Quick question:-Sometimes when you have a principal $G$-bundle $p:P\rightarrow B$ and another topological space $F$ upon which $G$ acts, it's said that one can define the associated bundle $P\times_{B}F$, but what does that mean when you have the base space like that?
I know how $P\times_{G}F$ is defined, but when it's written $P\times_{B}F$ how should I interpret it?
EDIT: Maybe comes from assuming that $B$ must be a Lie group? In that case the definition is always the same... are there any particular other definitions?
I'm saying that cause the definition of $Spinc^{\mathbb{C}}$-structure is the following:
"Let $p:P\rightarrow B$ be a principal $SO(p,q)$-bundle. A $Spin^{\mathbb{C}}$-structure is a principal $Spin^{\mathbb{C}}(p,q)$-bundle $\widetilde{p}:\widetilde{P}\rightarrow B$ and a principal $U(1)$-bundle $p':P'\rightarrow B$ such that
(i)There is a bundle map $\widetilde{r}:\widetilde{P}\rightarrow P\times_{B}P'$ that is a double cover;
(ii)The diagram
$$\begin{array}{ccc} \widetilde{P}&\times& Spin^{\mathbb{C}}(p,q) & \longrightarrow &\widetilde{P}\\ \downarrow{\widetilde{r}} & & \downarrow{r^{\mathbb{C}}} & &\downarrow{\widetilde{r}}\\ (P\times_{B}P')&\times&SO(p,q)\times U(1)&\longrightarrow&P\times_{B}P'\\ \end{array}$$ commute. As you can see we're assuming nothing about B, unless the definition I have is not complete.
Say $B$ is a group (say a Lie group, so its both a nice topological space (a manifold) and a group ) and $P \to B$ is a principal $B$-bundle. That means the base space is $B$ but also the group acting on $P$ is $B$.
Then there is an action of $B$ on both $P$ and $F$ and then the associated bundle $P \times_B F \to B$ is defined as the set of equivalence classes $$ (p,f) \sim (bp,b \cdot F) $$ for $b \in B$, which is also a group. By $bp$ I refer to the action of $B$ on $P$ and by $b \cdot F$ I mean the action of $b$ on $F$.
That being said, I wouldnt be surprised if all such bundles end up being trivial, but thats another questions.