100 Sasaki groups
that Y is equipped with a symplectic form ω representing an integral class rωs, see Remark 3.23.
After possibly replacingYby its symplectic blow-up at a point, there exists a spher-ical class inH2pY;Zqon whichrωsevaluates to 1. Therefore, the Boothby-Wang fibra-tion overY with Euler classrωsis a compact 2n`1-dimensional K-contact manifoldN
withπ1pNq “∆.
Remark6.8. It was shown in [10] that all finitely presentable groups are K-contact, cf.
Theorem 4.27. However, the proof only realizes these groups as fundamental groups of K-contact manifolds of dimension 2n`1ě7 because it relies on the join construction.
Remark 6.9. Theorem 3.94 implies that K-contact groups in dimension 3 are Sasaki groups. Moreover, three-dimensional Sasaki manifolds were classified by Geiges [48].
As a result their fundamental groups are lattices inS Op4q,S Lp2,r Rqor the real Heisen-berg group H3. This shows that the bounds on the dimension in Theorem 6.5 and Theorem 6.7 are optimal.
As a motivation to discuss the quasi-regular setting, let us rephrase the above results.
Consider the central extension (6.1) associated to a regular Sasaki structure:
0ÝÑC ÝÑΓÝÑπ1pXq ÝÑ0.
Hereπ1pXq is the ordinary fundamental group of a compact projective manifold. The Boothby-Wang Theorem ensures that all projective groups figure as the last term of the extension (6.1) associated to some regular Sasaki structure. Moreover, Theorem 6.5 shows that every projective group is the fundamental group of a compact regular Sasaki manifold. We can then rephrase Theorem 6.5 as: every group which appears at the right of a central extension associated to a regular Sasaki structure also figures in the middle of some other such extension.
It is then natural to ask whether the analogous statement holds true in the quasi-regular case. Our next theorem gives an affirmative answer to this question.
Theorem 6.10. Let M be a2n`1-dimensional compact quasi-regular Sasaki manifold and let
0ÝÑC ÝÑΓÝÑπorb1 pXq ÝÑ0 be the associated central extension. Thenπorb1 pXqis a Sasaki group.
Proof. The proof is a generalization to the orbifold setting of the argument in Theo-rem 6.5.
Let π: M Ñ Xbe the S1-orbibundle with Euler classrωs associated to the quasi-regular structure on M. Consider the blow-up of X at a smooth point. This gives a projective orbifoldYwith underlying topological spaceY “X#CPn. Clearlyπorb1 pXq “ πorb1 pYq. Moreover, we can endowYwith an integral Kähler classe“krωs `Ewhere E is the Poincaré dual to the exceptional divisor D – CPn´1 andk is a positive, large enough integer.
6.2 Realizability 101
We make use of Theorem 2.53 to ensure that we can pickk such that the principal S1-bundle associated to e P H2orbpY;Zq is smooth. For instance, it is enough to take k relatively prime to the orders of the uniformizing groups of X. Let N denote this bundle. By the Structure Theorem 3.59, N is a Sasaki manifold. We want to prove that π1pNq “πorb1 pYq.
Now consider the map p˚: πorb2 pYq ÝÑ πorb2 pXq induced by the blow-down map p: Y ÝÑ X. Let f: S2 ÝÑ BY be a representative of the generator of kerpp˚q. By construction the pullback of N under f is the Hopf bundle S3 ÝÑ S2. Therefore, f induces the following map of long exact sequences
¨ ¨ ¨ π2pBYq π1pS1q ¨ ¨ ¨
H2pBY;Zq
¨ ¨ ¨ π2pS2q π1pS1q ¨ ¨ ¨
H2pS2;Zq
δ
h x¨,ey
f˚
–
h
f˚
x¨,xy f˚
where x is the generator of H2pS2;Zq given by the orientation. Since x “ f˚peq, it follows that the map δ: πorb2 pYq ÝÑ π1pS1qis surjective. We conclude that π1pNq “
πorb1 pYq “πorb1 pXq.
