We prove the “Sullivan conjecture” on the classification of
-dimensional
complete intersections up to diffeomorphism. Here an
-dimensional
complete intersection is a smooth complex variety formed by the transverse intersection of
hypersurfaces
in
.
Previously Kreck and Traving proved the
-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
-spheres
.
Our proof involves several new ideas, including the use of the Hambleton–Madsen theory
of degree-
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
and
the Sullivan conjecture.
Keywords
complete intersection, diffeomorphism classification,
Sullivan conjecture, degree-d normal map
This article is currently available only to
readers at paying institutions. If enough institutions subscribe to
this Subscribe to Open journal for 2025, the
article will become Open Access in early 2025. Otherwise, this
article (and all 2025 articles) will be available only to paid
subscribers.