  $Z_{2}$ and Chern topological phases such as newly discovered quantum spin
Hall and original quantum Hall states hardly both co--exist in a single
material due to their contradictory requirement on the time--reversal symmetry
(TRS). We show that although the TRS is broken in systems with a periodically
driving ac-field, an effective TRS can still be defined provided the ac--field
is linearly polarized or certain other conditions are satisfied. The
controllable TRS provides us with a route to manipulate $Z_{2} $ and Chern
topological phases in a single material by tuning the polarization of the
ac--field. To demonstrate the idea, we consider a generic honeycomb lattice
model as a benchmark system that is relevant to electronic structures of
several monolayered materials. Our calculation shows that not only the
transitions between $Z_{2}$ and Chern phases can be induced but also features
such as the dispersion of the edge states can be controlled. This opens the
possibility of manipulating various topological phases in a single material and
can be a promising approach to engineer some new electronic states of matter.
