Section 5 of chapter 6 of M.A.Armstrong's "Basic Topology" is about triangulating orbit spaces. It says that if $K$ is a simplicial complex homeomorphic to space $X$ and $G$ acts on $X$ as a group of homeomorphisms, then we can find a simplicial complex homeomorphic to $X/G$.
In the following diagram, $p$ is the natural projection map and $s$ is the map induced from action of $G$ on simplicial complex $|K|$.
But the map $\psi$ is not a homeomorphism because it fails to be injective and we should replace $|K|$ with $|K^2|$ which is second barycentric subdivision of $|K|$.
There are two problems at the end of the section which are subject of my question:
- Show that the map $\psi: |K| \to |K/G|$ is a homeomorphism iff the action of $G$ on $|K|$ satisfies:
(a) The vertices of a 1-simplex of $K$ never lie in the same orbit.
(b) If the sets of vertices $v_0, ... v_k, a$ and $v_0, ... ,v_k, b$ span simplexes of K, and if a, b lie in the same orbit, then there exists $g\in G$ such that $g(v_i) = vi$ for $0 \le i \le k$ and $g(a) = b$.
- Check that conditions (a) and (b) of Problem 28 are always satisfied if we replace $K$ by its second barycentric subdivision.
I have some idea about the first condition of problem 28:
If we identify first and last point of a closed interval, we get $S^1$. But if we identify two $0$-simplexes of a 1-simplex, then we get a $0$-simplex and intuitively we can fix this by replacing the second barycentric subdivision of $|K|$ because this time we will get a 2-simplex instead of a 0-simplex.
How can we explicitly express this? What can we say about condition (b) and problem 29?
Edited:
Clarification of what "acting like a group of homeomorphisms"(I borrowed it from Armstrong) means :
A topologcial group $G$ is said to act as a group of homeomorphisms on a space $X$ if each group element induces a homeomorphism of the space in such a way that:
(a) $hg(x) = h(g(x))$ for all $g, h \in G$, for all $x \in X$.
(b) $e(x) = x$ for all $x \in X$, where $e$ is the identity element of $G$.
(c) the function $G \times X \to X$ defined by $(g,x) \mapsto g(x)$ is continuous.
Also Armstrong introduces the action of a group on a simplicial complex in this way:
Using the projection $p : |K|\to |K|/G$, we define a pair ${V,S}$ as follows: the elements of $V$ are the orbits (projections) of the vertices of $K$, and a subset $u_0, .. u_k$ of $V$ lies in $S$ iff there exist vertices $v_0, ... v_k$ of $K$ which span a simplex of $K$ and satisfy $p(v) = u_i$ for $0 \le i \le k$. The hypotheses of the realization theorem are easily checked; realizing ${V,S}$ in some euclidean space produces a complex which we shall denote by $K/G$. Now $p$ sends vertices of $K$ to vertices of $K/G$, and if $v_0, ... , v_k$ span a simplex of K, then $p(v_0), ... ,p(v_k)$ span a simplex of $K/G$. So $p$ determines a simplicial map $s: |K| \to |K/G|$. Also, for any $x \in |K|, g \in G$ we have $sg(x) = s(x)$, so that $s$ induces a function $\psi: |K|/G \to |K/G|$.