# Tagged Questions

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### Is $GL_2(\mathbb Z)\cdot X$ a dense subset of $\mathbb R^2$?

We know that the set $D=\{a+b\sqrt{2} \mid a,b\in \mathbb Z\}$ is dense in $\mathbb R$ because $D$ is a subgroup of $(\mathbb R,+)$ that is not of the form $\alpha \mathbb Z$. So, the following set ...
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### Direct sums, tensor products etc. of $G$-vector bundles are again $G$-spaces

Given two $G$-vector bundles $E$ and $F$ over a $G$-space $X$ ($G$ some finite group), I am interested in the vector bundles $E \oplus F$, $E \otimes F$, $\operatorname{Hom}(E,F)$ etc. I am familiar ...
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### Topologies of flag manifolds

I'm currently reading an article discussing flag manifolds and the action of $\mathrm{PSL}(n,\mathbb{C})$ on them. A flag (in my view at least) is a nested sequence $(y^1,\ldots,y^{n-1})$ of subspaces ...
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### Why care about group actions?

Let X be a space (topological space, manifold, etc) and let the group G act continuously on X. What extra (homotopical, homological, cohomological, diffeomorphical etc) data can extracted from X when ...
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### Dimension of a $G$-variety $X$ that is a finite union of $G$-orbits

Suppose that $G$ is an algebraic group acting on a variety $X$, and $X$ is a finite disjoint union of $G$-orbits $\mathcal{O}_i$, $i=1,\ldots,n$, under this action. Is it true that the dimension of ...
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### Determining whether this is a group action

I'm having trouble with an exercise we were given. I have to determine for which values $a,b\in\mathbb{R}$ $$n\cdot t=\phi_n(t)=2^nt+a^n+b$$ defines a group action of the group $(\mathbb{Z},+)$ on ...
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### Relationship between decompositions of a $G$-variety $V$

Let $V$ be a variety over a field $k$, and let $G$ be an algebraic group over $k$ which acts morphically on $V$. $V$ has three canonical decompositions, and I'm interested in the relationships ...
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### Find the orbit space $T^2 / \mathbb Z_2$

Let $T^2$ be the unit torus $$T^2 = \left\{ (\lambda, \lambda') \in \mathbb C^2 \mid |\lambda| = |\lambda'| = 1 \right\}.$$ Then the group $\mathbb Z_2$ is acting on $T^2$ by the rule ...
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### When the $(\mathbb R,+)$-action defined by the flow of a vector field on a manifold is proper ?

I'm interested in the conditions that a vector field has to satisfy in order for its flow to define a proper action. Technically, let $\xi$ be a smooth vector field. For each $m\in M$, there is a ...
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### Function spaces and transitive group actions

Note: this question is really a subquestion of this one, but I decided to ask it separately since it seems it might be attacked first. Let $B$ be a topological space and $G$ a topological group ...
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### A question about quotient under group action

Let $X$ be a Hausdorff space, and $G$ a group acting on $X$ by homeomorphisms. Let $H$ be a normal subgroup of $G$. Is it true that $X/G$ is homeomorphic to $(X/H)/(G/H)$ ? If so, can you please ...
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### Quotient of a locally compact Hausdorff space by a proper action is Hausdorff

I am trying to prove the following: Let $G$ be a topological group acting properly on a Hausdorff locally compact space $X$, i.e. preimages of compacts sets by the map G\times X\to X\times ...
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### Properly discontinuous action on a non-locally compact space

Let me begin with some definitions in order to avoid confusion. An action of a group $G$ on a space $X$ is proper if the map $G \times X \to X \times X$ given by $(g, x) \mapsto (x, gx)$ is proper, ...
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