  This paper states a definition of homotopic rotation set for higher genus
surface homeomorphisms, as well as a collection of results that justify this
definition. We first prove elementary results: we prove that this rotation set
is star-shaped, we discuss the realisation of rotation vectors by orbits or
periodic orbits and we prove the creation of new rotation vectors for some
configurations.Then we use the theory developped by Le Calvez and Tal in
[LCT18a] to obtain two deeper results:-- If the homotopical rotation set
contains the direction of a closed geodesic which has a self-intersection, then
there exists a rotational horseshoe and hence infinitely many periodic orbits
in many directions.-- If the homotopical rotation set contains the directions
of two closed geodesics that meet, there exists infinitely many periodic orbits
in many directions.
