76 Group extensions and Kähler groups
One sees immediately that the Sasakian analogue of Proposition 4.8 holds true.
Namely, the set S2n`1 is closed under taking finite index subgroups. One is then en-couraged to translate Proposition 4.9 in the Sasakian setting. This is where the first obstacles arise. In fact, the product of Sasakian manifolds is not Sasakian for trivial reasons. We will see in Sections 6.2 and 6.3 that Sasaki groups do not satisfy the prop-erties in Proposition 4.9. Nevertheless, Chen [30] showed that projective groups form a subset of Sasaki groups. In that sense, these properties are satisfied by a subclass of Sasaki groups.
Theorem 4.26 ([30]). Every projective group is the fundamental group of a Sasakian manifold of dimension2n`1ě7.
Proof. Let Γ “ π1pXq be a projective group with X a smooth projective variety of complex dimensionną1. LetMbe the Boothby-Wang bundle overX. Then taking the join with the standard 3-sphere yields a manifoldM‹S3diffeomorphic to anS3bundle overX. This is a Sasakian manifold withπ1pM‹S3q “Γ. In [10] the authors discussed formality of Sasakian manifolds. Their examples are also built as joins of a given Sasaki or K-contact manifold with S3. With the same methods they proved the analogous result in the K-contact setting.
Theorem 4.27([10]). Every finitely presentable group is the fundamental group of a K-contact manifold of dimension2n`1ě7.
Proof. Given a finitely presented groupΓthere exists a symplectic 2n-manifoldXwith π1pXq “Γandną1, see [50]. By Remark 3.23 we can construct a Botthby-Wang bun-dleM over X. Now the same argument in the proof of Theorem 4.26 applies. Namely, the joinM‹S3is ap2n`1q-dimensional K-contact manifold with fundamental group Γ.
In [30] Chen proved several results on Sasaki groups which can be considered as the Sasakian analogues of results from the Kähler case. Let us review them:
Theorem 4.28. SupposeΓis a Sasaki group. ThenΓis the fundamental group of some compact three-manifold M if and only if M has geometry modelled on S3, the three-dimensional Heisenberg group orS Lp2,ČRq.
Remark4.29. Theorem 4.28 can be considered the Sasakian analogue of a theorem of Dimca and Suciu [35] which we discussed in Section 4.2. In Section 6.5 we prove a sharpening of this result which can be regarded as the analogue of Theorem 4.23, see Theorem 6.21.
Theorem 4.30. SupposeΓis a Sasaki group.
1. Then Γ has either zero or one end. In particular,Γ cannot split as a non-trivial free product.
4.3 Analogues in the Sasakian setting 77
2. IfΓis solvable, it contains a nilpotent subgroup of finite index.
Remark4.31. Theorem 4.30 relies on the work of Campana [21, 22] on orbifold funda-mental groups of compact Kähler orbifolds, see [30, Lemma 3.1] and references therein.
However, the only Kähler orbifolds that are considered in [21, 22] are those whose un-derlying topological space is smooth. This assumption is not always satisfied for the orbifolds associated to quasi-regular Sasaki manifolds, since these can have genuine singularities. It is not clear to the author how it is possible to apply Campana’s results in this situation.
The authors of [10] discuss torsion in orbifold fundamental groups to prove the following:
Theorem 4.32 ([10]). LetΓ be an irreducible arithmetic lattice in a semi-simple real Lie group G of rank at least two with no co-compact factors and with trivial center. If Γis Sasaki, then it must be isomorphic to the groupπorb1 pXqof some Kähler orbifoldX. Moreover,Γcannot be a cocompact arithmetic lattice in S Op1,nqfor ně3.
We will see later that this is a special case of a more general statement, see Proposi-tion 6.24.
The results that we have seen rely only on the Structure Theorem and the join con-struction. On the other hand, the authors of [23] used topological properties of Sasakian manifolds to constrain their fundamental groups. Namely, in [23] is given a characteri-zation of Sasaki nilmanifolds using the results of Tievsky [113] on minimal models for Sasakian manifolds.
Theorem 4.33 ([23]). A compact nilmanifold of dimension 2n`1is Sasakian if and only if it is a quotient of the real Heisenberg group H2n`1 by a co-compact latticeΓ.
Later Kasuya [68] extended this result to solvmanifolds. The generalization follows from the study of the representations of Sasaki groups in GLp1,Cq. With the same methods Kasuya proved the following:
Theorem 4.34([68]). A polycyclic Sasaki group is virtually nilpotent.
We have already seen that Proposition 4.17 does not hold in the Sasakian setting.
This is the first instance of a feature of Kähler groups which is not enjoyed by Sasaki groups. On the other hand, the results in this section seem to suggest that Sasaki and Kähler groups have very similar behaviour. In Chapter 6 we will see that, although Sasaki groups are deeply related to Kähler groups and enjoy many of their properties, they also display antithetical behaviour in certain aspects.
