12
votes
1answer
127 views

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 ...
2
votes
0answers
20 views

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 ...
2
votes
0answers
34 views

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 ...
10
votes
3answers
408 views

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 ...
2
votes
2answers
83 views

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 ...
0
votes
1answer
33 views

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 ...
2
votes
1answer
56 views

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 ...
1
vote
0answers
53 views

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 ...
1
vote
0answers
42 views

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 ...
2
votes
1answer
114 views

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 ...
5
votes
1answer
125 views

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 ...
5
votes
1answer
513 views

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 ...
2
votes
0answers
183 views

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, ...
3
votes
1answer
176 views

Elliptic Points of Modular Group in Upper Half Plane

This is a very small question. Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq ...
2
votes
2answers
342 views

Complex projective line hausdorff as quotient space

I was wondering if there is a simple argument showing that the complex projective line defined as $\mathbb{CP^1} = \big(\mathbb{C}^2 \setminus \{0\}\big)/{\mathbb{C}^{\times}}$ is hausdorff when ...
3
votes
1answer
124 views

Circle Acting on Circle/Ball

Is much known about $S^1$-actions on the following simple spaces?: 1) $D^2$ the disk 2) More generally $D^n$ the n-ball 3) $S^1$ the circle In particular, does every $S^1$-action on the disk (or ...
4
votes
1answer
135 views

Continuous Actions and Homomorphisms

I am learning about the compact-open topology and have a small proposition I am struggling to prove. Let $G$ be a topological group, $X$ a compact, Hausdorff space, and $H(X)$, the homeomorphisms of ...