# Exotic $G_{2}$-manifolds

Diarmuid Crowley  Institute of Mathematics, University of Aberdeen, Aberdeen
AB24 3UE, UK<EMAIL_ADDRESS>and  Johannes Nordström  Department of
Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
<EMAIL_ADDRESS>
###### Abstract.

We exhibit the first examples of closed 7-dimensional Riemannian manifolds
with holonomy $G_{2}$ that are homeomorphic but not diffeomorphic. These are
also the first examples of closed Ricci-flat manifolds that are homeomorphic
but not diffeomorphic. The examples are generated by applying the twisted
connected sum construction to Fano 3-folds 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.

##  1\. 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 general strategy is to apply a topological classification theorem to a
plentiful supply of examples for which the classifying invariants are
computable. In this case we use the “twisted connected sum” construction of
closed $G_{2}$-manifolds [11]; it was shown in [4] that this construction
yields large numbers of closed $G_{2}$-manifolds that are _2-connected_ , with
torsion-free cohomology, and how to compute the invariants required to apply
homeomorphism classification results of Wilkens.

The diffeomorphism classification of 2-connected 7-manifolds was recently
completed in [6], which in particular introduced a generalised Eells-Kuiper
invariant that can distinguish between different smooth structures on the same
closed topological spin 7-manifold. In the present paper we show how to
compute this invariant for twisted connected sums, and use that to identify
examples of closed 2-connected manifolds with holonomy $G_{2}$ that are
homeomorphic but not diffeomorphic. To the best of our knowledge, these are
also the first examples of closed Ricci-flat 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
[16] or Joyce [10]. The main technical work of the paper is to examine in
detail the properties of some examples of Fano 3-folds and their anticanonical
divisors.

###  1.1. Twisted connected sums

Given a pair of Fano 3-folds $Y_{+},Y_{-}$ with smooth anticanonical K3
divisors $\Sigma_{\pm}\subset Y_{\pm}$ and a _matching_ diffeomorphism
$\matheur{r}:\Sigma_{+}\to\Sigma_{-}$ (Definition 4.3), the twisted connected
sum construction yields a closed 7-manifold $M$ with metrics of holonomy
$G_{2}$. The procedure is summarised in §2.3 and §4.1.

We categorise the matching as either _perpendicular_ or _non-perpendicular_ ,
depending on the action of $\matheur{r}$ on the images of $H^{2}(Y_{\pm})$ in
$H^{2}(\Sigma_{\pm})$. For most pairs $\mathcal{Y}_{+},\mathcal{Y}_{-}$ among
the 105 deformation types of Fano 3-folds, general deformation theory results
make it possible to find a perpendicular matching of some
$Y_{\pm}\in\mathcal{Y}_{\pm}$ resulting in a 2-connected twisted connected sum
$G_{2}$-manifold. As we explain below, such perpendicular matchings can never
be homeomorphic without being diffeomorphic.

Whether there is any non-perpendicular matching of a pair of members of
$\mathcal{Y}_{\pm}$ is in general a more difficult question. There are
necessary conditions of a lattice-arithmetical nature, but, as we discuss in
§5.1, showing that matchings exist also requires some detailed information
about the deformation theory of anticanonical divisors in $\mathcal{Y}_{\pm}$
that is hard to obtain systematically.

###  1.2. The classifying invariants

Let us recall the relevant topological classification results. Given a closed
2-connected 7-manifold $M$, 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 torsion-free, then
this data can be reduced to the third Betti number $b_{3}(M)$ and the greatest
integer $d(M)$ dividing $p_{M}$ in $H^{4}(M)$ (set $d=0$ if $p_{M}=0$). In
fact, the pair $(b_{3}(M),d(M))$ classifies such $M$ up to homeomorphism (by
Wilkens [17], see also Theorem 2.2).

For spin manifolds with $H^{4}(M)$ torsion-free, the _generalised Eells-Kuiper
invariant_ introduced in [6] reduces to a constant

$\mu(M)\in\mathbb{Z}/\widehat{d},$

and distinguishes between
$\widehat{d}:=\gcd(28,\textup{Num}\left(\frac{d}{4}\right))$ different smooth
structures on the topological manifold underlying $M$ (where
$\textup{Num}\left(\frac{a}{b}\right):=\frac{a}{\gcd(a,b)}$). If $p_{M}=0$ (so
$\widehat{d}=28$) then $\mu(M)$ coincides with the invariant introduced by
Eells and Kuiper [7].

In §2.2 we recall how $\mu(M)$ can be defined in terms of a
$\textrm{spin}^{\textrm{c}}$ coboundary of $M$. In §3 we then construct
explicit $\textrm{spin}^{\textrm{c}}$ coboundaries of twisted connected sums,
and use that to obtain a formula for their generalised Eells-Kuiper invariant
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 3.5).

###  1.3. The main examples

To have any chance of obtaining homeomorphic but non-diffeomorphic twisted
connected sums we must therefore search for non-perpendicular matchings. In
that case, both Fanos used must have Picard rank $b_{2}(Y_{\pm})\geq 2$ (_cf._
Remark 5.3). We are therefore led to study systematically the possible
matchings of Fanos with Picard rank 2.

######  Theorem 1.1.

  1. (i)

Any twisted connected sum $M$ of Fanos of rank 1 or 2 has $H^{4}(M)$ torsion-
free.

  2. (ii)

There are precisely six (unordered) pairs $\mathcal{Y}_{+},\mathcal{Y}_{-}$ of
deformation types of Fano 3-folds of Picard rank 2 with members that can be
matched in the sense of Definition 4.3 in such a way that the resulting
twisted connected sum $M$ has $\mu(M)\not=0$.

  3. (iii)

Those twisted connected sums are all 2-connected. Each of the six pairs gives
rise to a unique diffeomorphism type of $M$ with $\mu(M)\not=0$.

  4. (iv)

In total, they realise four distinct diffeomorphism types of $M$ with
$\mu(M)\not=0$.

  5. (v)

Precisely two of those are homeomorphic to some twisted connected sum
$M^{\prime}$ of Fanos of Picard rank $\leq 2$ such that $\mu(M^{\prime})=0$.

In particular, we obtain two pairs $(M,M^{\prime})$ of $G_{2}$-manifolds that
are homeomorphic but not diffeomorphic.

For each of the examples highlighted in Theorem 1.1, we indicate below the
pair of deformation types used, whether the matching is perpendicular, and the
classifying diffeomorphism invariants of the resulting twisted connected sum.
Recall that a del Pezzo 3-fold is a Fano 3-fold $Y$ whose anticanonical class
$-K_{Y}\in\operatorname{Pic}Y$ is even.

$\begin{array}[]{cccccc}\hline\cr\hline\cr\;\mathcal{Y}_{+}&\;\mathcal{Y}_{-}&\perp&b_{3}&d&\mu\\\ \hline\cr\ref{it:dp3}&\ref{it:dp5}&\textrm{yes}&101&8&0\\\ \ref{it:mm27}&\ref{it:mm27}&\textrm{no}&101&8&1\\\ \ref{it:dp5}&\ref{it:d16}&\textrm{yes}&89&8&0\\\ \ref{it:mm9}&\ref{it:mm27}&\textrm{no}&89&8&1\\\ \ref{it:mm17}&\ref{it:mm27}&\textrm{no}&89&8&1\\\ \ref{it:mm9}&\ref{it:mm9}&\textrm{no}&77&8&1\\\ \ref{it:mm17}&\ref{it:mm9}&\textrm{no}&77&8&1\\\ \ref{it:mm17}&\ref{it:mm17}&\textrm{no}&77&24&1\\\ \hline\cr\hline\cr\end{array}$ |  (i) Del Pezzo 3-folds of degree 3, _i.e._ cubic hypersurfaces in $\mathbb{P}^{4}$ (Picard rank 1). (ii) Del Pezzo 3-folds of degree 5 (Picard rank 1). (iii) Picard rank 1 Fanos of degree 16. (iv) Number 9 in the Mori-Mukai list of Picard rank 2 Fano 3-folds: $\mathbb{P}^{3}$ blown up in a curve of degree 7 and genus 5. (v) Number 17 in the Mori-Mukai list of Picard rank 2 Fano 3-folds: a smooth quadric hypersurface in $\mathbb{P}^{4}$ blown up in an elliptic curve of degree 5. (vi) Number 27 in the Mori-Mukai list of Picard rank 2 Fano 3-folds: $\mathbb{P}^{3}$ blown up in a twisted cubic curve.
---|---

In §4 we compute detailed topological data for all 36 types in the Mori-Mukai
list of rank 2 Fano 3-folds. In §5 we identify all pairs that satisfy the
necessary arithmetic conditions for existence of a non-perpendicular matching
resulting in a twisted connected sum with $\mu\not=0$. The only candidate
pairs are among the types (iv), (v) and (vi) above. By studying the
anticanonical divisors in those types we show that non-perpendicular matchings
do indeed exist in those six cases, 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 3. For two of the four twisted
connected sums with $\mu\not=0$ we can identify a perpendicular matching with
the same homeomorphism invariants, and those are the ones included in the
table above.

In the process 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 4).
Such matchings are of interest for the problem of constructing examples of
$G_{2}$-instantons by gluing, _cf._ Sá Earp and Walpuski [13].

###  1.4. 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
[12]) and 3-Sasakian metrics (Chinburg-Escher-Ziller [2]). 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 Ricci-flat manifolds have special
holonomy: $2n$-manifolds with holonomy $SU(n)$ ($n\geq 2$), $4n$-manifolds
with holonomy $Sp(n)$ ($n\geq 2$), 7-manifolds with holonomy $G_{2}$ and
8-manifolds with holonomy $Spin(7)$.

In real dimension 4, the only smooth manifold with holonomy $SU(2)$ is the K3
surface. In real dimension 6, simply-connected manifolds have unique smooth
structures, while there is no general classification result for finite but
non-trivial fundamental group. Manifolds with holonomy $SU(n)$ or $Sp(n)$
necessarily have $b_{2}>0$, as do the known examples of manifolds of holonomy
$Spin(7)$, putting manifolds with Ricci-flat holonomy in dimension $\geq 8$
out of reach current topological classification results. Thus 7-manifolds of
holonomy $G_{2}$ are the only kinds of closed Ricci-flat manifolds where
homeomorphic non-diffeomorphic examples are accessible with the current
technology.

### Acknowledgements

We thank Alessio Corti and Mark Haskins for useful discussions. DC
acknowledges the support of the Leibniz Prize of Wolfgang Lück, granted by the
Deutsche Forschungsgemeinschaft.

##  2\. Background

We begin with some more detailed explanation of the generalised Eells-Kuiper
invariant and the twisted connected sum construction, in order to clarify the
meaning of the statement of the main theorem, Theorem 1.1.

###  2.1. Spin and spin${}^{\textrm{c}}$ characteristic classes

Recall that the classifying space $BSpin$ is $3$-connected and
$\pi_{4}(BSpin)\cong\mathbb{Z}$. 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 $\pi\colon BSpin\to BSO$ 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

$p_{X}:=\frac{p_{1}}{2}(TX)\in H^{4}(X).$

A $\textrm{spin}^{\textrm{c}}$ structure on a vector bundle $E$ is equivalent
to a choice of complex line bundle $L$ such that $E\oplus L$ is spin. Given a
$\textrm{spin}^{\textrm{c}}$ structure on the tangent bundle of $X$ we can
therefore define characteristic classes $z:=c_{1}(L)\in H^{2}(X)$, and
$\widehat{p}_{X}:=\frac{p_{1}}{2}(TX\oplus L)+z^{2}\in H^{4}(X)$. Then
$2\widehat{p}_{X}=p_{1}(X)+z^{2}$. If $X$ is spin then we can of course take
$L$ to be trivial, and $\widehat{p}_{X}=p_{X}$.

######  Lemma 2.1 (_cf._ [6, 2.29–2.30]).

  1. (i)

If $TX$ has an almost complex structure then
$\widehat{p}_{X}=-c_{2}(X)+c_{1}(X)^{2}$.

  2. (ii)

$\widehat{p}_{X}=w_{4}(X)+w_{2}(X)^{2}\\!\\!\mod 2$.

  3. (iii)

Suppose $X$ is closed.

     * •

If $\dim X\leq 7$ 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 $x\in
H^{4}(X)$.

###  2.2. The generalised Eells-Kuiper invariant

We now describe the generalised Eells-Kuiper invariant of a closed spin
7-manifold $M$, in the special case when $H^{4}(M)$ is torsion-free. As in the
introduction, let $d$ be the greatest integer dividing $p_{M}$ (so $d$ is even
by Lemma 2.1(iii)), and
$\widehat{d}:=\gcd(28,\textup{Num}\left(\frac{d}{4}\right))$.

Let $W$ a $\textrm{spin}^{\textrm{c}}$ 8-manifold 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
$p_{M}$ [6, §2.6]. Then

$\mu(M):=\frac{\widehat{\alpha}^{2}-\sigma(W)}{8}-\frac{5\bar{z}^{2}\widehat{p}_{W}}{12}+\frac{z^{4}}{4}\in\mathbb{Z}/\widehat{d},$
(2.1)

where $\widehat{\alpha}:=\widehat{p}_{W}-dn\in H^{4}(W)$ for $n$ as above, and
$\bar{z}\in H^{2}(W,M)$ is any pre-image 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 $\textrm{spin}^{\textrm{c}}$ 8-manifold.

When $M$ is in addition 2-connected, 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._ [6, Theorem 1.3]).

Let $M_{0}$ and $M_{1}$ be closed 2-connected 7-manifolds such that
$H^{4}(M_{i})$ are torsion-free. Then

  1. (i)

$M_{0}$ and $M_{1}$ are homeomorphic if and only if
$b_{3}(M_{0})=b_{3}(M_{1})$ and $d(M_{0})=d(M_{1})$.

