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We show that links of isolated hypersurface singularities defined by invertible polynomials coming from the Johnson and Kollár list of Kähler-Einstein 3-folds that are rational homology 7-spheres remain rational homology 7-spheres under the so-called Berglund-Hübsch transpose rule coming from classical mirror symmetry constructions. Actually, this rule produces twins, that is, links with same degree, Milnor number and homology H 3 , with the exception of iterated Thom-Sebastiani sums of singularities of chain and cycle type, where the torsion and the Milnor number vary. The Berglund-Hübsch transpose rule not only gives a framework to better understand the existence of Sasaki-Einstein twins but also gives a mechanism for producing new examples of Sasaki-Einstein twins in the rational homology 7-sphere setting. We also give reasonable conditions for a Sasaki-Einstein rational homology 7-sphere to remain Sasaki-Einstein under the Berglund-Hübsch transpose rule. In particular, we found 75 new examples of Sasaki-Einstein rational homology 7-spheres arising as links of not well-formed hypersurface singularities. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalization of classical coKähler geometry from the point of view of generalized contact metric geometry. This allows us to generalize a theorem of Capursi (1984), Goldberg (1968) and show that the product M 1 × M 2 of generalized contact metric manifolds ( M i , Φ i , E ± , i , G i ) , i = 1 , 2 , where M 1 × M 2 is endowed with the product (twisted) generalized complex structure induced from Φ 1 and Φ 2 , is (twisted) generalized Kähler if and only if ( M i , Φ i , E ± , i , G i ) , i = 1 , 2 are (twisted) generalized coKähler structures. As an application of our theorem we construct new examples of twisted generalized Kähler structures on manifolds that do not admit a classical Kähler structure and we give examples of twisted generalized coKähler structures on manifolds which do not admit a classical coKähler structure. [ABSTRACT FROM AUTHOR]

In this article, we give ten examples of 2-connected seven dimensional Sasaki-Einstein manifolds for which the third homology group is completely determined. Using the Boyer-Galicki construction of links over particular Kähler-Einstein orbifolds, we apply a valid case of Orlik's conjecture to the links so that one is able to explicitly determine the entire third integral homology group. We give ten such new examples, all of which have the third Betti number satisfy 10 ≤ b 3 (L f) ≤ 20 . [ABSTRACT FROM AUTHOR]