Abstract
We exhibit the first examples of closed 7dimensional Riemannian manifolds with holonomy \(G_2\) that are homeomorphic but not diffeomorphic. These are also the first examples of closed Ricciflat manifolds that are homeomorphic but not diffeomorphic. The examples are generated by applying the twisted connected sum construction to Fano 3folds of Picard rank 1 and 2. The smooth structures are distinguished by the generalised Eells–Kuiper invariant introduced by the authors in a previous paper.
Introduction
Given a type of special geometric structure, it is often interesting to ask: do there exist manifolds with such structures that are homeomorphic but not diffeomorphic? In this paper we consider the case of Riemannian metrics with holonomy \(G_2\) on closed manifolds of dimension 7. The Lie group \(G_2\) can be described as the automorphism group of the octonion algebra \(\mathbb {O}\), and its natural action on \({{\,\mathrm{Im}\,}}\mathbb {O}\cong \mathbb {R}^7\) appears as an exceptional case in Berger’s classification of Riemannian holonomy [3]. Metrics with holonomy \(G_2\) are always Ricciflat [5].
A general strategy to address the question of the first paragraph is to apply a smooth classification theorem to a plentiful supply of examples for which the classifying invariants are computable. In this paper we make use of the “twisted connected sum” construction of closed \(G_{2}\)manifolds introduced by Kovalev [27]; it was shown in [10] that this construction yields large numbers of closed \(G_{2}\)manifolds that are 2connected (i.e. the homotopy groups \(\pi _1\) and \(\pi _2\) are trivial) with torsionfree cohomology, and how to compute the invariants required to apply homeomorphism classification results of Wilkens [42].
The diffeomorphism classification of 2connected 7manifolds was recently completed in [13], which in particular introduced a generalised Eells–Kuiper invariant that distinguishes all the different smooth structures on the same closed 2connected topological spin 7manifold. While this invariant can be difficult to compute for interesting examples of manifolds, in the present paper we show how to compute it for twisted connected sums, and use that to identify examples of closed 2connected manifolds with holonomy \(G_2\) that are homeomorphic but not diffeomorphic.
Using the diffeomorphism classification, the manifolds can be described explicitly as follows. Real vector bundles of rank 4 over \(S^4\) are classified by their Euler class e and first Pontrjagin class \(p_1\) in \(H^4(S^4) \cong \mathbb {Z}\). Let N and \(\Sigma _\mathrm{Mi}\) be the total space of the unit sphere bundle in the vector bundle with \((e,p_1) =(0, 16)\) and (1, 6), respectively. Then \(\Sigma _\mathrm{Mi}\) is an exotic 7sphere; indeed \(\Sigma _\mathrm{Mi}\) and \(S^7\) were among the first discovered examples of homeomorphic but nondiffeomorphic manifolds (Milnor [31]). Meanwhile for any \(k \ge 1\) the connected sum of k copies of N is a closed 2connected 7manifold with \(b_3(N^{\#k}) = k\) and torsionfree cohomology. The manifolds \(N^{\#k}\) and \(N^{\#k}\#\Sigma _\mathrm{Mi}\) are homeomorphic but not diffeomorphic (and in fact these are the only two diffeomorphism types with that underlying homeomorphism type).
Theorem 1.1
For \(k = 89\) and 101, both \(N^{\#k}\) and \(N^{\#k} \# \Sigma _\mathrm{Mi}\) admit a Riemannian metric with holonomy \(G_2\).
To the best of our knowledge, these are also the first examples of closed Ricciflat manifolds (of any dimension) that are homeomorphic but not diffeomorphic.
Since the paper is primarily devoted to the topological analysis of a particular class of examples of \(G_{2}\)manifolds, it makes practically no use of results about \(G_2\) geometry in general. Background on the definition of \(G_2\) and Riemannian holonomy can be found e.g. in the books by Salamon [36] or Joyce [26]. The main technical work of the paper is to examine in detail the properties of some examples of Fano 3folds and their anticanonical divisors.
Twisted connected sums
There are two known sources of examples of closed \(G_{2}\)manifolds. The first examples were constructed by Joyce in 1995 by desingularising quotients of flat tori [25]. In this paper we will make use of the later twisted connected sum construction. While this can be used to produce a large number of examples, it is still not known whether there exist infinitely many different topological types of closed 7manifolds that admit holonomy \(G_2\) metrics.
A Fano manifold is a smooth projective variety with ample anticanonical bundle, or in more differentialgeometric terms, a closed complex manifold whose first Chern class is a Kähler class. They have been studied extensively, and in complex dimension 3 they have been classified by Iskovskih [20,21,22] and Mori–Mukai [32, 33].
Given a pair of Fano 3folds \(Y_+, Y_\) with smooth anticanonical K3 divisors \(\Sigma _\pm \subset Y_\pm \) and a matching diffeomorphism (Definition 2.5), the twisted connected sum construction yields a closed simplyconnected 7manifold M with metrics of holonomy \(G_2\). The procedure is summarised in Sects. 2.3 and 4.1. Part of the usefulness of the twisted connected sum construction is that many geometric and topological features of the resulting \(G_{2}\)manifolds can be understood in terms of the relatively wellknown algebraic input data. On the other hand, the challenge is that a thorough understanding of the algebraic data is required to find any matchings at all.
We categorise the matching as either perpendicular or nonperpendicular, depending on the action of on the images of \(H^2(Y_\pm )\) in \(H^2(\Sigma _\pm )\) (Definition 5.1). It is shown in [10] that for most pairs \(\mathcal {Y}_+, \mathcal {Y}_\) among the 105 deformation types of Fano 3folds, general deformation theory results make it possible to find a perpendicular matching of some \(Y_\pm \in \mathcal {Y}_\pm \) resulting in a 2connected twisted connected sum \(G_{2}\)manifold. As we explain below, such perpendicular matchings can never be homeomorphic without being diffeomorphic.
Whether there is any nonperpendicular matching of a pair of members of \(\mathcal {Y}_\pm \) is in general a more difficult question, which has not previously been studied systematically. There are necessary conditions of a latticearithmetical nature, but, as we discuss in Sect. 5, showing that matchings exist also requires some detailed information about the deformation theory of anticanonical divisors in \(\mathcal {Y}_\pm \), which needs to be worked out separately for each individual deformation type of Fanos.
The classifying invariants
Let us recall the relevant smooth classification results. Given a closed 2connected 7manifold M, two obvious topological invariants are its cohomology ring \(H^*(M)\) and its spin characteristic class \(p_M \in H^4(M)\) (satisfying \(p_1(M) = 2p_M\)). If \(H^4(M)\) is torsionfree, then this data can be reduced to the third Betti number \(b_3(M)\) and the greatest integer d(M) dividing \(p_M\) in \(H^4(M)\) (we set \(d = 0\) if \(p_M = 0\)). In fact, the pair \((b_3(M), d(M))\) classifies such M up to homeomorphism (by Wilkens [42], see also Theorem 2.2).
In [13], we introduced the generalised Eells–Kuiper invariant of a closed spin 7manifold M. If \(H^4(M)\) is torsionfree then this invariant reduces to a constant
and distinguishes between \(\widehat{d}:= \gcd \big (28, Num \left( \frac{d}{4}\right) \big )\) different diffeomorphism classes of smooth structures on the topological manifold underlying M (where \(Num \left( \frac{a}{b}\right) := \frac{a}{\gcd (a,b)}\)). If \(p_M = 0\) (so \(\widehat{d}= 28\)) then \(\mu (M)\) coincides with the invariant introduced by Eells and Kuiper [16], which in particular distinguishes between the 28 classes of smooth structures on \(S^7\).
In Sect. 2.2 we recall how \(\mu (M)\) can be defined in terms of a \(\text {spin}^\text {c}\) coboundary of M. The challenge with this definition is that while the existence of a suitable coboundary is guaranteed, there is no algorithm for finding one, especially not one with a simple enough description that evaluating the formula (2.1) is tractable. However, we are able to construct explicit \(\text {spin}^\text {c}\) coboundaries of twisted connected sums, and in Sect. 3 we use those to express the generalised Eells–Kuiper invariant of a twisted connected sum in terms of data for the Fanos used and the matching. In particular, it turns out that any perpendicularly matched twisted connected sum has \(\mu = 0\) (Corollary 3.7).
The main examples
To have any chance of obtaining homeomorphic but nondiffeomorphic twisted connected sums we must therefore search for nonperpendicular matchings. In that case, both Fanos used must have Picard rank \(\ge 2\), cf. Remark 5.4. (The Picard group \({{\,\mathrm{Pic}\,}}Y\) of a Fano 3fold Y coincides with its integral second cohomology \(H^2(Y)\), so the Picard rank simply means its second Betti number \(b_2(Y)\).) We are therefore led to study systematically the possible matchings of Fanos with Picard rank 2, and in this setting we can obtain decisive results.
Theorem 1.2
 (i)
Any twisted connected sum M of Fano 3folds of Picard rank 1 or 2 has \(H^4(M)\) torsionfree.
 (ii)
There are precisely six (unordered) pairs \(\mathcal {Y}_+, \mathcal {Y}_\) of deformation types of Fano 3folds of Picard rank 2 with members that can be matched in the sense of Definition 2.5 in such a way that the resulting twisted connected sum M has \(\mu (M) \not = 0\).
 (iii)
Each of those six pairs gives rise to a single diffeomorphism type of M with \(\mu (M) \not = 0\); those M are all 2connected.
 (iv)
In total, they realise four distinct diffeomorphism types of M with \(\mu (M) \not = 0\).
 (v)
Precisely two of those are homeomorphic to some twisted connected sum \(M'\) of Fano 3folds of Picard rank \(\le 2\) such that \(\mu (M') = 0\).
In particular, we obtain two pairs \((M,M')\) of manifolds that are homeomorphic but not diffeomorphic and both admit metrics with holonomy \(G_2\). These are the manifolds identified in Theorem 1.1.
For each of the examples highlighted in Theorem 1.2, we list in Table 1 the pair of deformation types used, from among the following six. (Recall that a del Pezzo 3fold is a Fano 3fold Y whose anticanonical class \(K_Y \in {{\,\mathrm{Pic}\,}}Y\) is even.) (a) Del Pezzo 3folds of degree 3, i.e. cubic hypersurfaces in \(P^4\) (Picard rank 1) (b) Del Pezzo 3folds of degree 5 (Picard rank 1) (c) Picard rank 1 Fanos of degree 16 (d) Number 9 in the MoriMukai list of Picard rank 2 Fano 3folds: \(P^3\) blown up in a curve of degree 7 and genus 5 (e) Number 17 in the MoriMukai list of Picard rank 2 Fano 3folds: a smooth quadric hypersurface in \(P^4\) blown up in an elliptic curve of degree 5 (f) Number 27 in the MoriMukai list of Picard rank 2 Fano 3folds: \(P^3\) blown up in a twisted cubic curve For each example in Table 1 we list whether the matching is perpendicular, and the classifying diffeomorphism invariants of the resulting twisted connected sum. As seen in the table, the two pairs that are homeomorphic but not diffeomorphic.
In Sect. 4 we compute detailed topological data for all 36 types in the MoriMukai list of rank 2 Fano 3folds. In Sect. 7 we identify all pairs that satisfy the necessary arithmetic conditions for existence of a nonperpendicular matching resulting in a twisted connected sum with \(\mu \not = 0\). The only candidate pairs are among the types (d), (e) and (f) above.
The key difficulty in finding nonperpendicular matchings is to understand precisely which K3 surfaces \(\Sigma \) appear as anticanonical divisors in a given type of Fanos, identifying conditions in terms of the Picard lattice of \(\Sigma \) (i.e. \({{\,\mathrm{Pic}\,}}\Sigma = H^2(\Sigma ; \mathbb {Z}) \cap H^{1,1}(\Sigma ; \mathbb {C})\) equipped with the intersection form). Having at least reduced our list of candidates, we carry out this intricate work only for the types (d), (e) and (f) . We find in Theorem 7.8 that nonperpendicular matchings do in fact exist for each of the six pairs of those types, leading to the examples with \(\mu \not = 0\) above.
We then compare the homeomorphism invariants \((b_3, d)\) of the realised manifolds with the invariants realised by perpendicular matchings of the 1378 pairs of rank 1 and 2 Fanos, listed in Table 4 of Sect. 6. For two of the four twisted connected sums with \(\mu \not = 0\) we can identify some perpendicular matching with the same homeomorphism invariants, and two of those are included in the table above.
In another application of the classification results, we can also exhibit examples of holonomy \(G_2\) metrics on total spaces of smooth \(S^1\)bundles, while the last entry of Table 1 is an example of a \(G_{2}\)manifold that is a topological \(S^1\)bundle but not a smooth one (Remark 6.3). Further, as a byproduct of our analysis in Sect. 6 we identify all pairs of rank 2 Fanos that can be matched to define twisted connected sums M with \(H^2(M) \cong \mathbb {Z}\) (Table 5). Such matchings are of interest for the problem of constructing examples of \(G_2\)instantons by gluing, cf. Sá Earp and Walpuski [37], Walpuski [41] and Menet, the second author and Sá Earp [30].
Context
In dimension 7, the problem of finding special metrics on manifolds that are homeomorphic but not diffeomorphic has been considered for instance in the case of Riemannian metrics with positive sectional curvature (Kreck–Stolz [29]) and 3Sasakian metrics (Chinburg–Escher–Ziller [7]). In both cases, the examples exhibited have finite \(H^4\), and the smooth structures can be distinguished by the ordinary Eells–Kuiper invariant.
All known (irreducible) examples of closed Ricciflat manifolds have special holonomy: they are 2nmanifolds with holonomy SU(n) (\(n \ge 2\)), 4nmanifolds with holonomy Sp(n) (\(n \ge 2\)), 7manifolds with holonomy \(G_2\) and 8manifolds with holonomy \(Spin (7)\).
In real dimension 4, the only smooth manifold with holonomy SU(2) is the K3 surface. In real dimension 6, simplyconnected manifolds have unique smooth structures by Zhubr [43, Theorem 6.3], while there is no general classification result for finite but nontrivial fundamental group.
Manifolds with holonomy SU(n) or Sp(n) necessarily have \(b_2 \ge 1\), and in real dimension \(\ge 8\) the case \(b_2 > 1\) is out of reach of current smooth classification results. Complete intersections with \(c_1 = 0\) provide examples with holonomy SU(n) and \(b_2 = 1\), many of which have been smoothly classified by Traving [38] (see also [28, Theorem A]); however Traving’s results imply that homeomorphic complete intersections with \(c_1 = 0\) are diffeomorphic in real dimension \(\ge 6\).
Joyce [26, Theorem 15.4.3, 15.5.2, 15.5.6, 15.6.2 & 15.7.2] provides five examples of 8manifolds of holonomy \(Spin (7)\) with \(b_2 = 0\), and four of these also have \(b_3 = 0\). Computing the torsion in the cohomology of these examples is intricate, and even if they prove to be 3connected, then the underlying topological manifold admits two diffeomorphism classes of smooth structure if and only if 8 divides the first Pontrjagin class (if 8 does not divide the first Pontrjagin class, then the smooth structure is unique up to diffeomorphism) [11]. However, there is not currently a sufficiently large supply of such examples to be hopeful of finding exotic pairs of 3connected \(Spin (7)\) 8manifolds.
Thus 7manifolds of holonomy \(G_2\) are the only kinds of closed Ricciflat manifolds where homeomorphic nondiffeomorphic examples are accessible with the current technology.
In the context of nonRicciflat holonomy groups, an early application of Donaldson invariants was to give examples of closed manifolds with holonomy U(2), i.e. Kähler manifolds of complex dimension 2, that are homeomorphic but not diffeomorphic [15].
Background
We begin with further explanations of the generalised Eells–Kuiper invariant and the twisted connected sum construction, in order to elucidate the meaning of the main theorem, Theorem 1.2.
Spin and spin\(^\text {c}\) characteristic classes
Recall that \(BSpin \), the classifying space for stable spin vector bundles, is 3connected with \(\pi _4(BSpin ) \cong \mathbb {Z}\). It follows that \(H^4(BSpin ) \cong \mathbb {Z}\) is infinite cyclic. A generator is denoted \(\pm \frac{p_1}{2}\) and the notation is justified since for the canonical map \(\pi :BSpin \rightarrow BSO\) we have \(\pi ^*p_1 = 2 \frac{p_1}{2}\), where \(p_1\) is the first Pontrjagin class. Given a spin manifold X we write
A \(\text {spin}^\text {c}\) structure on a vector bundle E has an associated complex line bundle L such that \(E \oplus L\) is spin. Given a \(\text {spin}^\text {c}\) structure on the tangent bundle of X we can therefore define characteristic classes \(z := c_1(L) \in H^2(X)\), and \(\widehat{p}_X := \frac{p_1}{2}(TX \oplus L) \in H^4(X)\). Then \(2\widehat{p}_X = p_1(X) + z^2\). (If X is spin then the induced \(\text {spin}^\text {c}\) structure of course has L trivial, and \(\widehat{p}_X = p_X\).)