  2. (ii)

$M_{0}$ and $M_{1}$ are diffeomorphic if and only if in addition
$\mu(M_{0})=\mu(M_{1})$.

###  2.3. Definition of twisted connected sums

We now explain the construction of the twisted connected sum of a matching
pair of building blocks, using the set-up from [4, §3]. Like in the original
application of the construction by Kovalev [11], the present paper uses
building blocks obtained from Fano 3-folds (in our case of rank 1 or 2) by a
procedure explained in §4.1. The matching problem is elaborated on in §5.1.

######  Definition 2.3.

Let $Z$ be a non-singular algebraic 3-fold and $\Sigma\subset Z$ a non-
singular K3 surface. Let $N$ be the image of $H^{2}(Z)\to H^{2}(\Sigma)$. We
call $(Z,\Sigma)$ a _building block_ if

  1. (i)

the anticanonical class $-K_{Z}\in H^{2}(Z)$ is indivisible,

  2. (ii)

$\Sigma\in|{-}K_{Z}|$, and there is a projective morphism $f\colon
Z\to\mathbb{P}^{1}$ with $\Sigma=f^{\star}(\infty)$,

  3. (iii)

The inclusion $N\hookrightarrow H^{2}(\Sigma)$ is primitive, that is,
$H^{2}(\Sigma)/N$ is torsion-free.

  4. (iv)

The group $H^{3}(Z)$—and thus also $H^{4}(Z)$—is torsion-free.

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, $N\subseteq H^{1,1}(\Sigma)$ [4, Lemma 3.6], so that
$\Sigma$ is ‘$N$-polarised’.)

######  Remark.

The class $[\Sigma]\in H^{2}(Z)$ is always in the kernel of $H^{2}(Z)\to
H^{2}(\Sigma)$. In this paper we will only consider blocks where $[\Sigma]$ in
fact generates the kernel.

######  Definition 2.4.

Given a complex K3 surface $\Sigma$, we call a triple of 2-forms
$(\omega^{I},\omega^{J},\omega^{K})$ a _hyper-Kähler triple_ if $\omega^{I}$
is a Kähler form with respect to the complex structure of $\Sigma$,
$\omega^{J}+i\omega^{K}$ is a holomorphic (2,0)-form, and
$(\omega^{I})^{2}=(\omega^{J})^{2}=(\omega^{K})^{2}$.

Given hyper-Kähler triples
$(\omega^{I}_{\pm},\omega^{J}_{\pm},\omega^{K}_{\pm})$ on a pair of complex K3
surfaces $\Sigma_{\pm}$, we call a diffeomorphism
$\matheur{r}:\Sigma_{+}\to\Sigma_{-}$ a _hyper-Kähler rotation_ if
$\matheur{r}^{*}\omega^{I}_{-}=\omega^{J}_{+}$,
$\matheur{r}^{*}\omega^{J}_{-}=\omega^{I}_{+}$ and
$\matheur{r}^{*}\omega^{K}_{-}=-\omega^{K}_{+}$.

######  Definition 2.5.

Given a pair of building blocks $(Z_{\pm},\Sigma_{\pm})$, call a
diffeomorphism $\matheur{r}:\Sigma_{+}\to\Sigma_{-}$ a _matching_ if there
exist hyper-Kähler triples
$(\omega^{I}_{\pm},\omega^{J}_{\pm},\omega^{K}_{\pm})$ on $\Sigma_{\pm}$ such
that $\omega^{I}_{\pm}$ is the restriction of a Kähler form on $Z_{\pm}$, and
$\matheur{r}$ is a hyper-Kähler rotation.

Say that $\matheur{r}$ is a _perpendicular_ matching if the subspaces
$\matheur{r}^{*}N_{-}$ and $N_{+}$ are perpendicular with respect to the
intersection form on $H^{2}(\Sigma_{+})$.

Given a building block $(Z,\Sigma)$, let $\Delta\subset\mathbb{P}^{1}$ be a
disc that is a trivialising neighbourhood of $\infty$ for the fibration
$f:Z\to\mathbb{P}^{1}$, and $U:=f^{-1}(\Delta)$. Then $V:=Z\setminus U$ is a
manifold with boundary diffeomorphic to $S^{1}\times\Sigma$.

######  Definition 2.6.

Given a pair of building blocks $(Z_{\pm},\Sigma_{\pm})$ with a matching
$\matheur{r}$, their _twisted connected sum_ $M$ is the smooth 7-manifold
defined by gluing $S^{1}\times V_{+}$ and $S^{1}\times V_{-}$ by the
diffeomorphism

$\displaystyle S^{1}\times S^{1}\times\Sigma_{+}$ $\displaystyle\to
S^{1}\times S^{1}\times\Sigma_{-},$ (2.2) $\displaystyle(u,v,x)$
$\displaystyle\mapsto(v,u,\matheur{r}(x))$

of their boundaries.

######  Theorem 2.7.

The twisted connected sum $M$ admits metrics with holonomy $G_{2}$.

###### Proof (_cf._ [4, Corollary 6.4])..

By [8, Theorem D], the interiors 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 there exists a parallel
$SU(3)$-structure, and the asymptotic limit of that can be characterised by a
hyper-Kähler triple on $\Sigma_{\pm}$. We may prescribe it to be the hyper-
Kähler triple appearing in the Definition 2.5. Then $\matheur{r}$ is a hyper-
Kähler rotation of the asymptotic limits, so according to Kovalev [11, Theorem
5.34] the map (2.2) can be used to glue together torsion-free product
$G_{2}$-structures on $S^{1}\times V_{+}$ and $S^{1}\times V_{-}$ to a
torsion-free $G_{2}$-structure on $M$. ∎

###  2.4. Topology of twisted connected sums

Let $(Z_{\pm},\Sigma_{\pm})$ be a pair of building blocks with a matching
$\matheur{r}:\Sigma_{+}\to\Sigma_{-}$. We think of $\matheur{r}$ as
identifying 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 ([4, Theorem 4.8]).

Suppose $M$ is a twisted connected sum of building blocks
$(Z_{\pm},\Sigma_{\pm})$ such that the kernel of $H^{2}(Z_{\pm})\to
H^{2}(\Sigma_{\pm})$ is generated by $[\Sigma_{\pm}]$. Then

  1. (i)

$\pi_{1}(M)=0$ and $H^{1}(M)=0$;

  2. (ii)

$H^{2}(M)\cong N_{+}\cap N_{-}$;

  3. (iii)

$H^{3}(M)\cong\mathbb{Z}\oplus(L/_{N_{+}+N_{-}})\oplus(N_{-}\cap
T_{+})\oplus(N_{+}\cap T_{-})\oplus H^{3}(Z_{+})\oplus H^{3}(Z_{-})$;

  4. (iv)

$H^{4}(M)\cong H^{4}(S)\oplus(T_{+}\cap
T_{-})\oplus(L/_{N_{-}+T_{+}})\oplus(L/_{N_{+}+T_{-}})\oplus
H^{3}(Z_{+})\oplus H^{3}(Z_{-})$.

This implies in particular that if the matching $\matheur{r}$ is perpendicular
then $H^{4}(M)$ is torsion-free, and

$b_{2}(M)=0,\quad b_{3}(M)=b_{3}(Z_{+})+b_{3}(Z_{-})+23.$ (2.3)

If in addition the perpendicular direct sum $N_{+}\perp N_{-}\subset L$ is
primitive then $M$ is 2-connected.

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

$H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})\subset H^{4}(Z_{+})\oplus H^{4}(Z_{-})$
(2.4)

be the subgroup of pairs with equal image in $H^{4}(\Sigma)$. We define a
homomorphism

$Y:H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})\to H^{4}(M)$ (2.5)

as follows (_cf._ [4, Definition 4.13]). For $([\alpha_{+}],[\alpha_{-}])\in
H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})$, let $[\beta]$ be their common image in
$H^{4}(S)$. Then we may choose the cocycles $\alpha_{\pm}\in
C^{4}(Z_{\pm};\mathbb{Z})$ so that the restriction of $\alpha_{\pm}$ to
$\partial V_{\pm}\cong S^{1}\times\Sigma\subset Z_{\pm}$ is the pull-back of
$\beta$ by projection to the $\Sigma$ factor. Then the pull-back 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

$\flat^{\pm}:N_{\mp}\to N_{\pm}^{*}$ (2.6)

denote the homomorphism induced by the intersection form of $L$, and
$N_{\mp}^{\prime}\subseteq N_{\pm}^{*}$ the image of $\flat^{\pm}$. Since
$N_{\pm}^{*}\subset H^{4}(Z_{\pm})$, we can also regard $N^{\prime}_{\mp}$ as
a subset of $H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})$.

######  Lemma 2.9.

The image of (2.5) is the direct summand
$H^{4}(S)\oplus(L/_{N_{-}+T_{+}})\oplus(L/_{N_{+}+T_{-}})$ of $H^{4}(M)$, and
the kernel is $N^{\prime}_{-}\oplus N^{\prime}_{+}\subset
H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})$.

######  Remark 2.10.

The image of $Y$ contains all the torsion of $H^{4}(M)$, which we can identify
as

$\operatorname{Tor}H^{4}(M)\cong\operatorname{Tor}N_{+}^{*}/N^{\prime}_{-}\oplus\operatorname{Tor}N_{-}^{*}/N^{\prime}_{+}.$
(2.7)

######  Proposition 2.11 ([4, Proposition 4.20/Remark 4.21]).

$p_{M}=-Y(c_{2}(Z_{+}),c_{2}(Z_{-})).$

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 $N^{\prime}_{\mp}\subset H^{4}(Z_{\pm})$ (which depends on the
matching).

##  3\. The generalised Eells-Kuiper invariant of twisted connected sums

The main result of this section is Theorem 3.4, 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. We compute $\mu(M)$ using
explicit $\textrm{spin}^{\textrm{c}}$ coboundaries.

Throughout this section, let $M$ be the twisted connected sum of a pair of
building blocks $(Z_{\pm},\Sigma_{\pm})$, using a matching
$\matheur{r}\colon\Sigma_{+}\to\Sigma_{-}$. We assume for simplicity that the
kernel of $H^{2}(Z_{\pm})\to H^{2}(\Sigma_{\pm})$ is generated by
$[\Sigma_{\pm}]$.

As in Definition 2.6, we use the tubular neighbourhood
$U_{\pm}\cong\Sigma_{\pm}\times\Delta\subset Z_{\pm}$ of $\Sigma_{\pm}$;
denote the coordinate on the unit disc $\Delta$ by $z$. Consider
$B_{\pm}:=Z_{\pm}\times\Delta$, and denote the coordinate on its unit disc
factor by $w$. Form an 8-manifold $\mathring{W}$ by gluing $B_{+}$ and $B_{-}$
along $U_{\pm}\times\Delta$, using the map

$\displaystyle\Sigma_{+}\times\Delta\times\Delta$
$\displaystyle\to\Sigma_{-}\times\Delta\times\Delta,$ $\displaystyle(x,z,w)$
$\displaystyle\mapsto(\matheur{r}(x),w,z).$

We can form a smooth compact manifold $W$ with boundary by attaching to
$\mathring{W}$ the result of gluing the boundaries
$\Sigma_{\pm}\times\partial\overline{\Delta}\times\partial\overline{\Delta}$
of $(Z_{\pm}\setminus U_{\pm})\times\partial\overline{\Delta}$ by
$(x,z,w)\mapsto(\matheur{r}(x),w,z)$ — which is precisely the twisted
connected sum $M$. Hence $W$ is a coboundary of $M$.

Since $Z_{\pm}$ are not spin, nor are $B_{\pm}$ or $W$. However, they are all
$\textrm{spin}^{\textrm{c}}$.

###  3.1. Cohomology

We compute the integral cohomology of $W$ 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 $k\leq 4$. One can of course
recover the remaining cohomology groups from Poincaré duality and universal
coefficients, but what we care about is a description for $k\leq 4$ that lets
us determine the characteristic classes and the intersection form.

Obviously $B_{+}\cap B_{-}\simeq\Sigma$ and $B_{\pm}\simeq Z_{\pm}$, so there
is a long exact sequence

$H^{k-1}(\Sigma)\to H^{k}(W)\to H^{k}(Z_{+})\oplus H^{k}(Z_{-})\to
H^{k}(\Sigma).$

Hence, using that $H^{2}(Z_{\pm})\to H^{2}(\Sigma)=L$ has image $N_{\pm}$ by
definition and is assumed to have kernel $\mathbb{Z}$, we find

$\displaystyle H^{1}(W)$ $\displaystyle=0$ $\displaystyle H^{2}(W)$
$\displaystyle\cong(N_{+}\cap N_{-})\oplus\mathbb{Z}^{2}$ $\displaystyle
H^{3}(W)$ $\displaystyle\cong L/(N_{+}+N_{-})\oplus H^{3}(Z_{+})\oplus
H^{3}(Z_{-})$ $\displaystyle H^{4}(W)$ $\displaystyle\cong
H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-}).$

where $H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})$ is defined as in (2.4). There is a
natural short exact sequence

$0\to N_{+}^{*}\oplus N_{-}^{*}\to
H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})\to\mathbb{Z}\to 0.$ (3.1)

We can describe the isomorphism $H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})\to
H^{4}(W)$ 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 pull-back of $\alpha_{0}$ on $D_{\pm}\times\Delta$, and form a
cocycle on $W$ by patching the pull-backs of $\alpha_{\pm}$ to $B_{\pm}$. It
is clear that the composition of this isomorphism with the restriction map
$H^{4}(W)\to H^{4}(M)$ equals the map $Y$ defined in (2.5).

