Intersections of Quadrics, Moment-angle Manifolds and Connected Sums

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.Comment: We have included many clarifying suggestions and minor corrections from some colleagues who read the manuscript carefully. The only change in content from the previous version is the suppression a special case (item 3) of Theorem 1.3 because we have not been able to fill in the details of any of the known sketched proofs (including ours

Hexagonal Tilings and Locally C6 Graphs

We give a complete classification of hexagonal tilings and locally C6 graphs, by showing that each of them has a natural embedding in the torus or in the Klein bottle. We also show that locally grid graphs are minors of hexagonal tilings (and by duality of locally C6 graphs) by contraction of a perfect matching and deletion of the resulting parallel edges, in a form suitable for the study of their Tutte uniqueness.Comment: 14 figure