Lemma 2.1
(cf. [13, 2.39–2.40])
 (i)
If TX has an almost complex structure then \(\widehat{p}_X = c_2(X) + c_1(X)^2\).
 (ii)
\(\widehat{p}_X = w_4(X) + w_2(X)^2 \!\!\mod 2\).
 (iii)
Suppose X is closed.
If \(\dim X \le 7\) then \(\widehat{p}_X\) is even.
If \(\dim X = 8\) then \(\widehat{p}_X\) is characteristic for the intersection form of X, i.e. \(x^2 = x \cup \widehat{p}_X \!\!\mod 2\) for any \(x \in H^4(X)\).
The generalised Eells–Kuiper invariant
We now describe the invariant \(\mu (M)\) of a closed spin 7manifold M, in the special case when \(H^4(M)\) is torsionfree. As in the introduction, let d be the greatest integer dividing \(p_M\) (so d is even by Lemma 2.1(iii)), and \(\widehat{d}:= \gcd \big (28, Num \left( \frac{d}{4}\right) \big )\).
Let \(W\) be a \(\text {spin}^\text {c}\) 8manifold with \(\partial W= M\). To use \(W\) to compute the generalised Eells–Kuiper invariant \(\mu (M)\) we need an element \(n \in H^4(W)\) such that the image of dn in \(H^4(M)\) equals \(p_M\) [13, §2.7]. Then
where \(\widehat{\alpha }:= \widehat{p}_W d n \in H^4(W)\) for n as above, \(\sigma (W)\) denotes the signature of the intersection form of W, and \({\bar{z}} \in H^2(W,M)\) is any preimage of z (\({\bar{z}}^2 \in H^4(W,M)\) is independent of the choice of \({\bar{z}}\)). The integrals of \(\widehat{\alpha }^2\) and \(z^4\) make sense since they are squares of classes that vanish on the boundary. That \(\mu (M)\) is independent of the choice of \(W\) is a consequence of the index formula for the Dirac operator on a closed \(\text {spin}^\text {c}\) 8manifold.
When M is in addition 2connected, there are precisely \(\widehat{d}\) different smooth structures on the topological manifold underlying M, and they are distinguished by \(\mu (M)\).
Theorem 2.2
(cf. [13, Theorems 1.2 and 1.3]) Let \(M_0\) and \(M_1\) be closed 2connected 7manifolds such that \(H^4(M_i)\) are torsionfree. Then
 (i)
\(M_0\) and \(M_1\) are homeomorphic if and only if \(b_3(M_0) = b_3(M_1)\) and \(d(M_0) = d(M_1)\).
 (ii)
\(M_0\) and \(M_1\) are diffeomorphic if and only if in addition \(\mu (M_0) = \mu (M_1)\).
Definition of twisted connected sums
We now explain the construction of the twisted connected sum of a matching pair of building blocks, using the setup from [10, §3]. Like in the original application of the construction by Kovalev [27], the present paper uses building blocks obtained from Fano 3folds (in our case of rank 1 or 2) by a procedure explained in Sect. 4.1. The matching problem is elaborated on in Sect. 5.
Definition 2.3
Let Z be a nonsingular algebraic 3fold and \(\Sigma \subset Z\) a nonsingular K3 surface. Let N be the image of \(H^2(Z) \rightarrow H^2(\Sigma )\). We call \((Z,\Sigma )\) a building block if
 (i)
the class in \(H^2(Z)\) of the anticanonical line bundle \(K_Z\) is indivisible;
 (ii)
\(\Sigma \in {}K_Z\) (i.e. \(\Sigma \) is an anticanonical divisor), and there is a projective morphism \(f:Z\rightarrow \mathbb {P}^1\) with \(\Sigma =f^\star (\infty )\);
 (iii)
the inclusion \(N\hookrightarrow H^2(\Sigma )\) is primitive, that is, \(H^2(\Sigma )/N\) is torsionfree;
 (iv)
the group \(H^3(Z)\)—and thus also \(H^4(Z)\)—is torsionfree.
We call N, equipped with the restriction of the intersection form on \(H^2(\Sigma )\), the polarising lattice of the block. (Because \(H^{2,0}(Z)\) is automatically trivial, \(N \subseteq H^{1,1}(\Sigma )\) [10, Lemma 3.6], so that \(\Sigma \) is ‘Npolarised’.)
Remark 2.4
The class \([\Sigma ] = K_Z \in H^2(Z)\) is always in the kernel of \(H^2(Z) \rightarrow H^2(\Sigma )\). In this paper we will only consider blocks where \([\Sigma ]\) in fact generates the kernel.
Definition 2.5
Let \(Z_\pm \) be complex 3folds, \(\Sigma _\pm \subset Z_\pm \) smooth anticanonical K3 divisors and Kähler classes. Let \(\Pi _\pm \subset H^2(\Sigma _\pm ;\mathbb {R})\) denote the type \((2,0)+(0,2)\) part. We call a diffeomorphism a matching of and if and , while is nontrivial.
We also say that is a matching of \(Z_+\) and \(Z_\) if there are Kähler classes so that the above holds.
Given a building block \((Z, \Sigma )\), let \(\Delta \subset \mathbb {P}^1\) be an open disc that is a trivialising neighbourhood of \(\infty \) for the fibration \(f : Z \rightarrow \mathbb {P}^1\), and \(U := f^{1}(\Delta )\). Then \(V := Z \setminus U\) is a manifold with boundary diffeomorphic to \(S^1 \times \Sigma \).
Construction 2.6
(Twisted connected sum) Given a pair of building blocks \((Z_\pm , \Sigma _\pm )\) with a matching , their twisted connected sumM is the smooth 7manifold defined by gluing \(S^1 \times V_+\) and \(S^1 \times V_\) by the diffeomorphism
of their boundaries.
Theorem 2.7
The twisted connected sum M admits metrics with holonomy \(G_2\).
Proof sketch
(cf [10, Corollary 6.4]) By [19, Theorem D], the interiors of the manifolds \(V_\pm \) admit metrics with holonomy SU(3) that are asymptotically cylindrical, i.e. in a collar neighbourhood \(\cong \mathbb {R}\times S^1 \times \Sigma _\pm \) of the boundary they are close to a product cylinder metric. Then \(V_\pm \) admit parallel SU(3)structures, defining torsionfree product \(G_{2}\)structures on \(S^1 \times V_\pm \). The definition of what it means for to be a matching ensures that the SU(3)structures can be chosen so that is a ‘hyperKähler rotation’ of the SU(2)structures defining the asymptotic limit. According to Kovalev [27, Theorem 5.34] the map (2.2) can then be used to glue together the product \(G_{2}\)structures on \(S^1 \times V_+\) and \(S^1 \times V_\) to a torsionfree \(G_{2}\)structure on M. \(\square \)
Topology of twisted connected sums
Let \((Z_\pm , \Sigma _\pm )\) be a pair of building blocks with a matching . We think of as allowing us to identify both \(\Sigma _+\) and \(\Sigma _\) with a standard smooth K3 surface \(\Sigma \). Letting \(L := H^2(\Sigma )\), we can then think of the polarising lattices \(N_\pm \) of the two blocks as a pair of sublattices of L.
Let \(T_\pm \) denote the orthogonal complement of \(N_\pm \) in L. The following result summarises the cohomology of the twisted connected sum M.
Theorem 2.8
[10, Theorem 4.8] Suppose M is a twisted connected sum of building blocks \((Z_\pm , \Sigma _\pm )\) such that the kernel of \(H^2(Z_\pm ) \rightarrow H^2(\Sigma _\pm )\) is generated by \([\Sigma _\pm ]\). Then
 (i)
\(\pi _1(M)=0\) and \(H^1(M)=0\);
 (ii)
\(H^2(M) \cong N_+\cap N_\);
 (iii)
\(H^3(M) \cong \mathbb {Z}\oplus (L/_{N_+ + N_}) \oplus (N_\cap T_+) \oplus (N_+\cap T_)\oplus H^3(Z_+)\oplus H^3(Z_)\);
 (iv)
\(H^4(M) \cong H^4(\Sigma ) \oplus (T_+\cap T_)\oplus (L/_{N_ + T_+})\oplus (L/_{N_+ + T_}) \oplus H^3(Z_+) \oplus H^3(Z_)\).
This implies in particular that if the matching is perpendicular (i.e. if \(N_+\) is perpendicular to \(N_\) in L, cf. Definition 5.1) then \(H^4(M)\) is torsionfree, and
If in addition the perpendicular direct sum \(N_+ \perp N_ \subset L\) is primitive then M is 2connected.
We need to devote some extra attention to describing \(H^4(M)\), since that contains the spin characteristic class \(p_M\), whose greatest divisor d(M) is one of our classifying invariants from Theorem 2.2. Let
be the subgroup of pairs with equal image in \(H^4(\Sigma )\). We define a homomorphism
as follows (cf. [10, Definition 4.13]). For \(([\alpha _+], [\alpha _]) \in H^4(Z_+) \oplus _0 H^4(Z_)\), let \([\beta ]\) be their common image in \(H^4(\Sigma )\). Then we may choose the cocycle \(\alpha _\pm \in C^4(Z_\pm ;\mathbb {Z})\) so that the restriction of \(\alpha _\pm \) to \(\partial V_\pm \cong S^1 \times \Sigma \subset Z_\pm \) is the pullback of \(\beta \) by projection to the \(\Sigma \) factor. Then the pullbacks of \(\alpha _\pm \) to \(S^1 \times V_\pm \) patch to a cocycle on M under the gluing (2.2), and we may set \(Y([\alpha _+], [\alpha _])\) to be the class represented by that cocycle.
Let
denote the homomorphism induced by the intersection form of L, and \(N_\mp ' \subseteq N_\pm ^*\) the image of \(\flat ^{\pm }\). Since \(N_\pm ^* \subset H^4(Z_\pm )\), we can also regard \(N'_\mp \) as a subset of \(H^4(Z_+) \oplus _0 H^4(Z_)\).
Lemma 2.9
[10, Lemma 4.14] The image of the map Y defined in (2.5) is the direct summand \(H^4(\Sigma ) \oplus (L/_{N_ + T_+})\oplus (L/_{N_+ + T_})\) of \(H^4(M)\), and the kernel is \(N'_ \oplus N'_+ \subset H^4(Z_+) \oplus _0 H^4(Z_)\).
Remark 2.10
The image of Y contains all the torsion of \(H^4(M)\), which we can identify as
Proposition 2.11
[10, Proposition 4.20/Remark 4.21]
Combining the previous two results, the greatest divisor d(M) of \(p_M\) can be determined from \(c_2(Z_\pm ) \in H^4(Z_\pm )\) (which depends purely on the blocks) and \(N'_\mp \subset H^4(Z_\pm )\) (which depends on the matching).
The generalised Eells–Kuiper invariant of twisted connected sums
The main result of this section is Theorem 3.6, which gives a formula for the generalised Eells–Kuiper invariant \(\mu (M)\) of a twisted connected sum M in terms of data for the pair of building blocks used and data for the matching. We compute \(\mu (M)\) using an explicit \(\text {spin}^\text {c}\) coboundary.
Throughout this section, let M be the twisted connected sum of a pair of building blocks \((Z_\pm , \Sigma _\pm )\), using a matching . To simplify the description of the cohomology, we assume that the kernel of \(H^2(Z_\pm ) \rightarrow H^2(\Sigma _\pm )\) is generated by \([\Sigma _\pm ]\).
The \(\text {spin}^\text {c}\) coboundaries we construct to compute the generalised Eells–Kuiper invariant for twisted connected sum manifolds can be viewed as parametrised plumbing of trivial disc bundles. By way of context we recall the plumbing of bundles; see Browder [6, V §2]. If \(\pi _0 :W_0 \rightarrow M_0\) is a \(D^p\)bundle over a qmanifold and \(\pi _1 :W_1 \rightarrow M_1\) is a \(D^q\)bundle over a pmanifold, then the plumbing of \(\pi _0\) and \(\pi _1\) is the manifold \(W = (W_0 \sqcup W_1)/{\sim }\) obtained from the disjoint union of \(W_0\) and \(W_1\) by trivialising \(\pi _0\) over \(D^q \subset M_0\) and \(\pi _1\) over \(D^p \subset M_1\), identifying the resulting spaces \(D^p \times D^q\) and \(D^q \times D^p\) by exchanging coordinates and finally smoothing corners. The case of a pair of trivial \(D^1\)bundles over \(S^1\) is illustrated below.
Construction 3.1
(Parametric plumbing) As in Construction 2.6, we use the tubular neighbourhood \(U_\pm \cong \Delta \times \Sigma _\pm \subset Z_\pm \) of \(\Sigma _\pm \); denote the coordinate on the open disc \(\Delta \) by z. Consider \(B_\pm := \Delta \times Z_\pm \), and denote the coordinate on its open disc factor by w. Form an 8manifold \(\mathring{W}\) by gluing \(B_+\) and \(B_\) along \(\Delta \times U_\pm \), using the map
We can form a smooth compact manifold W with boundary by attaching to \(\mathring{W}\) the result of gluing the boundaries \(\partial {\overline{\Delta }}\times \partial {\overline{\Delta }}\times \Sigma _\pm \) of \(\partial {\overline{\Delta }}\times (Z_\pm \setminus U_\pm )\) by . Comparing with (2.2), we see that the boundary is precisely the twisted connected sum M. Hence \(W\) is a coboundary of M.
Since \(Z_\pm \) are not spin, \(B_\pm \) and W are not spin either. However, \(Z_\pm \) and \(B_\pm \) have complex structures, inducing \(\text {spin}^\text {c}\) structures. While the complex structures of \(B_+\) and \(B_\) do not agree on the overlap region, their \(\text {spin}^\text {c}\) structures do agree, so W is \(\text {spin}^\text {c}\) too.
Cohomology
We compute the integral cohomology of the coboundary W from Construction 3.1 by Mayer–Vietoris. More precisely, we compute \(H^k(\mathring{W}) \cong H^k(W)\) and \(H^k_{cpt}(\mathring{W}) \cong H^k(W,M)\) for \(k \le 4\). One can of course recover the remaining cohomology groups from Poincaré duality and universal coefficients, but what we care about is a description for \(k \le 4\) that lets us determine the characteristic classes and the intersection form.
There are obvious homotopy equivalences \(B_+ \cap B_ \simeq \Sigma \) and \(B_\pm \simeq Z_\pm \), so there is a long exact sequence
Hence, using that \(H^2(Z_\pm ) \rightarrow H^2(\Sigma ) = L\) has image \(N_\pm \) by definition and is assumed to have kernel \(\mathbb {Z}\), we find
where \(H^4(Z_+) \oplus _0 H^4(Z_)\) is defined as in (2.4). There is a natural short exact sequence
We can describe the isomorphism \(H^4(Z_+) \oplus _0 H^4(Z_) \rightarrow H^4(W)\) as follows: given \((c_+, c_) \in H^4(Z_+) \oplus _0 H^4(Z_)\), choose a cocycle \(\alpha _0 \in C^4(\Sigma )\) representing the common image of \(c_\pm \) in \(H^4(\Sigma )\), let \(\alpha _\pm \in C^4(Z_\pm )\) be cocycles representing \(c_\pm \) that equal the pullback of \(\alpha _0\) on \(\Delta \times \Sigma _\pm \), and form a cocycle on W by patching the pullbacks of \(\alpha _\pm \) to \(B_\pm \). It is clear that the composition of this isomorphism with the restriction map \(H^4(W) \rightarrow H^4(M)\) equals the map Y defined in (2.5).
Remark 3.2
Only the isomorphism for \(H^3(W)\) involves making an arbitrary choice for a splitting of a short exact sequence—below it will turn out to be important that the isomorphisms for \(H^2(W)\) and \(H^4(W)\) are natural.
Similarly, we can compute \(H^*(W,M)\) from a long exact sequence
finding
Note there is a short exact sequence
Characteristic classes
As emphasised in Remark 3.2, the isomorphisms for \(H^2(W)\) and \(H^4(W)\) above are natural. One consequence of this is that we can pin down the characteristic classes of W by considering their restrictions to the open subsets \(B_+ = \Delta \times Z_+\) and \(B_ = \Delta \times Z_\).