######  Remark 3.1.

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 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

$H^{k-4}(\Sigma)\to H^{k-2}(Z_{+})\oplus H^{k-2}(Z_{-})\to H^{k}(W,M)\to
H^{k-3}(\Sigma),$

finding

$\displaystyle H^{1}(W,M)$ $\displaystyle=0$ $\displaystyle H^{2}(W,M)$
$\displaystyle\cong\mathbb{Z}^{2}$ $\displaystyle H^{3}(W,M)$
$\displaystyle=0$ $\displaystyle H^{4}(W,M)$
$\displaystyle\cong(H^{2}(Z_{+})\oplus H^{2}(Z_{-}))/\mathbb{Z}.$

Note there is a short exact sequence

$0\to\mathbb{Z}\to(H^{2}(Z_{+})\oplus H^{2}(Z_{-}))/\mathbb{Z}\to N_{+}\oplus
N_{-}\to 0.$

###  3.2. Characteristic classes

As emphasised in Remark 3.1, 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 $B_{+}$, $B_{-}$.

The restriction of $w_{2}(W)\in H^{2}(W)$ to $B_{\pm}$ is the pull-back of the
Poincaré dual of $\Sigma_{\pm}$ in $Z_{\pm}$. The $\textrm{spin}^{\textrm{c}}$
structures of $B_{\pm}$ patch up to a unique $\textrm{spin}^{\textrm{c}}$
structure on $W$: this is essentially just saying that we can specify an
integral pre-image $z\in H^{2}(W)$ 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

$z^{2}=0\in H^{4}(W).$

Similarly we can pin down the $\textrm{spin}^{\textrm{c}}$ characteristic
class $\widehat{p}_{W}$. The restriction to $B_{\pm}$ is $-c_{2}+c_{1}^{2}$ of
the relevant $U(4)$-structure, so 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 [4, Proposition 4.20].

The image of $\widehat{p}_{W}$ in $H^{4}(\Sigma)\cong\mathbb{Z}$ is
$p_{\Sigma}=-c_{2}(\Sigma)=-24$, so in view of the short exact sequence (3.1)
the interesting information about $\widehat{p}_{W}$ is the pre-image 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
$H^{4}(Z)\to H^{4}(\Sigma)$ 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

$\bar{c}_{2}(Z)\in N^{*}\otimes\mathbb{Z}/24.$ (3.2)

The calculation above implies that $\widehat{p}_{W}\\!\\!\mod 24$ is
determined by $\bar{c}_{2}(Z_{\pm})$.

######  Lemma 3.2.

$\widehat{p}_{W}\\!\\!\mod 24$ is the image of
$(-\bar{c}_{2}(Z_{+}),-\bar{c}_{2}(Z_{-}))$.

######  Remark 3.3.

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.

###  3.3. Intersection form

The intersection pairing between $H^{4}(W)$ and $H^{4}(W,M)$ is simply the
natural duality between and $H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})$ and
$(H^{2}(Z_{+})\oplus H^{2}(Z_{-}))/\mathbb{Z}$. To understand the intersection
form we also need to describe the map $H^{4}(W,M)\to H^{4}(W)$.

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 $H^{4}(W,M)\to H^{4}(W)$
factors through the natural map $(H^{2}(Z_{+})\oplus
H^{2}(Z_{-}))/\mathbb{Z}\to N_{+}\oplus N_{-}$. Dually, the composition
$H^{4}(W)\cong H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})\to\mathbb{Z}$ 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 $N_{+}\oplus N_{-}\to N_{+}^{*}\oplus
N_{-}^{*}$.

The intersection form on each $B_{\pm}$ is obviously trivial, so there are no
diagonal terms. We end up with

$\displaystyle N_{+}\oplus N_{-}$ $\displaystyle\to N_{+}^{*}\oplus N_{-}^{*}$
(3.3) $\displaystyle(x_{+},\;x_{-})$
$\displaystyle\mapsto(\flat^{+}(x_{-}),\;\flat^{-}(x_{+})),$

where $\flat^{\pm}$ is the inner product homomorphism as in (2.6). The image
equals $N_{-}^{\prime}\oplus N_{+}^{\prime}\subset N_{+}^{*}\oplus N_{-}^{*}$,
where $N_{\mp}^{\prime}$ is the image of $\flat^{\pm}:N_{\mp}\to N_{\pm}^{*}$
as before. (This is consistent with the claim from Lemma 2.9 that the kernel
of $Y$ is precisely $N_{-}^{\prime}\oplus N_{+}^{\prime}$, and also with
Remark 3.3.) Both summands of $N^{\prime}_{-}\oplus N^{\prime}_{+}$ are
isotropic, so the signature of $W$ equals 0.

###  3.4. Computing the generalised Eells-Kuiper invariant

Suppose that the twisted connected sum $M$ has $H^{4}(M)$ torsion-free, so
that the description of the generalised Eells-Kuiper invariant from §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
$\widehat{d}:=\gcd(28,\textup{Num}\left(\frac{d}{4}\right))$ 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 non-trivial if $d$ is 8
or 24.

The $z^{4}$ term in (2.1) obviously makes no contribution since $z^{2}=0$, and
the signature term vanishes. 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 $Z_{\pm}\times\\{0\\}$. Thus $\bar{z}^{2}$
is twice the Poincaré dual to the K3 divisor (generator for the copy of
$\mathbb{Z}$ in $(H^{2}(Z_{+})\oplus H^{2}(Z_{-}))/\mathbb{Z}$). Since the K3
divisor has trivial normal bundle,
$\bar{z}^{2}\widehat{p}_{W}=2p_{\Sigma}=-48$, and
$\frac{5\bar{z}^{2}\widehat{p}_{W}}{12}$ is divisible by 4. Since the modulus
$\widehat{d}$ is 1 or 2, the only possible non-trivial contribution to the RHS
of (2.1) is $\frac{\widehat{\alpha}^{2}}{8}$.

The definition of $d$ means that $\widehat{p}_{W}\in N_{-}^{\prime}\oplus
N_{+}^{\prime}\\!\\!\mod d$. By Lemma 3.2, we can determine the elements
$[x_{\pm}]\in{N_{\pm}\otimes\mathbb{Z}/d}$ such that
$\widehat{p}_{W}=(\flat^{+}([x_{-}]),\flat^{-}([x_{+}]))\\!\\!\mod d$ from
$\bar{c}_{2}(Z_{\pm})$ and the configuration of embeddings $N_{\pm}\subset L$
of the matching. We may then take

$\widehat{\alpha}=(\flat^{+}(x_{-}),\flat^{-}(x_{+}))\in N_{-}^{\prime}\oplus
N_{+}^{\prime}.$

The pre-image under (3.3) of this $\widehat{\alpha}$ in $N_{+}\oplus N_{-}$ is
$(x_{+},\;x_{-})$, so $\widehat{\alpha}^{2}=2x_{+}.x_{-}$. Hence

######  Theorem 3.4.

Let $x_{\pm}\in N_{\pm}$ such that
$\flat^{\mp}(x_{\pm})=\bar{c}_{2}(Z_{\mp})\\!\\!\mod d$. Then

$\mu(M)\;=\;\frac{x_{+}.x_{-}}{4}\in\mathbb{Z}/\widehat{d}.$ (3.4)

$x_{\pm}$ 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 well-defined
$n:=\frac{1}{d}\widehat{p}_{W}\in H^{4}(Z_{+})\oplus_{0}H^{4}(Z_{-})$, and
that choice gives $\widehat{\alpha}=0$.

######  Corollary 3.5.

$\mu(M)=0$ 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 non-perpendicular matchings that can happen even when
$d$ is divisible by 8, in which case $\mu(M)$ is non-zero.

###  3.5. 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 $H^{4}(M)$ and its quadratic
refinement (_cf._ [6, Definition 2.15]).

Since the building blocks $Z_{\pm}$ are assumed to have no torsion in
$H^{3}(Z_{\pm})$, $W$ resolves all the torsion in $H^{4}(M)$. [4, Remark 4.12]
claims that the two summands $\operatorname{Tor}N_{\pm}^{*}/N^{\prime}_{\mp}$
of $\operatorname{Tor}H^{4}(M)$ 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 quadratic refinement of the
linking form. Because $\widehat{p}_{W}$ is even by Remark 3.3, we can
characterise the quadratic linking family on $\operatorname{Tor}H^{4}(M)$ 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 3.6.

$\operatorname{Tor}N_{\pm}^{*}/N^{\prime}_{\mp}\subseteq\operatorname{Tor}H^{4}(M)$
is isotropic for the torsion linking form, and Lagrangian for $q$.

###### Proof.

We can write any
$c\in\operatorname{Tor}\operatorname{Tor}N_{\pm}^{*}/N^{\prime}_{\mp}\subset
H^{4}(M)$ 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 $N_{-}^{\prime}$. 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 $q(c)=0$. ∎

##  4\. Fano-type blocks

We now describe how to construct building blocks for the twisted connected
sums from closed Fano 3-folds, and list detailed data for blocks obtained from
Fanos of Picard rank 1 or 2.

###  4.1. Construction of building blocks from Fanos

Recall that a Fano 3-fold is a smooth closed complex 3-fold $Y$ such that
$-K_{Y}$ is ample, or equivalently, $c_{1}(Y)$ can be represented by a Kähler
class. Any Fano 3-fold is simply connected, so $\operatorname{Pic}Y\cong
H^{2}(Y)$. Together with a natural form, this is a key deformation invariant
of $Y$.

######  Definition 4.1.

The _Picard lattice_ of a Fano 3-fold $Y$ is $N:=H^{2}(Y)$ equipped with the
quadratic form $(D_{1},D_{2})\mapsto D_{1}\cdot D_{2}\cdot(-K_{Y})$ (where
$\cdot$ is the cup product on $H^{2}(Y)$).

There are 105 deformation types of Fano 3-folds. Except for two of those, any
Fano 3-fold $Y$ has a pencil in $|{-}K_{Y}|$ with smooth base locus $C$.

######  Proposition 4.2 ([3, Propositions 4.24 and 5.7], _cf._ [4, Proposition
3.17]).

The blow-up $Z$ of $Y$ at the smooth base locus $C$ of an anticanonical
pencil, together with the proper transform $\Sigma$ of a divisor in said
pencil, is a building block. Further

  1. (i)

The image of $H^{2}(Z)\to H^{2}(\Sigma)$ is isomorphic to $H^{2}(Y)$, and the
kernel is generated by $[\Sigma]$.

  2. (ii)

The image in $H^{1,1}(\Sigma)$ of the Kähler cone of $Z$ contains the image of
the Kähler cone of $Y$.

We call such $(Z,\Sigma)$ _Fano-type_ blocks. Some of the important data of
the block can be read off immediately from well-known data about $Y$. For
instance (i) implies that the polarising lattice of the block is isometric to
the Picard lattice of $Y$, and (_cf._ [3, Lemma 5.6])

$b_{3}(Z)=-K_{Y}^{3}+b_{3}(Y)+2.$ (4.1)

In view of Lemma 3.2, the interesting characteristic class information about
$Z$ is $\bar{c}_{2}(Z)\in N^{*}\otimes\mathbb{Z}/24$. It is explained before
Lemma 5.18 of [3] that
$c_{2}(Z)=g(c_{2}(Y)+c_{1}(Y)^{2}))+24[\mathbb{P}^{1}]$, where
$\mathbb{P}^{1}$ is a fibre in the exceptional divisor of the blow-up $Z\to
Y$, and $g$ is the composition of the isomorphisms $H^{4}(Y)\cong N^{*}\cong
H^{4}(Z,\Sigma)$. Thus we can think of $\bar{c}_{2}(Z)\in N^{*}\otimes 24$
simply as

$\bar{c}_{2}(Z)=c_{2}(Y)+c_{1}(Y)^{2}\\!\\!\mod 24.$ (4.2)

######  Definition 4.3.

A _matching_ of a pair of Fano 3-folds $Y_{\pm}$ is a diffeomorphism
$\matheur{r}:\Sigma_{+}\to\Sigma_{-}$ of smooth K3 divisors
$\Sigma_{\pm}\in|{-}K_{Y_{\pm}}|$ that is a hyper-Kähler rotation (in the
sense of Definition 2.4) of some hyper-Kähler triples
$(\omega^{I}_{\pm},\omega^{J}_{\pm},\omega^{K}_{\pm})$ on $\Sigma_{\pm}$ such
that $\omega^{I}_{\pm}$ is the restriction of a Kähler form on $Y_{\pm}$.

Say that $\matheur{r}$ is a _perpendicular_ matching if the subspaces
$\matheur{r}^{*}H^{2}(Y_{-})$ and $H^{2}(Y_{+})$ of $H^{2}(\Sigma_{+})$ are
perpendicular with respect to the intersection form on $H^{2}(\Sigma_{+})$.

Proposition 4.2(ii) immediately implies that a matching of Fano 3-folds gives
rise to a matching (in the sense of Definition 2.5) of the resulting Fano-type
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 Fano-type blocks.

$\begin{array}[t]{l c c c c c}\hline\cr\hline\cr
Y&r&-K_{Y}^{3}&b_{3}(Y)&b_{3}(Z)&\bar{c}_{2}(Z)\\\
\hline\cr\mathbb{P}^{3}&4&4^{3}&0&66&22\\\ Q&3&3^{3}\cdot 2&0&56&2\\\
V_{1}&2&2^{3}&42&52&16\\\ V_{2}&2&2^{3}\cdot 2&20&38&20\\\ V_{3}&2&2^{3}\cdot
3&10&36&24\\\ V_{4}&2&2^{3}\cdot 4&4&38&4\\\ V_{5}&2&2^{3}\cdot 5&0&42&8\\\
&1&2&104&108&2\\\ &1&4&60&66&4\\\ &1&6&40&48&6\\\ &1&8&28&38&8\\\
&1&10&20&32&10\\\ &1&12&14&28&12\\\ &1&14&10&26&14\\\ &1&16&6&24&16\\\
&1&18&4&24&18\\\ &1&22&0&24&22\\\ \hline\cr\hline\cr\end{array}$ Table 1. Rank
1 Fano blocks

###  4.2. Rank 1 blocks

Table 1 summarises the key data of Fano 3-folds of rank 1 and the resulting
building blocks (_cf._ [3, Table 1]). The data included is the index $r$
(_i.e._ the largest integer such that $-K_{Y}=rH$ for some
$H\in\operatorname{Pic}Y$), the anticanonical degree $-K_{Y}^{3}$, $b_{3}(Y)$,
$b_{3}(Z)$, and the evaluation of $\bar{c}_{2}(Z)$ (the mod 24 reduction of
$c_{2}(Z)$) on the positive generator $H\in\operatorname{Pic}Y$.