Chapter 5
Invariants and underlying structures
Sasaki structures have a variety of underlying structures. When defining invariants of Sasaki structures it is natural to ask which of the underlying structures they depend on. In this chapter we will focus on invariants of the transverse Kähler structure. In particular, we will discuss basic Hodge numbers, basic Chern classes and the type of Sasaki structures. Moreover, we relate these invariants to the topology of the under-lying almost contact and contact structures. Specifically, we will provide examples of Sasaki structures on smooth manifolds whose invariants disagree and discuss which of the underlying structures are equivalent.
In Section 5.1 we present the results in [15, 49] on invariance of basic Betti and Hodge numbers. This will serve as motivation for the results proven in Section 5.3.
5.1 Invariance of basic Betti and Hodge numbers
In Chapter 3 we have introduced several invariants of Sasaki manifolds. In this section we will focus on basic Betti an Hodge numbers. We begin by showing that the former are topological invariants of the Sasakian manifold. In order to prove this we need a lemma of Tachibana.
Lemma 5.1([109]). LetpM, η, φ,R,gqbe a compact Sasaki manifold of dimension2n` 1. Letαbe a harmonic p-form on M with1 ď p ďn. ThenιRα “ 0and the formφα given by
φαpX1, . . . ,Xpq “
p
ÿ
i“1
αpX1, . . . , φXi, . . . ,Xpq is harmonic.
We can now prove the following:
Theorem 5.2([15, Theorem 7.4.14]). LetpM, η, φ,R,gqbe a compact Sasaki manifold of dimension2n`1. Then the basic cohomology HB˚pFqonly depends on the topology of M. In particular, the basic Betti numbers of any two Sasaki structures on M agree.
79
80 Invariants and underlying structures
Proof. Let αbe a harmonic p-form on M with 1 ď p ď n. By Lemma 5.1 ιRα “ 0.
Moreover,LRα “ 0 becauseR is Killing andαis harmonic. Henceα is basic. From the definition of the basic Hodge star operator ¯‹it follows that‹α“η^‹α, see (3.11).¯ This yields
dp‹αq “dη^‹α¯ ´η^dp‹αq “¯ L‹α¯ ´η^dB‹α¯ (5.1) where L is the operator defined in (3.12). Now the left-hand side vanishes since α is coclosed. The two forms on the right-hand side are not proportional so they both must vanish. The equation L‹α¯ “ 0 means that α is primitive while the vanishing of the second term shows that αis basic harmonic. We conclude that α must be the unique harmonic representative of its class in the basic primitive cohomology group PrpFq. ThusPrpFqis a topological invariant because it is isomorphic to the de Rham cohomology ofM. Now the basic cohomology groupsHrBpFqare topological invariants because Theorem 3.78 gives a decomposition
HrBpFq “ à
kě0
LkPr´2kpFq.
It is natural to ask whether or not basic Hodge numbers can distinguish Sasaki struc-tures on the same smooth manifold. This is indeed the case as shown by the following example due to Boyer and explained in [49].
Example5.3 ([49, Example 3.4]). We present here two Sasaki structures with different Hodge numbers are given onM “#21pS2ˆS3q, the 21-fold connected sum ofS2ˆS3. The first such structurepη1, φ1,R1,g1qis the Boothby-Wang fibration on a K3 surface.
Therefore, the basic Hodge numbers are the Hodge numbers of the K3 surface. In particularh2,0pF1q “ 1. On the other hand, M supports the following positive Sasaki structure pη2, φ2,R2,g2q; cf. Example 3.66 and [15, page 356]. The connected sum
#21pS2ˆS3qcan be realized as the linkLf “Vf
ŞS7ĂC4where fpz0,z1,z2,z3q “z220 `z221 `z222 `z0z3
is a weighted homogeneous polynomial of degreed “22 with weightw“ p1,1,1,21q.
That is,M is theS1-orbibundle over the hypersurface Xf ĂCP3w. This Sasaki structure is positive becauseř
wi ´d “ 2 ą 0, see Proposition 3.85. Then a vanishing result proved independently in [95] and [51] implies thath2,0pF2q “0.
It is then natural to ask whether an example of Sasaki structures of the same type but with different Hodge numbers exists, see Section 5.3.
The fact that the Reeb foliation of a Sasaki manifold is transversally Kähler imposes some rigidity on the transverse geometry. Namely, the basic Dolbeault cohomology groups, and therefore the basic Hodge numbers, are invariant under deformations of type II, see Definition 3.51, because these deformations preserve the Reeb foliation and the transverse holomorphic structure. It turns out that basic Hodge numbers are invariant