The restriction of \(w_2(W) \in H^2(W; \mathbb {Z}/2)\) to \(B_\pm \) is \(w_2(B_\pm )\), which equals the pullback of \(w_2(Z_\pm )\). That is the mod 2 reduction of \(c_1(Z_\pm ) = [\Sigma _\pm ] \in H^2(Z_\pm )\) and so \(w_2(Z_\pm )\) is the mod 2 Poincaré dual of \(\Sigma _\pm \) in \(Z_\pm \). The \(\text {spin}^\text {c}\) structures of \(B_\pm \) patch up to a unique \(\text {spin}^\text {c}\) structure on W: this is essentially just saying that we can specify an integral preimage \(z \in H^2(W)\) of \(w_2(W)\) uniquely by setting the restriction to \(B_\pm \) to be \(c_1(Z_\pm )\). Note that the restriction of \(z^2\) to each of \(B_\pm \) is 0; hence
Similarly we can pin down the \(\text {spin}^\text {c}\) characteristic class \(\widehat{p}_W\). The restriction to \(B_\pm \) is \(\widehat{p}_{B_\pm }\), which by Lemma 2.1(i) equals \(c_2 + c_1^2\) of the relevant U(4)structure. Using again that \(c_1(Z_\pm )^2 = 0\), we thus have
Lemma 3.3
The image of \(\widehat{p}_W\) in \(H^4(Z_+) \oplus _0 H^4(Z_)\) is \((c_2(Z_+), c_2(Z_))\).
The restriction of \(\widehat{p}_W\) to M is \(p_M\), which recovers Proposition 2.11, and arguably makes for a nicer proof than that in [10, Proposition 4.20].
Since the normal bundle of \(\Sigma \subset W\) is trivial, the image of \(\widehat{p}_W\) in \(H^4(\Sigma ) \cong \mathbb {Z}\) is given by \(p_\Sigma = c_2(\Sigma ) = 24\), so in view of the short exact sequence (3.1) most of the interesting information about \(\widehat{p}_W\) is captured by the preimage of \(\widehat{p}_W \!\!\mod 24\) in \((N_+^* \oplus N_^*) \otimes \mathbb {Z}/24\). Similarly, for any building block Z, the image of \(c_2(Z)\) under \(H^4(Z) \rightarrow H^4(\Sigma )\) is 24, so \(c_2(Z) \!\!\mod 24\) has a preimage in \(H^4(Z, \Sigma ; \mathbb {Z}/24) \cong N^* \otimes \mathbb {Z}/24\). Denote that by
The calculation above implies that \(\widehat{p}_W \!\!\mod 24\) is determined by \({\bar{c}}_2(Z_\pm )\).
Corollary 3.4
\(\widehat{p}_W \!\!\mod 24 \in H^4(W; \mathbb {Z}/24) \cong H^4(Z_+;\mathbb {Z}/24) \oplus _0 H^4(Z_; \mathbb {Z}/24)\) is the image of \(({\bar{c}}_2(Z_+), {\bar{c}}_2(Z_))\) under the map in (3.1).
Remark 3.5
Since \(c_2(Z_\pm )\) are even, so is \(\widehat{p}_W\). Therefore Lemma 2.1(iii) implies that the intersection form of W must be even.
Intersection form
The intersection pairing between \(H^4(W)\) and \(H^4(W,M)\) is simply the natural duality between \(H^4(Z_+) \oplus _0 H^4(Z_)\) and \((H^2(Z_+) \oplus H^2(Z_))/\mathbb {Z}\). To understand the intersection form we also need to describe the map \(H^4(W,M) \rightarrow H^4(W)\).
First note that the \(\mathbb {Z}\) term in \(H^4(W,M) \cong (H^2(Z_+) \oplus H^2(Z_))/\mathbb {Z}\) is the Poincaré dual to the K3 divisors, and obviously has trivial image in \(H^4(W)\). Hence \(H^4(W,M) \rightarrow H^4(W)\) factors through the natural map \((H^2(Z_+) \oplus H^2(Z_))/\mathbb {Z}\rightarrow N_+ \oplus N_\). Dually, the composition \(H^4(W) \cong H^4(Z_+) \oplus _0 H^4(Z_) \rightarrow \mathbb {Z}\) corresponds to restriction to the K3 divisor, which is trivial for any class with compact support in \(\mathring{W}\). So the map also factors through the inclusion \(N_+^* \oplus N_^* \hookrightarrow H^4(Z_+) \oplus _0 H^4(Z_)\), and is characterised by a homomorphism \(N_+ \oplus N_ \rightarrow N_+^* \oplus N_^*\).
The intersection form on each \(B_\pm \) is obviously trivial, so there are no diagonal terms. In summary, the map \(H^4(W,M) \rightarrow H^4(M)\) is therefore the composition of \(H^4(W,M) \rightarrow N_+ \oplus N_\),
and \(N_+^* \oplus N_^* \rightarrow H^4(W)\), where \(\flat ^{\pm }\) is the inner product homomorphism as in (2.6). The image equals \(N_' \oplus N_+' \subset N_+^* \oplus N_^*\), where \(N_\mp '\) is the image of \(\flat ^\pm : N_\mp \rightarrow N_\pm ^*\) as before. (This is consistent with the claim from Lemma 2.9 that the kernel of Y is precisely \(N_' \oplus N_+'\), and also with Remark 3.5) Both summands of \(N'_ \oplus N'_+\) are isotropic and so \(\sigma (W)\), the signature of W vanishes:
Computing the generalised Eells–Kuiper invariant
Suppose that the twisted connected sum M has \(H^4(M)\) torsionfree, so that the description of the generalised Eells–Kuiper invariant from Sect. 2.2 is valid. Let d denote the greatest integer dividing the spin characteristic class \(p_M \in H^4(M)\). Since M contains a K3 with trivial normal bundle, \(d \mid 24\) a priori. Hence \(\widehat{d}:= \gcd (28, Num \left( \frac{d}{4}\right) )\) is either 1 or 2, depending on whether d is divisible by 8 or not. In particular \(\mu (M) \in \mathbb {Z}/\widehat{d}\) can only possibly be nontrivial if d is 8 or 24.
We now compute \(\mu (M)\) by evaluating (2.1) for the parametric plumbing coboundary of Construction 3.1. The \(z^4\) term in (2.1) obviously makes no contribution since \(z^2 = 0\), and the signature term vanishes by (3.4). We can take \({\bar{z}}\) to be the sum of the generators of \(H^2_{cpt}(\Delta ) \subset H^2_{cpt}(B_\pm )\), or equivalently the Poincaré duals of \(\{0\} \times Z_\pm \). This is because the intersection of \(\{0\} \times Z_+\) with \(\{0\} \times Z_\) is \(\{0\} \times \Sigma _\), which is the Poincaré dual of \(z_{B_}\), and similarly for \(z_{B_+}\). Thus \({\bar{z}}^2\) is twice the Poincaré dual to the K3 divisor (the intersection of \(\{0\} \times Z_+\) and \(\{0\} \times Z_\), generating the copy of \(\mathbb {Z}\) in \((H^2(Z_+) \oplus H^2(Z_))/\mathbb {Z}\)). It follows that \({\bar{z}}^2 \widehat{p}_W = 2 \widehat{p}_{W\Sigma }\). As above, the triviality of the normal bundle of \(\Sigma \) implies that \(\widehat{p}_{W\Sigma } = p_{\Sigma } = 24\), and hence \(\frac{5{\bar{z}}^2 \widehat{p}_W}{12}\) is divisible by 4. Since the modulus \(\widehat{d}\) is 1 or 2, the only possible nontrivial contribution to the RHS of (2.1) is \(\frac{\widehat{\alpha }^2}{8}\).
The definition of d means that \(\widehat{p}_W \in N_' \oplus N_+' \!\!\mod d\), where \(N_' \oplus N_+'\) is the image of \(H^4(W,M) \rightarrow H^4(W)\) identified in Sect. 3.3. By Corollary 3.4, we can find elements \([x_\pm ] \in {N_\pm \otimes \mathbb {Z}/d}\) such that \(\widehat{p}_W = (\flat ^+([x_]), \flat ^([x_+])) \!\!\mod d\) from \({\bar{c}}_2(Z_\pm ) \in N_\pm ^* \otimes \mathbb {Z}/24\) and the configuration of embeddings \(N_\pm \subset L\) of the matching. We may then take
One preimage under (3.3) of this \(\widehat{\alpha }\) in \(N_+ \oplus N_\) is \((x_+, \; x_)\), so \(\widehat{\alpha }^2 = 2x_+.x_\). Hence we have
Theorem 3.6
Let \(x_\pm \in N_\pm \) such that \(\flat ^\mp (x_\pm ) = {\bar{c}}_2(Z_\mp ) \!\!\mod d\). Then
The elements \(x_\pm \) must be even because \(c_2(Z_\pm )\) is, so the RHS is indeed integral.
If \(N_+ \perp N_\), i.e. if the matching used is perpendicular, then d is the greatest common divisor of \({\bar{c}}_2(Z_+)\) and \({\bar{c}}_2(Z_)\), so we can trivially take \(x_\pm = 0\). Indeed, \(H^4(Z_+) \oplus _0 H^4(Z_) \hookrightarrow H^4(M)\), so since there is no torsion in \(H^3(Z_\pm )\), there is a welldefined \(n := \frac{1}{d} \widehat{p}_W \in H^4(Z_+) \oplus _0 H^4(Z_)\), and that choice gives \(\widehat{\alpha }= 0\).
Corollary 3.7
\(\mu (M) = 0\) for all twisted connected sums obtained by perpendicular matching.
But if neither of \(c_2(Z_\pm )\) are divisible by 4, then \(x_+.x_\) can be \(4 \!\!\mod 8\). For nonperpendicular matchings that can happen even when d is divisible by 8, in which case \(\mu (M)\) is nonzero.
Quadratic refinement of the torsion linking form
Although in this paper we only consider examples of twisted connected sums with no torsion in \(H^4(M)\), let us briefly use the coboundary W to analyse in general the torsion linking form of M and its quadratic refinement (cf. [13, Definition 2.23]).
Since the blocks \(Z_\pm \) are assumed to have no torsion in \(H^3(Z_\pm )\), \(H^4(W)\) resolves all the torsion in \(H^4(M)\). Corti et al. [10, Remark 4.12] claims that the two summands \({{\,\mathrm{Tor}\,}}N_\pm ^*/N'_\mp \) of \({{\,\mathrm{Tor}\,}}H^4(M)\) in (2.7) are isotropic for the torsion linking form, and naturally dual. This can be checked by considering cocycles representing the classes, but it is nicer to do it by computing cup products on W. Indeed, this way we can also compute the refinement of the linking form. Because \(\widehat{p}_W\) is even by Remark 3.5, we can characterise the quadratic linking family on \({{\,\mathrm{Tor}\,}}H^4(M)\) as follows: it assigns to the image in \(H^4(M)\) of (the unique) \({\textstyle \frac{1}{2}}\widehat{p}_W\) the discriminant form q of the even lattice \(H^4(W,M)/\mathrm {rad}\).
Proposition 3.8
\({{\,\mathrm{Tor}\,}}N_\pm ^*/N'_\mp \subseteq {{\,\mathrm{Tor}\,}}H^4(M)\) is isotropic for the torsion linking form, and Lagrangian for q.
Proof
We can write any \(c \in {{\,\mathrm{Tor}\,}}N_+^*/N'_ \subset H^4(M)\) as the restriction of an element of the form \((\alpha ,0) \in N_+^* \oplus N_^* \subset H^4(W)\), where \(\alpha \) is in the rational span of \(N_'\). Then \((\alpha , 0)\) is the image under (3.3) of some \((0, y) \in (N_+ \oplus N_) \otimes \mathbb {Q}\). This pairs trivially with \((\alpha , 0)\), so \(q(c) = 0\). \(\square \)
Fanotype blocks
We now describe how to construct building blocks for twisted connected sums from closed Fano 3folds, and list detailed data for blocks obtained from Fanos of Picard rank 1 or 2.
Construction of building blocks from Fanos
Recall that a Fano 3fold is a smooth closed complex 3fold Y whose anticanonical bundle \(K_Y\) is ample, or equivalently, \(c_1(Y)\) can be represented by a Kähler form. Any Fano 3fold is simplyconnected with \(H^{2,0}(Y) = 0\), so \({{\,\mathrm{Pic}\,}}Y \cong H^2(Y)\). Together with a natural form, this is a key deformation invariant of Y.
Definition 4.1
For a closed complex 3fold Y, the anticanonical form on \(H^2(Y)\) is the symmetric bilinear form \((D_1, D_2) \mapsto D_1 \cdot D_2 \cdot (K_Y)\) (where \(\cdot \) is the cup product on \(H^2(Y)\)).
If \({{\,\mathrm{Pic}\,}}Y \cong H^2(Y)\) we call \({{\,\mathrm{Pic}\,}}Y\) equipped with the anticanonical form the Picard lattice of Y.
There are 105 deformation types of Fano 3folds. Except for two of those, any Fano 3fold Y has a pencil in \({}K_Y\) with smooth base locus C.
Construction 4.2
Given a Fano 3fold Y and \(C \subset Y\) a smooth curve that is the base locus of an anticanonical pencil, let Z be the blowup of Y in C. If \(\Sigma \subset Z\) is the proper transform of a smooth element of said pencil, then we call \((Z, \Sigma )\) a Fanotype building block.
Proposition 4.3
([9, Propositions 4.24 and 5.7], cf. [10, Proposition 3.17]) \((Z, \Sigma )\) of Construction 4.2 is indeed a building block. Further
 (i)
the image of \(H^2(Z) \rightarrow H^2(\Sigma )\) is isomorphic to \(H^2(Y)\), and the kernel is generated by \([\Sigma ]\);
 (ii)
the image in \(H^{1,1}(\Sigma )\) of the Kähler cone of Z contains the image of the Kähler cone of Y.
Some of the important data of the block can be read off immediately from wellknown data about Y. For instance (i) implies that the polarising lattice of the block is isometric to the Picard lattice of Y, and (cf. [9, Lemma 5.6])
Meanwhile, the second Chern class can be described as follows. The pullback \(\pi ^* : H^2(Y) \rightarrow H^2(Z)\) of the blowup map \(\pi : Z \rightarrow Y\) is injective, and \(H^2(Z)\) is the direct sum of \(\pi ^* H^2(Y)\) and \(\mathbb {Z}c_1(Z)\). Thus an element of \(H^4(Z)\) can be characterised in terms of its product with \(c_1(Z)\) and its image under the Poincaré adjoint \(\pi _! : H^4(Z) \rightarrow H^4(Y)\) of \(\pi ^*\), i.e. the map characterised by equality of the intersection products \((\pi _! x)y\) and \(x(\pi ^*y) \in \mathbb {Z}\) for any \(x \in H^4(Z)\) and \(y \in H^2(Y)\). We have \(c_2(Z)c_1(Z) = 24\) and \(c_1(Z)^2 = 0\) for any building block, so for a Fanotype block \(c_2(Z)\) is completely determined by the following result.
Lemma 4.4
[9, (5–13)] For any complex 3fold Y, \(C \subset Y\) a smooth curve contained in a smooth anticanonical divisor and \(\pi : Z \rightarrow Y\) the blowup of Y in C,
In view of Corollary 3.4, the main interesting information about \(c_2(Z)\) is the mod 24 reduction \({\bar{c}}_2(Z) \in N^* \otimes \mathbb {Z}/24\), which we can thus think of simply as
When we tabulate data for the blocks, we will record \(\pi _! c_2(Z)\) rather than \({\bar{c}}_2(Z)\) for completeness.
Note that Definition 2.5 makes sense for Fano 3folds as well as building blocks. Proposition 4.3(ii) implies that a matching of a pair of Fano 3folds gives rise to a matching of the resulting Fanotype blocks. We use the phrase twisted connected sum of Fanos (e.g. in the introduction) to mean the twisted connected sum that arises from such a matching of Fanotype blocks.
Rank 1 blocks
Table 2 summarises the key data of Fano 3folds of rank 1 and the resulting building blocks (cf. [9, Table 1]). The data included is the index r (i.e. the largest integer such that \(K_Y = rH\) for some \(H \in {{\,\mathrm{Pic}\,}}Y\)), the anticanonical degree \(K_Y^3\), \(b_3(Y)\), \(b_3(Z)\), and the pairing of \(\pi _!c_2(Z)\) and the positive generator \(H \in {{\,\mathrm{Pic}\,}}Y\) (equivalently, the product of \(c_2(Z)\) and \(\pi ^* H\)).
\(b_3(Z)\) is simply obtained from the preceding data by (4.1). In the rank 1 case, \(\pi _! c_2(Z)\) is also easily determined as follows: For any Fano one has \(c_2(Y)(K_Y) = 24\), so if \(K_Y = rH\) then (4.3) implies that
N is generated by the image of H. Its selfintersection (with respect to the quadratic form on N) is not included in the table, but it is simply \(\frac{K_Y^3}{r^2}\).
The table of rank 2 Fano blocks
In Table 3 we collect the following data for building blocks obtained from rank 2 Fano 3folds Y.
The number of the corresponding entry in the MoriMukai list of rank 2 Fanos.
The anticanonical degree \(K_Y^3\) of the Fano Y.
The quadratic form of the Picard lattice N presented with respect to a basis \(G\), \(H\) that spans the nef cone (i.e. the ample classes are exactly the linear combinations of \(G\) and \(H\) where both coefficients are positive); we do not know a general reason why the extremal rays should generate all of N and not just a finite index sublattice, but it does for all entries in the list.
