  The topology of complex classical paths is investigated to discuss quantum
tunnelling splittings in one-dimensional systems. Here the Hamiltonian is
assumed to be given as polynomial functions, so the fundamental group for the
Riemann surface provides complete information on the topology of complex paths,
which allows us to enumerate all the possible candidates contributing to the
semiclassical sum formula for tunnelling splittings. This naturally leads to
action relations among classically disjoined regions, revealing entirely
non-local nature in the quantization condition. The importance of the proper
treatment of Stokes phenomena is also discussed in Hamiltonians in the normal
form.
