-set topology, so the first part of the book will serve as a nice review with some new theorems every now and then. My main ...
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### Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$

Can the fundamental group of an open connected subset $X$ of $\mathbb{R}^n$ have a torsion element?
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### Topological Meaning of semi-direct product

I know that the amalgamated free product of two groups $G\star_K H$ has a certain topological meaning. What about a semi-direct product $H \rtimes G$ ?
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### Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
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### Why isn't this free product of groups abelian?

I'm trying to prove that the free group $A=A_1*A_2$, where $A_1, A_2\neq 1$ is not abelian. Following the hints below: Let $x,y\in A_1*A_2$, where $x\neq y$. Suppose now $A_1=F(S)$ and $A_2=F(T)$, ...
I know that the fundamental group of the double torus is $\pi_1(M)=\langle a,b,c,d;a^{-1}b^{-1}abc^{-1}d^{-1}cd\rangle$. How can I calculate its center subgroup $C$? Is $C$ trivial? Let $p$ be the ...