The rational cohomology of a $p$-local compact group

Abstract:

For any prime $p$, the theory of $p$-local compact groups is modelled on the $p$-local homotopy theory of classifying spaces of compact Lie groups and $p$-compact groups and generalises the earlier concept of $p$-local finite groups. These objects have maximal tori and Weyl groups, although the Weyl groups need not be generated by pseudoreflections. In this paper, we study the rational $p$-adic cohomology of the classifying space of a $p$-local compact group and prove that just as for compact Lie groups, it is isomorphic to the ring of invariants of the Weyl group action on the cohomology of the classifying space of the maximal torus. This is applied to show that unstable Adams operations on $p$-local compact groups are determined in the appropriate sense by the map they induce on rational cohomology.