  The universal properties of (2 + 1)D topological phases of matter enriched by
a symmetry group G are described by G-crossed extensions of unitary modular
tensor categories (UMTCs). While the fusion and braiding properties of
quasiparticles associated with the topological order are described by a UMTC,
the G-crossed extensions further capture the properties of the symmetry action,
fractionalization, and defects arising from the interplay of the symmetry with
the topological order. We describe the relation between the G-crossed UMTC and
the topological state spaces on general surfaces that may include symmetry
defect branch lines and boundaries that carry topological charge. We define
operators in terms of the G-crossed UMTC data that represent the mapping class
transformations for such states on a torus with one boundary, and show that
these operators provide projective representations of the mapping class groups.
This allows us to represent the mapping class group on general surfaces and
ensures a consistent description of the corresponding symmetry-enriched
topological phases on general surfaces. Our analysis also enables us to prove
that a faithful G-crossed extension of a UMTC is necessarily G-crossed modular.
