facts later in the proof of Lemma \[lem:jointrivial\].

In the rest of this section we focus on the fiber join construction performed on $G$-fibrations. We use fiber joins to kill obstructions that occur in the construction of $G$-fibrations. The fiber join of two $G$-fibrations is defined in the following way: Let $p_1: E_1\to B$ and $p_2: E_2\to B$ be two fibrations. We define a fibration $E_1\times _B E_2$ and maps $E_1\times _B E_2\to E_i$ for $i=1,2$ by the following pullback diagram $$\xymatrix{
E_1\times _B E_2 \ar[r]\ar[d] & E_{1}  \ar[d]^{p_{1}} \\
E_2 \ar[r]^{p_2}             &  B . }$$ Then the $G$-space $E_1*_B E_2$ is defined as the homotopy pushout of the following diagram $$\xymatrix{
E_1\times _B E_2 \ar[r]\ar[d] & E_{1}  \ar[d] \\
E_2 \ar[r] & E_1*_B E_2.  }$$ By the universal property of homotopy pushouts we get a $G$-fibration $$p_1*p_2:E_1*_B E_2\to B$$ called the [*fiber join of $p_1$ and $p_2$*]{}. Iterating this construction, we obtain a $G$-fibration $$\underset{k}{\ast} \, p: \underbrace{E*_B E*_B
\dots *_B E}_{k\text{-many}}\to B$$ which we call the $k$-fold fiber join of $p$ with itself. Note that if $p$ is a $G$-fibration with fiber type $\{F_H\}$, then the fiber type of the $k$-fold join $\ast _k p$ is $\{\ast _k 