Contents 1. Introduction 1 2. Terminology 6 3. Deformation theory of twisted covers 9 4. Twisted covers and admissible covers 11 5. Rigidification and Teichmüller structures 14 6. Abelian twisted level structures 19 7. Automorphisms of twisted G-covers 22 Appendix A. Some remarks on étale cohomology of Deligne-Mumford stacks 33 References 35 [W] S. Wewers, Construction of Hurwitz spaces, Institut für Experimentelle Mathematik preprint No. 21 (1998).

Abstract. We provide a significant extension of the twisted connected sum construction of G2-manifolds, ie Riemannian 7-manifolds with holonomy group G2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Our extension allows us to prove many new results about compact G2-manifolds and leads to new perspectives for future research in the area. Some of the main contributions of the paper are:(i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. (ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a subclass of weak Fano 3-folds) can be used as components in the twisted connected sum construction. (iii) We construct many new topological types of compact G2-manifolds by applying the twisted connected sum to asymptotically Calabi-Yau 3-folds of semi-Fano type studied in [18]. (iv) We obtain much more precise topological information about twisted connected sum G2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G2-manifolds. (v) We describe "geometric transitions" between G2-metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds. (vi) We construct many G2-manifolds that contain rigid compact associative 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 G2-metrics on a given 7-manifold may consist of many different connected components, and opens up many further questions for future study. For instance, the higher-dimensional gauge theory invariants proposed by Donaldson may provide ways to detect G2-metrics on a given 7-manifold that are not deformation equivalent.

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Abstract. Twisted Gromov-Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold X which depend in addition on a vector bundle over X and an invertible multiplicative characteristic class. Special cases are closely related to local Gromov-Witten invariants of the bundle, and to genus-zero one-point invariants of complete intersections in X . We develop tools for computing genus-zero twisted Gromov-Witten invariants of orbifolds and apply them to several examples. We prove a "quantum Lefschetz theorem" which expresses genus-zero one-point Gromov-Witten invariants of a complete intersection in terms of those of the ambient orbifold X . We determine the genus-zero Gromov-Witten potential of the type A surface singularityˆC 2 /Zn˜. We also compute some genus-zero invariants ofˆC 3 /Z 3˜, verifying predictions of Aganagic-Bouchard-Klemm. In a self-contained Appendix, we determine the relationship between the quantum cohomology of the An surface singularity and that of its crepant resolution, thereby proving the Crepant Resolution Conjectures of Ruan and Bryan-Graber in this case.

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