The absolute value \(\Delta \) of the discriminant of the quadratic form on N.
The anticanonical class \(K_Y\) in terms of the basis \(G\), \(H\).
The element \(\pi _!c_2(Z) = c_2(Y) + c_1(Y)^2 \in N^*\) (whose mod 24 reduction is \({\bar{c}}_2(Z) \in N^* \otimes \mathbb {Z}/24\) defined in (3.2)) presented in the dual basis of \(G\), \(H\). In other words, the entries of the row vector are the pairing of \(c_2(Z)\) with \(\pi ^*G\) and \(\pi ^*H\).
The third Betti number \(b_3(Y)\) of the Fano.
The third Betti number \(b_3(Z)\) of the resulting block.
The final four columns include data relevant for nonperpendicular matching, about ample \(A \in N\) such that \(A^2\) is not too large compared with \(\Delta \). In view of Lemmas 6.4 and 7.1, nonperpendicular matchings of a pair of rank 2 Fanos \(Y_+, Y_\) are only possible if
The largest ratio \(\frac{\Delta }{A^2}\) occurs for #18, which has \(\Delta = 16\) and an ample A with \(A^2 = 10\). Accordingly, we list all ample classes with \(A^2 \le 1.6\Delta \). (In some examples there is more than one such class. We do not write out A itself in terms of the basis G, H, but there is never any ambiguity.) For each such A we list the following data.
The result of evaluating \(\pi _! c_2(Z)\) on A (equivalently: \(c_2(Z) \pi ^*A\)).
\(A^2\), the product of A with itself in the Picard lattice.
\(B^2\), where B is a generator for the orthogonal complement to A in the Picard lattice.
\(h := \log _2 \frac{\Delta }{A^2}\).
Annotated MoriMukai list of rank 2 Fano 3folds
We now indicate how the data in Table 3 is assembled. The anticanonical degree \(K_Y^3\) and \(b_3(Y)\) are taken from Iskovskih–Prokhorov [23, Table 12.3], and \(b_3(Z)\) is obtained from (4.1) as before. For the computation of the basis of the nef cone, the quadratic form on the Picard lattice N and \(\pi _! c_2(Z)\) we divide the list (except the last two entries) into three groups.
No fewer than 27 of the 36 classes of Fanos can be described explicitly as blowups in a smooth curve of a rank 1 Fano \(Y_0\) of index \(r \ge 2\) (so \({{\,\mathrm{Pic}\,}}Y_0\) is generated by \({}\frac{1}{r}K_{Y_0}\)). Then one edge of the nef cone of Y is clearly generated by \(H:= \pi ^*({}\frac{1}{r}K_{Y_0})\).
The hypothesis of the next lemma can be read as an elementary formulation of “C is cut out schemetheoretically by sections of L”, i.e. the tensor product \(\mathcal {I}_C \otimes \mathcal {L}\) of the ideal sheaf of C and the sheaf of sections of L being globally generated.
Lemma 4.5
Let L be a line bundle on a closed complex manifold \(Y_0\), and let \(\pi : Y \rightarrow Y_0\) be the blowup of a smooth curve C in \(Y_0\). Let \(E \subset Y\) be the exceptional divisor of \(\pi \).
Suppose that for every trivialising neighbourhood \(U \subset Y_0\) of L, the ideal \(\mathcal {I}_C(U) := \{ f : f_{C\cap U} \equiv 0\}\) in the ring \(\mathcal {O}_{Y_0}(U) := \{ \text {holomorphic functions } U \rightarrow \mathbb {C}\}\) is generated by coordinate representatives of global sections of L that vanish identically on C. Then \(\pi ^*L  E\) is basepointfree.
Proof
Note that if \(D \in L\) contains C, then \(\pi ^*D  E\) is effective (if D is smooth then this is simply the proper transform) and belongs to \(\pi ^*L  E\). Therefore if \(\pi ^*L  E\) has a base point \(p \in Y \setminus E\), then any global section of L vanishes at the corresponding point \(\pi (p) \in Y_0\). The hypothesis forces the contradiction \(\pi (p) \in C\).
Recall that E can be identified with \(\mathbb {P}(N_{C/Y_0})\), the projectivisation of the normal bundle of C. If \(\pi ^*L  E\) has a basepoint \(p \in E\), then that corresponds to a nonzero normal vector v to C at \(\pi (p) \in C\) such that v is tangent to every element of L that contains C. Then the derivative of every local defining function of C near \(\pi (p)\) vanishes on v, contradicting that C is smooth. \(\square \)
Returning to the setting of a blowup in a rank 1 Fano \(Y_0\), Lemma 4.5 implies that if C is cut out by sections of \({}\frac{n}{r}K_{Y_0}\) then \(G= \pi ^*({}\frac{n}{r}K_{Y_0})  E\) is nef, so the cone spanned by \(G\) and \(H\) is certainly contained in the nef cone of Y.
Lemma 4.6
If the blowup Y of \(Y_0\) in C is Fano, and n is the minimal integer such C is cut out by sections of \({}\frac{n}{r}K_{Y_0}\), then the nef cone of Y is exactly the cone spanned by \(G= \pi ^*({}\frac{n}{r}K_{Y_0})  E\) and \(H\).
Proof
This can be verified for each of the 27 rank 2 Fano 3folds of this type in the classification, aided e.g. by the descriptions of Coates–Corti–Galkin–Kasprzyk [8, §18–§53] of each Fano 3fold as the zero locus of a general section of a vector bundle (often a complete intersection) of a “key variety”, a GIT quotient of a vector space (often a toric variety) whose nef cone is evident. \(\square \)
Note that Y being Fano means that \(K_Y = G+ (rn)H\) is in the interior of the nef cone, corresponding to \(1 \le n \le r1\) in all cases. Indeed, according to [20, Corollary 7.1.2(iii)], the blowup of a rank 1 Fano 3fold \(Y_0\) in C is Fano if and only if \(1 \le n \le r1\); this too is regarded as a consequence of the classification, similarly to how we do not know any uniform proof of Lemma 4.6.
Staying in the context of Lemma 4.6, the matrix representing the quadratic form on N with respect to the basis \(G\), \(H\) can be computed from
We can apply (4.4) to read off \(\pi _!c_2(Z)(K_Y)\) from the other data in the table. To control the other half of \(\pi _! c_2(Z)\), we apply Lemma 4.4 to \(\pi _0 : Y \rightarrow Y_0\) to deduce that the product of \(c_2(Y) + c_1(Y)^2\) with \(\pi _0^*(K_{Y_0})\) equals the product of \(c_2(Y_0) + c_1(Y_0)^2\) with \(K_{Y_0}\). We can thus apply (4.4) again to obtain
(and the RHS is contained in Table 2).
Another four entries in the list are divisors in \(\mathbb {P}^2 \times \mathbb {P}^2\) (including \(\mathbb {P}^1 \times \mathbb {P}^2\)). Then we can take the basis \(G, H\) of N to correspond to the restrictions of the hyperplane bundles from the two factors. For a divisor \(Y \in aG+bH\), we can readily compute the quadratic form on N from \(K_Y = (3a)G+ (3b)H\). Further, \(c(Y) = (1+3G+ 3G^2)(1+3H+3H^2)(1+[Y])^{1}\) gives
To evaluate the product with \(G\) we compute
and the product with \(H\) is analogous.
Another 3 cases are branched double covers over other rank 2 Fanos, \(p: Y \rightarrow X\). Then we can take G and H to be pullbacks of the previously identified edges of the nef cone in \({{\,\mathrm{Pic}\,}}X\). If the branch locus is a smooth divisor in 2L, then \(TY \oplus p^*(2L) \cong p^*(TX \oplus L)\) implies \(c_1(Y) = p^*(c_1(X)  [L])\) and \(c_2(Y) = p^*(c_2(X) c_1(X)[L] + 2[L]^2)\), so
For each entry Y in the MoriMukai list, we repeat below the description from [23, Table 12.3]. For blowups of rank 1 Fanos in a smooth curve C we only indicate in addition the smallest integer n such that C is cut out by sections of \({}\frac{n}{r}K_{Y_0}\). For the remaining 9 cases we provide some additional explanation.
 #1
Blowup of \(V_1\) (degree 1 del Pezzo 3fold, degree 6 hypersurface in \(\mathbb {P}^4(3,2,1,1,1)\)) in an elliptic curve that is the intersection of two divisors in \({}{\textstyle \frac{1}{2}}K_{V_1}\) (i.e. the hyperplane class). \(n = 1\).
This is the only rank 2 Fano where the linear system \({}K_Y\) is not free; the base locus is the preimage \(\mathbb {P}^1\) over the basepoint of \({}{\textstyle \frac{1}{2}}K_{V_1}\). It is therefore the only case where Construction 4.2 does not produce an associated “Fanotype” building block Z (though one could define a building block by blowing up in several steps [9, Proposition 5.9]).
 #2
Double cover of \(\mathbb {P}^1 \times \mathbb {P}^2\) branched over (2, 4) divisor. \(G\) and \(H\) are the pullbacks of \(\mathcal {O}_{\mathbb {P}^1}(1)\) and \(\mathcal {O}_{\mathbb {P}^2}(1)\) respectively. Use (4.8) (and result from #34) to compute
$$\begin{aligned} c_2(Y) + c_1(Y)^2 = 18GH+ 12H^2 +3(G+ 2H)(GH) = 9GH+ 6H^2 , \end{aligned}$$and hence \(\pi _! c_2(Z) = \left( {\begin{matrix} \!12&18\! \end{matrix}} \right) \).
 #3
Blowup of \(V_2\) (degree 2 del Pezzo 3fold, a double cover of \(\mathbb {P}^3\) branched over a quartic hypersurface, or equivalently a degree 4 hypersurface in \(\mathbb {P}^3(2, 1, 1, 1)\)) in an elliptic curve that is the intersection of two hyperplane divisors. \(n = 1\).
 #4
Blowup of \(\mathbb {P}^3\) along the intersection of two cubic hypersurfaces. \(n = 3\).
 #5
Blowup of \(V_3\) (cubic hypersurface of \(\mathbb {P}^4\)) along the intersection of two hyperplane divisors (a plane cubic curve). \(n = 1\).
 #6
Generic members of the family are (2,2) divisor in \(\mathbb {P}^2 \times \mathbb {P}^2\), but some are double cover of a (1, 1) divisor \(W \subset \mathbb {P}^2 \times \mathbb {P}^2\) branched over smooth divisor \(B \in {}K_W\). \(G\) and \(H\) are pullbacks of \(\mathcal {O}(1)\) from the two \(\mathbb {P}^2\) factors, and \(\pi _! c_2(Z)\) can be computed from (4.7) (or from (4.8) and the result in #32).
 #7
Blowup of Q (quadric hypersurface in \(\mathbb {P}^4\)) in intersection of two sections by quadrics. \(n = 2\).
 #8
Double cover of \(V_7\) (\(\mathbb {P}^3\) blown up in a point, #35) whose branch locus is a divisor \(B \in {}K_{V_7}\) (for generic members of the intersection \(B \cap E\) with the exceptional divisor of \(V_7 \rightarrow \mathbb {P}^3\) is smooth, but for some members of the family B is reduced but not smooth). \(G\) and \(H\) are the pullbacks of the respective classes on \(V_7\).
By (4.8) we have \(c_2(Y) + c_1(Y)^2 = p^*\left( c_2(V_7) + c_1(V_7)^2  3 ({\textstyle \frac{1}{2}}K_{V_7})^2)\right) \). We compute in #35 that \(c_2(V_7) + c_1(V_7)^2 = \left( {\begin{matrix} \!18&22\! \end{matrix}} \right) \) in terms of the basis for \(V_7\), and we can read off from the Picard lattice that \(({\textstyle \frac{1}{2}}K_{V_7})^2 = \left( {\begin{matrix} \!3&4\! \end{matrix}} \right) \). Hence \(c_2(Y) + c_1(Y)^2 = p^*\!\left( {\begin{matrix} \!9&10\! \end{matrix}} \right) \), which in terms of the basis for \({{\,\mathrm{Pic}\,}}Y\) is written as \(\left( {\begin{matrix} \!18&20\! \end{matrix}} \right) \).
 #9
Blowup of \(\mathbb {P}^3\) in a curve C of degree 7 and genus 5, which is an intersection of a twoparameter family of cubic hypersurfaces. \(n = 3\).
 #10
Blowup of \(V_4\) (complete intersection of two quadrics in \(\mathbb {P}^5\)) in an elliptic curve that is the intersection of two hyperplane sections. \(n = 1\).
 #11
Blowup of \(V_3\) (cubic hypersurface in \(\mathbb {P}^4\)) along a line. \(n = 1\).
 #12
Blowup of \(\mathbb {P}^3\) along a curve of degree 6 and genus 3 which is an intersection of cubic hypersurfaces. \(n = 3\).
 #13
Blowup of Q (quadric hypersurface in \(\mathbb {P}^4\)) along a curve of degree 6 and genus 2. \(n = 2\).
 #14
Blowup of \(V_5\) (section of Plückerembedded Grassmannian \(Gr(2,5) \subset \mathbb {P}^9\) by a subspace of codimension 3) in an elliptic curve that is the intersection of two hyperplane sections. \(n = 1\).
 #15
Blowup of \(\mathbb {P}^3\) along the intersection of a quadric A and a cubic B (for generic members of the family A is smooth, but for some members of the family A is reduced but not smooth). \(n = 3\).
 #16
Blowup of \(V_4\) (complete intersection of two quadrics in \(\mathbb {P}^5\)) in a conic. \(n = 1\).
 #17
Blowup of Q along an elliptic curve of degree 5. \(n = 2\).
 #18
Double cover of \(\mathbb {P}^1 \times \mathbb {P}^2\) branched over (2, 2) divisor. Compute \(\pi _! c_2(Z) = \left( {\begin{matrix} \!12&18\! \end{matrix}} \right) \) using (4.8) like in #2.
 #19
Blowup of \(V_4\) along a line. \(n = 1\).
 #20
Blowup of \(V_5\) along a twisted cubic. \(n = 1\).
 #21
Blowup of Q along a twisted quartic (a rational degree 4 curve, isomorphic to the image of \((s : t) \mapsto (s^4 : s^3t : s^2t^2 : st^3 : t^4)\)). \(n = 2\).
 #22
Blowup of \(V_5\) along a conic. \(n = 1\).
 #23
Blowup of Q along an intersection of two divisors \(A \in \mathcal {O}_Q(1)\) and \(B \in \mathcal {O}_Q(2)\) (A may be smooth or singular). \(n = 2\).
 #24
A (1, 2) divisor in \(\mathbb {P}^2 \times \mathbb {P}^2\). Compute \(\pi _! c_2(Z)\) by applying (4.7) with \(a = 1, b = 2\).
 #25
Blowup of \(\mathbb {P}^3\) along an elliptic curve that is the intersection of two quadrics. \(n = 2\).
 #26
Blowup of \(V_5\) along a line. \(n = 1\).
 #27
Blowup of \(\mathbb {P}^3\) along a twisted cubic. \(n = 2\).
 #28
Blowup of \(\mathbb {P}^3\) along a plane cubic (an elliptic curve). \(n = 3\).
 #29
Blowup of Q along a conic (complete intersection of two hyperplane sections). \(n = 1\).
 #30
Blowup of \(\mathbb {P}^3\) along a conic. \(n = 2\).
 #31
Blowup of Q along a line. \(n = 1\).
 #32
A (1, 1)divisor on \(\mathbb {P}^2 \times \mathbb {P}^2\). Compute \(\pi _! c_2(Z)\) by applying (4.7) with \(a = b = 1\).
 #33
Blowup of \(\mathbb {P}^3\) along a line. \(n = 1\).
 #34
\(Y = \mathbb {P}^1 \times \mathbb {P}^2\). \(G= \mathcal {O}_{\mathbb {P}^1}(1)\) and \(H= \mathcal {O}_{\mathbb {P}^2}(1)\). Compute \(\pi _! c_2(Z) = \left( {\begin{matrix} \!12&18\! \end{matrix}} \right) \) by applying (4.7) with \(a = 1\), \(b = 0\).
 #35
\(\mathbb {P}^3\) blown up in one point, or equivalently \(\mathbb {P}(\mathcal {O}\oplus \mathcal {O}(1))\) over \(\mathbb {P}^2\). \(G\) is the proper transform of a plane passing through the blowup point, and \(H\) is a plane not passing through the blowup point. The product of \(c_2(Y) + c_1(Y)^2\) with \(H\) is 22, just as it is for \(\mathbb {P}^3\).
