# Tagged Questions

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### Representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely ...
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### General Linear Group is(not) compactly generated

We know that any connected Lie Group is compactly generated. I have a feeling that $\mathbb{GL}_n\mathbb{R}$ is not compactly generated. Is it true? If it is how can I prove this? If not, what ...
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### Definition of Lie Groups

In the definition of Lie Group, we require that $$(x,y)\rightarrow x*y \text{ and } x\rightarrow x^{-1}$$ both be smooth. Are there any examples of groups that satisfy only one of these and not the ...
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### Symmetry group of a non abelian group manifold

Reading an article ...$B$ is itself the manifold of some group $H$. It should be noted that, if $H$ is a non abelian group, the symmetry group $G$ of the group manifold is not $H$ but ...
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### Isotropy group for the subset of Grassmannian

Consider a complex Grassmannian $Gr_{k}(C^{n})$, which is a symmetric space with symmetry group $U(n)$ (i.e. unitary group). Consider a subspace $S_{0}$ of the Grassmannian determined by the canonical ...
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### Tangent space, a group and a manifold

Let G be a group with a smooth manifold structure and let $u:G\times G\rightarrow G$ be the smooth multiplication defined by $(x, y)\mapsto xy$. Question: Why is the Tangent map $T_{(e, e)}u$ given ...
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### Visualize $\mathbb{S}^3/\Gamma$!

I thought the only 3-manifold with positive constant curvature is $\mathbb{S}^3$. But I faced $\mathbb{S}^3/\Gamma$, where $\Gamma$ is a finite subgroup of $SO(4)$ and surprised! My problem is that I ...
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### Spherical Grid Identifcation

I'm trying to see how the lower half of these grids look like when I make the following identification onto the unit sphere: CLICK HERE TO SEE IMAGE In notation, how would one represent this ...
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### Prove that the group $\langle U\rangle\leq G$ is open and closed in $G \subset GL_n\mathbb R$.

Let $G \subset GL_n\mathbb{R}$ be a closed subgroup and $U \subset G$ open with respect to the subspace topology. Prove that the group $\langle U \rangle$ generated by $U$, i.e. the smallest subgroup ...
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### Conditions for a group to admit the structure of a Lie group

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
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### What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...
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### How much does $\operatorname{Aut}(H_1(S))$ determine a homeomorphism $S \to S$?

Let $S$ be an orientable compact surface. A homeomorphism $f: S \to S$ induces an isomorphism $f_{*}: H_1(S) \to H_1(S)$. How much can we say the converse? Namely, if we are given an element of ...
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### Fundamental Domain of Manifold Reference Request

I am interested in learning about the fundamental domain of a manifold and I am wondering if anyone know of any papers or descriptions online other than Wikipedia and the linked articles? I am looking ...
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### Fundamental groups of certain 3-manifolds

I'm starting my master's thesis on geometry/topology & group theory. I'd like to know examples of fundamental groups of 3-manifolds having geometric structure of the following types: ...
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### Fundamental group of a compact manifold

In an article I am currently reading, the author tells us that for compact (finite dimensional topological) manifolds X and finite groups $\Gamma$, the set $$\mathrm{Hom}(\pi_1X,\Gamma)/\Gamma$$ where ...
Let $G=C_{19}\rtimes C_9=\langle a,b\ |\ a^{19}=b^9=1, a^b=a^7\rangle$ be a nonabelian group of order $171$. Is there a (compact) 3-manifold $M$ with $\pi_1(M)\cong G$? Thanks for any help!
I'm trying to understand why, if $S$ is a Seifert fibered space, then $\pi_1(S)$ is residually finite. From theorems 12.2 and 11.10 in Hempel's "3-manifolds", we can work with a finite-sheeted ...