Remark 6.11. The orbifold fundamental group of a compact orbifold is finitely pre-sentable. Since every such group is the fundamental group of a symplectic manifold, the K-contact analogue of Theorem 6.10 is, in fact, weaker than Theorem 6.7.
A natural question to ask on Sasaki groups is the following:
Question:In which dimension does there exist a Sasakian manifold M withπ1pMq “Γ for a given Sasaki groupΓ?
Theorem 6.5 together with Remark 6.9 provide an answer to this question whenΓis also a projective group. In the Kähler case the analogous problem has a partial answer in Proposition 4.9. However, in the Sasakian setting taking products with simply con-nected manifolds does not yield Sasaki manifolds in a natural way. Moreover, consider a quasi-regular Sasaki structure such that the mapδ of Lemma 6.2 is surjective. Then the Euler classeof the associated group extension pulls back to the Kähler class ofXby Proposition 6.1. Hence the powersek of the Euler class cannot vanish fork ďdimCpXq.
This suggests that some dimensional restrictions may apply. Our next result exploits this observation to prove that the Sasakian analogue of ii) in Proposition 4.9 is false.
102 Sasaki groups
Theorem 6.12. The setS2n`1is not contained inŤ
mąnS2m`1for n “1,2,3.
Proof. FixnP t1,2,3u. We want to find elements ofS2n`1, i.e., fundamental groups of a compact Sasakian manifold of dimension 2n`1, which cannot be realized asπ1pMq forMa Sasakian manifold of dimension larger than 2n`1. In particular we will show that this property is satisfied by the integral Heisenberg groups of dimension 3,5 and 7.
We have seen in Example 4.20 that the Heisenberg group H2n`1 arises as the fun-damental group of the Sasaki manifold given by the Boothby-Wang fibration over the 2n-dimensional torusT2n. ThusH2n`1 PS2n`1.
Now supposeH2n`1“π1pMqwhereMis a Sasakian manifold of dimension 2m`1.
A quasi-regular structure on Myields a central extension of the form:
0ÝÑC ÝÑϕ H2n`1 ÝÑπorb1 pXq ÝÑ0. (6.6) First we prove by contradiction thatC ‰ 0 andC ‰ Zk leaving as the only pos-sibilityC “ Z, see (6.2). Since H2n`1 is torsion-free,C cannot be a non-trivial finite group. Now suppose thatC is the trivial group. By Lemma 6.3πorb1 pXqsurjects onto π1pXqwith kernel generated by elements of finite order. HenceH2n`1 – πorb1 pXq and torsion-freeness yields an isomorphismH2n`1–π1pXq. ThusH2n`1is a Kähler group, contradicting the results of [27].
The only possibility left isC – Z. We will first treat the case where ϕp1q “ z is a generator of the center ofH2n`1. In this case the central extension (6.6) reads
0ÝÑZÝÑH2n`1 ÝÑZ2nÝÑ0
with Euler classeP H2pZ2n;Zq. LetBZ2nbe the classifying space obtained by attaching cells to BX as explained in Section 4.1. By Proposition 6.1 the Euler class e of the extension above is mapped to the Euler classrωsof theS1-orbibundleπ: M ÝÑ Xby the injectionι˚: H2pBZ2n;ZqãÑHorb2 pX;Zq. Thus
0“ι˚pen`1q “ι˚peqn`1 “ rωsn`1.
On the other hand, ω is a Kähler form on a 2m-dimensional orbifold so thatrωsl ‰ 0 for alllďm. We conclude thatmďn.
If we are not in the preceding case, thenϕp1q “zk so that the sequence (6.6) reads
0ÝÑZÝÑH2n`1ÝÑ Gk ÝÑ0, (6.7)
where, in turn,Gkfits in the short exact sequence
0ÝÑZk ÝÑGk ÝÑZ2nÝÑ0.
Again by Proposition 6.1, ι˚pekq “ rωs where ek is the Euler class of the central extension (6.7). The class ek is not torsion because rωs is a Kähler class and ι˚ is injective. Moreover, the Lyndon–Hochschild–Serre spectral sequence forZk ĂGkgives