  We study the generic band structures of the five-dimensional (5D) Weyl
semimetal, in which the band degeneracies are 2D Weyl surfaces in the momentum
space, and may have non-trivial linkings with each other if they carry nonzero
second Chern numbers. We prove a number of theorems constraining the
topological linking configurations of the Weyl surfaces, which can be viewed as
a 5D generalization of the celebrated Doubling Theorem for 3D Weyl semimetal.
As a direct physical consequence of these constraints, the 5D Weyl semimetal
hosts a rich structure of topological boundary states. We show that on the 4D
boundary of the 5D Weyl semimetal, there are 3D chiral Fermi hypersurfaces
protected by bulk Weyl surfaces. On top of that, for bulk Weyl surfaces that
are linked and carry nonzero second Chern numbers, the associated boundary 3D
Fermi hypersurfaces will shrink to singularities at certain energies, which
trace out a protected 1D Weyl nodal arc, in analogy to the Fermi arc on the 3D
Weyl semimetal surface.
