Unstable Adams operations on \(p\)-local compact groups (Q422084)

\textit{C. Broto, R. Levi} and \textit{B. Oliver} introduced \(p\)-local compact groups in [Geom. Topol. 11, 315--427 (2007; Zbl 1135.55008)]. This algebraic concept generalizes that of \(p\)-local finite groups and, in particular, allows one to study compact Lie groups \(p\)-locally, extending also \textit{W. G. Dwyer} and \textit{C. W. Wilkerson}'s \(p\)-compact groups [Ann. Math. (2) 139, No. 2, 395--442 (1994; Zbl 0801.55007)]. It is thus a natural question to look for Adams operations in this setting. The authors of this article define first what an Adams operation on a \(p\)-local compact group should be, namely a fusion preserving automorphism \(\phi\) of the ``Sylow \(p\)-subgroup'' and a lift to the linking system in a rigid sense. The degree of the Adams operation is the \(p\)-adic unit \(\zeta\) such that the restriction of \(\phi\) to the discrete torus is the \(\zeta\)-power automorphism. Such Adams operations are then constructed in any \(p\)-local compact group for large values of \(k\), where \(p^k \mid \zeta -1\).NEWLINENEWLINEIn contrast to what happens with connected \(p\)-compact groups, not all values of \(k\) can occur and the degree does not determine the operation in general. Such Adams operations have already found an application in the same volume [\textit{A. González}, Algebr. Geom. Topol. 12, No. 2, 867--908 (2012; Zbl 1259.55005)], relating \(p\)-local compact groups to \(p\)-local finite groups via fixed points.