Mapping spaces of compact Lie groups and $p$-adic completion

Abstract:

If ${\mathbf {BG}},\;{\mathbf {BH}}$ are the classifying spaces of compact Lie groups, with ${\mathbf {H}}$ connected, then the mapping space functor ${\mathbf {map}}({\mathbf {BG}}, - )$ commutes with $p$-completion on ${\mathbf {BH}}$: i.e., for each $f:{\mathbf {BG}} \to {\mathbf {BH}}$ the component $({\mathbf {map}}{({\mathbf {BG}},{\mathbf {BH}})_f})_p^ \wedge$ is $p$-complete, and is homotopy equivalent to ${\mathbf {map}}{({\mathbf {BG}},{\mathbf {BH}}_p^ \wedge )_{i \circ f}}$.