I'm currently reading "Bordism of Elementary Abelian Groups via Inessential Brown-Peterson Homology" by Hanke ([arXiv:1503.04563](https://arxiv.org/abs/1503.04563)) and have come across some notation that I'm not familiar with. Let $\phi: \mathbb{Z}/3 \rightarrow (\mathbb{Z}/3)^2$ be a group homomorphism and $S^5$ the five dimensional sphere in $\mathbb{C}^4$. $\phi$ induces an action of $\mathbb{Z}/3$ on $(\mathbb{Z}/3)^2$ and $S^5$ comes with a natural $\mathbb{Z}/3$ action, the standard action from the lens space construction. The paper then writes
$$(\mathbb{Z}/3)^2 \times\_{\mathbb{Z}/3} S^5$$
How is this defined? Furthermore, it is stated that this comes with a $(\mathbb{Z}/3)^2$ action.


As a check, it should be defined so that
$$((\mathbb{Z}/3)^2 \times\_{(\mathbb{Z}/3)^2} (S^{2m\_1+1}\times S^{2m\_2+1})/(\mathbb{Z}\_3)^2 \cong L^{2m\_1+2}\_3\times L^{2m\_2+1}\_3$$
is a product of lens spaces.



Understanding $(\mathbb{Z}/3)^2 \times_{\mathbb{Z}/3} M$
This has been more or less said in the comments, but it seems like someone should write an answer.


If $H$ is a subgroup of $G$ and $H$ has a left action on a space $X$, then a space $G\times \_HX$ is defined, a quotient space of $G\times X$, by declaring $(gh,x)\sim (g,hx)$. Note that we are using the right action of $H$ on the space $G$. (If we like, we can create a left action on $G\times X$ using the right action on $G$ and the left action on $X$, writing $h(g,x)= (gh^{-1},hx)$. But this is unnecessary.)


Let's denote the equivalence class of $(g,x)$ by, say, $[g,x]$. The quotient space gets a left $G$-action, defined by $g\_1[g\_2,x]=[g\_1g\_2,x]$. (This is well-defined because the left action of $G$ on (the space) $G$ commutes with the right action: $[g\_1(g\_2h),x]=[(g\_1g\_2)h,x]=[g\_1g\_2,hx]$.


In categorical terms: The subgroup $H\subset G$ (or the homomorphism $H\to G$) is giving us a functor $X\mapsto G\times\_HX$ from the category of spaces with left $H$-action to the category of spaces with left $G$-action. This functor is left adjoint to the forgetful functor.


This is very similar to how a homomorphism $R\to S$ of (associative) rings gives a functor $X\mapsto S\otimes\_R X$ from $R$-modules to $S$-modules, again left adjoint to the forgetful functor.


