  We prove the "Sullivan Conjecture" on the classification of 4-dimensional
complete intersections up to diffeomorphism. Here an $n$-dimensional complete
intersection is a smooth complex variety formed by the transverse intersection
of $k$ hypersurfaces in $CP^{n+k}$.
  Previously Kreck and Traving proved the 4-dimensional Sullivan Conjecture
when 64 divides the total degree (the product of the degrees of the defining
hypersurfaces) and Fang and Klaus proved that the conjecture holds up to the
action of the group of homotopy 8-spheres $\Theta_8 = Z/2$.
  Our proof involves several new ideas, including the use of the
Hambleton-Madsen theory of degree-$d$ normal maps, which provide a fresh
perspective on the Sullivan Conjecture in all dimensions. This leads to an
unexpected connection between the Segal Conjecture for $S^1$ and the Sullivan
Conjecture.
