  We study topological properties of families of Hamiltonians which may contain
degenerate energy levels aka. band crossings. The primary tool are Chern
classes, Berry phases and slicing by surfaces. To analyse the degenerate locus,
we study local models. These give information about the Chern classes and Berry
phases. We then give global constraints for the topological invariants. This is
an hitherto relatively unexplored subject. The global constraints are more
strict when incorporating symmetries such as time reversal symmetries. The
results can also be used in the study of deformations. We furthermore use these
constraints to analyse examples which include the Gyroid geometry, which
exhibits Weyl points and triple crossings and the honeycomb geometry with its
two Dirac points.
