Intersections of quadrics, moment-angle manifolds and connected sums

Abstract

For the intersections of real quadrics in ${ℝ}^n$ and in ${ℂ}^n$ 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.