$b_{3}(Z)$ is simply obtained from the preceding data by (4.1). In the rank 1
case, $\bar{c}_{2}(Z)$ is also easily determined as follows: For any Fano one
has $c_{2}(Y)(-K_{Y})=24$, so if $-K_{Y}=rH$ then (4.2) implies that

$\bar{c}_{2}(Z)H=\frac{24-K_{Y}^{3}}{r}\mod 24.$ (4.3)

The self-intersection of the generator of $N$ (which is not mentioned in the
table) is simply $\frac{-K_{Y}^{3}}{r^{2}}$.

###  4.3. The table of rank 2 Fano blocks

In Table 2 we collect the following data for building blocks obtained from
rank 2 Fanos $Y$.

  * •

The number of the corresponding entry in the Mori-Mukai 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. (We do not know a general reason for 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 $\bar{c}_{2}(Z)\in N^{*}\otimes\mathbb{Z}/24$ defined in (3.2),
presented in the dual basis of $G$, $H$.

  * •

The third Betti number $b_{3}(Y)$ of the Fano.

  * •

The third Betti number $b_{3}(Z)$ of the resulting block.

$\begin{array}[b]{c@{\hspace{8pt}}cccccccc@{\hspace{5pt}}crr}\hline\cr\hline\cr\text{MM\\#}\hfil\hskip
8.0&-K_{Y}^{3}&N&\Delta&-K_{Y}&\bar{c}_{2}(Z)&b_{3}(Y)&b_{3}(Z)&\bar{c}_{2}(Z)A\hfil\hskip
5.0&\;A^{2}&B^{2}&h\\\ \hline\cr 1\hfil\hskip
8.0&4&\left(\begin{smallmatrix}0&1\\\
1&2\end{smallmatrix}\right)&1&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&*&44&*\\\\[3.00003pt] 2\hfil\hskip
8.0&6&\left(\begin{smallmatrix}0&2\\\
2&2\end{smallmatrix}\right)&4&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&18\\!\end{smallmatrix}\right)&40&48&6\hfil\hskip
5.0&6&-6&-0.58\\\\[3.00003pt] 3\hfil\hskip
8.0&8&\left(\begin{smallmatrix}0&2\\\
2&4\end{smallmatrix}\right)&4&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&20\\!\end{smallmatrix}\right)&22&32\\\\[3.00003pt]
4\hfil\hskip 8.0&10&\left(\begin{smallmatrix}0&3\\\
3&4\end{smallmatrix}\right)&9&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&22\\!\end{smallmatrix}\right)&20&32&10\hfil\hskip
5.0&10&-90&-0.15\\\\[3.00003pt] 5\hfil\hskip
8.0&12&\left(\begin{smallmatrix}0&3\\\
3&6\end{smallmatrix}\right)&9&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&0\\!\end{smallmatrix}\right)&12&26&12\hfil\hskip
5.0&12&-12&-0.42\\\\[3.00003pt] 6\hfil\hskip
8.0&12&\left(\begin{smallmatrix}2&4\\\
4&2\end{smallmatrix}\right)&12&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&18\\!\end{smallmatrix}\right)&18&32&12\hfil\hskip
5.0&12&-4&0\\\\[3.00003pt] 7\hfil\hskip 8.0&14&\left(\begin{smallmatrix}0&4\\\
4&6\end{smallmatrix}\right)&16&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&2\\!\end{smallmatrix}\right)&10&26&14\hfil\hskip
5.0&14&-56&0.19\\\\[3.00003pt] 8\hfil\hskip
8.0&14&\left(\begin{smallmatrix}2&4\\\
4&4\end{smallmatrix}\right)&8&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&20\\!\end{smallmatrix}\right)&18&34\\\\[3.00003pt]
9\hfil\hskip 8.0&16&\left(\begin{smallmatrix}2&5\\\
5&4\end{smallmatrix}\right)&17&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&22\\!\end{smallmatrix}\right)&10&28&16\hfil\hskip
5.0&16&-272&0.09\\\\[3.00003pt] 10\hfil\hskip
8.0&16&\left(\begin{smallmatrix}0&4\\\
4&8\end{smallmatrix}\right)&16&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&4\\!\end{smallmatrix}\right)&6&24&16\hfil\hskip
5.0&16&-16&0\\\ \hfil\hskip 8.0&&&&&&&&4\hfil\hskip
5.0&24&-24&-0.58\\\\[3.00003pt] 11\hfil\hskip
8.0&18&\left(\begin{smallmatrix}2&5\\\
5&6\end{smallmatrix}\right)&13&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&0\\!\end{smallmatrix}\right)&10&30&18\hfil\hskip
5.0&18&-234&-0.47\\\\[3.00003pt] 12\hfil\hskip
8.0&20&\left(\begin{smallmatrix}4&6\\\
6&4\end{smallmatrix}\right)&20&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!22&22\\!\end{smallmatrix}\right)&6&28&20\hfil\hskip
5.0&20&-4&0\\\\[3.00003pt] 13\hfil\hskip
8.0&20&\left(\begin{smallmatrix}2&6\\\
6&6\end{smallmatrix}\right)&24&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&2\\!\end{smallmatrix}\right)&4&26&20\hfil\hskip
5.0&20&-30&0.26\\\\[3.00003pt] 14\hfil\hskip
8.0&20&\left(\begin{smallmatrix}0&5\\\
5&10\end{smallmatrix}\right)&25&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&8\\!\end{smallmatrix}\right)&2&24&20\hfil\hskip
5.0&20&-20&0.32\\\ \hfil\hskip 8.0&&&&&&&&8\hfil\hskip 5.0&30&-30&-0.26\\\
\hfil\hskip 8.0&&&&&&&&20\hfil\hskip 5.0&40&-40&-0.68\\\\[3.00003pt]
15\hfil\hskip 8.0&22&\left(\begin{smallmatrix}6&6\\\
6&4\end{smallmatrix}\right)&12&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!0&22\\!\end{smallmatrix}\right)&8&32\\\\[3.00003pt]
16\hfil\hskip 8.0&22&\left(\begin{smallmatrix}2&6\\\
6&8\end{smallmatrix}\right)&20&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&4\\!\end{smallmatrix}\right)&4&28&22\hfil\hskip
5.0&22&-110&-0.14\\\\[3.00003pt] 17\hfil\hskip
8.0&24&\left(\begin{smallmatrix}4&7\\\
7&6\end{smallmatrix}\right)&25&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!22&2\\!\end{smallmatrix}\right)&2&28&24\hfil\hskip
5.0&24&-600&0.06\\\\[3.00003pt] 18\hfil\hskip
8.0&24&\left(\begin{smallmatrix}0&4\\\
4&2\end{smallmatrix}\right)&16&\left(\begin{smallmatrix}1\\\
2\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&18\\!\end{smallmatrix}\right)&4&30&6\hfil\hskip
5.0&10&-40&0.68\\\ \hfil\hskip 8.0&&&&&&&&18\hfil\hskip 5.0&18&-72&-0.17\\\
\hfil\hskip 8.0&&&&&&&&24\hfil\hskip 5.0&24&-6&-0.58\\\\[3.00003pt]
19\hfil\hskip 8.0&26&\left(\begin{smallmatrix}4&7\\\
7&8\end{smallmatrix}\right)&17&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!22&4\\!\end{smallmatrix}\right)&4&32&2\hfil\hskip
5.0&26&-442&-0.61\\\\[3.00003pt] 20\hfil\hskip
8.0&26&\left(\begin{smallmatrix}2&7\\\
7&10\end{smallmatrix}\right)&29&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&8\\!\end{smallmatrix}\right)&0&28&2\hfil\hskip
5.0&26&-754&0.16\\\\[3.00003pt] 21\hfil\hskip
8.0&28&\left(\begin{smallmatrix}6&8\\\
8&6\end{smallmatrix}\right)&28&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!2&2\\!\end{smallmatrix}\right)&0&30&4\hfil\hskip
5.0&28&-4&0\\\\[3.00003pt] 22\hfil\hskip
8.0&30&\left(\begin{smallmatrix}4&8\\\
8&10\end{smallmatrix}\right)&24&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!22&8\\!\end{smallmatrix}\right)&0&32&6\hfil\hskip
5.0&30&-20&-0.32\\\\[3.00003pt] 23\hfil\hskip
8.0&30&\left(\begin{smallmatrix}8&8\\\
8&6\end{smallmatrix}\right)&16&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!4&2\\!\end{smallmatrix}\right)&2&34\\\\[3.00003pt]
24\hfil\hskip 8.0&30&\left(\begin{smallmatrix}2&5\\\
5&2\end{smallmatrix}\right)&21&\left(\begin{smallmatrix}2\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&18\\!\end{smallmatrix}\right)&0&32&12\hfil\hskip
5.0&14&-6&0.58\\\ \hfil\hskip 8.0&&&&&&&&6\hfil\hskip
5.0&30&-70&-0.51\\\\[3.00003pt] 25\hfil\hskip
8.0&32&\left(\begin{smallmatrix}0&4\\\
4&4\end{smallmatrix}\right)&16&\left(\begin{smallmatrix}1\\\
2\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&22\\!\end{smallmatrix}\right)&1&34&10\hfil\hskip
5.0&12&-12&0.42\\\ \hfil\hskip 8.0&&&&&&&&22\hfil\hskip
5.0&20&-20&-0.32\\\\[3.00003pt] 26\hfil\hskip
8.0&34&\left(\begin{smallmatrix}6&9\\\
9&10\end{smallmatrix}\right)&21&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!2&8\\!\end{smallmatrix}\right)&0&36\\\\[3.00003pt]
27\hfil\hskip 8.0&38&\left(\begin{smallmatrix}2&5\\\
5&4\end{smallmatrix}\right)&17&\left(\begin{smallmatrix}1\\\
2\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&22\\!\end{smallmatrix}\right)&0&40&16\hfil\hskip
5.0&16&-272&0.09\\\\[3.00003pt] 28\hfil\hskip
8.0&40&\left(\begin{smallmatrix}18&9\\\
9&4\end{smallmatrix}\right)&9&\left(\begin{smallmatrix}1\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&22\\!\end{smallmatrix}\right)&2&44\\\\[3.00003pt]
29\hfil\hskip 8.0&40&\left(\begin{smallmatrix}0&4\\\
4&6\end{smallmatrix}\right)&16&\left(\begin{smallmatrix}2\\\
1\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&2\\!\end{smallmatrix}\right)&0&42&14\hfil\hskip
5.0&14&-56&0.19\\\\[3.00003pt] 30\hfil\hskip
8.0&46&\left(\begin{smallmatrix}6&6\\\
6&4\end{smallmatrix}\right)&12&\left(\begin{smallmatrix}1\\\
2\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!2&22\\!\end{smallmatrix}\right)&0&48\\\\[3.00003pt]
31\hfil\hskip 8.0&46&\left(\begin{smallmatrix}2&5\\\
5&6\end{smallmatrix}\right)&13&\left(\begin{smallmatrix}1\\\
2\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&2\\!\end{smallmatrix}\right)&0&48&20\hfil\hskip
5.0&18&-234&-0.47\\\\[3.00003pt] 32\hfil\hskip
8.0&48&\left(\begin{smallmatrix}2&4\\\
4&2\end{smallmatrix}\right)&12&\left(\begin{smallmatrix}2\\\
2\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&18\\!\end{smallmatrix}\right)&0&50&12\hfil\hskip
5.0&12&-4&0\\\\[3.00003pt] 33\hfil\hskip
8.0&54&\left(\begin{smallmatrix}0&3\\\
3&4\end{smallmatrix}\right)&9&\left(\begin{smallmatrix}1\\\
3\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&22\\!\end{smallmatrix}\right)&0&56&10\hfil\hskip
5.0&10&-90&-0.15\\\\[3.00003pt] 34\hfil\hskip
8.0&54&\left(\begin{smallmatrix}0&3\\\
3&2\end{smallmatrix}\right)&9&\left(\begin{smallmatrix}2\\\
3\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!12&18\\!\end{smallmatrix}\right)&0&56&6\hfil\hskip
5.0&8&-72&0.17\\\ \hfil\hskip 8.0&&&&&&&&18\hfil\hskip
5.0&14&-126&-0.64\\\\[3.00003pt] 35\hfil\hskip
8.0&56&\left(\begin{smallmatrix}2&4\\\
4&4\end{smallmatrix}\right)&8&\left(\begin{smallmatrix}2\\\
2\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&22\\!\end{smallmatrix}\right)&0&58\\\\[3.00003pt]
36\hfil\hskip 8.0&62&\left(\begin{smallmatrix}2&5\\\
5&10\end{smallmatrix}\right)&5&\left(\begin{smallmatrix}1\\\
2\end{smallmatrix}\right)&\left(\begin{smallmatrix}\\!18&10\\!\end{smallmatrix}\right)&0&64\\\\[3.00003pt]
\hline\cr\hline\cr\end{array}$ Table 2. Rank 2 Fano blocks

The final three columns include data relevant for non-perpendicular matching,
about $A\in N$ such that $A^{2}$ is not too large compared with $\Delta$. In
view of Lemmas 5.8 and 5.11, non-perpendicular matchings of a pair of rank 2
Fanos $Y_{+},Y_{-}$ are only possible if

$A_{+}^{2}A_{-}^{2}\leq\Delta_{+}\Delta_{-}.$ (4.4)

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
$A^{2}\leq 1.6\Delta$. (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 $\bar{c}_{2}(Z)$ on $A$.