 #36
\(\mathbb {P}(\mathcal {O}\oplus \mathcal {O}(2))\) over \(\mathbb {P}^2\). \(G\) is the pullback of \(\mathcal {O}_{\mathbb {P}^2}(1)\), while \(H\) is the dual of the tautological bundle. The intersection form on \(H^2(Y)\) is given by \(G^3 = 0\), \(G^2H= 1\), \(GH^2 = 2\), \(H^3 = 4\) (use that the section \(\mathbb {P}(\mathcal {O}(2))\) is a divisor representing \(H\)). Because TY is stably isomorphic to \(f^*T\mathbb {P}^2 \oplus \left( H\otimes f^*(\mathcal {O}\oplus \mathcal {O}(2))\right) \), where \(f : Y \rightarrow \mathbb {P}^2\) is the fibration, we find that
$$\begin{aligned} c(Y) = (1 + 3G+ 3G^2)(1+H)(12G+H), \end{aligned}$$so \(c_1(Y) = G+ 2H\) and \(c_2(Y) = 3G^2 + 4GH+T^2\). Hence \(c_2(Y) + c_1(Y)^2 = 2G^2 +8GH+5H^2\), which has product 18 with \(G\) and 34 with \(H\). Thus \(\pi _! c_2(Z) = \left( {\begin{matrix} \!18&34\! \end{matrix}} \right) \).
Remark 4.7
In each case we have described a basis for the Picard lattice N, which is tantamount to specifying an Nmarking in the sense of Definition 5.6. On an elementary level, we could therefore interpret each entry in the list as defining a set \(\mathcal {Y}\) of Nmarked Fano 3folds.
The matching problem
Combining the results described in Sects. 2.3 and 4.1, we can produce twisted connected sum \(G_{2}\)manifolds from matching pairs of Fano 3folds. We will apply the methods developed in [10] to the problem of finding matchings between Fanos of rank 1 and 2. In this section, we summarise the results from [10, §6] on finding matchings with a prescribed “configuration” of the Picard lattices of a pair of Fano 3folds, reducing the problem to a combination of problems in lattice arithmetic and deformation theory. The main result here—Proposition 5.8—improves on [10, Proposition 6.18] to deal more clearly with skew configurations.
Configurations and matching
Let \(\Sigma _\pm \subset Y_\pm \) be smooth anticanonical divisors of a pair of Fanos, and a matching. Let \(h_+\) be a marking of \(\Sigma _+\), i.e. an isometry \(h_+ : H^2(\Sigma _+) \rightarrow L\) where L is a copy of the K3 lattice (the unique unimodular lattice of signature (3,19)). Then is a marking of \(\Sigma _\). The images of \(H^2(Y_\pm ) \subset H^2(\Sigma _\pm )\) under \(h_\pm \) are a pair of primitive sublattices \(N_\pm \subset L\), isometric to the Picard lattices. This pair is welldefined up to the action of the isometry group O(L), and plays a crucial role.
Definition 5.1
Given a pair of Fanos with Picard lattices \(N_+\) and \(N_\), call a pair of primitive embeddings \(N_+, N_ \hookrightarrow L\) a configuration. Two such pairs of embeddings are considered equivalent if they are related by the action of O(L).
We call a configuration orthogonal if the reflections of \(L(\mathbb {R})\) in \(N_+\) and \(N_\) commute. If in addition \(N_+ \cap N_\) is trivial then we call the configuration perpendicular. If the configuration is not orthogonal then we call it skew.
We saw in Sect. 2.4 that the homeomorphism invariants of the twisted connected sum M resulting from the matching depend on the configuration (e.g. \(H^2(M) = N_+ \cap N_\)), and in Theorem 3.6 that the generalised Eells–Kuiper invariant \(\mu (M)\) does too. We therefore ask:
Given a pair\(\mathcal {Y}_+\), \(\mathcal {Y}_\)of deformation types of Fano 3folds, which configurations of embeddings\(N_\pm \subset L\)of their Picard lattices arise from some matching of elements of\(\mathcal {Y}_+\)and \(\mathcal {Y}_\)?
We see below that it is not too hard to answer this when one of the types has Picard rank 1, and we will be able to say quite a lot when both types have Picard rank 2. In general the question is quite difficult, but in any case a first step in simplifying it is to rephrase it as a problem of finding suitable triples of classes in \(L(\mathbb {R}) := L \otimes \mathbb {R}\). Recall that the period of a marked K3 surface \((\Sigma , h)\) is an oriented twoplane \(\Pi \subset L(\mathbb {R})\), the image under \(h : H^2(\Sigma ; \mathbb {R}) \rightarrow L(\mathbb {R})\) of the real and imaginary parts of classes in \(H^{2,0}(\Sigma ; \mathbb {C})\).
Lemma 5.2
Let \(Y_\pm \) be a pair of Fano 3folds, and let \(N_\pm \subset L\) be the images of primitive isometric embeddings of the respective Picard lattices. Then the pair \((N_+, N_)\) is the configuration of some matching of \(Y_+\) and \(Y_\) if and only if there exist
an orthonormal triple \((k_+, k_, k_0)\) of positive classes in \(L(\mathbb {R})\),
anticanonical divisors \(\Sigma _\pm \subset Y_\pm \),
markings \(h_\pm \) of \(\Sigma _\pm \),
such that the oriented plane \(\langle k_\mp , \pm k_0\rangle \) is the period of \((\Sigma _\pm , h_\pm )\), \(h^{1}_\pm (k_\pm )\) is the restriction of a Kähler class on \(Y_\pm \), and \(N_\pm \) is the image of the composition \(H^2(Y_\pm ) \rightarrow H^2(\Sigma _\pm ) \rightarrow L\).
Proof
Necessity is trivial, setting for the Kähler classes that appear in Definitions 2.5, and \(k_0\) corresponding to a generator of , all normalised to unit length. Sufficiency relies on the Torelli theorem, cf. [10, Proposition 6.2]. \(\square \)
To study how the matching problem depends on the choice of configuration, let us first set up some notation for various lattices.
\(W := N_+ + N_\) (this need not be primitive in L),
\(T_\pm \subset L\) the perpendicular of \(N_\pm \),
\(T := T_+ \cap T_\), or equivalently the perpendicular of W,
\(W_\pm := T_\mp \cap N_\pm \), and
\(\Lambda _\pm \subset L\) the perpendicular to \(T \oplus W_\mp \), or equivalently the perpendicular to \(W_\mp \) in the primitive overlattice of W.
Remark 5.3
\(N_\pm \subseteq \Lambda _\pm \), with equality if and only if \(N_+\) and \(N_\) “intersect orthogonally”, i.e. when \(W(\mathbb {R}) = W_+(\mathbb {R}) \oplus W_(\mathbb {R}) \oplus (N_+(\mathbb {R}) \cap N_(\mathbb {R}))\); equivalently the configuration is orthogonal in the sense of Definition 5.1.
Necessary conditions
Note that in Lemma 5.2 we must obviously have \(k_\pm \in N_\pm (\mathbb {R})\). On the other hand, \(N_\pm \) is contained in the Picard group of the marked K3 \((\Sigma _\pm , h_\pm )\), which is the subgroup of L orthogonal to the period; the marked K3 is automatically \(N_\pm \)polarised. Thus \(k_\mp \) and \(k_0\) must both lie in \(T_\pm (\mathbb {R})\). Hence
Now we come to the heart of how the difficulty of the matching problem depends on the configuration one tries to achieve: (5.1) implies that the period \(\langle k_\mp , \pm k_0\rangle \) is orthogonal to all of \(\Lambda _\pm \), so the marked K3 divisors used in a matching with the given configuration are forced to be\(\Lambda _\pm \)polarised.
The significance is that the Picard group of a generic K3 divisor in a generic member of a deformation type \(\mathcal {Y}_\pm \) of Fano 3folds will be precisely the Picard lattice \(N_\pm \) of that type. To find matchings for a configuration where \(\Lambda _\pm \) is strictly bigger than \(N_\pm \), we therefore require nongeneric K3 divisors in members of \(\mathcal {Y}_\pm \) (the moduli space of \(\Lambda _\pm \)polarised marked K3 surfaces forms a subspace of the \(N_\pm \)polarised K3s, whose codimension is \({{\,\mathrm{rk}\,}}\Lambda _\pm  {{\,\mathrm{rk}\,}}N_\pm \)).
For configurations where \(\Lambda _\pm = N_\pm \), we deduce in Sect. 6 the existence of matchings between some elements of \(\mathcal {Y}_+\) and \(\mathcal {Y}_\) from a general fact (due to Beauville [2]) that a generic \(N_\pm \)polarised K3 appears as an anticanonical divisor in some member of \(\mathcal {Y}_\pm \). In view of Remark 5.3, this comparatively easy case corresponds to orthogonal configurations. To apply a similar argument for skew configurations (where \(\Lambda _\pm \supset N_\pm \)), we first need to show for those specific \(\Lambda _\pm \) that generic \(\Lambda _\pm \)polarised K3s appear as anticanonical divisors in members of \(\mathcal {Y}_\pm \). Even when it is true, it is something that we can so far only verify case by case. We refer to this process as ‘handcrafting’.
Remark 5.4
Before moving on to existence results for matchings with a prescribed configuration, let us point out some necessary conditions.
 (i)
Since \(W(\mathbb {R})\) contains a twodimensional positivedefinite subspace (spanned by the orthogonal classes \(k_+\) and \(k_\)), while its orthogonal complement in the signature (3, 19) lattice L contains a class \(k_0\) with \(k_0^2 > 0\), the quadratic form on W must be nondegenerate of signature \({(2, {{\,\mathrm{rk}\,}}(W)2)}\).
 (ii)
\(W_\pm \subset N_\pm \) must contain some ample classes of \(Y_\pm \).
 (iii)
Since \(\Lambda _+ \cap \Lambda _ \subset {{\,\mathrm{Pic}\,}}\Sigma _\pm \) and is orthogonal to an ample class of \(\Sigma _\pm \), it cannot contain any \((2)\)classes.
Remark 5.5
In particular, (ii) implies that any matching involving a Fano with Picard rank \({{\,\mathrm{rk}\,}}N= 1\) must be perpendicular. Moreover, for a configuration of lattices \(N_+\) and \(N_\) where at least one has rank 2, if the intersection \(N_+ \cap N_\) is nontrivial then (ii) forces the configuration to be orthogonal in the sense of Definition 5.1. For configurations of Picard lattices of Fanos of rank \(\le 2\) that satisfy the necessary conditions to be realised by a matching, we therefore have the following trichotomy:
Perpendicular configurations, i.e. every element of \(N_+\) is orthogonal to every element of \(N_\).
Orthogonal configurations with nontrivial intersection. Then \(N_+ \cap N_\) must have rank 1.
Skew configurations. Then \(N_+ \cap N_\) must be trivial, but \(N_+\) is not perpendicular to \(N_\) (the maps \(N_\pm \rightarrow N_\mp ^*\) must have rank 1).
In Sects. 6 and 7, we will consider these cases in turn.
Sufficient conditions
In order to describe the ‘genericity properties’ we require for anticanonical K3 divisors in families of Fano 3folds, we recall some further terminology. The period domain is the space of oriented positivedefinite 2planes in \(L(\mathbb {R})\). It can be identified with \({\{ \Pi \in \mathbb {P}(L(\mathbb {C})) : \Pi ^2 = 0, \; \Pi \, {\overline{\Pi }} > 0\}}\) in order to exhibit a natural complex structure. Given \(\Lambda \subset L\), the period domain of \(\Lambda \)polarised K3 surfaces is \(D_\Lambda := \{ \Pi \in \mathbb {P}(\Lambda ^\perp (\mathbb {C})) : \Pi ^2 = 0,\; \Pi \, {\overline{\Pi }} > 0\}\).
Definition 5.6
Given a nondegenerate lattice N, an Nmarking of a closed 3fold Y is a surjective homomorphism \(i_Y : H^2(Y) \rightarrow N\) that is isometric for the anticanonical form of Definition 4.1.
We avoid calling \(i_Y\) an “Npolarisation” since we do not impose any conditions on ample classes. If Y is Fano then the Picard lattice is nondegenerate so \(i_Y\) is simply an isometry.
Definition 5.7
Let \(N \subseteq \Lambda \subset L\) be primitive nondegenerate sublattices of L, and let \({{\,\mathrm{Amp}\,}}_\mathcal {Y}\) be a nonempty open subcone of the positive cone in \(N(\mathbb {R})\). We say that a set \(\mathcal {Y}\) of Nmarked 3folds is \((\Lambda , {{\,\mathrm{Amp}\,}}_\mathcal {Y})\)generic if there is \(U_\mathcal {Y}\subseteq D_\Lambda \) with complement a countable union of complex analytic submanifolds of positive codimension with the property that: for any \(\Pi \in U_\mathcal {Y}\) and \(k \in {{\,\mathrm{Amp}\,}}_\mathcal {Y}\) there is \(Y \in \mathcal {Y}\), a smooth anticanonical divisor \(\Sigma \subset Y\) and a marking \(h: H^2(\Sigma ) \rightarrow L\) such that \(\Pi \) is the period of \((\Sigma , h)\), the composition \(H^2(Y) \rightarrow H^2(\Sigma ) \rightarrow L\) equals the marking \(i_Y\), and \(h^{1}(k)\) is the image of the restriction to \(\Sigma \) of a Kähler class on Y.
To be able to prove that a set \(\mathcal {Y}\) of Fano 3folds satisfies the definition we typically take \(\mathcal {Y}\) to be a deformation type, but to make sense of the definition we do not need to remember any additional structure on \(\mathcal {Y}\) (cf. Remark 4.7).
Meanwhile, when applying the next proposition we typically want all elements of the sets \(\mathcal {Y}_\pm \) to be Fano 3folds (or building blocks) that are topologically the same, so that we have some control over the topology of the \(G_{2}\)manifolds resulting from the matchings produced; essentially this means that all elements of \(\mathcal {Y}_\pm \) should belong to the same deformation type.
Proposition 5.8
Consider a configuration of primitive nondegenerate sublattices \(N_+, N_ \subset L\), and let \(\mathcal {Y}_\pm \) be a pair of sets of \(N_\pm \)marked 3folds. Define W, \(W_\pm \) and \(\Lambda _\pm \) as above. Suppose that there exist nonempty open cones \({{\,\mathrm{Amp}\,}}_{\mathcal {Y}_\pm } \subseteq N_\pm (\mathbb {R})^+\) such that
 (i)
the sets \(\mathcal {Y}_\pm \) are \((\Lambda _\pm , {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_\pm })\)generic,
 (ii)
\(W_\pm \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_\pm } \ne \emptyset \).
Then there is an open dense subcone \(\mathcal {W} \subseteq (W_+(\mathbb {R}) \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_+}) \times (W_(\mathbb {R}) \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_})\) such that for every \((k_+, k_) \in \mathcal {W}\) with \(k_+^2 = k_^2\) there exist \(Y_\pm \in \mathcal {Y}_\pm \), anticanonical K3 divisors \(\Sigma _\pm \subset Y_\pm \) and Kähler classes such that , with a matching of and whose configuration is the given pair of embeddings \(N_\pm \subset L\).
Proof
The argument is essentially the same as [10, Proposition 6.18], even though the conclusion stated here is slightly stronger.
Let \(T = W^\perp \) as before. Denote the ranks of W and \(W_\pm \) by r and \(r_\pm \). Then \(W_\pm (\mathbb {R})\) and \(T(\mathbb {R})\) are real vector spaces of signature \((1, \, r_\pm  1)\) and \((1, \, 21 r)\) respectively
In view of Lemma 5.2 and (5.1), matchings correspond to certain triples of classes \((k_+, k_, k_0)\) such that \(k_\pm \) and \(k_0\) belong to the positive cones \(W_\pm (\mathbb {R})^+\) and \(T(\mathbb {R})^+\) respectively. Consider therefore the real manifold
Below, we need the open subset \(\mathcal {A}= \mathcal {A}_+ \times \mathcal {A}_ \times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr )\), where \(\mathcal {A}_\pm := \mathbb {P}(W_\pm (\mathbb {R}) \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_\pm })\) is nonempty by hypothesis (ii). We have two Griffiths period domains
and projections
As stated before Definition 5.7, \(D_{\Lambda _\pm }\) can be regarded as an open subset of \(\mathbb {P}(C_\pm )\), where \(C_\pm \) is the null cone in \(\Lambda _\pm ^\perp \otimes \mathbb {C}\); if \(\alpha , \beta \) is an oriented orthonormal basis of \(\Pi \in D_{\Lambda _\pm }\) then \(\Pi \mapsto \langle \alpha +i\beta \rangle \in \mathbb {P}(C_\pm )\). Given a choice \(\alpha \) and \(\beta \), we can identify \(T_\Pi D_{\Lambda _\pm }\) with pairs (u, v) of vectors in the orthogonal complement of \(\Pi \) in \(\Lambda _\pm ^\perp (\mathbb {R})\). Then the complex structure on \(T_\Pi D_{N_\pm }\) is given by \(J : (u, v) \mapsto (v, u)\).
