  We establish the topology of the spin-singlet superconducting states in the
bare hyperhoneycomb lattice and derive analytically the full phase diagram
using only symmetry and topology in combination with simple energy arguments.
The phase diagram is dominated by two states preserving time-reversal symmetry.
We find that the line-nodal state dominating at low doping levels is
topologically nontrivial and exhibits surface Majorana flat bands, which we
show perfectly match the bulk-boundary correspondence using Berry phase
approach. At higher doping levels we find a fully gapped state with trivial
topology. By analytically calculating the topological invariant of the line
nodes, we derive the critical point between the line-nodal and fully gapped
states as a function of both pairing parameters and doping. We find that the
line-nodal state is favored not only at lower doping levels but also if
symmetry-allowed deformations of the lattice is present. Adding simple energy
arguments we establish that a fully gapped state with broken time-reversal
symmetry likely appears covering the actual phase transition. We find this
time-reversal symmetry broken state to be topologically trivial, while we find
an additional point nodal state at very low doping levels to have nontrivial
topology with associated Fermi surface arcs. We eventually address the
robustness of the phase diagram to generalized models also including adiabatic
spin-orbit coupling, and show how all but the point nodal state are reasonably
stable.
