  We construct the Wess-Zumino-Witten (WZW) term in lattice gauge theory by
using a Dirac operator which obeys the Ginsparg-Wilson relation. Topological
properties of the WZW term known in the continuum are reproduced on the lattice
as a consequence of a non-trivial topological structure of the space of
admissible lattice gauge fields. In the course of this analysis, we observe
that the gauge anomaly generally implies that there is no basis of a Weyl
fermion which leads to a single-valued expectation value in the fermion sector.
The lattice Witten term, which carries information of a gauge path along which
the gauge anomaly is integrated, is separated from the WZW term and the
multivaluedness of the Witten term is shown to be related to the homotopy group
$\pi_{2n+1}(G)$. We also discuss the global $\SU(2)$ anomaly on the basis of
the WZW term.
