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Dimension functions for spherical fibrations. (English) Zbl 1417.55004
It has been conjectured in [D. Benson and 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 [R. G. Swan, Ann. Math. (2) 72, 267–291 (1960; Zbl 0096.01701)]. The case $$k=2$$ has been proved by A. Adem and 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 $$p >2$$.
The Euler class of a fibration is said to be $$p$$-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 $$\mathbb{S}^n \times\mathbb{S}^n$$.

##### MSC:
 55M35 Finite groups of transformations in algebraic topology (including Smith theory) 55S10 Steenrod algebra 55S37 Classification of mappings in algebraic topology
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