  We consider the cohomological classification of the 4+2-dimensional
topological field, which is proposed by L\"uscher, for SU(2)_L \times U(1)_Y
electroweak theory. The dependence on the admissible abelian gauge field of
U(1)_Y is determined through topological argument, with SU(2)_L gauge field
fixed as background. We then show the exact cancellation of the local gauge
anomaly of the mixed type {SU(2)_L}^2 \times U(1)_Y at finite lattice spacing,
as well as {U(1)_Y}^3, using the pseudo reality of SU(2)_L and the anomaly
cancellation conditions in the electroweak theory given in terms of the
hyper-charges of U(1)_Y.
