For an action of a group $G$ on a topological space $Y$ where each element of $G$ corresponds to a homeomorphism of $Y$, Hatcher defines the orbit space $Y/G$ of the action to be the quotient of $Y$ by the equivalence relation of being in the same orbit of the action. This confuses me. The notation $Y/G$ mentions the space and the group, but not the action. From visualising, I’m pretty sure two $Y/G$’s would not even have to be homeomorphic. Consider the subspace $X$ of $\mathbb{R}^2$ given by the union of all lines $x=z$ and $y=z$ with $z\in\mathbb{Z}$. Here are two actions of $\mathbb{Z}$ on $X$. In the first, a generator maps to the function $x+(1,0)$, in the second it maps to $x+(1,1)$. The quotient by the first action is a line with a bunch of circles glued to it at a discrete set of points, whereas the quotient by the second is a bunch of circles indexed by the integers with the North pole of one glued to the South pole of the next. These are homotopy equivalent but not homeomorphic. So why the notation?
Ambiguity in orbit space notation
algebraic-topologygroup-actionsnotation
Best Answer
You are right, a (left) group action is a function $\alpha : G \times Y \to Y$ with suitable properties, thus it would be correct to write $Y/\alpha$. However, it is common use (or, if you want, common abuse of notation) to write it in the form $Y/G$. Unfortunately one cannot say much more than that.