  In the presence of both space and time reversal symmetries, an s-wave A1g
superconducting state is usually topologically trivial. Here we demonstrate
that an exception can take place in a type of nonsymmorphic lattice structures.
We specify the demonstration in a system with a centrosymmetric space group
P4/nmm, the symmetry group that governs iron-based superconductors, by showing
the existence of a second-order topological state protected by a mirror
symmetry. The topological superconductivity is featured by 2Z degenerate Dirac
cones on the (1,0) edge, and Z pairs of Majorana modes at the intersection
between the (1,1) and (1,-1) edges. The topological invariance and Fermi
surface criterion for the topological state are provided. Moreover, we point
out that the previously proposed s-wave state in iron-based superconductors,
which features a sign-changed superconducting order parameter between two
electron pockets, is such a topological state. Thus, these results not only
open a new route to pursue topological superconductivity, but also establish a
measurable quantity to settle one long-lasting debate on the pairing nature of
iron-based superconductors.
