  The ground state subspace of a topological phase of matter forms a
representation of the mapping class group of the space on which the state is
defined. We show that elements of the mapping class group of a surface of genus
$g$ can be obtained through a sequence of topological charge projections along
at least three mutually intersecting non-contractible cycles. We demonstrate
this both through the algebraic theory of anyons and also through an analysis
of the topology of the space-time manifold. We combine this result with two
observations: (i) that surfaces of genus $g$ can be effectively simulated in
planar geometries by using bilayer, or doubled, versions of the topological
phase of interest, and inducing the appropriate types of gapped boundaries; and
(ii) that the required topological charge projections can be implemented as
adiabatic unitary transformations by locally tuning microscopic parameters of
the system, such as the energy gap. These observations suggest a possible path
towards effectively implementing modular transformations in physical systems.
In particular, they also show how the $\text{Ising} \otimes
\overline{\text{Ising}}$ state, in the presence of disconnected gapped
boundaries, can support universal topological quantum computation.
