  We investigate the non-Abelian Bianchi identities in pure SU(2) lattice
Yang-Mills theory in three and four dimensions. The non-Abelian Stokes theorem
proposed recently allows to formulate the Bianchi identities in terms of local
physical fluxes. Then the violation of Bianchi identities becomes a well
defined concept ultimately related to chromomagnetic fields degeneracy points.
We present numerical evidences that in D=4 the suppression of the Bianchi
identities violation destroys confinement while the removal of the degeneracy
points drives the theory to the topologically non-trivial sector.
