Intersections of Quadrics, Moment-angle Manifolds and Connected Sums

TL;DR

This paper advances the understanding of the topology of intersections of multiple quadrics, showing they are often diffeomorphic to connected sums of sphere products, and explores their properties and modifications through topological operations.

Contribution

It generalizes previous results to intersections of more than two quadrics, identifies new families of such manifolds, and provides a simplified proof for the case of two quadrics, also analyzing topological changes and cohomology rules.

Findings

Intersections of k>2 quadrics can be diffeomorphic to connected sums of sphere products.

New families of manifolds are constructed via topological operations on associated polytopes.

Modified rules for cohomology products of moment-angle manifolds are proposed.

Abstract

The topology of the intersection of two real homogeneous coaxial quadrics was studied by the second author who showed that its intersection with the unit sphere is in most cases diffeomorphic to a connected sum of sphere products. Combining that approach with a recent one (due to Antony Bahri, Martin Bendersky, Fred Cohen and the first author) we study here the intersections of k>2 quadrics and we identify very general families of such manifolds that are diffeomorphic to connected sums of sphere products. These include those moment-angle manifolds for which the result was conjectured by Frederic Bosio and Laurent Meersseman. As a byproduct, a simpler and neater proof of the result for the case k=2 is obtained. Two new sections contain results not included in the first version of this article: Section 2 describes the topological change on the manifolds after the operations of cutting off a vertex or an edge of the associated polytope, which can be combined in a special way with the previos results to produce new infinite families of manifolds that are connected sums of sphere products. In other cases we get slightly more complicated manifolds: with this we solve another question by Bosio-Meersseman about the manifold associated to the truncated cube. In Section 3 we use this to show that the known rules for the cohomology product of a moment-angle manifold have to be drastically modified in the general situation. We state the modified rule, but leave the details of this for another publication. Section 0 recalls known definitions and results and in section 2.1 some elementary topological constructions are defined and explored. In the Appendix we state and prove some results about specific differentiable manifolds, which are used in sections 1 and 2.