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Intersections of quadrics, moment-angle manifolds and connected sums

Samuel Gitler and Santiago López de Medrano

Geometry & Topology 17 (2013) 1497–1534

For the intersections of real quadrics in n and in 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.

To Sylvain Cappell, on his 65^\textth birthday

Mathematical Subject Classification 2010
Primary: 14P25, 57R19
Secondary: 57S25, 57R65
Received: 7 June 2012
Revised: 11 December 2012
Accepted: 14 January 2013
Published: 8 June 2013
Proposed: Steve Ferry
Seconded: Simon Donaldson, Leonid Polterovich
Samuel Gitler
Department of Mathematics
El Colegio Nacional
Centro de Investigacion IPN
Apartado Postal 14740
14620 Mexico, DF
Santiago López de Medrano
Instituto de Matemáticas
Universidad Nacional Autónoma de México
04510 Mexico