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.

We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular attention to a subclass of weak Fano 3-folds that we call semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike Fanos they often contain P 1 s with normal bundle O(−1) ⊕ O(−1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds.We introduce some general methods to compute the basic topological invariants of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples.All the features of the ACyl Calabi-Yau 3-folds studied here find application in [17] where we construct many new compact G 2 -manifolds using Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds are particularly well-adapted for this purpose.

Let M be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to [0, ∞) × X for some compact connected Ricci-flat manifold X. We begin by proving general structure theorems for M ; in particular we show that there is no loss of generality in assuming that M is simply-connected and irreducible with Hol(M ) = SU(n), where n is the complex dimension of M . If n > 2 we then show that there exists a projective orbifold M and a divisor D ∈ |−K M | with torsion normal bundle such that M is biholomorphic to M \ D, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting. We give examples where M is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair (M , D) we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on M \ D.

We develop a powerful new analytic method to construct complete non-compact Ricciflat 7-manifolds, more specifically G2-manifolds, i.e. Riemannian 7-manifolds (M, g) whose holonomy group is the compact exceptional Lie group G2. Our construction gives the first general analytic construction of complete non-compact Ricci-flat metrics in any odd dimension and establishes a link with the Cheeger-Fukaya-Gromov theory of collapse with bounded curvature.The construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M → B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics gǫ on M that collapses with bounded curvature as ǫ → 0 to the original Calabi-Yau metric on the base B. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics; these are the natural higher-dimensional analogues of the ALF metrics that are well known in 4-dimensional hyperkähler geometry.We give two illustrations of the strength of our method. Firstly we use it to construct infinitely many diffeomorphism types of complete non-compact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Secondly we use it to prove the existence of continuous families of complete non-compact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete non-compact G2-metrics were known.

We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G 2 -structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically cylindrical G 2 -manifold is exactly G 2 if and only if the fundamental group π 1 (M) is finite and neither M nor any double cover of M is homeomorphic to a cylinder.

We define a Z 48 -valued homotopy invariant ν(ϕ) of a G 2 -structure ϕ on the tangent bundle of a closed 7-manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)-structure. For manifolds of holonomy G 2 obtained by the twisted connected sum construction, the associated torsion-free G 2 -structure always has ν(ϕ) = 24. Some holonomy G 2 examples constructed by Joyce by desingularising orbifolds have odd ν.We define a further homotopy invariant ξ(ϕ) such that if M is 2-connected then the pair (ν, ξ) determines a G 2 -structure up to homotopy and diffeomorphism. The class of a G 2 -structure is determined by ν on its own when the greatest divisor of p 1 (M ) modulo torsion divides 224; this sufficient condition holds for many twisted connected sum G 2 -manifolds.We also prove that the parametric h-principle holds for coclosed G 2 -structures. Proposition 1.10. Let f : M ∼ = M be a spin diffeomorphism with mapping torus T f . Then D(ϕ, f * ϕ) = 24 A(T f ) ∈ Z.

We present a comprehensive classification of closed smooth 2‐connected manifolds of dimension 7. This builds on the almost‐smooth classification from the first author's thesis. The main new ingredient is a generalisation of the Eells–Kuiper invariant that is defined for any closed spin 7‐manifold M, regardless of whether the spin characteristic class pM∈H4false(Mfalse) is torsion. We also determine the inertia group of 2‐connected M — equivalently the number of oriented smooth structures on the underlying topological manifold — in terms of pM and the torsion linking form.

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