Ambiguity in orbit space notation

Question

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?

Answer 1

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.

Answer 2

Just to augment the answer of @PaulFrost a bit, there is a alternate method for denoting orbit spaces that sidesteps this abuse of notation.

As you probably know, an action of $G$ on a topological space $Y$ is the same as a homomorphism $$\alpha : G \to \text{Homeo}(Y) $$ where $\text{Homeo}(Y)$ denotes the group of self-homeomorphisms of $Y$ under the operation of function composition. If we let $\Gamma = \text{image}(\alpha) < \text{Homeo}(Y)$ then $\alpha$ becomes an isomorphism between $G$ and $\Gamma$, and we then write $Y/\Gamma$ for the quotient. This removes all ambiguity of the quotient space itself. It nonetheless loses a bit in that the original group $G$ has been lost, as has its homomorphism to $\Gamma$.

This alternate notation is particularly common in geometric situations. For example, if $X$ is some kind of geometric space such as the Euclidean plane or the hyperbolic plane, one is interested in isometric actions on $X$, which means actions that preserve the geometric structure (i.e. the Riemannian metric). The subgroup of $\text{Homeo}(X)$ consisting of isometries is denoted $\text{Isom}(X)$. It is then very common to denote (and study) quotient spaces $X / \Gamma$ of subgroups $\Gamma < \text{Isom}(X)$.