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**Unstable Adams operations acting on \(p\)-local compact groups and fixed points.**
*(English)*
Zbl 1259.55005

F. Junod, R. Levi and A. Libman defined unstable Adams operations on \(p\)-local compact groups in [Algebr. Geom. Topol. 12, No. 1, 49–74 (2012; Zbl 1258.55010)]. One typical application of Adams operations for compact Lie groups \(G\) has been to use fixed points to approximate \(BG\) by classifying spaces of finite groups, see E. M. Friedlander and G. Mislin [Invent. Math. 83, 425–436 (1986; Zbl 0566.55011)]. Recently, C. Broto and J. M. Møller have performed this for \(p\)-compact groups [Algebr. Geom. Topol. 7, 1809–1919 (2007; Zbl 1154.55009)] and the aim of the article under review is to study to which extent this approach can be applied to \(p\)-local compact groups.

It turns out that each Adams operation defines a whole family of them – by iteration – and these determine a so-called “strong fixed points” transporter system. It is however difficult to check whether or not they correspond to saturated fusion systems. This is achieved in the case of \(p\)-local compact groups of rank one and for those coming from the unitary compact Lie group \(U(n)\). Here, the \(p\)-local compact group is indeed approximated by \(p\)-local finite groups and such an approximation yields a stable element theorem for computing the mod \(p\) cohomology.

It turns out that each Adams operation defines a whole family of them – by iteration – and these determine a so-called “strong fixed points” transporter system. It is however difficult to check whether or not they correspond to saturated fusion systems. This is achieved in the case of \(p\)-local compact groups of rank one and for those coming from the unitary compact Lie group \(U(n)\). Here, the \(p\)-local compact group is indeed approximated by \(p\)-local finite groups and such an approximation yields a stable element theorem for computing the mod \(p\) cohomology.

Reviewer: Jérôme Scherer (Lausanne)

### MSC:

55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |

55S25 | \(K\)-theory operations and generalized cohomology operations in algebraic topology |

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\textit{A. González}, Algebr. Geom. Topol. 12, No. 2, 867--908 (2012; Zbl 1259.55005)

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