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I met the following notation in a book I was reading which I've never seen before

$$A \times_{G}B$$

Here $G$ is a group, $A,B$ are topological spaces. How do we define this notation?

P.S. I have a guess according to the context but I'm not sure if it is right. In the context there is a group action of $G$ on both $A$ and $B$. Is this space defined to be the orbit space of the product space $A \times B$?

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Here $A,B$ are $G$-spaces and $$A\times_GB=(A\times B)/G$$ is the quotient by the diagonal $G$-action on $A\times B$. To make sense of this you should assume that $G$ acts on both $A,B$ either from the left, or from the right.

On the other hand this notation is most frequently encountered when $A$ is a right $G$-space and $B$ is a left $G$-space. Then $A$ is turned into a left $G$-space by letting $g\cdot a=ag^{-1}$, and the construction is the same. Thus in this case $A\times_GB$ is the quotient of $A\times B$ by the $G$-action $$g\cdot (a,b)=(ag^{-1},gb).$$

Tyrone
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  • In the latter case, it can also be considered the quotient $(A\times B)/\sim$ by the equivalence relation $(ag,b)\sim(a,gb)$. I like that this illustrates the similarity to extension-of-scalars for rings (constructed using tensor products). – anon Feb 04 '21 at 04:56