  * •

$A^{2}$, 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.

  * •

$h:=\log_{2}\frac{\Delta}{A^{2}}$.

###  4.4. Annotated Mori-Mukai list of rank 2 Fanos

We now indicate how the data in Table 2 is assembled. The anticanonical degree
$-K_{Y}^{3}$ and $b_{3}(Y)$ are taken from Iskovskih-Prokhorov [9, 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
$\bar{c}_{2}(Z)$ we divide the list (except the last two entries) into three
groups.

No fewer than 27 of the 36 classes of Fanos are described explicitly as blow-
ups of a rank 1 Fano $Y_{0}$ of index $r\geq 2$ in a smooth curve. Let $n$ be
the smallest integer such that $C$ is cut out by divisors in
$|{-}\frac{n}{r}K_{Y_{0}}|$ (always $1\leq n\leq r-1$). Let $E$ denote the
exceptional divisor. Then we can take $G=\pi^{*}({-}\frac{n}{r}K_{Y_{0}})-E$
and $H:=\pi^{*}({-}\frac{1}{r}K_{Y_{0}})$ as the generators for the nef cone
of $Y$, _i.e._ $G$ is the proper transform of a divisor in
$|{-}\frac{n}{r}K_{Y_{0}}|$ that contains $C$, and $H$ is the pull-back of the
hyperplane bundle $-\frac{1}{r}K_{Y_{0}}$. The quadratic form on $N$ can be
computed from

$\displaystyle H^{2}\;$ $\displaystyle=\;(-K_{Y_{0}})^{3},$
$\displaystyle(nH{-}G).H\;$ $\displaystyle=\;\deg C,$
$\displaystyle(nH{-}G)^{2}\;$ $\displaystyle=\;-\chi(C)$

We can apply (4.3) to read off $c_{2}(Z)(-K_{Y})$ from the other data in the
table. To control the other half of $\bar{c}_{2}(Z)$, note that
$c_{2}(Y)+c_{1}(Y)^{2}$ evaluated on $\pi^{*}(-K_{Y_{0}})$ equals
$c_{2}(Y_{0})+c_{1}(Y_{0})^{2}$ evaluated on $-K_{Y_{0}}$. We can thus apply
(4.3) again to obtain

$\bar{c}_{2}(Z)H=\frac{24-K_{Y_{0}}^{3}}{r_{0}}$ (4.5)

(and the RHS is contained in Table 1).

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 $-K_{Y}=(3-a)G+(3-b)H$. Further,
$c(Y)=(1+3G+3G^{2})(1+3H+3H^{2})(1+[Y])^{-1}$ gives

$\displaystyle c_{2}(Y)+c_{1}(Y)^{2}$
$\displaystyle=3G^{2}+9HG+3G^{2}-(3G+3H)[Y]+[Y]^{2}+(3G+3H-[Y])^{2}$
$\displaystyle=12G^{2}+27HG+12H^{2}-(9G+9H)[Y]+2[Y]^{2}.$

To evaluate it on $G$ we compute

$(c_{2}(Y)+c_{1}(Y)^{2})[Y]G=3((2a-3)(b-1)(b-2)+6)G^{2}H^{2},$ (4.6)

and the evaluation on $H$ is analogous.

Another 3 cases are branched double covers over other rank 2 Fanos, $\pi:Y\to
X$. Then we can take $G$ and $H$ to be pull-backs of the basis for
$\operatorname{Pic}X$. If the branch locus is a smooth divisor in $|2L|$, then
$TY\oplus\pi^{*}(2L)\cong\pi^{*}(TX\oplus L)$ implies
$c_{1}(Y)=\pi^{*}(c_{1}(X)-[L])$ and
$c_{2}(Y)=\pi^{*}(c_{2}(X)-c_{1}(X)[L]+2[L]^{2})$, so

$c_{2}(Y)+c_{1}(Y)^{2}=\pi^{*}\left(c_{2}(X)+c_{1}(X)^{2}+3[L]([L]-c_{1}(X))\right).$
(4.7)

For each entry $Y$ in the Mori-Mukai list, we repeat below the description
from [9, Table 12.3]. For blow-ups 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. #1

Blow-up of $V_{1}$ (degree 1 del Pezzo 3-fold, 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 pre-image $\mathbb{P}^{1}$ over the base point of
$|{-}{\textstyle\frac{1}{2}}K_{V_{1}}|$. It is therefore the only case where
we cannot apply Proposition 4.2 to construct an associated “Fano-type”
building block $Z$ (though one could define a building block by blowing up in
several steps [3, Proposition 5.9]).

  2. #2

Double cover of $\mathbb{P}^{1}\times\mathbb{P}^{2}$ branched over $(2,4)$
divisor. $G$ and $H$ are the pull-backs of $\mathcal{O}_{\mathbb{P}^{1}}(1)$
and $\mathcal{O}_{\mathbb{P}^{2}}(1)$ respectively. Use (4.7) (and result from
#34) to compute

$c_{2}(Y)+c_{1}(Y)^{2}=18GH+12H^{2}+3(G+2H)(-G-H)=9GH+6H^{2},$

and hence
$\bar{c}_{2}(Z)=\left(\begin{smallmatrix}\\!12&18\\!\end{smallmatrix}\right)$.

  3. #3

Blow-up of $V_{2}$ (degree 2 del Pezzo 3-fold, 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. #4

Blow-up of $\mathbb{P}^{3}$ along the intersection of two cubic hypersurfaces.
$n=3$.

  5. #5

Blow-up of $V_{3}$ (cubic hypersurface of $\mathbb{P}^{4}$) along the
intersection of two hyperplane divisors (a plane cubic curve). $n=1$.

  6. #6

(2,2) divisor in $\mathbb{P}^{2}\times\mathbb{P}^{2}$, or double cover of $W$
($(1,1)$ divisor in $\mathbb{P}^{2}\times\mathbb{P}^{2}$) branched over smooth
divisor $B\in|{-}K_{W}|$. $G$ and $H$ are pull-backs of $\mathcal{O}(1)$ from
the two $\mathbb{P}^{2}$ factors, and $\bar{c}_{2}(Z)$ is computed from (4.6).

  7. #7

Blow-up of $Q$ (quadric hypersurface in $\mathbb{P}^{4}$) in intersection of
two sections by quadrics. $n=2$.

  8. #8

Double cover of $V_{7}$ ($\mathbb{P}^{3}$ blown up in a point, #35) whose
branch locus is a divisor $B\in|{-}K_{V_{7}}|$ such that the intersection
$B\cap E$ with the exceptional divisor of $V_{7}\to\mathbb{P}^{3}$ is smooth,
or reduced but not smooth. $G$ and $H$ are the pull-backs of the respective
classes on $V_{7}$. By (4.7) we have
$c_{2}(Y)+c_{1}(Y)^{2}=\pi^{*}(c_{2}(V_{7})+c_{1}(V_{7})^{2}-3(-{\textstyle\frac{1}{2}}K_{V_{7}})^{2}$.
We compute in #35 that
$c_{2}(V_{7})+c_{1}(V_{7})^{2}=\left(\begin{smallmatrix}\\!18&22\\!\end{smallmatrix}\right)$
in terms of the basis for $V_{7}$, and we can read off from the Picard lattice
that
$(-{\textstyle\frac{1}{2}}K_{V_{7}})^{2}=\left(\begin{smallmatrix}\\!3&4\\!\end{smallmatrix}\right)$.
Hence
$c_{2}(Y)+c_{1}(Y)^{2}=\pi^{*}\\!\left(\begin{smallmatrix}\\!9&10\\!\end{smallmatrix}\right)$,
which in terms of the basis for $\operatorname{Pic}Y$ is written as
$\left(\begin{smallmatrix}\\!18&20\\!\end{smallmatrix}\right)$.

  9. #9

Blow-up of $\mathbb{P}^{3}$ in a curve $C$ of degree 7 and genus 5, which is
an intersection of a two-parameter family of cubic hypersurfaces. $n=3$.

  10. #10

Blow-up 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. #11

Blow-up of $V_{3}$ (cubic hypersurface in $\mathbb{P}^{4}$) along a line.
$n=1$.

  12. #12

Blow-up of $\mathbb{P}^{3}$ along a curve of degree 6 and genus 3 which is an
intersection of cubic hypersurfaces. $n=3$.

  13. #13

Blow-up of $Q$ (quadric hypersurface in $\mathbb{P}^{4}$) along a curve of
degree 6 and genus 2. $n=2$.

  14. #14

Blow-up of $V_{5}$ (section of Plücker-embedded Grassmannian
$Gr(2,5)\subset\mathbb{P}^{9}$ by a subspace of codimension 3) in elliptic
curve that is the intersection of two hyperplane sections. $n=1$.

  15. #15

Blow-up of $\mathbb{P}^{3}$ along the intersection of a quadric $A$ and a
cubic $B$ such that $A$ is smooth, or $A$ is reduced but not smooth. $n=3$.

  16. #16

Blow-up of $V_{4}$ (complete intersection of two quadrics in $\mathbb{P}^{5}$)
in a conic. $n=1$.

  17. #17

Blow-up of $Q$ along an elliptic curve of degree 5. $n=2$.

  18. #18

Double cover of $\mathbb{P}^{1}\times\mathbb{P}^{2}$ branched over $(2,2)$
divisor. Compute $\bar{c}_{2}(Z)=(12,18)$ using (4.7) like in #2.

  19. #19

Blow-up of $V_{4}$ along a line. $n=1$.

  20. #20

Blow-up of $V_{5}$ along a twisted cubic. $n=1$.

  21. #21

Blow-up of $Q$ along a twisted quartic (a rational degree 4 curve, isomorphic
to the image of $(s:t)\mapsto(s^{4}:s^{3}t:s^{2}t^{2}:st^{3}:t^{4})$). $n=2$.

  22. #22

Blow-up of $V_{5}$ along a conic. $n=1$.

  23. #23

Blow-up 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. #24

A $(1,2)$ divisor in $\mathbb{P}^{2}\times\mathbb{P}^{2}$. Compute
$\bar{c}_{2}(Z)$ by applying (4.6) with $a=1,b=2$.

  25. #25

Blow-up of $\mathbb{P}^{3}$ along an elliptic curve that is the intersection
of two quadrics. $n=2$.

  26. #26

Blow-up of $V_{5}$ along a line. $n=1$.

  27. #27

Blow-up of $\mathbb{P}^{3}$ along a twisted cubic. $n=2$.

  28. #28

Blow-up of $\mathbb{P}^{3}$ along a plane cubic (an elliptic curve). $n=3$.

  29. #29

Blow-up of $Q$ along a conic (complete intersection of two hyperplane
sections). $n=1$.

  30. #30

Blow-up of $\mathbb{P}^{3}$ along a conic. $n=2$.

  31. #31

Blow-up of $Q$ along a line. $n=1$.

  32. #32

A $(1,1)$-divisor on $\mathbb{P}^{2}\times\mathbb{P}^{2}$. Compute
$\bar{c}_{2}(Z)$ by applying (4.6) with $a=b=1$.

  33. #33

Blow-up of $\mathbb{P}^{3}$ along a line. $n=1$.

  34. #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
$\bar{c}_{2}(Z)=\left(\begin{smallmatrix}\\!12&18\\!\end{smallmatrix}\right)$
by applying (4.6) with $a=1$, $b=0$.

  35. #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 blow-up point, and $H$ is
a plane not passing through the blow-up point. $c_{2}(Y)+c_{1}(Y)^{2}$
evaluates to 22 on $H$, just as it does on a plane in $\mathbb{P}^{3}$.

  36. #36

$\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(2))$ over $\mathbb{P}^{2}$. $G$ is
the pull-back 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
$G^{3}=0$, $G^{2}H=1$, $GH^{2}=2$, $H^{3}=4$ (use that the section
$\mathbb{P}(\mathcal{O}(2))$ is a divisor representing $H$). Because $TY$ is
stably isomorphic to
$\pi^{*}T\mathbb{P}^{2}\oplus\left(H\otimes\pi^{*}(\mathcal{O}\oplus\mathcal{O}(-2))\right)$,
find that $c(Y)=(1+3G+3G^{2})(1+H)(1-2G+H)$, 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 evaluates to 18 on $G$ and
34 on $H$, so
$\bar{c}_{2}(Z)=\left(\begin{smallmatrix}\\!18&10\\!\end{smallmatrix}\right)$.

##  5\. Twisted connected sums of Fanos of rank 1 and 2

Combining the results described in §2.3 and §4.1, we can produce twisted
connected sum $G_{2}$-manifolds from matching pairs of Fano 3-folds. We now
apply the methods developed in [4] to the problem of _finding_ matchings
between Fanos of rank 1 and 2. While it is easy to find perpendicular
matchings, we also investigate systematically the possible non-perpendicular
matchings in order to exhibit examples with non-zero Eells-Kuiper invariant,
and to prove Theorem 1.1.

###  5.1. The matching problem

Let us first summarise the results from [4, §6] concerning how to reduce the
problem of finding matchings between Fano 3-folds to a combination of problems
in lattice arithmetic and deformation theory.

Let $\Sigma_{\pm}\subset Y_{\pm}$ be smooth anticanonical divisors of a pair
of Fanos, and $\matheur{r}:\Sigma_{+}\to\Sigma_{-}$ a matching. Let $h_{+}$ be
a _marking_ of $\Sigma_{+}$, _i.e._ an isometry $h_{+}:H^{2}(\Sigma_{+})\to L$
where $L$ is an abstract copy of the K3 lattice (the unique unimodular lattice
of signature (3,19)). Then $h_{-}:=h_{+}\circ\matheur{r}^{*}$ 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 well-defined up to the action of the
isometry group $O(L)$. Let us call it the _configuration_ of $\matheur{r}$.

