Induced topological invariants by lattice fermions in odd dimensions

We study the fermionic contribution to external gauge fields in odd dimensional space-time with lattice Wilson fermion to get the topological invariant, Chern-Simons class. We also discuss the coefficient, consistent with the topological quantization, is stable against the small variation of the parameters owing to the topological arguments with respect to the momentum space. Especially in three dimensions, we show that the coefficient is free from higher order corrections. (author)