Observe that the real analytic embedded submanifold \(\mathbb {P}\bigl (W_\mp (\mathbb {R})^+\bigr )\times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr ) \hookrightarrow D_{\Lambda _\pm }\) is totally real: for \(w \in W_\mp \) and \(t \in T(\mathbb {R})\), the tangent space \(\mathcal {T}\) to \(\mathbb {P}\bigl (W_\mp (\mathbb {R})^+\bigr )\times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr )\) at \(\Pi = \langle w,t\rangle \) corresponds to (u, v) such that \(u \in w^\perp \subseteq W_\mp (\mathbb {R})\) and \(v \in t^\perp \subseteq T(\mathbb {R})\), so \(J(\mathcal {T})\) is transverse to \(\mathcal {T}\).
Crucially, this totally real submanifold has maximal dimension:
Consequently, its intersection with any positivecodimensional complex analytic submanifold of \(D_{\Lambda _\pm }\) is a positivecodimensional real analytic submanifold of \(\mathbb {P}\bigl (W_\mp (\mathbb {R})^+\bigr )\times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr )\). Hence the preimage in \(\mathbb {P}\bigl (W_\mp (\mathbb {R})^+\bigr )\times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr )\) of the subset \(U_{\mathcal {Z}_\pm } \subset D_{\Lambda _\pm }\) from Definition 5.7 is open and dense. Because \(\text {pr}_\pm \) is a projection of a product manifold onto a factor the same is true for \(\text {pr}_\pm ^{1}(U_{\mathcal {Z}_\pm }) \subset D\). In turn,
is open and dense in \(\mathcal {A}_+ \times \mathcal {A}_ \times \mathbb {P}(T(\mathbb {R})^+)\), and hence the image \(\mathcal {W}'\) of this subset under projection to \(\mathcal {A}_+ \times \mathcal {A}_\) is open and dense in \(\mathcal {A}_+ \times \mathcal {A}_\).
If we let \(\mathcal {W} = \{(k_+, k_) \in (W_+(\mathbb {R}) \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_+}) \times (W_(\mathbb {R}) \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_}) : ([k_+], [k_]) \in \mathcal {W}'\}\), then for every \((k_+, k_) \in \mathcal {W}\) such that \(k_+^2 = k_^2\) there is a \(k_0 \in T(\mathbb {R})^+\) such that Lemma 5.2 applies to \((k_+, k_, k_0)\). \(\square \)
Orthogonal matching
We now consider the problem of finding matchings of Fano 3folds of Picard rank 1 or 2, with prescribed configuration that is orthogonal in the sense of Definition 5.1. As pointed out in Remark 5.3, this corresponds to the Picard lattices \(N_\pm \) being equal to the lattices \(\Lambda _\pm \) that are used in the hypothesis of Proposition 5.8. Therefore the following genericity result is enough to let us apply Proposition 5.8 for these configurations.
Proposition 6.1
( [9, Proposition 6.9], based on Beauville [2]) Let \(\mathcal {Y}\) be a deformation type of Fano 3folds, and embed its Picard lattice N primitively in L. Then there exists an open cone \({{\,\mathrm{Amp}\,}}_\mathcal {Y}\subset N(\mathbb {R})\) such that \(\mathcal {Y}\) is \((N, {{\,\mathrm{Amp}\,}}_\mathcal {Y})\)generic.
In the trichotomy of Remark 5.5, orthogonal configurations encompass the cases of perpendicular configurations and orthogonal configurations with nontrivial intersection. Now we study in turn the twisted connected sum \(G_{2}\)manifolds that result from matchings of these kinds, using Fano 3folds of Picard rank 1 or 2.
Perpendicular matching
The simplest way to find a matching between elements of two deformation types \(\mathcal {Y}_\pm \) of Fano 3folds is to consider perpendicular configurations, i.e. where the images of \(N_+\) and \(N_\) in L are perpendicular to each other. One reason is that we do not need any genericity results beyond Proposition 6.1, but a further reason is arithmetic: for nearly all pairs \(\mathcal {Y}_\pm \) one has \({{\,\mathrm{rk}\,}}N_+ + {{\,\mathrm{rk}\,}}N_ \le 11\), in which case Nikulin [34, Theorem 1.12.4] guarantees that there does in fact exist a primitive embedding of the perpendicular direct sum \(N_+ \perp N_\) into L.
In particular, we can apply this to find perpendicular matchings among the 53 types of Fanos of Picard rank 1 and 2. However, we ignore #1 in the list of rank 2 Fanos, since that does not have an associated Fanotype building block. Given one of the 1378 unordered pairs \(\mathcal {Y}_+\), \(\mathcal {Y}_\) among the other 52 deformation types, we carry out the following procedure:
Apply [34, Theorem 1.12.4] to find a primitive embedding into L of the perpendicular direct sum \(N_+ \perp N_\).
Apply Proposition 5.8 to find a matching between some \(Y_\pm \in \mathcal {Y}_\pm \) with the given configuration \(N_+, N_ \subset L\).
Apply Proposition 4.3 to produce a pair of Fanotype building blocks with a perpendicular matching.
Apply Theorem 2.7 to construct a twisted connected sum \(G_{2}\)manifold M.
Now Theorem 2.8 shows that M is 2connected, with \(H^4(M)\) torsionfree, and
Proposition 2.11 implies that d(M), the greatest divisor of \(p_M\), equals the greatest common divisor of \(\pi _! c_2(Z_+)\) and \(\pi _! c_2(Z_)\), while Corollary 3.7 shows that \(\mu (M) = 0\). Thus all the classifying diffeomorphism invariants of M can be determined from the data in Tables 2 and 3 .
Table 4 lists the invariants of the 1378 twisted connected sums obtained this way. A total of 131 different 2connected manifolds are realised, with 60 different values of \(b_3(M)\). For comparison, twisted connected sums involving only rank 1 Fanos realise 82 different manifolds, and 46 different values of \(b_3(M)\) [10, Table 3].
Remark 6.2
According to Theorem 1.7 and Corollary 1.13 of [12], the torsionfree \(G_{2}\)structures of diffeomorphic 2connected twisted connected sums with d not divisible by 3 are automatically homotopic (if one chooses the diffeomorphism correctly). While Table 4 shows that there are also numerous instances of diffeomorphic twisted connected sums with \(d = 6\) (e.g. \((b_3, d) = (103,6)\) is realised by 7 different twisted connected sums), Wallis [40, Corollary 6.5.2] builds on the methods of this paper to compute the \(G_{2}\)structures’ \(\xi \)invariant introduced in [12, Definition 6.8], showing that the \(G_{2}\)structures are homotopic for these examples too. However, [40, Table 8.1] exhibits other examples (involving skew matchings) where the \(G_{2}\)structures of twisted connected sums can be distinguished by the \(\xi \)invariant.
Remark 6.3
Recall from Wall’s classification of closed simplyconnected spin 6manifolds with torsionfree cohomology [39, Theorem 5] that for d divisible by 12, there is such a 6manifold X with the cohomology ring of \(S^2 \times S^4\) and greatest divisor of \(p_X\) equal to d.
The total space M of the circle bundle over \(X \# (S^3 \times S^3)^{\# k}\) with Euler class equal to the generator of \(H^2\) is then a closed 2connected 7manifold with \(b_3(M) = 2k+1\), torsionfree \(H^4(M)\) and \(d(M) = d\) (and this is the only way to obtain such total spaces). One can use the corresponding closed disc bundle as a coboundary for M to compute that \(\mu (M) = 0\). (The more general problem of deciding which 2connected 7manifolds can be realised as the total spaces of circle bundles is analysed by Jiang [24].)
Thus the classification Theorem 2.2 implies that the 6 entries in Table 4 that have \(d = 12\) or 24 are diffeomorphic to total spaces of circle bundles (in contrast to the result of Baraglia [1, Proposition 6.2.1] that closed \(G_{2}\)manifolds cannot be smoothly fibred by 4manifolds), while the last entry of Table 1 is a topological \(S^1\)bundle but not a smooth one.
Orthogonal matching with nontrivial intersection
Next we consider matchings with configurations such that \(N_+ \cap N_\) is nontrivial. Then both Fanos must have Picard rank \(\ge 2\). If we restrict attention to the case when both Fanos have Picard rank precisely 2, then as pointed out in Remark 5.5 the only configurations with \(N_+ \cap N_\) nontrivial for which we can possibly find a matching are the ones that are orthogonal, in the sense of Definition 5.1. Such configurations have \(\Lambda _\pm = N_\pm \), so to apply Proposition 5.8 we essentially do not need any genericity result beyond Proposition 6.1 that we applied to find perpendicular matchings—the only extra data we need is to actually determine the cone \({{\,\mathrm{Amp}\,}}_\mathcal {Y}\), but for rank 2 Fano 3folds we have done that in Sect. 4.4.
Compared with the perpendicular matching problem, the difficulty of finding matchings of rank 2 blocks with nontrivial intersection \(N_+ \cap N_\) is therefore one of latticearithmetic: there must exist some integral lattice W of rank 3, containing \(N_+\) and \(N_\), such that the orthogonal complement of \(W_\pm \subset N_\pm \) of \(N_\mp \) is nontrivial, and contains a class \(A_\pm \in {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_\pm }\).
Lemma 6.4
Let \(N_\pm \) be integral lattices of rank 2 with signature (1, 1), and let \(A_\pm \in N_\pm \). Let \(\Delta _\pm \) be the discriminant of \(N_\pm \), and let \(B_\pm \) be a generator of the orthogonal complement. Then there exists a rank 3 integral lattice W with primitive isometric embeddings \(N_\pm \hookrightarrow W\) such that \(A_\pm \perp N_\mp \) if and only if \(B_+^2 = B_^2\) and \(\Delta _+\Delta _ = k^2 A_+^2A_^2\) for some integer k. Then k is the generator of the image of the product \(N_+ \times N_ \rightarrow \mathbb {Z}\) in W.
Proof
If such a W exists, then the images of \(B_+\) and \(B_\) are both primitive vectors, and both perpendicular to both \(A_+\) and \(A_\). Thus, up to sign, \(B_+\) and \(B_\) must have the same image \(B \in W\), and hence \(B_+^2 = B^2 = B_^2\).
Note that for a pair \((v_+, v_) \in N_+ \times N_\), the product of their images in W is
In particular, if we let \(a_\pm \) be the positive generator of the image of \(N_\pm \rightarrow \mathbb {Z}, \; v \mapsto v.B_\pm \), then \(a_+a_ = kB^2\). Conversely if \(a_+a_\) is divisible by \(B^2\), then we can use (6.1) (together with the given forms on \(N_+\) and \(N_\)) to define the desired integral quadratic form on \(W := (N_+ \oplus N_)/\langle B_+  B_\rangle \).
Now observe that the index of the sublattice \(\langle A_+, B_+\rangle \subseteq N_\pm \) is \(\frac{B^2}{a_\pm }\). Letting \(\Delta _\pm \) be the discriminant of \(N_\pm \), we must therefore have \(A_+^2 B^2 = (\Delta _\pm )\left( \frac{B^2}{a_\pm }\right) ^2\). Hence
and \(a_+a_\) is divisible by \(B^2\) if and only if \(\frac{\Delta _+\Delta _}{A_+^2 A_^2}\) is a perfect square. \(\square \)
Next we summarise the relevant topological calculations.
Lemma 6.5
Let \((Z_\pm , \Sigma _\pm )\) be a pair of building blocks whose polarising lattices \(N_\pm \) have rank 2, such that the kernel of \(H^2(Z_\pm ) \rightarrow H^2(\Sigma _\pm )\) is generated by \([\Sigma _\pm ]\). Let be a matching whose configuration \(N_+, N_ \subset L\) has \(N_+ \cap N_\) of rank 1, and let M be the resulting twisted connected sum.
Let \(W := N_+ + N_\), and let \(A_\pm \in N_\pm \) be a primitive vector spanning the orthogonal complement of \(N_\mp \) in \(N_\pm \). Then
 (i)
\(H^2(M) \cong \mathbb {Z}\);
 (ii)
\(b_3(M) = b_3(Z_+) + b_3(Z_) + 22\);
 (iii)
\({{\,\mathrm{Tor}\,}}H^3(M) \cong {{\,\mathrm{Tor}\,}}L/W\);
 (iv)
\({{\,\mathrm{Tor}\,}}H^4(M) \cong (\mathbb {Z}/k)^2\), for k as in Lemma 6.4;
 (v)
if \(k = 1\) then d(M) (which divides 24) is the greatest common divisor of \({\bar{c}}_2(Z_+)A_+\) and \({\bar{c}}_2(Z_)A_ \in \mathbb {Z}/24\), for \({\bar{c}}_2(Z_\pm )\) as in (4.3);
 (vi)
if \(k = 1\) and d(M) is divisible by 8 (so that the Eells–Kuiper invariant \(\mu (M)\) takes values in \(\mathbb {Z}/2\)) then
$$\begin{aligned} \mu (M) = \frac{{{\,\mathrm{gd}\,}}({\bar{c}}_2(Z_+)){{\,\mathrm{gd}\,}}({\bar{c}}_2(Z_))}{4} \in \mathbb {Z}/2 , \end{aligned}$$where \({{\,\mathrm{gd}\,}}({\bar{c}}_2(Z_\pm )) \in \{2,4,6,8,12,24\}\) is the greatest divisor of \({\bar{c}}_2(Z_\pm )\).
Proof
(i)–(iii) are immediate consequences of Theorem 2.8.
The image \(N'_\mp \) of the product homomorphism \(\flat ^{\pm } : N_\mp \rightarrow N_\pm ^*\) from (2.6) has rank 1, and Lemma 6.4 implies that it has cotorsion \(\mathbb {Z}/k\) in \(N_\pm ^*\). (iv) now follows from (2.7).
When \(k = 1\), so that \(N'_\mp \subset N_\pm ^*\) is primitive, the isomorphism \(N_\pm ^*/N'_\mp \cong \mathbb {Z}\) is realised by evaluation on \(A_\pm \). Therefore (v) follows from Proposition 2.11 and Lemma 2.9.
If \(x_\pm \in N_\pm \) such that \(\flat ^\pm (x_\pm ) = {\bar{c}}_2(Z_\mp ) \mod d(M)\), then the image of \(\frac{x_\pm }{2}\) in \(N_\pm /\langle A_\pm \rangle \) has the same parity as \(\frac{{{\,\mathrm{gd}\,}}({\bar{c}}_2(Z_\pm ))}{2}\). Since \(k = 1\), we find
and (vi) follows from Theorem 3.6. \(\square \)
Six examples of matchings of rank 2 Fanos with nontrivial intersection \(N_+ \cap N_\) are listed in [10, Example No 9]. However, Lemma 6.4 and the data in Table 5 allow us to be more decisive.
Theorem 6.6
There are precisely 19 pairs of rank 2 Fanos that can be matched to give twisted connected sums with \(H^2(M) \cong \mathbb {Z}\). In all cases \(H^4(M)\) is torsionfree and \(\mu (M) = 0\). For each of the pairs there is at least one matching such that \(\pi _2(M) \cong \mathbb {Z}\).
In Table 5 we list the following data about the 19 pairs:
the numbers \(\#_\pm \) of the entries in the MoriMukai list used,
the square of the generator B of the intersection \(N_+ \cap N_\),
the squares of the ample classes \(A_\pm \in N_\pm \),
the topological invariants \(b_3(M)\) and d(M) of the resulting twisted connected sums.
Proof of Theorem 6.6
In Table 3 we have listed \(h := \log _2 \frac{\Delta }{A^2}\) and \(B^2\) for all ample classes A in the Picard lattices of rank 2 Fanos, such that h is not too small. Therefore it is easy to read off all cases where the criterion of Lemma 6.4 is satisfied, and they are the ones listed in Table 5. Indeed, because \(\Delta \) is never greater than \(2A^2\) for any entry in the table, matchings of rank 2 Fano 3folds are only possible when \(\Delta _+\Delta _ = A_+^2A_^2\), or equivalently \(h_+ + h_ = 0\).
For each of the 19 pairs, we can apply Nikulin [34, Theorem 1.12.4] to embed the rank 3 lattice W from Lemma 6.4 in L, thus defining a configuration of primitive embeddings \(N_+, N_ \subset L\). Because \(\Lambda _\pm = N_\pm \) we can apply Propositions 5.8 and 6.1 to find a compatible matching.
Since we always have \(k = 1\), any twisted connected sum arising from matchings with nontrivial \(N_+ \cap N_\) must have \({{\,\mathrm{Tor}\,}}\pi _2(M) \cong {{\,\mathrm{Tor}\,}}H^3(M) = 0\) by Lemma 6.5(iii). For the one pair (row 14 in Table 5) where d(M) is divisible by 8, so that \(\mu (M)\) could possibly be nonzero, Lemma 6.5(v) shows that \(\mu (M) = 0\) anyway. \(\square \)
There are two reasons why Theorem 6.6 does not claim that there is a unique diffeomorphism type of twisted connected sum arising from each of the 19 pairs. The first is that for some of the pairs we can also embed \(W \subset L\) nonprimitively, giving a twisted connected sum M where \({{\,\mathrm{Tor}\,}}\pi _2(M) \cong {{\,\mathrm{Tor}\,}}H^3(M)\) is nontrivial; we do not study this further at this point.
