  We prove stability results of the Morse index plus nullity of Yang-Mills
connections in dimension 4 under weak convergence. Precisely we establish that
the sum of the Morse indices and the nullity of a bounded sequence of
Yang-Mills connections is asymptotically bounded above by the sum of the Morse
index and the nullity of the weak limit and the bubbles while the Morse indices
are asymptotically bounded below by the sum of the Morse index of the weak
limit and the bubbles.
