  We provide a manifestly topological classification scheme for generalised
Weyl semimetals, in any spatial dimension and with arbitrary Weyl surfaces
which may be non-trivially linked. The classification naturally incorporates
that of Chern insulators. Our analysis refines, in a mathematically precise
sense, some well-known 3D constructions to account for subtle but important
global aspects of the topology of semimetals. Using a fundamental locality
principle, we derive a generalized charge cancellation condition for the Weyl
surface components. We analyse the bulk-boundary correspondence under a duality
transformation, which reveals explicitly the topological nature of the
resulting surface Fermi arcs. We also analyse the effect of moving Weyl points
on the bulk and boundary topological semimetal invariants.
