Dimension functions for spherical fibrations (Q1713045)

It has been conjectured in [\textit{D. Benson} and \textit{J. Carlson}, Math. Z. 195, 221--238 (1987; Zbl 0593.20062)] that a finite group $G$ has a free action on a finite CW-complex $X$ with the homotopy type of a product of spheres $\mathbb{S}^{n_1} \times \mathbb{S}^{n_2} \times \dots \times \mathbb{S}^{n_k}$ with trivial action on homology if and only if the maximal rank of an elementary abelian $p$-group contained in $G$ is at most $k$. The case $k=1$ in the conjecture is known to be true according to [\textit{R. G. Swan}, Ann. Math. (2) 72, 267--291 (1960; Zbl 0096.01701)]. The case $k=2$ has been proved by \textit{A. Adem} and \textit{J. H. Smith} [ibid. 154, No. 2, 407--435 (2001; Zbl 0992.55011)] for finite groups that do not involve $\text{Qd}(p) = (\mathbb{Z}/p)^2 \rtimes \text{SL}_2 (\mathbb{Z}/p)$ for any prime $p >2$. \par The Euler class of a fibration is said to be \textit{$p$-effective} if its restriction to elementary abelian $p$-subgroups of maximal rank is not nilpotent. Let $X_{\widehat p}$ be the Bousfield-Kan $p$-completion, and $X^{hK}: = \text{Map}(EK, X)^K$, the space of homotopy fixed points. Let \[ X[m] = (\underbrace{X \ast X \ast \cdots \ast X}_{m-\text{times}} )_{\widehat p} \] be the $p$-completion of the $m$-fold join. In this paper, the authors show that if $P$ is a finite $p$-group and $X \simeq (\mathbb{S}^n )_{\widehat p}$ is a $P$-space, then there is a positive integer $m$ such that $(X[m])^{hP} \simeq (\mathbb{S}^r )_{\widehat p}$ for some $r$, and that if $p$ is an odd prime, then there is no mod-$p$ spherical fibration $\xi : E \rightarrow B \text{Qd}(p)$ with a $p$-effective Euler class. They also show that if $G = \text{Qd}(p)$, then there is no finite free $G$-CW-complex $X$ homotopy equivalent to a product of two spheres $\mathbb{S}^n \times\mathbb{S}^n$.