Intersections of quadrics, moment-angle manifolds and connected sums

Abstract

The geometric topology of the generic intersection of two homogeneous coaxial quadrics in R n was studied in [LdM1] where it was shown that its intersection with the unit sphere is in most cases diffeomorphic to a triple product of spheres or to the connected sum of sphere products. The proof involved a geometric description of the group actions on them and of their polytope quotients as well as a splitting of the homology groups of those manifolds. It also relied heavily on a normal form for them and many related computations. The part about group actions, polytopes and homology splitting was equally valid for the intersection of any number of such quadrics, but the obstacle to extending the main result for more than two turned out to be the hopeless-looking problem of finding their normal forms, close to that of classifying all simple polytopes. 1) Whether they can always be built up from spheres by repeatedly taking products or connected sums: they produced new examples for any k which are so, but also showed how to construct many cases which are not. Many interesting questions arose, including a specific conjecture.2) The transition between different topological types when the generic condition is broken at some point of a deformation (wall-crossing).3) A product rule of their cohomology ring (in the spirit of the description of the homology of Z given in [LdM1]) and its applications to question 1).Meanwhile, and independently, in [D-J] essentially the same manifolds were constructed in a more abstract way, where the main objective was to study the algebraic topology of some important quotients of them called initially toric manifolds and now quasitoric manifolds. This article originated an important development through the work of many authors, and there is a vast and deep literature along those lines for which the reader is referred to [B-P]. Yet for a long time no interchange occurred between the two lines of research involving the same objects, until small connections appeared in the final version of [B-M] and in [D-S]. In particular, it turned out that examples relevant to question 1) above were known to these authors (see [B2]), and in [B1] Baskakov had a product rule for the cohomology ring, similar but dual to that of [B-M] mentioned in 3) above. All those examples were independent and more or less simultaneous, yet both product rules followed from an earlier computation by Buchstaber and Panov of the cohomology ring 1 .One recent expression of the line of research derived from [D-J] is the article [B-B-C-G] where a far-reaching generalization is made and a general geometric splitting formula is derived that is, in particular, a very good tool for understanding the relations among the homology groups of different spaces. This understanding turned out to be fundamental for us in tracing a way through the abstract situation of the intersection of k > 2 quadrics, thus combining efficiently both approaches to the subject as expressed in [LdM1] and [B-B-C-G] to obtain the results in the present article. ...

Stanley Reisner rings of generalized truncation polytopes and their moment angle manifolds

“…For truncation polytopes P = vc k (∆ n ) with n 2, k 0 this was done in [8]. Our results on the bigraded Betti numbers agree with partial description of the diffeomorphism types of Z P for generalized truncation polytopes with r = 1, 2 from [6]. In Section 3 we prove a criterion of when a face ring of an arbitrary generalized truncation polytope is minimally non-Golod, the latter property was proved for truncation polytopes in [1,Theorem 6.19].…”

“…According to [6,Theorem 2.2], the moment-angle manifold Z P corresponding to a generalized truncation polytope P with r = 2, n 1 n 2 > 1, k 0 is diffeomorphic to a connected sum of sphere products with 2 spheres in each product, if m = n 1 + n 2 + 2 + k < 3(n 1 + n 2 ) = 3d, that is, for all 0 ≤ k < 2(n 1 + n 2 − 1). It is easy to see that Betti numbers of Z P obtained from this description are equal to those calculated using Theorem 2.1 and formula (1.3).…”

“…The author doesn't know any example of a simple polytope P s.t. Z P is topologically a nontrivial connected sum of sphere products including a product of three spheres (see also [6]). 2.…”

“…2. It is proved in [6,Theorem 1.3] that a moment-angle manifold corresponding to an even dimensional dual neighbourly polytope is diffeomorhic to a connected sum of sphere products. Below we prove minimal non-Golodness for the Stanley-Reisner rings in the case of even dimensional dual neighbourly polytopes.…”

Toric Homotopy Theory

“…This generalizes to K being the dual of the boundary of certain higher dimensional polytopes. Further generalizations were proved in [15]. This raises several questions.…”