2
votes
1answer
97 views

Actions of compact Lie groups on (possibly but hopefully not) non-regular spaces

Suppose $G$ is a compact Lie group acting freely on a topological space $Q$ (about whose separation conditions I make no assumptions) and the qoutient $Q/G$ is known to be completely regular Hausdorff ...
1
vote
1answer
269 views

Free and cellular G-action implies free G-complex?

Recall that a CW-complex $X$ with an action of a group $G$ which permutes the cells (i.e., for any $g \in G$ and any cell $\sigma \subseteq X$, $g\sigma$ is a cell) is called a $G$-complex. If the ...
3
votes
1answer
478 views

Abstract definition of properly discontinuous action

A discrete group $G$ acts properly discontinuously on a manifold $M$ if the set $\{g\in G\mid gK\cap K\neq \emptyset \}$ is finite for every compact $K\subset M$. Is there a more abstract ...
3
votes
1answer
312 views

Action on a compact group

If $G$ is an infinite compact group, how many orbits can $G$ have under the group action of its continuous automorphisms ?
1
vote
0answers
171 views

Faithful actions of finite groups on topological spaces

Suppose that $G$ is a finite group acting faithfully on a topological space $X$. In the smooth setting, one can deduce that for each $x$ in $M$, the induced map $$G_x \to Diff_x\left(M\right)$$ from ...
10
votes
4answers
883 views

When do isometric actions exist?

Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...
25
votes
3answers
7k views

Properly Discontinuous Action

When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...
4
votes
1answer
271 views

Fixed points sets of pushouts

Let $G$ be a group and $X \to Y, X \to Z$ morphisms of $G$-sets with pushout $P=Y \cup_X Z$. Is then $P^G$ the pushout of $X^G \to Y^G, X^G \to Z^G$? This is not clear from general category theory, ...