We saw in §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 3.4 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
3-folds, 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 two-plane
$\Pi\subset L(\mathbb{R})$, the image under $h:H^{2}(\Sigma;\mathbb{R})\to
L(\mathbb{R})$ of the real and imaginary parts of classes in
$H^{2,0}(\Sigma;\mathbb{C})$.

######  Lemma 5.1.

Let $Y_{\pm}$ be a pair of Fano 3-folds, 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
$H^{2}(Y_{\pm})\to H^{2}(\Sigma_{\pm})\to L$.

###### Proof.

Necessity is trivial, setting
$k_{\pm}=h_{\pm}[\omega^{I}_{\pm}]=h_{\mp}[\omega^{J}_{\mp}]$ and $k_{0}=\pm
h_{\pm}[\omega^{K}_{\pm}]$ for the hyper-Kähler triples appearing in
Definition 4.3. Sufficiency relies on the Torelli theorem, _cf._ [4,
Proposition 6.2]. ∎

Let us now consider how this problem depends on the choice of configuration.
We first set up some notation for various lattices. Let

  * •

$W:=N_{+}+N_{-}$ (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 5.2.

$N_{\pm}\subseteq\Lambda_{\pm}$, with equality if and only if $N_{+}$ and
$N_{-}$ “intersect orthogonally”, _i.e._ when
$W(\mathbb{R})=W_{+}(\mathbb{R})\oplus
W_{-}(\mathbb{R})\oplus(N_{+}(\mathbb{R})\cap N_{-}(\mathbb{R}))$, or
equivalently, when reflections of $L(\mathbb{R})$ in the subspaces
$N_{+}(\mathbb{R})$ and $N_{-}(\mathbb{R})$ commute.

Note that 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

$k_{\pm}\in W_{\pm}(\mathbb{R}),\quad k_{0}\in T(\mathbb{R}).$ (5.1)

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 3-folds 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 _non-generic_ 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
$\operatorname{rk}\Lambda_{\pm}-\operatorname{rk}N_{\pm}$).

For configurations where $\Lambda_{\pm}=N_{\pm}$, we deduce below the
existence of matchings between some elements of $\mathcal{Y}_{+}$ and
$\mathcal{Y}_{-}$ from a general fact (due to Beauville [1]) that a generic
$N_{\pm}$-polarised K3 appears as an anticanonical divisor in some member of
$\mathcal{Y}_{\pm}$. To apply a similar argument for a configuration 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 will
refer to this process as ‘handcrafting’.

######  Remark 5.3.

Before we state our main existence result for matchings with a prescribed
configuration, let us point out some necessary conditions.

  1. (i)

Since $W(\mathbb{R})$ contains two positive classes $k_{+}$ and $k_{-}$, and
its orthogonal complement contains the positive class $k_{0}$, the quadratic
form on $W$ must be non-degenerate of signature $(2,w-2)$.

  2. (ii)

$W_{\pm}\subset N_{\pm}$ must contain some ample classes of $Y_{\pm}$.

  3. (iii)

Since $\Lambda_{+}\cap\Lambda_{-}\subset\operatorname{Pic}\Sigma_{\pm}$ and is
orthogonal to an ample class of $\Sigma_{\pm}$, it cannot contain any
$(-2)$-classes.

In particular, (ii) implies that any matching involving a Fano with Picard
rank $\operatorname{rk}N=1$ must be perpendicular.

In order to describe the required genericity property, we recall some further
terminology. The period domain is the space of oriented positive 2-planes in
$L(\mathbb{R})$. It can be identified with
$\\{\Pi\in\mathbb{P}(L(\mathbb{C})):\Pi\,\overline{\Pi}>0\\}$ in order to
exhibit a natural complex structure. Given $N\subset L$, the period domain of
$N$-polarised K3 surfaces is
$D_{N}:=\\{\Pi\in\mathbb{P}(N^{\perp}(\mathbb{C})):\Pi\,\overline{\Pi}>0\\}$.

######  Definition 5.4.

Let $\mathcal{Y}$ be a deformation family of Fano 3-folds, and pick a
primitive isometric embedding $N\subset L$ of its Picard lattice. Let
$\operatorname{Amp}_{\mathcal{Y}}$ be an open subcone of the positive cone in
$N(\mathbb{R})$, and $\Lambda\subset L$ a primitive sublattice that contains
$N$.

We say that $\mathcal{Y}$ is _$(\Lambda,\operatorname{Amp}_{\mathcal{Y}})$
-generic_ if there is $U_{\mathcal{Y}}\subseteq D_{N}$ with complement a
locally finite union of complex analytic submanifolds of positive codimension
with the property that: for any $\Pi\in U_{\mathcal{Y}}$ and
$k\in\operatorname{Amp}_{\mathcal{Y}}$ there is $Y\in\mathcal{Y}$, an
anticanonical divisor $\Sigma\subset Y$ and a marking $h:H^{2}(\Sigma)\to L$
such that $\Pi$ is the period of $(\Sigma,h)$, and $h^{-1}(k)$ is the image of
the restriction to $\Sigma$ of a Kähler class on $Y$. (These conditions entail
that $N$ is the image of $H^{2}(Y)\to H^{2}(\Sigma)\to L$.)

######  Proposition 5.5.

Let $\mathcal{Y}_{\pm}$ be a pair of deformation families of Fano 3-folds, and
pick a configuration of primitive isometric embeddings $N_{\pm}\subset L$ of
their Picard lattices. Define $W$, $W_{\pm}$ and $\Lambda_{\pm}$ as above.
Suppose that there exist open cones
$\operatorname{Amp}_{\mathcal{Y}_{\pm}}\subset N_{\pm}(\mathbb{R})$ such that

  * •

$\mathcal{Y}_{\pm}$ are
$(\Lambda_{\pm},\operatorname{Amp}_{\mathcal{Y}_{\pm}})$-generic,

  * •

$W_{\pm}\cap\operatorname{Amp}_{\mathcal{Y}_{\pm}}\neq\emptyset$.

Then there exist $Y_{\pm}\in\mathcal{Y}_{\pm}$ and anticanonical K3 divisors
$\Sigma_{\pm}\subset Y_{\pm}$ with a matching
$\matheur{r}:\Sigma_{+}\to\Sigma_{-}$ whose configuration is the given pair of
embeddings $N_{\pm}\subset L$.

###### Proof.

Like [4, Proposition 6.18]; note that $W$ is automatically an orthogonal push-
out of $\Lambda_{+}$ and $\Lambda_{-}$, in the terminology of [4]. ∎

###  5.2. Perpendicular matching

The simplest way to find a matching between elements of two deformation types
$\mathcal{Y}_{\pm}$ of Fano 3-folds is to consider configurations
$N_{+},N_{-}\subset L$ such that $N_{+}$ and $N_{-}$ are perpendicular. There
are two reasons for this. The first is arithmetic: for nearly all pairs
$\mathcal{Y}_{\pm}$ one has $\operatorname{rk}N_{+}+\operatorname{rk}N_{-}\leq
11$, in which case Nikulin [14, Theorem 1.12.4] guarantees that there does in
fact exist a primitive embedding of the perpendicular direct sum $N_{+}\perp
N_{-}$ into $L$. The second is that such configurations have
$\Lambda_{\pm}=N_{\pm}$, so the following genericity result is enough to let
us apply Proposition 5.5.

######  Proposition 5.6 ([3, Proposition 6.9], based on Beauville [1]).

Let $\mathcal{Y}$ be a deformation type of Fano 3-folds, and embed its Picard
lattice $N$ primitively in $L$. Then there exists an open cone
$\operatorname{Amp}_{\mathcal{Y}}\subset N(\mathbb{R})$ such that
$\mathcal{Y}$ is $(N,\operatorname{Amp}_{\mathcal{Y}})$-generic.

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 Fano-type building block.
Given one of the 1378 unordered pairs $\mathcal{Y}_{+}$, $\mathcal{Y}_{-}$
among the other 52 deformation types, we carry out following procedure:

  * •

Apply [14, Theorem 1.12.4] to find a primitive embedding $N_{+}\perp
N_{-}\subset L$.

  * •

Apply Proposition 5.5 to find a matching between some
$Y_{\pm}\in\mathcal{Y}_{\pm}$ with the given configuration $N_{+},N_{-}\subset
L$.

  * •

Apply Proposition 4.2 to produce a pair of Fano-type 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 2-connected, with $H^{4}(M)$ torsion-free,
and

$b_{3}(M)=b_{3}(Z_{+})+b_{3}(Z_{-})+23.$

Proposition 2.11 implies that $d(M)$, the greatest divisor of $p_{M}$, is the
greatest common divisor of $\bar{c}_{2}(Z_{+})$ and $\bar{c}_{2}(Z_{-})$,
while Corollary 3.5 shows that $\mu(M)=0$. Thus all the classifying
diffeomorphism invariants of $M$ can be determined from the data in Tables 1
and 2.

Table 3 lists the invariants of the 1378 twisted connected sums obtained this
way. A total of 131 different 2-connected 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 $b_{3}(M)$ [4, Table 3].

######  Remark 5.7.

According to Theorem 1.7 and Corollary 1.13 of [5], the torsion-free
$G_{2}$-structures of diffeomorphic 2-connected twisted connected sums with
$d$ not divisible by 3 are automatically homotopic (if one chooses the
diffeomorphism correctly). However, the table 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. In that
case it is not clear whether the associated torsion-free $G_{2}$-structures
are homotopic.

$b_{3}$ |  $\\#$ | d
---|---|---
|  |  2 |  4 |  6 |  8 |  12 |  24
71 |  15 |  8 |  5 |  1 |  1 |  0 |  0
73 |  20 |  16 |  3 |  1 |  0 |  0 |  0
75 |  40 |  35 |  3 |  1 |  0 |  1 |  0
77 |  39 |  36 |  0 |  2 |  0 |  1 |  0
79 |  73 |  65 |  3 |  4 |  0 |  1 |  0
81 |  65 |  60 |  1 |  4 |  0 |  0 |  0
83 |  76 |  66 |  3 |  6 |  1 |  0 |  0
85 |  65 |  51 |  8 |  4 |  1 |  1 |  0
87 |  74 |  66 |  4 |  3 |  0 |  1 |  0
89 |  62 |  54 |  5 |  2 |  1 |  0 |  0
91 |  50 |  46 |  2 |  2 |  0 |  0 |  0
93 |  49 |  45 |  4 |  0 |  0 |  0 |  0
95 |  52 |  49 |  0 |  2 |  0 |  0 |  1
97 |  49 |  42 |  3 |  3 |  1 |  0 |  0
99 |  55 |  43 |  7 |  3 |  2 |  0 |  0
101 |  32 |  25 |  1 |  5 |  1 |  0 |  0
103 |  65 |  54 |  3 |  7 |  1 |  0 |  0
105 |  45 |  42 |  0 |  3 |  0 |  0 |  0
107 |  45 |  40 |  1 |  3 |  1 |  0 |  0
109 |  34 |  31 |  0 |  3 |  0 |  0 |  0
111 |  42 |  39 |  0 |  2 |  1 |  0 |  0
113 |  43 |  37 |  5 |  0 |  1 |  0 |  0
115 |  30 |  28 |  1 |  1 |  0 |  0 |  0
117 |  29 |  27 |  1 |  0 |  1 |  0 |  0
119 |  31 |  28 |  0 |  3 |  0 |  0 |  0
121 |  30 |  27 |  1 |  2 |  0 |  0 |  0
123 |  18 |  17 |  0 |  1 |  0 |  0 |  0
125 |  9 |  8 |  1 |  0 |  0 |  0 |  0
127 |  20 |  14 |  3 |  2 |  1 |  0 |  0
129 |  11 |  10 |  0 |  1 |  0 |  0 |  0
131 |  9 |  8 |  1 |  0 |  0 |  0 |  0

$b_{3}$ |  $\\#$ | d
---|---|---
|  |  2 |  4 |  6 |  8 |  12 |  24
133 |  3 |  3 |  0 |  0 |  0 |  0 |  0
135 |  10 |  9 |  0 |  1 |  0 |  0 |  0
137 |  12 |  12 |  0 |  0 |  0 |  0 |  0
139 |  4 |  4 |  0 |  0 |  0 |  0 |  0
141 |  2 |  1 |  1 |  0 |  0 |  0 |  0
143 |  3 |  3 |  0 |  0 |  0 |  0 |  0
145 |  7 |  7 |  0 |  0 |  0 |  0 |  0
147 |  2 |  2 |  0 |  0 |  0 |  0 |  0
151 |  1 |  1 |  0 |  0 |  0 |  0 |  0
153 |  2 |  2 |  0 |  0 |  0 |  0 |  0
155 |  8 |  7 |  1 |  0 |  0 |  0 |  0
157 |  4 |  4 |  0 |  0 |  0 |  0 |  0
159 |  6 |  6 |  0 |  0 |  0 |  0 |  0
161 |  3 |  3 |  0 |  0 |  0 |  0 |  0
163 |  8 |  8 |  0 |  0 |  0 |  0 |  0
165 |  3 |  3 |  0 |  0 |  0 |  0 |  0
167 |  2 |  2 |  0 |  0 |  0 |  0 |  0
169 |  3 |  3 |  0 |  0 |  0 |  0 |  0
171 |  1 |  1 |  0 |  0 |  0 |  0 |  0
173 |  2 |  2 |  0 |  0 |  0 |  0 |  0
175 |  1 |  1 |  0 |  0 |  0 |  0 |  0
179 |  4 |  4 |  0 |  0 |  0 |  0 |  0
181 |  1 |  1 |  0 |  0 |  0 |  0 |  0
183 |  1 |  1 |  0 |  0 |  0 |  0 |  0
187 |  3 |  3 |  0 |  0 |  0 |  0 |  0
189 |  1 |  1 |  0 |  0 |  0 |  0 |  0
195 |  1 |  1 |  0 |  0 |  0 |  0 |  0
197 |  2 |  2 |  0 |  0 |  0 |  0 |  0
239 |  1 |  1 |  0 |  0 |  0 |  0 |  0
Total |  1378 |  1215 |  71 |  72 |  14 |  5 |  1

Table 3. Twisted connected sums from perpendicular matching of Fanos with
Picard rank 1 or 2

###  5.3. Orthogonal matching with non-trivial intersection

Next we consider matchings with configurations such that $N_{+}\cap N_{-}$ is
non-trivial. Then both Fanos must have Picard rank $\geq 2$. If we restrict
attention to the case when both Fanos have Picard rank precisely 2, then
Remark 5.3(ii) means that the only configurations with $N_{+}\cap N_{-}$ non-
trivial for which we can possibly find a matching are those where $N_{+}$ and
$N_{-}$ intersect orthogonally, in the sense of Remark 5.2. Such
configurations have $\Lambda_{\pm}=N_{\pm}$, so to apply Proposition 5.5 we do
not need any genericity result beyond Proposition 5.6 that we applied to find
perpendicular matchings.

