For the intersections of real quadrics in
associated to simple polytopes (also known as
universal abelian covers and
moment-angle manifolds, respectively) we obtain the following results:
(1) Every such manifold of dimension greater than or equal to 5, connected up
to the middle dimension and with free homology, is diffeomorphic to a connected sum
of sphere products. The same is true for the manifolds in infinite families stemming
from each of them. This includes the moment-angle manifolds for which the result
was conjectured by F Bosio and L Meersseman.
(2) The topological effect on the manifolds of cutting off vertices and edges from
the polytope is described. Combined with the result in (1), this gives the same result
for many more natural, infinite families.
(3) As a consequence of (2), the cohomology rings of the two manifolds
associated to a polytope need not be isomorphic, contradicting published results
about complements of arrangements.
(4) Auxiliary but general constructions and results in geometric topology.