)=0.$$ So, $\ast _N p$ is $H$-fiber homotopy equivalent to the trivial fibration.

Equivariant Federer spectral sequence {#sect:Federer}
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The main purpose of this section is to prove the following theorem which is due to M. Klaus [@klaus]. We give a different proof here using the equivariant Federer spectral sequence which was introduced by Møller in [@moller].

\[thm:finiteness of homotopy\] Let $G$ be a finite group and $V$ be a complex representation of $G$. Then, for every $n>0$, there is an $m\geq 1$ such that $\pi_n (\operatorname{Aut}_G ({\mathbb S}(V ^{\oplus k})) $ is finite for all $k \geq m$.

Before the proof, we first introduce the standard definitions about Bredon cohomology and local coefficients systems that we are going to use to describe the equivariant Federer spectral sequence. For more details on Bredon cohomology we refer the reader to [@bredon].

Let $X$ be a topological space. A [*local coefficient system*]{} of $X$ is a functor $L:\Pi (X)\to {\mathcal Ab}$ where $\Pi (X)$ denotes the fundamental groupoid of $X$ and ${\mathcal Ab}$ denotes the category of abelian groups. Let ${\mathcal L}$ denote the category whose objects are pairs $(X,L)$ where $X$ is a topological space and $L$ is a local coefficient 