  The fermionic topological charge of lattice gauge fields, given in terms of a
spectral flow of the Hermitian Wilson--Dirac operator, or equivalently, as the
index of Neuberger's lattice Dirac operator, is shown to have analogous
properties to L\"uscher's geometrical lattice topological charge. The main new
result is that it reduces to the continuum topological charge in the classical
continuum limit. (This is sketched here; the full proof will be given in a
sequel to this paper.) A potential application of the ideas behind fermionic
lattice topological charge to deriving a combinatorial construction of the
signature invariant of a 4-manifold is also discussed.
