Fermionic topological charge of families of lattice gauge fields
  Topological charge of families of lattice gauge fields is defined fermionically via families index theory for the overlap Dirac operator. Certain obstructions to gauge invariance of the overlap chiral fermion determinant, as well as the lattice analogues of certain obstructions to gauge fixings without the Gribov problem, have natural descriptions in this context. 
David H. Adams