Compared with the perpendicular matching problem, the difficulty of finding
matchings of rank 2 blocks with non-trivial intersection $N_{+}\cap N_{-}$ is
therefore one of lattice-arithmetic: 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 non-trivial, and
contains a class $A_{\pm}\in\operatorname{Amp}_{\mathcal{Y}_{\pm}}$.

######  Lemma 5.8.

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 $\Delta_{+}\Delta_{-}=k^{2}A_{+}^{2}A_{-}^{2}$ for
some integer $k$. Then $k$ is the generator of the image of the product
$N_{+}\times N_{-}\to\mathbb{Z}$ in $W$.

###### Proof.

Up to sign, $B_{+}$ and $B_{-}$ must have the same image $B\in W$, so
$B_{+}^{2}=B^{2}=B_{-}^{2}$.

Note that for a pair $(v_{+},v_{-})\in N_{+}\times N_{-}$, the product of
their images in $W$ is

$v_{+}.v_{-}=\frac{(v_{+}.B_{+})(v_{-}.B_{-})}{B^{2}}.$

In particular, if we let $a_{\pm}$ be the positive generator of the image of
$N_{\pm}\to\mathbb{Z},\;v\mapsto v.B_{\pm}$, then $a_{+}a_{-}=-kB^{2}$.
Conversely if $a_{+}a_{-}$ is divisible by $B^{2}$, then we can 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

$(a_{+}a_{-})^{2}=\frac{\Delta_{+}\Delta_{-}(B^{2})^{2}}{A_{+}^{2}A_{-}^{2}},$

and $a_{+}a_{-}$ is divisible by $B^{2}$ if and only if
$\frac{\Delta_{+}\Delta_{-}}{A_{+}^{2}A_{-}^{2}}$ is a perfect square. ∎

Next we summarise the relevant topological calculations.

######  Lemma 5.9.

Let $(Z_{\pm},\Sigma_{\pm})$ be a pair of building blocks whose polarising
lattices $N_{\pm}$ have rank 2, such that the kernel of $H^{2}(Z_{\pm})\to
H^{2}(\Sigma_{\pm})$ is generated by $[\Sigma_{\pm}]$. Let
$\matheur{r}:\Sigma_{+}\to\Sigma_{-}$ 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 vectors
spanning the orthogonal complement of $N_{\mp}$ in $N_{\pm}$. Then

  1. (i)

$H^{2}(M)\cong\mathbb{Z}$,

  2. (ii)

$b_{3}(M)=b_{3}(Z_{+})+b_{3}(Z_{-})+22$,

  3. (iii)

$\operatorname{Tor}H^{3}(M)\cong\operatorname{Tor}L/W$,

  4. (iv)

$\operatorname{Tor}H^{4}(M)\cong(\mathbb{Z}/k)^{2}$, for $k$ as in Lemma 5.8,

  5. (v)

If $k=1$ then $d(M)$ (which divides 24) is the greatest common divisor of
$\bar{c}_{2}(Z_{\pm})A_{\pm}\in\mathbb{Z}/24$.

  6. (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

$\mu(M)=\frac{\operatorname{gd}(\bar{c}_{2}(Z_{+}))\operatorname{gd}(\bar{c}_{2}(Z_{-}))}{4}\in\mathbb{Z}/2,$

where $\operatorname{gd}(\bar{c}_{2}(Z_{\pm}))\in\\{2,4,6,8,12,24\\}$ is the
greatest divisor of $\bar{c}_{2}(Z_{\pm})$.

###### Proof.

(i)-(iii) are immediate consequences of Theorem 2.8.

The image $N^{\prime}_{\mp}$ of the product homomorphism
$\flat^{\pm}:N_{\mp}\to N_{\pm}^{*}$ from (2.6) has rank 1, and Lemma 5.8
implies that it has cotorsion $\mathbb{Z}/k$ in $N_{\pm}^{*}$. (iv) now
follows from (2.7).

When $k=1$, so that $N^{\prime}_{\mp}\subset N_{\pm}^{*}$ is primitive, the
isomorphism $N_{\pm}^{*}/N^{\prime}_{\mp}\cong\mathbb{Z}$ 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
$\frac{\operatorname{gd}(\bar{c}_{2}(Z_{\pm}))}{2}$. Since $k=1$, we find

$\frac{x_{+}}{2}.\frac{x_{-}}{2}\;=\;\frac{\operatorname{gd}(\bar{c}_{2}(Z_{+}))}{2}\,\frac{\operatorname{gd}(\bar{c}_{2}(Z_{-}))}{2}\mod
2,$

and (vi) follows from Theorem 3.4. ∎

Six examples of matchings with of rank 2 Fanos with non-trivial intersection
$N_{+}\cap N_{-}$ are listed in [4, Example No 9]. However, Lemma 5.8 and the
data in Table 4 allow us to be more decisive.

######  Theorem 5.10.

There are precisely 18 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 torsion-free 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 4 we list the following data about the 18 pairs:

  * •

the numbers $\\#_{\pm}$ of the entries in the Mori-Mukai 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.

$\begin{array}[b]{rr<EMAIL_ADDRESS>12.0&\makebox[0.0pt]{$B^{2}$}&\makebox[0.0pt]{$A_{+}^{2}$}&\makebox[0.0pt]{$A_{-}^{2}$}&\makebox[0.0pt]{$b_{3}(M)$}\hskip
18.0&\makebox[0.0pt]{$d(M)$}\\\ \hline\cr 6&6\hskip 12.0&-4&12&12&86\hskip
18.0&12\\\ 6&12\hskip 12.0&-4&12&20&82\hskip 18.0&4\\\ 6&21\hskip
12.0&-4&12&28&84\hskip 18.0&4\\\ 6&32\hskip 12.0&-4&12&12&104\hskip 18.0&12\\\
12&12\hskip 12.0&-4&20&20&78\hskip 18.0&4\\\ 12&21\hskip
12.0&-4&20&28&80\hskip 18.0&4\\\ 12&32\hskip 12.0&-4&20&12&100\hskip 18.0&4\\\
21&21\hskip 12.0&-4&28&28&82\hskip 18.0&4\\\ 21&32\hskip
12.0&-4&28&12&102\hskip 18.0&4\\\ 32&32\hskip 12.0&-4&12&12&122\hskip
18.0&4\\\ 2&24\hskip 12.0&-6&6&14&102\hskip 18.0&6\\\ 18&24\hskip
12.0&-6&24&14&84\hskip 18.0&12\\\ 10&10\hskip 12.0&-16&16&16&70\hskip
18.0&8\\\ 14&22\hskip 12.0&-20&20&30&78\hskip 18.0&2\\\ 14&25\hskip
12.0&-20&20&20&80\hskip 18.0&2\\\ 13&14\hskip 12.0&-30&20&30&72\hskip
18.0&4\\\ 14&18\hskip 12.0&-40&40&10&76\hskip 18.0&2\\\ 18&34\hskip
12.0&-72&18&8&108\hskip 18.0&6\\\ \hline\cr\hline\cr\end{array}$ Table 4.
Twisted connected sums from matchings of rank 2 blocks with non-trivial
intersection $N_{+}\cap N_{-}$

###### Proof of Theorem 5.10..

In Table 2 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 5.8 is satisfied, and they are the ones listed in Table 4. Indeed,
because $\Delta$ is never greater than $2A^{2}$ for any entry in the table,
matchings of rank 2 Fano 3-folds are only possible when
$\Delta_{+}\Delta_{-}=A_{+}^{2}A_{-}^{2}$, or equivalently $h_{+}+h_{-}=0$.

For each of the 18 pairs, we can apply Nikulin [14, Theorem 1.12.4] to embed
the rank 3 lattice $W$ from Lemma 5.8 in $L$, thus defining a configuration of
primitive embeddings $N_{+},N_{-}\subset L$. Because $\Lambda_{\pm}=N_{\pm}$
we can apply Propositions 5.6 and Proposition 5.5 to find a compatible
matching.

Since we always have $k=1$, any twisted connected sum arising from matchings
with non-trivial $N_{+}\cap N_{-}$ must have
$\operatorname{Tor}\pi_{2}(M)\cong\operatorname{Tor}H^{3}(M)=0$ by Lemma
5.9(iii). For the one pair (row 13 in Table 4) where $d(M)$ is divisible by 8,
so that $\mu(M)$ could possibly be non-zero, 5.9(v) shows that $\mu(M)=0$
anyway. ∎

There are two reasons why Theorem 5.10 does not claim that there is a unique
diffeomorphism type of twisted connected sum arising from each of the 18
pairs. The first is that for some of the pairs we can also embed $W\subset L$
non-primitively, giving a twisted connected sum $M$ with
$\operatorname{Tor}\pi_{2}M\cong\operatorname{Tor}H^{3}(M)$ is non-trivial; we
do not study this further at this point. The second 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.

Similarly, it is unclear whether the two entries in Table 4 that have the same
values of $b_{3}(M)$ and $d(M)$ (rows 2 and 8) must be diffeomorphic or even
homeomorphic. 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 [12] for 7-manifolds
with $\pi_{2}M\cong\mathbb{Z}$ and _finite_ $H^{4}(M)$.

###  5.4. Arithmetic conditions for handcrafting

Having dealt with matchings of rank 2 Fanos where the configuration
$N_{+},N_{-}\subset L$ is perpendicular or where $N_{+}\cap N_{-}$ is trivial,
it remains only to consider the ‘skew’ case: $N_{+}\cap N_{-}$ is trivial, but
$N_{+}$ is not perpendicular to $N_{-}$. To find skew matchings we have to
deal with both lattice-arithmetical problems, and gain some detailed
understanding of the deformation theory of the Fanos involved. Because of the
case-by-case checking required, we think of this task as ‘handcrafting’.

When the Fanos have Picard rank 2, at least the arithmetic part of the problem
can be dealt with systematically. Remark 5.3(ii) implies that 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 $W_{k}:=N_{+}\oplus N_{-}$ 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 5.11.

Let $-\Delta_{\pm}$ be the discriminant of $N_{\pm}$. Then $W_{k}$ has
signature $(2,2)$ if and only if

$k^{2}A_{+}^{2}A_{-}^{2}<\Delta_{+}\Delta_{-}.$

###### 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

$\begin{pmatrix}A_{+}^{2}&0&0&0\\\ 0&B_{+}^{2}&0&kn_{+}n_{-}\\\
0&0&A_{-}^{2}&0\\\ 0&kn_{+}n_{-}&0&B_{-}^{2}\end{pmatrix}$

and discriminant
$A_{+}^{2}A_{-}^{2}(B_{+}^{2}B_{-}^{2}-k^{2}n_{+}^{2}n_{-}^{2})=n_{+}^{2}n_{-}^{2}D$.
Since $A_{\pm}^{2}B_{\pm}^{2}=-\Delta_{\pm}n_{\pm}^{2}$, we find

$D=\Delta_{+}\Delta_{-}-k^{2}A_{+}^{2}A_{-}^{2}.\qed$

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 $h_{+}+h_{-}>0$, for
$h_{\pm}:=\log_{2}\frac{\Delta_{\pm}}{A_{\pm}^{2}}$. We can readily identify
pairs satisfying this necessary condition from Table 2.

In terms of the notation from §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 5.12.

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 5.11, 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

$H^{2}(-B_{\mp}^{2})(\Delta_{+}\Delta_{-}-k^{2}A_{+}^{2}A_{-}^{2})\;\geq\;\Delta_{\pm}\Delta_{+}\Delta_{-}.$
(5.2)

Then

$(v.H)^{2}-v^{2}H^{2}\;\geq\;\Delta_{\pm}$

for any $v\in\Lambda_{\pm}$ linearly independent of $H$.