The second reason is that even if we consider only the matchings with \(\pi _2(M) \cong \mathbb {Z}\), we cannot automatically deduce that the resulting diffeomorphism type is independent of the matching. This is because we do not have a classification theorem for this type of manifold. We hope to return to the problem of classifying such manifolds elsewhere. We expect that to determine the diffeomorphism type one must further compute the square of a generator of \(H^2(M)\), and also some generalisation of the invariants used by Kreck and Stolz [29] for 7manifolds with \(\pi _2(M) \cong \mathbb {Z}\) and finite\(H^4(M)\).
Similarly, the computations presented here are not sufficient to decide whether the two entries in Table 5 that have the same values of \(b_3(M)\) and d(M) (rows 2 and 8) must be diffeomorphic or even homeomorphic. However, according to Wallis [40, §B.3] these two manifolds are distinguished by the square of the generator of \(H^2(M)\).
Skew matching
Having dealt with matchings of rank 2 Fano 3folds where the configuration \(N_+, N_ \subset L\) is orthogonal (whether \(N_+ \cap N_\) is trivial or not), we now consider the skew case. To find skew matchings we have to deal with latticearithmetical problems, and also to gain some detailed understanding of the deformation theory of the Fanos involved. Because of the casebycase checking required, we think of this task as ‘handcrafting’.
Arithmetic conditions for skew matching
For Fano 3folds of Picard rank 2, at least the arithmetic part of the problem of finding matchings with a skew configuration \(N_+, N_ \subset L\) can be dealt with systematically. As pointed out in Remark 5.5, \(N_+ \cap N_\) must be trivial in this case, and the subgroup \(W_\pm \subset N_\pm \) orthogonal to \(N_\mp \) has rank exactly 1, and is generated by some ample class \(A_\pm \). Then W is isometric to \(W_k\) for some integer k, where the quadratic form on
is characterised as follows: \(A_\pm \perp N_\mp \), and \(H_+.H_ = k\) for some \(H_\pm \in N_\pm \) such that \(H_\pm , A_\pm \) is a basis for \(N_\pm \). (The choice of \(H_\pm \) only affects the sign of k.)
Lemma 7.1
Let \(\Delta _\pm \) be the discriminant of \(N_\pm \). Then \(W_k\) has signature (2, 2) if and only if
Proof
Since \(W_k\) contains the positive definite subspace \(\langle A_+, A_\rangle \) and some negative elements, its signature is (2,2) if and only if its discriminant D is positive. Let \(B_\pm \in N_\pm \) be a generator for the orthogonal complement of \(A_\pm \) in \(N_\pm \), and let \(n_\pm \) be the index of \(\langle A_\pm , B_\pm \rangle \subseteq N_\pm \). The index \(n_+n_\) sublattice of W spanned by \(A_+, B_+, A_, B_\) has intersection form
and discriminant \(A_+^2 A_^2 (B_+^2B_^2  k^2n_+^2n_^2) = n_+^2n_^2D\). Since \(A_\pm ^2 B_\pm ^2 = \Delta _\pm n_\pm ^2\), we find
\(\square \)
In particular, a necessary condition for finding a matching of rank 2 Fanos \(Y_+, Y_\) with skew configuration \(N_+, N_ \subset L\) is that there are ample classes \(A_\pm \in N_\pm \) with
for \(h_\pm := \log _2 \frac{\Delta _\pm }{A_\pm ^2}\). We can readily identify pairs satisfying this necessary condition from Table 3.
In terms of the notation from Sect. 5.1, \(W_\pm \) is generated by \(A_\pm \), and if\(W_k \subset L\) is primitive then \(\Lambda _\pm \) is the orthogonal complement of \(A_\pm \) in \(W_k\). Let us make an observation about the form on \(\Lambda _\pm \) that will prove useful.
Lemma 7.2
Let \(N_\pm \) be a pair of rank 2 lattices of signature (1, 1), and discriminant \(\Delta _\pm \). Given positive classes \(A_\pm \in N_\pm \), define \(W_k\) as in Lemma 7.1, with \(k > 0\) such that \(\Delta _+\Delta _ > k^2 A_+^2 A_^2\). Let \(B_\pm \) be a generator for the orthogonal complement of \(A_\pm \) in \(N_\pm \), and let \(\Lambda _\pm \) be the orthogonal complement of \(A_\mp \) in \(W_k\).
Suppose \(H\in N_\pm \) has the property that
Then
for any \(v \in \Lambda _\pm \) linearly independent of \(H\).
Proof
The inequality certainly holds if \(v \in N_\pm \). If H is a multiple of \(A := A_\pm \), then the inequality follows easily for any \(v \in \Lambda _\pm \).
\(\Lambda _\pm \) is generated by \(N_\pm \) together with \(B := B_\mp \). If H is linearly independent of A, then the projection \(B'\) of B to the orthogonal complement of \(N_\pm \) in \(\Lambda _\pm \) can be written as
whose square is
It suffices to show that
If we let m denote the index of \(\langle A,H\rangle \) in \(N_\pm \), and let n be the index of \(\langle A_\mp , B\rangle \) in \(N_\mp \), then \((A.H)^2  A^2 H^2 = m^2\Delta _\pm \), while \(B.H = \pm k mn\). Therefore \((B.H)^2\Delta _\mp = k^2 m^2 A_\mp ^2 B^2\), and
\(\square \)
Topology of skew matchings
We now explain how to determine the topology of twisted connected sums obtained from a skew matching of rank 2 blocks. In particular, we identify all pairs of deformation types of rank 2 Fano 3folds that could possibly be matched to produce twisted connected sums with nonzero generalised Eells–Kuiper invariant.
Proposition 7.3
Let \((Z_\pm , \Sigma _\pm )\) be a pair of building blocks whose polarising lattices \(N_\pm \) have rank 2, such that the kernel of \(H^2(Z_\pm ) \rightarrow H^2(\Sigma _\pm )\) is generated by \([\Sigma _\pm ]\). Let be a matching whose configuration \(N_+, N_ \subset L\) has \(N_+ \oplus N_\) isometric to \(W_k\) from Lemma 7.1, and let M be the resulting twisted connected sum.
Let \(A_\pm \in N_\pm \) be a generator for the orthogonal complement of \(N_\mp \) in \(N_\pm \). Then
 (i)
\(H^2(M) = 0\);
 (ii)
\(b_3(M) = b_3(Z_+) + b_3(Z_) + 21\);
 (iii)
\({{\,\mathrm{Tor}\,}}H^3(M) \cong L/W_k\);
 (iv)
\({{\,\mathrm{Tor}\,}}H^4(M) \cong (\mathbb {Z}/k)^2\);
 (v)
if \(k = 1\) then d(M) (which divides 24) is the greatest common divisor of \({\bar{c}}_2(Z_+)A_\pm \) and \({\bar{c}}_2(Z_)A_\pm \in \mathbb {Z}/24\), for \({\bar{c}}_2(Z_\pm )\) as in (4.3);
 (vi)
if \(k = 1\) and d(M) is divisible by 8 (so that the Eells–Kuiper invariant \(\mu (M)\) takes values in \(\mathbb {Z}/2\)) then
$$\begin{aligned} \mu (M) = \frac{{{\,\mathrm{gd}\,}}({\bar{c}}_2(Z_+)){{\,\mathrm{gd}\,}}({\bar{c}}_2(Z_))}{4} \in \mathbb {Z}/2 , \end{aligned}$$where \({{\,\mathrm{gd}\,}}({\bar{c}}_2(Z_\pm )) \in \{2,4,6,8,12,24\}\) is the greatest divisor of \({\bar{c}}_2(Z_\pm )\).
Proof
(i)–(iii) are immediate from Theorem 2.8. (iv)–(vi) are entirely analogous to the proof of Lemma 6.5. \(\square \)
Remark 7.4
We see in Table 3 that \(h < 1\) (i.e. \(\Delta <2A^2\)) for any ample class A on any rank 2 Fano 3fold. Therefore Lemma 7.1 implies that \(W_k\) can only have signature (2, 2) for \(k = 1\). Therefore, whenever we do find a matching of rank 2 Fanos with a skew configuration, the resulting twisted connected sum M always has \(H^4(M)\) torsionfree by Proposition 7.3(iv).
Remark 7.5
\(\mu (M)\) is nonzero if and only if both \({\bar{c}}_2(Z_\pm )A_\pm \) are divisible by 8 while neither \({\bar{c}}_2(Z_\pm )\) is divisible by 4. Consulting Table 3, we find that the only rank 2 Fano type blocks with such ample classes A are
 #9
with \(A := G+ H\) \((h = 0.09)\),
 #17
with \(A := G+ H\) \( (h = 0.06)\),
 #18
with \(A := G+ 2H\) \((h = 0.58)\),
 #27
with \(A := G+ H\) \( (h = 0.09)\).
Since #18 has h quite negative, the condition (7.1) implies that the only ways to match rank 2 Fanos with a skew configuration to construct a twisted connected sum M with \(\mu (M)\) is to use a pair among #9, #17 and #27.
Handcrafting examples with nonzero generalised Eells–Kuiper invariant
Let \(\mathcal {Y}_+\), \(\mathcal {Y}_\) be a pair of deformation types of rank 2 Fanos. For a skew configuration \(N_+, N_ \subset L\) of their Picard lattices, \(\Lambda _\pm \) has rank 3. Because \(\Lambda _\pm \) is strictly bigger than \(N_\pm \), the genericity result Proposition 6.1 does not suffice for applying Proposition 5.8 to find a matching with the prescribed configuration. The laborious part of finding skew matchings is to prove that generic \(\Lambda _\pm \)polarised K3 surfaces still appear as anticanonical K3 divisors in some members of \(\mathcal {Y}_\pm \), despite being more special than the generic, \(N_\pm \)polarised K3 divisors in the family.
In the present paper we go to that effort only in cases that lead to twisted connected sums with \(\mu (M) \not = 0\). In Theorem 7.8 we show that all skew configurations of the Picard lattices of rank 2 Fanos identified in Remark 7.5 are in fact realised by a matching. The genericity results needed to find those matchings are provided by Lemma 7.7. To prove that we use
Lemma 7.6
Let \(\Sigma \) be a K3 surface, and \(H\in {{\,\mathrm{Pic}\,}}\Sigma \) a primitive nef class with \(H^2 \ge 4\). Then \(H\) is a very ample class (i.e. the linear system \(H\) defines an embedding \(\Sigma \hookrightarrow \mathbb {P}^{\frac{H^2}{2} + 1})\) unless there is some \(v \in {{\,\mathrm{Pic}\,}}\Sigma \) such that
 (i)
\(d = 2\) and \(v^2 = 0\), or
 (ii)
\(d = 0\) and \(v^2 = 2\),
for \(d := v.H\).
Proof
According to [10, Lemma 7.15] (based on [35, Chapter 3]), the arithmetic conditions rule out all possible ways that \(H\) can fail to be very ample. (i) rules out the existence of any classes with \(d = 1\) and \(v^2 = 0\), and hence \(H\) being monogonal. Therefore \(H\) has no fixed part, and defines a morphism. \(H^2 \not = 2\), \(H\) primitive and (i) rule out the three different ways that \(H\) could be hyperelliptic, so it defines a birational morphism onto its image. (ii) rules out any \((2)\)curves being contracted, so \(H\) defines an isomorphism. \(\square \)
Using Lemma 7.6 and various results about curves on K3 surfaces, it is for many families \(\mathcal {Y}\) of Fano 3folds possible to obtain conditions on a lattice \(\Lambda \) (containing the Picard lattice N of \(\mathcal {Y}\)) that ensure that any K3 with Picard lattice isometric to \(\Lambda \) can be embedded as an anticanonical divisor in some element of \(\mathcal {Y}\). The conditions are to exclude the existence in \(\Lambda \) of elements v with certain combinations of \(v^2\) and inner products of v with elements of N. Once these ‘handcrafting’ conditions have been proved for some collection of blocks, then it would be feasible to get a computer to generate candidate configurations involving those blocks, and to verify whether the handcrafting conditions hold.
However, for the purposes of checking a few examples by hand, it is expedient to instead organise the argument by checking that in the examples we are concerned with, the sort of inequality produced by Lemma 7.2 rules out the presence in \(\Lambda \) of vectors v with the relevant properties.
Lemma 7.7
Let \(\mathcal {Y}\) be the deformation type of #27, #9 or #17 in the MoriMukai list of rank 2 Fanos. Let \(N \subset L\) be a primitive embedding of its Picard lattice, and let \(\Lambda \subset L\) be a primitive lattice containing N. Let \(G, H\) be the basis of N described in Sect. 4.4, and \({{\,\mathrm{Amp}\,}}_\mathcal {Y}\subset N(\mathbb {R})\) the interior of the cone spanned by \(G\) and \(H\). Suppose that
for all \(v \in \Lambda \) that are linearly independent of H, where \(\Delta \) is the discriminant of N. Then \(\mathcal {Y}\) is \((\Lambda , {{\,\mathrm{Amp}\,}}_\mathcal {Y})\)generic.
Proof
It is enough to prove that any K3 surface \(\Sigma \) with Picard lattice isometric to precisely \(\Lambda \) has a marking \(h : H^2(\Sigma ) \rightarrow L\) mapping \({{\,\mathrm{Pic}\,}}(\Sigma ) \rightarrow \Lambda \) and an embedding \(\Sigma \hookrightarrow Y\) of \(\Sigma \) as an anticanonical divisor in some \(Y \in \mathcal {Y}\), such that the image of the ample cone of Y in \(H^2(\Sigma )\) is \(h^{1}({{\,\mathrm{Amp}\,}}_\mathcal {Y})\).
 #27
This is the blowup of \(\mathbb {P}^3\) in a twisted cubic. In this case the Picard lattice is \(\left( {\begin{matrix} 2 &{} 5 \\ 5 &{} 4 \end{matrix}} \right) \), so H is a class of degree 4. (7.3) means that if \(v \in \Lambda \) is linearly independent of H and \(d := v.H\) then
$$\begin{aligned} d^2  4v^2 \ge 17, \end{aligned}$$(7.4)so neither of the cases in Lemma 7.6 can occur. Since H is not orthogonal to any \((2)\)class in \(\Lambda \), we can choose a marking of \(\Sigma \) that maps \({{\,\mathrm{Pic}\,}}(\Sigma )\) to \(\Lambda \), such that \(H\) is the image of a nef class. Lemma 7.6 then implies that \(H\) corresponds to a very ample class, embedding \(\Sigma \) as a quartic in \(\mathbb {P}^3\).
The class \(2H G\) has square \(2\), so by a standard application of the Riemann–Roch theorem for surfaces either \(2HG\) or \((2HG)\) is effective [35, Corollary 3.7(i)]. Since \(H.(2HG) = 3\) is positive, it must be \(2HG\) that is effective. Since any irreducible effective class v has \(v^2 \ge 2\), (7.4) implies that in fact there are no such classes of degree \(d \le 2\). Therefore \(2HG\) is irreducible. A further wellknown application of Riemann–Roch (see discussion after [35, Corollary 3.7]) implies that \(2HG\) is represented by a smooth rational curve \(\Gamma \). Its image in \(\mathbb {P}^3\) has degree 3, so must be a twisted cubic. By blowing up \(\Gamma \) we obtain a Fano \(Y \in \mathcal {Y}\).
 #9
This is the family of blowups of \(\mathbb {P}^3\) in smooth curves of degree 7 and genus 5. The Picard lattice is \(\left( {\begin{matrix} 2 &{} 5 \\ 5 &{} 4 \end{matrix}} \right) \) in this case too, so we have already proved that \(H\) embeds \(\Sigma \) as a quartic K3 in \(\mathbb {P}^3\), and that \(2HG\) is irreducible and represented by a twisted cubic \(\Gamma \).
We now want to prove that \(3HG= H+ [\Gamma ]\) can be represented by a smooth curve C. Because \(3HG\) is big and nef, by [35, Theorem 3.8(d) & 3.15] the only way it can fail to be basepointfree is if it is monogonal, i.e. if \(3HG= aE + F\) where \(E^2 = 0\) and F is the fixed part. Since \(H\) is very ample certainly \(H+ [\Gamma ]\) cannot have any nontrivial fixed part other than \(\Gamma \), and since \(H^2 \not = 0\) we cannot have \(F = \Gamma \) either.
Hence \(3HG\) is basepointfree, and a general member C is smooth by Bertini’s theorem. Now C has degree 7 and genus 5, so by Blanc–Lamy [4, Theorem 1.1] blowing up C defines a Fano in the family \(\mathcal {Y}\) unless there is a 4secant line \(\ell \) to C.
