summand with the $i$-th summand. Since $U(n)$ is connected, there is a path between $T(i,j)$ and the identity. Using this path, we can define a homotopy between $\gamma ^i$ and $\gamma ^j$.

For more general $H$-spaces $F_H$, there exists a swap map $$S(i,j): \ast _k F_H \to \ast _k F_H,$$ which swaps the $i$-th and $j$-th coordinates, similar to the linear transformation $T(i,j)$ in the proof of Lemma \[lem:joinswap\]. If $F_H$ is a free $H$-space homotopy equivalent to an odd dimensional sphere, then $S(i,j)$ will be homotopy equivalent to the identity map. If the $H$-action on $F_H$ is not free, then the swap map $S(i,j)$ is not homotopy equivalent to the identity in general even when $F_H$ is homotopy equivalent to an odd dimensional sphere. On the other hand, if $F_H$ is a homotopy representation with the property that all fixed point spheres are odd dimensional, then under certain conditions on $H$ or on the dimension function of $F_H$, one can prove that $S(i,j)$ is homotopy equivalent to the identity (see Proposition 20.12 in [@lueck]).

We end this section with the following observation.

\[lem:jointrivial\] Let $H$ be a finite group and $p: E \to {\mathbb S}^n $ be an $H$-fibration 