###### Proof.

Without loss of generality, $H\in N_{\pm}$ is primitive, and linearly
independent of $A:=A_{\pm}$. Then $A$, $H$ and $B:=B_{\mp}$ form a basis of
$\Lambda_{\pm}$. For $v=xG+yH+zB$

$\displaystyle(v.H)^{2}-v^{2}H^{2}$
$\displaystyle=(A.Hx+H^{2}y+B.Hz)^{2}-H^{2}(A^{2}x^{2}+H^{2}y^{2}+B^{2}z^{2}+2A.Hxy+2B.Hyz)$
$\displaystyle=\big{(}(A.H)^{2}-A^{2}H^{2}\big{)}x^{2}+2(A.H)(B.H)xz+\big{(}(B.H)^{2}-B^{2}H^{2}\big{)}z^{2}$
$\displaystyle=\Delta_{\pm}\left(x+\frac{(A.H)(B.H)}{\Delta_{\pm}}z\right)^{2}+\left((B.H)^{2}-B^{2}H^{2}-\frac{(A.H)^{2}(B.H)^{2}}{\Delta_{\pm}}\right)z^{2}.$

It therefore suffices to show that the coefficient of $z^{2}$ is
$\geq\Delta_{\pm}$. Now note that $B.H=\pm kn_{\mp}$, where $n_{\mp}$ is the
index of $\langle A_{\mp},B_{\mp}\rangle\subseteq N_{\mp}$. Therefore
$(B.H)^{2}\Delta_{\mp}=-A_{\mp}^{2}B_{\mp}^{2}$, so

$\displaystyle\Delta_{+}\Delta_{-}\left((B.H)^{2}-B^{2}H^{2}-\frac{(A.H)^{2}(B.H)^{2}}{\Delta_{\pm}}\right)$
$\displaystyle=k^{2}A_{\mp}^{2}B_{\mp}^{2}(\Delta_{\pm}-(A.H)^{2})-B^{2}H^{2}\Delta_{+}\Delta_{-}$
$\displaystyle=k^{2}A_{\mp}^{2}B_{\mp}^{2}A_{\pm}^{2}H^{2}-B_{-}^{2}H^{2}\Delta_{+}\Delta_{-}$
$\displaystyle=H^{2}(-B_{\mp})^{2}(\Delta_{+}\Delta_{-}-k^{2}A_{+}^{2}A_{-}^{2}).\qed$

###  5.5. Handcrafting examples with non-zero 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 5.6 does not suffice
for applying Proposition 5.5 to find a matching with the prescribed
configuration. The labourious 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 will lead to
twisted connected sums with $\mu(M)\not=0$. We therefore first explain how to
determine the topology of twisted connected sums obtained from a skew matching
of rank 2 blocks, and then prove in Theorem 5.17 that all skew configurations
of the Picard lattices of rank 2 Fanos that could possibly give twisted
connected sums are in fact realised by a matching.

######  Proposition 5.13.

Let $(Z_{\pm},\Sigma_{\pm})$ be a pair of building blocks whose polarising
lattices $N_{\pm}$ have rank 2, such that the kernel of $H^{2}(Z_{\pm})\to
H^{2}(\Sigma_{\pm})$ is generated by $[\Sigma_{\pm}]$. Let
$\matheur{r}:\Sigma_{+}\to\Sigma_{-}$ be a matching whose configuration
$N_{+},N_{-}\subset L$ has $N_{+}\oplus N_{-}$ isometric to $W_{k}$ from Lemma
5.11, 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

  1. (i)

$M$ is 2-connected,

  2. (ii)

$b_{3}(M)=b_{3}(Z_{+})+b_{3}(Z_{-})+21$,

  3. (iii)

$\operatorname{Tor}H^{3}(M)\cong L/W_{k}$,

  4. (iv)

$\operatorname{Tor}H^{4}(M)\cong(\mathbb{Z}/k)^{2}$,

  5. (v)

If $k=1$ then $d(M)$ (which divides 24) is the greatest common divisor of
$\bar{c}_{2}(Z_{\pm})A_{\pm}\in\mathbb{Z}/24$.

  6. (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

$\mu(M)=\frac{\operatorname{gd}(\bar{c}_{2}(Z_{+}))\operatorname{gd}(\bar{c}_{2}(Z_{-}))}{4}\in\mathbb{Z}/2,$

where $\operatorname{gd}(\bar{c}_{2}(Z_{\pm}))\in\\{2,4,6,8,12,24\\}$ 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 5.9. ∎

Now let us note two consequences for skew matchings of rank 2 Fanos.

######  Remark 5.14.

  1. (i)

Because $h<1$ for all rank 2 Fanos, $W_{k}$ 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
$H^{4}(M)$ torsion-free.

  2. (ii)

$\mu(M)$ is non-zero if and only if both $\bar{c}_{2}(Z_{\pm})A_{\pm}$
divisible by 8 while neither $\bar{c}_{2}(Z_{\pm})$ is divisible by 4.
Consulting Table 2, we find that the only rank 2 Fano type blocks with such
ample classes $A$ are

    1. #9

with $A:=G+H$ $\phantom{2}(h=0.09)$,

    2. #17

with $A:=G+H$ $\phantom{2}(h=0.06)$,

    3. #18

with $A:=G+2H$ $(h=-0.58)$,

    4. #27

with $A:=G+H$ $\phantom{2}(h=0.09)$.

Since #18 has $h$ quite negative, the only ways to match rank 2 Fanos to a
construct twisted connected sum $M$ with $\mu(M)$ is to use a pair among #9,
#17 and #27.

Lemma 5.16 provides the genericity results we need to find skew matchings of
these pairs. To prove it we use

######  Lemma 5.15.

Let $\Sigma$ be a K3 surface, and $H\in\operatorname{Pic}\Sigma$ a primitive
nef class with $H^{2}\geq 4$. 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
$v\in\operatorname{Pic}\Sigma$ such that

  1. (i)

$d=1$ and $v^{2}=0$, or

  2. (ii)

$d=2$ and $v^{2}=0$, or

  3. (iii)

$d=0$ and $v^{2}=-2$,

for $d:=v.H$.

###### Proof.

According to [4, Lemma 7.15] (based on [15, Chapter 3]), the arithmetic
conditions rule out all possible ways that $H$ can fail to be very ample. (i)
rules out $|H|$ being monogonal, so $|H|$ has no fixed part, and defines a
morphism. $H^{2}\not=2$, $H$ primitive and (ii) rule out the three different
ways that $|H|$ could be hyperelliptic, so it defines a birational morphism
onto its image. (iii) rules out any $(-2)$-curves being contracted, so $|H|$
defines an isomorphism. ∎

######  Lemma 5.16.

Let $\mathcal{Y}$ be the deformation type of #27, #9 or #17 in the Mori-Mukai
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 §4.4, and
$\operatorname{Amp}_{\mathcal{Y}}\subset N(\mathbb{R})$ the interior of the
cone spanned by $G$ and $H$. Suppose that

$(v.H)^{2}-H^{2}v^{2}\geq\Delta$ (5.3)

for all $v\in\Lambda$ that are linearly independent of $H$. Then $\mathcal{Y}$
is $(\Lambda,\operatorname{Amp}_{\mathcal{Y}})$-generic.

###### Proof.

It is enough to prove that any marked K3 surface $\Sigma$ with Picard lattice
_precisely_ $\Lambda$ can be embedded as an anticanonical divisor in some
$Y\in\mathcal{Y}$, so that the image of the ample cone of $Y$ in $L$ is
$\operatorname{Amp}_{\mathcal{Y}}$.

  1. #27

This is the family of blow-ups of $\mathbb{P}^{3}$ in twisted cubics. In this
case the Picard lattice is $\left(\begin{smallmatrix}2&5\\\
5&4\end{smallmatrix}\right)$, so $H$ is a class of degree 4. (5.3) means that
if $v\in\operatorname{Pic}\Sigma$ is linearly independent of $H$ and $d:=v.H$
then

$d^{2}-4v^{2}\geq 17,$ (5.4)

so none of the cases in Lemma 5.15 can occur. Thus $H$ is very ample, and
embeds $\Sigma$ a quartic in $\mathbb{P}^{3}$.

Moreover, since any irreducible effective class $v$ has $v^{2}\geq-2$, (5.4)
implies that in fact there are no such classes of degree $d\leq 2$. In
particular, the degree 3, effective $(-2)$-class $2G-H$ is irreducible. By
Riemann-Roch it 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 $Y\in\mathcal{Y}$.

  2. #9

This is the family of blow-ups of $\mathbb{P}^{3}$ in smooth curves of degree
7 and genus 5. The Picard lattice is $\left(\begin{smallmatrix}2&5\\\
5&4\end{smallmatrix}\right)$ in this case too, so we have already proved that
$H$ embeds $\Sigma$ as a quartic K3 in $\mathbb{P}^{3}$, and that $2H-G$ is
irreducible and represented by a twisted cubic $\Gamma$.

We now want to prove that $3H-G$ can be represented by a smooth curve $C$.
Since $H$ is ample, $3H-G$ is positive on all curves except possibly $\Gamma$.
Since in fact $(3H-G)\cdot\Gamma=1>0$, the class $3H-G$ is ample. In
particular $|3G-H|$ is base point free, and a general member $C$ is smooth.
Now $C$ has degree 7 and genus 5, so blowing up $C$ defines an element of the
family $\mathcal{Y}$.

  3. #17

This is the family of blow-ups of smooth quadrics $Q\subset\mathbb{P}^{4}$ in
smooth elliptic curves of degree 5. In this case the Picard lattice is
$\left(\begin{smallmatrix}4&7\\\ 7&6\end{smallmatrix}\right)$, so $H$ is a
degree 6 class. (5.3) means that

$d^{2}-6v^{2}\geq 25$ (5.5)

for any $v\in\Lambda$ not a multiple of $H$. In particular Lemma 5.15 implies
that $H$ is very ample, embeds $\Sigma$ as the intersection of a quadric $Q$
and a cubic in $\mathbb{P}^{4}$. 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 (5.5).

Now consider the class $E:=2H-G$, which has degree 5 and $E^{2}=0$. (5.5)
rules out the existence irreducible classes in $\operatorname{Pic}\Sigma$ with
$d\leq 3$, so $E$ is irreducible. In particular $E$ does not have any
$(-2)$-curve components, so $E$ is nef. Therefore [15, 3.8] implies that $|E|$
is base point free. A generic $C\in|E|$ is therefore a smooth elliptic curve
of degree 5. Blowing up $Q$ in $C$ yields a member $Y\in\mathcal{Y}$. ∎

######  Theorem 5.17.

Let $\mathcal{Y}_{+}$ and $\mathcal{Y}_{-}$ be a pair of deformation types of
rank 2 Fanos among #9, #17 and #27 on the Mori-Mukai list. Let $N_{\pm}$ be
their Picard lattices, and let $A_{\pm}\in N_{\pm}$ be the ample class listed
in Table 2. Define $W_{1}$ as in Lemma 5.11, embed $W_{1}\subset L$
primitively, and consider the resulting configuration $N_{+},N_{-}\subset L$

  1. (i)

There is a matching of some elements of $\mathcal{Y}_{\pm}$ with that
configuration.

  2. (ii)

This is the only non-perpendicular configuration of $N_{+}$ and $N_{-}$ for
which exists a matching.

###### Proof.

  1. (i)

For this configuration, the lattice $\Lambda_{\pm}\subset L$ defined in §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 §4.4. Looking up the values of
$H_{\pm}^{2}$, $A_{\pm}^{2}$, $B_{\pm}^{2}$ and $\Delta_{\pm}$ in Table 2,
Lemma 5.12 implies that

$(v.H_{\pm})^{2}-H_{\pm}^{2}v^{2}\geq\Delta_{\pm}$ (5.6)

for any $v\in\Lambda_{\pm}$ linearly independent of $H_{\pm}$. Therefore Lemma
5.16 implies that $\mathcal{Y}_{\pm}$ is
$(\Lambda_{\pm},\operatorname{Amp}_{\mathcal{Y}_{\pm}})$-generic, and the
desired matching exists by Proposition 5.5.

  2. (ii)

By Lemma 5.8 there can be no matchings of these types with a configuration
such that $N_{+}\cap N_{-}$ is non-trivial, so the non-perpendicular
configuration must be skew. We have explained that for a skew configuration to
satisfy the conditions of Remark 5.3, $N_{+}\oplus N_{-}$ must be isometric to
$W_{1}$. Thus it only remains to rule out any configurations where $W_{1}$ is
embedded non-primitively in $L$.

In Lemma 5.11 we computed that the discriminant of $W_{1}$ is

$D=\Delta_{+}\Delta_{-}-A_{+}^{2}A_{-}^{2}.$

When $\mathcal{Y}_{\pm}$ are both among #9 and #27, $D=33$ is square-free, 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
square-free. Hence there are no non-primitive 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

$\begin{pmatrix}4&11&1&0\\\ 11&24&0&0\\\ 1&0&4&11\\\ 0&0&11&24\end{pmatrix}.$

Because this matrix has rank 3 over $\mathbb{Z}/7$, the discriminant group
must be $\mathbb{Z}/49$ rather than $(\mathbb{Z}/7)^{2}$, and $W_{1}$ has an
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 $W_{1}\subset L$ non-primitively 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 $\widetilde{B}_{\pm}:=\pm\frac{24}{7}K+5A_{\mp}$.
With respect that basis, the quadratic form on $\Lambda_{\pm}$ is represented
by

$\begin{pmatrix}4&7&48\\\ 7&6&72\\\ 48&72&552\end{pmatrix}.$

Suppose that $\Sigma$ is an anticanonical K3 divisor in some rank 2 Fano of
type #27, with $\operatorname{Pic}\Sigma$ isometric to this $\Lambda$. Then
$\Sigma$ also embeds in a smooth quadric $Q\subset\mathbb{P}^{4}$. For
$v=xG+yH+z\widetilde{B}$ we can expand

$6v^{2}-(v.H)^{2}=25(x+\textstyle 8\frac{16}{25})^{2}-\textstyle
5\frac{19}{25}z^{2}.$

Hence (5.5) fails only for some $v\in\Lambda$ such that $v^{2}=0$ and $d=3$,
_e.g._ $E^{\prime}:=9G+H-B$. The argument from the proof of Lemma 5.16 for the
case #17 shows that $E^{\prime}$ is 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 non-singular. ∎

The topological invariants of the twisted connected sums $M$ that arise from
the six matchings found in Theorem 5.17 can now be computed from Proposition
5.13. They are 2-connected with $H^{4}(M)$ torsion-free, $\mu(M)=1$ (by
design), and $b(M)$ and $d(M)$ as listed in the introduction. 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 3 to see that for two of these four smooth
manifolds, there exist perpendicular twisted connected sums $M^{\prime}$ with
the same $(b_{3},d)$. Then $M$ and $M^{\prime}$ are homeomorphic by Theorem
2.2(i). However, $\mu(M^{\prime})=0$ by Corollary 3.5, so $M$ and $M^{\prime}$
are not diffeomorphic.

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