  Symmetries play an essential role in identifying and characterizing
topological states of matter. Here, we classify topologically two-dimensional
(2D) insulators and semimetals with vanishing spin-orbit coupling using
time-reversal ($\mathcal{T}$) and inversion ($\mathcal{I}$) symmetry. This
allows us to link the presence of edge states in $\mathcal{I}$ and
$\mathcal{T}$ symmetric 2D insulators, which are topologically trivial
according to the Altland-Zirnbauer table, to a $\mathbb{Z}_2$ topological
invariant. This invariant is directly related to the quantization of the Zak
phase. It also predicts the generic presence of edge states in Dirac
semimetals, in the absence of chiral symmetry. We then apply our findings to
bilayer black phosphorus and show the occurrence of a gate-induced topological
phase transition, where the $\mathbb{Z}_2$ invariant changes.
