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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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