homologically trivially on some finite CW-complex $X$ homotopy equivalent to a product of spheres

The paper is organized as follows: In Section \[sect:EquivariantFibrations\], we introduce $G$-fibrations and discuss the effects of taking fiber joins of $G$-fibrations. In Section \[sect:Federer\], we discuss the equivariant Federer spectral sequence introduced by Møller [@moller] and using it, we give another proof for a theorem by M. Klaus [@klaus] (see Theorem \[thm:finiteness of homotopy\]). In Section \[sect:construction\], we introduce our main construction method, and finally in Section \[sect:mainthm\], we prove Theorem \[thm:main\].

$G$-fibrations {#sect:EquivariantFibrations}
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In this section, we first give some preliminaries on $G$-fibrations and then prove some lemmas on fiber joins of $G$-fibrations. For more details on this material we refer the reader to [@lueck] and [@waner2]. Some of this material also appears in [@connolly-prassidis], [@guclukanthesis], [@klaus], and [@unlu-thesis].

\[defn:Gfibration\] A $G$-fibration is a $G$-map $p: E \to B$ which satisfies the following homotopy lifting property for every $G$-space $X$: Given a commuting diagram of $G$-maps $$\xymatrix{
X \times \{ 0\} \ar[d] \ar[r]^-{h}
& E \ar[d]^{p}  \\
X \times I \ar[r]^-{H}
&  B,}\\$$ there exists a $G$-map $\widetilde H: X\times I \to E$ such that $\widetilde H |_{X\times \{0\}}=h$ and $p \circ \widetilde 