  The subtle interplay between local and global charges for topological
semimetals exactly parallels that for singular vector fields. Part of this
story is the relationship between cohomological semimetal invariants, Euler
structures, and ambiguities in the torsion of manifolds. Dually, a topological
semimetal can be represented by Euler chains from which its surface Fermi arc
connectivity can be deduced. These dual pictures, and the link to topological
invariants of insulators, are organised using geometric exact sequences. We go
beyond Dirac-type Hamiltonians and introduce new classes of semimetals whose
local charges are subtle Atiyah-Dupont-Thomas invariants globally constrained
by the Kervaire semicharacteristic, leading to the prediction of torsion Fermi
arcs.
