What is $\times_G$ in this context?

Question

I've seen the definition for $P \times_\rho F$ in regards to the associated bundle.

See, for example, here: https://en.wikipedia.org/wiki/Associated_bundle

The paper I am reading from uses the notation $P \times_G G/H$ when $H$ is a subgroup when talking about bundle reductions.

Additionally, I know what the adjoint representation is but have no clue what $P\times_G ad$ means. Its as if we are treating it as a group?

What is the $G$-action meant by just $G$?

I've seen this question, but I still don't understand it in this context: What is $P\times_G E$?

The paper is here and the notation is used before page $7$. http://digitalassets.lib.berkeley.edu/etd/ucb/text/Appel_berkeley_0028E_16855.pdf

Answer 1

If $A$ is a right-$G$-space and $B$ a left $G$-space, $A\times_G B$ denotes the quotient of $A\times B$ by the relation $(ag,b)\sim (a,gb)$. Here $P$ is a $G$-bundle so a right $G$-space and $G/H$ is a left $G$-space, so $P\times_G G/H$ makes sense.

For a given right $G$-space $A$, $B\mapsto A\times_G B$ is actually a functor, and so you can evaluate it on maps of $G$-spaces $f:B\to C$ too, and you can denote it by $A\times_G f: A\times_G B\to A\times_G C$