Alessio Corti, Mark Haskins, Johannes Nordström, Tommaso Pacini
We broaden significantly the applicability of the Donaldson-Kovalev twisted connected sum construction and deepen the study of the resulting G_2-manifolds. Some of our main results are: (i) We construct many new topological types of compact G_2-manifolds. (ii) We obtain much more precise topological information about twisted connected sum G_2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G_2-manifolds. (iii) We construct many G_2-manifolds that contain rigid compact associative 3-folds. (iv) We clarify several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. (v) We show that the large number of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds in arXiv:1206.2277 can be used as components in the construction; semi-Fano 3-folds are a subclass of weak Fano 3-folds whose geometry is amenable to the solution of the matching problem via deformation theory arguments. (vi) We describe "geometric transitions" between G_2-metrics on different 7-manifolds mimicking "flopping" behaviour among weak Fano 3-folds and "conifold transitions" between Fano and weak Fano 3-folds. (vii) We prove that many smooth 2-connected 7-manifolds can be realised as twisted connected sums in numerous ways; by varying the building blocks matched we can vary the number of rigid associative 3-folds constructed therein. The latter result leads to speculation that the moduli space of G_2-metrics on a given 7-manifold may consist of many different connected components. The higher-dimensional gauge theory invariants proposed by Donaldson may provide ways to detect G_2-metrics on a given 7-manifold that are not deformation equivalent.
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http://arxiv.org/abs/1207.4470
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