  We study the Dancer-Fucik spectrum of the fractional p-Laplacian operator. We
construct an unbounded sequence of decreasing curves in the spectrum using a
suitable minimax scheme. For p=2, we present a very accurate local analysis. We
construct the minimal and maximal curves of the spectrum locally near the
points where it intersects the main diagonal of the plane. We give a sufficient
condition for the region between them to be nonempty, and show that it is free
of the spectrum in the case of a simple eigenvalue. Finally we compute the
critical groups in various regions separated by these curves. We compute them
precisely in certain regions, and prove a shifting theorem that gives a
finite-dimensional reduction in certain other regions. This allows us to obtain
nontrivial solutions of perturbed problems with nonlinearities crossing a curve
of the spectrum via a comparison of the critical groups at zero and infinity.
