  We compute the topological susceptibility chi_t in SU(3) lattice gauge theory
using fermionic methods based on the Atiyah-Singer index theorem. Near the
phase transition we find a smooth crossover behavior for chi_t with values
decreasing from (191(5) MeV)^4 to (100(5) MeV)^4 as we increase the temperature
from 0.88 T_c to 1.31 T_c, showing that topological excitations exist far above
T_c. Our study is the first large scale analysis of the topological
susceptibility at high temperature based on the index theorem and the results
agree well with field theoretical methods.
