  We investigate the properties of an impurity particle interacting with a
Fermi gas in a Chern-insulating state. The interaction leads to the formation
of an exotic polaron, which consists of a coherent superposition of the
topologically-trivial impurity and the surrounding topological cloud. We
characterize this intriguing topologically-composite object by calculating its
transverse (Hall) conductivity, using diagrammatic as well as variational
methods. The "polaronic Hall conductivity" is shown to be non-zero whenever the
surrounding cloud is prepared in a non-trivial Chern insulating state, which we
attribute to the transverse drag exerted by the dressing cloud on the impurity.
In this way, the polaron partially inherits the topological properties of the
Chern insulator through genuine interaction effects. This is also analysed at
the microscopic level of wave functions, by identifying a "composite Berry
curvature" for the polaron, which closely mimics the Berry curvature of the
Chern insulator's band structure. Finally, we discuss how this interplay
between topology and many-body correlations can be studied in cold-atom
experiments, using available technologies.
