  We unveil that the holey graphyne (HGY), a two-dimensional carbon allotrope
where benzene rings are connected by two $-$C$\equiv$C$-$ bonds fabricated
recently in a bottom-up way, exhibits topological electronic states. Using
first-principles calculations and Wannier tight-binding modeling, we discover a
higher-order topological invariant associated with $C_2$ symmetry of the
material, and show that the resultant corner modes appear in nanoflakes
matching to the structure of precursor reported previously, which are ready for
direct experimental observations. In addition, we find that a band inversion
between emergent $g$-like and $h$-like orbitals gives rise to a nontrivial
topology characterized by $\mathbb{Z}_2$ invariant protected by an energy gap
as large as 0.52 eV, manifesting helical edge states mimicking those in the
prominent quantum spin Hall effect, which can be accessed experimentally after
hydrogenation in HGY. We hope these findings trigger interests towards
exploring the topological electronic states in HGY and related future
electronics applications.