It thus remains only to show that no such \(\ell \) exists. If it did, then restricting the pencil of hyperplanes containing \(\ell \) to C defines a pencil of degree 3, a “complete \(g^1_3\)” in the terminology of Brill–Noether theory. The curve C therefore has Clifford index 1. By the main theorem of Green–Lazarsfeld [18] this implies the existence of a line bundle \(L \in \Lambda \) with \(L^2 = 0\) and \(L.C = 3\). Now (7.4) implies \(H.L \ge 5\). But then the span of \(H+L\) and \(3HG\) is positivedefinite, which is impossible.
 #17
This is the family of blowups of smooth quadrics \(Q \subset \mathbb {P}^4\) in smooth elliptic curves of degree 5. In this case the Picard lattice is \(\left( {\begin{matrix} 4 &{} 7 \\ 7 &{} 6 \end{matrix}} \right) \), so H is a degree 6 class. (7.3) means that
$$\begin{aligned} d^2  6v^2 \ge 25 \end{aligned}$$(7.5)for any \(v \in \Lambda \) not a multiple of H. In particular Lemma 7.6 implies that \(H\) is very ample, and by Riemann–Roch the embedded image of \(\Sigma \) in \(\mathbb {P}^4\) is the intersection of a quadric Q and a cubic. The quadric must be smooth, for if Q is singular then it contains planes. The section of \(\Sigma \) by such a plane would be a plane cubic, defining a class with \(v^2 = 0\) and \(d = 3\), which is ruled out by (7.5) (see Fukuoka [17, Lemma 2.4]).
Now consider the class \(E := 2H G\), which has degree 5 and \(E^2 = 0\). (7.5) rules out the existence of irreducible classes in \({{\,\mathrm{Pic}\,}}\Sigma \) with \(d \le 3\), so E is irreducible. In particular E does not have any \((2)\)curve components, so E is nef. Therefore [35, Theorem 3.8(b)] implies that E is basepointfree. A generic \(C \in E\) is therefore a smooth elliptic curve of degree 5.
If C had a 4secant line \(\ell \), then the pencil of planes through \(\ell \) would define a morphism from C to \(\mathbb {P}^1\), which is impossible. Therefore by Blanc–Lamy [4, Theorem 1.1], blowing up Q in C yields a member \(Y \in \mathcal {Y}\).
\(\square \)
For the three families of blocks under consideration, the sufficient conditions provided by Lemmas 7.2 and 7.7 turn out to be good enough to prove the existence of skew matching between any pair of families. Moreover, an ad hoc argument rules out any configurations beyond the ones realised.
Theorem 7.8
Let \(\mathcal {Y}_+\) and \(\mathcal {Y}_\) be a pair of deformation types of rank 2 Fanos among #9, #17 and #27 on the MoriMukai list. Let \(N_\pm \) be their Picard lattices, and let \(A_\pm \in N_\pm \) be the ample class listed in Table 3. Define \(W_1\) as in Lemma 7.1, embed \(W_1 \subset L\) primitively, and consider the resulting configuration \(N_+, N_ \subset L\).
 (i)
There is a matching of some elements of \(\mathcal {Y}_\pm \) with that configuration.
 (ii)
This is the only nonperpendicular configuration of \(N_+\) and \(N_\) for which a matching exists.
Proof

(i)
For this configuration, the lattice \(\Lambda _\pm \subset L\) defined in Sect. 5.1 is the orthogonal complement of \(A_\mp \) in \(W_1\). Let \(G_\pm , H_\pm \in N_\pm \) be the basis vectors described in Sect. 4.4. Looking up the values of \(H_\pm ^2\), \(A_\pm ^2\), \(B_\pm ^2\) and \(\Delta _\pm \) in Table 3, Lemma 7.2 implies that
$$\begin{aligned} (v.H_\pm )^2  H_\pm ^2v^2 \ge \Delta _\pm \end{aligned}$$(7.6)for any \(v \in \Lambda _\pm \) linearly independent of \(H_\pm \). Therefore Lemma 7.7 implies that \(\mathcal {Y}_\pm \) is \((\Lambda _\pm , {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_\pm })\)generic, and the desired matching exists by Proposition 5.8.

(ii)
By Lemma 6.4 there can be no matchings of these types with a configuration such that \(N_+ \cap N_\) is nontrivial, so a nonperpendicular configuration must be skew. We have explained that for a skew configuration to satisfy the conditions of Remark 5.4, \(N_+ \oplus N_\) must be isometric to \(W_1\). Thus it only remains to rule out configurations where \(W_1\) is embedded nonprimitively in L.
In Lemma 7.1 we computed that the discriminant of \(W_1\) is
$$\begin{aligned} D = \Delta _+\Delta _  A_+^2 A_^2 . \end{aligned}$$When \(\mathcal {Y}_\pm \) are both among #9 and #27, \(D = 33\) is squarefree, so \(W_1\) does not have any integral overlattice. When \(\mathcal {Y}_+\) is one of #9 and #27 while \(\mathcal {Y}_\) is #17 we get \(D = 41\), which is also squarefree. Hence there are no nonprimitive embeddings \(W_1 \subset L\) in these cases.
However, when \(\mathcal {Y}_\pm \) are both #17 we get \(D = 49\). In the basis \(G_+, A_+, G_, A_\), the quadratic form on \(W_1\) can be written as
$$\begin{aligned} \begin{pmatrix} 4 &{}\quad 11 &{}\quad 1 &{}\quad 0 \\ 11 &{}\quad 24 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 4 &{}\quad 11 \\ 0 &{}\quad 0 &{}\quad 11 &{}\quad 24 \end{pmatrix} . \end{aligned}$$This matrix has rank 3 over \(\mathbb {Z}/7\), so the discriminant group must be \(\mathbb {Z}/49\) rather than \((\mathbb {Z}/7)^2\), and \(W_1\) has a index 7 overlattice \(\widetilde{W}\) (which is in fact unimodular). Indeed, \(K := G_+ + A_+  G_  A_\) has \(K^2 = 98\) and its product with any element of W is divisible by 7. Therefore we can define \(\widetilde{W}\) by adjoining \(\frac{1}{7}K\) to W.
The only possible way to embed \(W_1 \subset L\) nonprimitively is via a primitive embedding \(\widetilde{W} \subset L\). We now check that there are no matchings with this configuration. Note that \(\Lambda _\pm \) is spanned by \(G_\pm \), \(H_\pm \) and \(\widetilde{B}_\pm := \pm \frac{24}{7} K + 5A_\mp \). In that basis, the quadratic form on \(\Lambda _\pm \) is represented by
$$\begin{aligned} \begin{pmatrix} 4 &{}\quad 7 &{}\quad 48 \\ 7 &{}\quad 6 &{}\quad 72 \\ 48 &{}\quad 72 &{}\quad 552 \end{pmatrix} . \end{aligned}$$We find that (7.5) fails for some \(v \in \Lambda _+\) such that \(v^2 = 0\) and \(d = 3\), e.g. \(v = E' :=  9G_+  H_+ + \widetilde{B}_+\).
Now suppose that \(\Sigma \) is an anticanonical K3 divisor in some rank 2 Fano of type #27, with \({{\,\mathrm{Pic}\,}}\Sigma \) isometric to this \(\Lambda _+\). Then \(H_+\) embeds \(\Sigma \) in a smooth quadric \(Q \subset \mathbb {P}^4\). The argument from the proof of Lemma 7.7 for the case #17 shows that because \((E')^2 = 0\) and \(E'.H_+ = 3\), the linear system \(E'\) on \(\Sigma \) is basepointfree, and represented by a smooth elliptic curve C. By Riemann–Roch, C is a plane cubic. Because Q contains C it must also contain the plane of C, contradicting that Q is nonsingular.
\(\square \)
Proof of main theorem
To prove Theorem 1.2 it now remains only to put together the pieces provided above.
Theorem 7.8 provides exactly one configuration with a matching for each pair of rank 2 Fano types among #9, #17 and #27 (referred to as (d), (e) and (f) in Table 1 in the introduction). Each of those six pairs produces a closed 7manifold M with holonomy \(G_2\) by Construction 2.6, whose topology can be computed from Proposition 7.3 and the data in Table 3. These M are 2connected with \(H^4(M)\) torsionfree, and \(b_3(M)\) and d(M) as listed in Table 1. By design \(\mu (M) = 1\), while all other matchings of rank 2 Fanos give \(\mu = 0\) by Corollary 3.7, Theorem 6.6 and Remark 7.5.
The six matchings realise four different pairs \((b_3,d)\), and hence four different diffeomorphism types by Theorem 2.2(ii). We can then consult Table 4 to see that for two of these four smooth manifolds, there exist perpendicular twisted connected sums \(M'\) with the same \((b_3, d)\). Then M and \(M'\) are homeomorphic by Theorem 2.2(i). However, \(\mu (M') = 0\) by Corollary 3.7, so M and \(M'\) are not diffeomorphic.
References
 1.
Baraglia, D.: \({G}_2\) geometry and integrable systems, Ph.D. thesis, University of Oxford (2009). arXiv:1002.1767
 2.
Beauville, A.: Fano Threefolds and K3 Surfaces, The Fano Conference, pp. 175–184. Univ. Torino, Turin (2004)
 3.
Berger, M.: Sur les groupes d’holonomie homogène des variétés à connexion affines et des variétés riemanniennes. Bull. Soc. Math. Fr. 83, 279–330 (1955)
 4.
Blanc, J., Lamy, S.: Weak Fano threefolds obtained by blowingup a space curve and construction of Sarkisov links. Proc. Lond. Math. Soc. 105, 1047–1075 (2012)
 5.
Bonan, E.: Sur les variétés riemanniennes à groupe d’holonomie \({G}_2\) ou \({S}pin(7)\). C. R. Acad. Sci. Paris 262, 127–129 (1966)
 6.
Browder, W.: Surgery on Simplyconnected Manifolds. Springer, New York, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65 (1972)
 7.
Chinburg, T., Escher, C., Ziller, W.: Topological properties of Eschenburg spaces and 3Sasakian manifolds. Math. Ann. 339, 3–20 (2007)
 8.
Coates, T., Corti, A., Galkin, S., Kasprzyk, A.: Quantum periods for 3dimensional Fano manifolds. Geom. Topol. 20, 103–256 (2016)
 9.
Corti, A., Haskins, M., Nordström, J., Pacini, T.: Asymptotically cylindrical Calabi–Yau 3folds from weak Fano 3folds. Geom. Topol. 17, 1955–2059 (2013)
 10.
Corti, A., Haskins, M., Nordström, J., Pacini, T.: \(G_2\)manifolds and associative submanifolds via semiFano 3folds. Duke Math. J. 164, 1971–2092 (2015)
 11.
Crowley, D., Nagy, Cs.: Inertia groups of \(3\)connected \(8\)manifolds, Research announcement, Melbourne 2019. http://www.dcrowley.net/research.html
 12.
Crowley, D., Nordström, J.: New invariants of \({G}_2\)structures. Geom. Topol. 19, 2949–2992 (2015)
 13.
Crowley, D., Nordström, J.: The classification of 2connected 7manifolds. Proc. Lond. Math. Soc. 119, 1–54 (2019)
 14.
da C. Guio, T., Jockers, H., Klemm, A., Yeh, H.Y.: Effective action from Mtheory on twisted connected sum \({G}_2\)manifolds. Commun. Math. Phys. 359, 535–601 (2018)
 15.
Donaldson, S.K.: La topologie différentielle des surfaces complexes. C. R. Acad. Sci. Paris 301, 317–320 (1985)
 16.
Eells Jr., J., Kuiper, N.: An invariant for certain smooth manifolds. Ann. Mat. Pura Appl. (4) 60, 93–110 (1962)
 17.
Fukuoka, T.: On the existence of almost Fano threefolds with del Pezzo fibrations. Math. Nachr. 290(8–9), 1281–1302 (2017)
 18.
Green, M., Lazarsfeld, R.: Special divisors on curves on a K3 surface. Invent. Math. 89, 357–370 (1987)
 19.
Haskins, M., Hein, H.J., Nordström, J.: Asymptotically cylindrical Calabi–Yau manifolds. J. Differ. Geom. 101, 213–265 (2015)
 20.
Iskovskih, V.A.: Fano threefolds. I. Izv. Akad. Nauk SSSR Ser. Mat. 41(3), 516–562 (1977)
 21.
Iskovskih, V.A.: Fano threefolds. II. Izv. Akad. Nauk SSSR Ser. Mat. 42(3), 506–549 (1978)
 22.
Iskovskih, V.A.: Anticanonical models of threedimensional algebraic varieties. Sovr. Prob. Mat. 12, 59–157 (1979)
 23.
Iskovskih, V.A., Prokhorov, Y.G.: Fano varieties, Algebraic geometry, V. Encyclopaedia Math. Sci., vol. 47. Springer, Berlin, pp. 1–247 (1999)
 24.
Jiang, Y.: Regular circle actions on 2connected 7manifolds. J. Lond. Math. Soc. 90(2), 373–387 (2014)
 25.
Joyce, D.: Compact Riemannian 7manifolds with holonomy \({G}_{2}\). I. J. Differ. Geom. 43, 291–328 (1996)
 26.
Joyce, D.: Compact manifolds with special holonomy, OUP Mathematical Monographs Series. Oxford U Press, Oxford (2000)
 27.
Kovalev, A.: Twisted connected sums and special Riemannian holonomy. J. Reine Angew. Math. 565, 125–160 (2003)
 28.
Kreck, M.: Surgery and duality. Ann. Math. (2) 149(3), 707–754 (1999)
 29.
Kreck, M., Stolz, S.: Some nondiffeomorphic homeomorphic homogeneous \(7\)manifolds with positive sectional curvature. J. Differ. Geom. 33(2), 465–486 (1991)
 30.
Menet, G., Nordström, J., Sá Earp, H.: Construction of \({G}_2\)instantons via twisted connected sums. Math. Res. Lett., arXiv:1510.03836(to appear) (2019)
 31.
Milnor, J.W.: On manifolds homeomorphic to the 7sphere. Ann. Math. (2) 64(2), 399–405 (1956)
 32.
Mori, S., Mukai, S.: Classification of Fano \(3\)folds with \(B_{2}\ge 2\). Manuscr. Math. 36(2), 147–162 (1981)
 33.
Mori, S., Mukai, S.: Erratum: “Classification of Fano 3folds with \(B_2\ge 2\)”. Manuscr. Math. 110(3), 407 (2003)
 34.
Nikulin, V.: Integer symmetric bilinear forms and some of their applications, Izv. Akad. Nauk SSSR Ser. Mat. 43: 111–177, 238. English translation: Math. USSR Izvestia 14(1980), 103–167 (1979)
 35.
Reid, M., Chapters on algebraic surfaces, Complex algebraic geometry (Park City, UT, 1993), IAS/Park City Math. Ser., vol. 3. Amer. Math. Soc., Providence, pp. 3–159 (1997)
 36.
Salamon, S.: Riemannian Geometry and Holonomy Groups, Pitman Reseach Notes in Mathematics, vol. 201. Longman, Harlow (1989)
 37.
Sá Earp, H., Walpuski, T.: \({G}_2\)instantons on twisted connected sums. Geom. Topol. 19, 1263–1285 (2015)
 38.
Traving, C.: Zur Diffeomorphieklassifikation vollständiger Durchschnitte. Diplomarbeit, Johannes Gutenberg Universität Mainz (1985)
 39.
Wall, C.T.C.: Classification problems in differential topology. V. On certain 6manifolds. Invent. Math. 1, 355–374 (1966)
 40.
Wallis, D.: Topological invariants of \({G}_2\)manifolds, Ph.D. thesis, University of Bath (2019)
 41.
Walpuski, T.: \({G}_2\)instantons over twisted connected sums: an example. Math. Res. Lett. 23, 529–544 (2016)
 42.
Wilkens, D.L.: Closed \((s{}1)\)connected \((2s{+}1)\)manifolds, \(s=3,\,7\). Bull. Lond. Math. Soc. 4, 27–31 (1972)
 43.
Zhubr, A.V.: Closed simply connected sixdimensional manifolds: proofs of classification theorems. Algebra i Analiz 12(4), 126–230 (2000). Translation in St. Petersburg Math. J. 12 (4), 605–680 (2001)
Acknowledgements
We thank Alastair Craw, Alessio Corti, Mark Haskins, Jesus MartinezGarcia, Matthew Turner and Dominic Wallis for useful discussions, and the referee for constructive comments. DC acknowledges the support of the Leibniz Prize of Wolfgang Lück, granted by the Deutsche Forschungsgemeinschaft. JN thanks the Simons Foundation for its support under the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics (Grant #488631, Johannes Nordström).
Author information
Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Ngaiming Mok.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Crowley, D., Nordström, J. Exotic \(G_2\)manifolds. Math. Ann. (2020). https://doi.org/10.1007/s00208020020091
Received:
Revised:
Published: