Proposition 3.90. Let M1 ‹k1,k2 M2 be the smooth join of two compact quasi-regular Sasaki or K-contact manifolds M1and M2. Then M1‹k1,k2 M2is a bundle overX1with fiber M2{Zk2 associated to the orbibundle M1 ÑX1.
Proof. M1 ‹k1,k2 M2 is the quotient of M1 ˆ M2 by the circle action (3.14). Let the subgroup Zk2 Ă S1 act first to obtain the quotient M1 ˆ M2{Zk2. Denote a point in M1ˆM2{Zk2 bypx,rysq. NowM1‹k1,k2M2is the quotient ofM1ˆM2{Zk2 by the diagonal S1-action whereS1 “S1{Zk2 acts bypx,rysqeiθ “ pxeiθ,rye´i
k1 k2θ
sq. By definition this is
theM2{Zk2-bundle associated toM1 ÑX1.
Corollary 3.91. Let M1 ‹M2 be the join of two compact regular Sasaki or K-contact manifolds M1and M2. Denote by M ÑX1and M2 ÑX2the associated Boothby-Wang fibrations. Then M1‹M2is a M2-bundle over X1.
Remark3.92. The roles of the compact regular Sasaki manifolds M1and M2 are inter-changeable. Hence, M1‹M2is a M1-bundle over X2.
Remark 3.93. The join construction for K-contact manifolds is a special case of the contact fiber bundles by Lerman, see [83].
3.9 Topology of Sasakian manifolds
We conclude this chapter by reviewing some of the topological properties of Sasakian manifolds.
We begin by showing that in dimension 3 every K-contact structure is Sasaki. This is a consequence of the fact that every almost complex structure on a surface is integrable.
Theorem 3.94. A3-dimensional K-contact manifoldpM, η, φ,R,gqis Sasaki.
Proof. By Corollary 3.47, we have to show that the induced almost CR structure is integrable. Let H “ tX ´ iJX|X P Du as in Section 3.5. We want to prove the integrability condition
rX´iφX,Y´iφYs PH for all X,Y PD. (3.15) By Proposition 3.41 the tensor Np2q vanishes. Thus we get Np2qpφX,Yq “ ηprφX,Ys ` rX, φYsq “0 forX,Y P D. Thus (3.15) is equivalent to
φrX,Ys ´φrφX, φYs ´ rφX,Ys ´ rX, φYs “0 for all X,Y PD.
The equation above is easily verified by choosing a basis of D of the form tX,Y “
φXu.
We now turn our attention to Sasaki manifolds of higher dimension. It is very nat-ural to ask which topological properties of Kähler manifolds are enjoyed by Sasakian manifolds.
64 Sasaki manifolds
We begin by investigating formality. For the basic material we refer the reader to [44]. Consider a commutative differential graded algebra pA,dAq(CDGA Afor short) overR. We denote by|a|the degree of an elementaP Aand byH˚pAqthe cohomology of the complexpA,dAq. The CDGA we mainly focus on is the de Rham complex of a Sasakian manifold.
A CDGAAis calledminimalif 1) Ais the free algebraŹ
V over a graded vector spaceV.
2) There exists a set of generatorstaiu, indexed by a well-ordered set, such that i) |ai| ď |aj|foriă jand
ii) dAaiis expressed in terms ofajfor jăi.
Amorphismof CDGA’s is then a morphism of algebras that commutes with the diff er-ential and respects the grading. Aquasi-isomorphismof CDGA’s is a morphism which induces an isomorphism in cohomology. More generally, we can define the notion of weak equivalence. Namely, two commutative differential graded algebras Aand Bare said to beweakly equivalentif there is a sequence of quasi-isomorphisms
AÐÝC1ÝÑC2 ÐÝ ¨ ¨ ¨ ÐÝCnÝÑ B.
Anelementary extensionof a CDGApA,dAqis a CDGA of the formpB“AbŹ V,dBq satisfying the following properties:
i) Vis finite-dimensional and all elements ofV have the same degree.
ii) dBpaq “dApaqfor allaP Aand dBpvq PAfor allvPV.
Aminimal modelfor a CDGAAis a minimal commutative differential graded algebra ŹV together with a quasi-isomorphismρ: Ź
V ÝÑA. A CDGA is calledconnected if its 0-th cohomology group is isomorphic toR. Every connected CDGA admits a min-imal model which is unique up to isomorphism, see [58]. Therefore, weakly equivalent connected CDGA’s have isomorphic minimal models.
Definition 3.95. Let Mbe a connected manifold andŹ
V the minimal model for its de Rham complexΩpMq. Consider the cohomologyH˚dRpMqas a CDGA with trivial diff er-ential. The manifoldMisformalifŹ
V is a minimal model forHdR˚ pMq. Equivalently, M is formal if there exists a morphism of CDGA’sψ: Ź
V ÝÑ HdR˚ pMq inducing an isomorphism in cohomology.
Kähler manifolds form a class of manifolds which enjoys formality [33]. It is then natural to ask whether Sasakian manifolds are formal. The first results in this direction were given by Tievsky in his Ph.D. thesis [113]. We briefly review his results.
LetpM, η, φ,R,gqbe a compact Sasaki manifold. Consider the operator dcB “ipB´Bq¯ on complex valued basic forms. Let ΩcBpFq be the complex of dc-closed forms with
3.9 Topology of Sasakian manifolds 65
differential given by dB and HcBpFq its cohomology. Let V “ xyy be a graded vector space with |y| “ 1. An elementary extension by Ź
V is then determined by defining dy. Now one can define morphisms relating an elementary extension of the basic dc -cohomology with the de Rham complex. Namely, consider
´ΩcBpFq bŹ
V,dy“dη¯ ´
HcBpFq bŹ
V,dy“ rdηs¯ α`βby rαs ` rβs by.
τ
Moreover, we can define the morphism
´ΩcBpFq bŹV,dy“dη¯
pΩpMq,dq
α`βby α`β^η .
σ
The main result in [113] is the following:
Theorem 3.96([113]). LetpM, η, φ,R,gqbe a compact Sasaki manifold. Then the dia-gram
´
HBcpFq bŹ
V,dy“ rdηs¯ τ ÐÝÝ
´ΩcBpFq bŹ
V,dy“dη¯ σ
ÝÑ pΩpMq,Ý dq is a weak equivalence of CDGA’s.
Moreover,`
HcBpFq bŹ
V,dy“ rdηs˘
is isomorphic to`
HBpFq bŹV,dy“ rdηs˘ . Therefore, the minimal model of`
HBpFq bŹ
V,dy “ rdηs˘
is isomorphic to the min-imal model ofpHdRpMq,0q. A compact contact manifold satisfying this condition will be called aTievsky typemanifold.
Generally, determining whether a manifold is formal is a difficult problem. An im-portant tool in detecting non-formality is given by Massey products. Massey products are indeed an obstruction to formality, a proof of this fact can be found in [33] in the dis-cussion after Theorem 4.1. Using this result, it is easy to see that there exist non-formal Sasakian manifolds. In fact the Boothby-Wang fibration over a torus pT2qn with stan-dard Kähler class gives such a manifold. In [10] Biswas, Fernández, Muñoz and Tralle proved that all higher order Massey products vanish on Sasakian manifolds. Hence only triple Massey products can detect non-formality.
Another important topological property of Sasakian manifolds is the Hard Lefschetz Theorem which was proven in the Sasaki setting by Cappelletti Montano, De Nicola and Yudin.
Theorem 3.97 ([25]). Let pM, η, φ,R,gq be a compact Sasaki manifold of dimension
66 Sasaki manifolds
2n`1. Then for0ďrďn there exists an isomorphism Lr: Hn´rpMq ÝÑHn`r`1pMq
rβs ÞÑ rη^ pdηqr^βs
whereβ is the unique harmonic representative of its cohomology class. Moreover, the isomorphism is independent of the Sasaki metric.
A compact contact manifold with such an isomorphism is called aLefschetz contact manifold. An example of Lefschetz K-contact manifold of Tievsky type which does not admit a Sasaki structure was given in [24]. In this case the obstruction is given by the fundamental group.
As in the Kähler case, the class of fundamental groups of compact Sasakian mani-folds satisfies several properties. For instance, we have the following:
Proposition 3.98. The abelianization H1pMqof the fundamental groupπ1pMqof a Sa-sakian manifold M has even rank.
Proof. This follows directly from partivqin Proposition 3.71 and Corollary 3.77 on the basic Hodge numbers of a Sasaki manifold M. Namely, for any Sasaki structure on M the basic Hodge numbers satisfyh1,0pFq “h0,1pFq “ 12b1pFq “ 12b1pMq.
Fundamental groups of compact Sasakian manifolds are known to meet further con-straints. Since such groups are the subject of Chapter 6, we review the literature on Sasaki groups in detail in Section 4.3.
Chapter 4
Group extensions and Kähler groups
As a consequence of the Structure Theorem 3.59, the topology of Sasakian manifolds is closely related to the topology of Kähler orbifolds. In particular, fundamental groups of compact Sasakian manifolds can be described in terms of projective orbifold fundamen-tal groups. In turn, these groups are related to projective groups. Hence, we explain here these relations in order to discuss fundamental groups of compact Sasakian manifolds.
4.1 Group cohomology and central extensions
In this section we recall the definition of group cohomology and discuss group exten-sions. The proofs which are omitted in this section, as well as a detailed discussion on these topics, can be found in [19, 81, 108].
Given a groupΓa connected aspherical space BΓwithπ1pBΓq “ Γis aclassifying space for Γ. The classifying space is determined by Γ up to homotopy equivalence.
There are several construction of classifying spaces. Since we are only interested in fundamental groups of compact manifolds, we present below one such construction which is particularly convenient for us.
The space BΓ is classifying forΓ in the following sense. Any homomorphism of groupsΓ ÝÑ ∆is induced by a map BΓ ÝÑ B∆unique up to homotopy. Therefore, the homotopy type ofBΓis determined uniquely byΓ.
We define thegroup cohomologyH˚pΓ;Rqof Γto be the cohomology H˚pBΓ;Rq for a ringR.
Example 4.1 (Finite cyclic groups). A classifying space for Γ “ Zn is given by the infinite dimensional lens space Lp8,nq. Namely, the quotient of S8 Ă C8 by the standard action of Zn. Clearly π1pLp8,nqq “ Zn. Moreover, Lp8,nq is aspherical
67
68 Group extensions and Kähler groups
because its universal covering spaceS8is contractible. Thus we have
HkpZn;Zq “
$
’&
’%
Z, ifk“0;
Zn, fork ą0 even;
0, otherwise.
Example 4.2 (Surface groups). Let Γg be the fundamental group of a closed oriented surfaceΣg of genusgě2. ThenΣg is the classifying space forΓg because its universal covering is a hyperbolic disk. We conclude thatH˚pΓg;Zq –H˚pΣg;Zq. The groupsΓg
are called surface groups.
Example4.3 (Free groups). Let Fn be the free group on ngenerators. Then Fn is the fundamental group of a wedge ofncirclesŽ
nS1. Moreover, Ž
nS1 has trivial higher homotopy groups. HenceŽ
nS1“ BFn. It follows that the cohomology ofFnis trivial in degree larger than 1.
We present now an alternative defintion of group cohomology. In certain situations this will turn out to be more suitable than the definition given above.
LetΓbe a group andRaΓ-module. Consider the group HompΓr,Rqof homogeneous homomorphisms from ther-fold direct productΓˆ¨ ¨ ¨ˆΓtoR. In other words, consider the group of homomorphisms
ϕ: Γˆ ¨ ¨ ¨ ˆΓÝÑR such that
γϕpγ1, . . . , γrq “ϕpγγ1, . . . , γγrq
forγ, γ1, . . . , γr P Γ. Now consider the map d : HompΓ‚,Rq ÝÑHompΓ‚`1,Rqdefined by
dϕpγ1, . . . , γr`1q “
r`1
ÿ
i“1
p´1qi`1ϕpγ1, . . . ,γˆi, . . . , γr`1q
where the notation ˆγi means that the i-th entry is omitted. Then pHompΓ‚,Rq,dq is a cochain complex. One can show that the group cohomologyH˚pΓ,Rqis isomorphic to the cohomology of the complexpHompΓ‚`1,Rq,dq. Let us present an instance in which this definition is more convenient for computations.
Example 4.4. From the above description follows that the cohomology of a torsion group of ordermvanishes if the coefficients are divisible bym. For simplicity let us con-sider real coefficients. Namely, letΓbe a group such thatγm“0 for allγP Γand con-siderRas a trivialΓ-module. It is clear that every homomorphismϕ: Γˆ ¨ ¨ ¨ ˆΓÝÑR satisfiesmϕ “0. Therefore, we have mHrpΓ;Rq “ 0 for allr ą0. On the other hand mHrpΓ;Rq “ HrpΓ;mRq “ HrpΓ;Rq “ 0. Hence we haveHr˚pΓ;Zq “0.
Consider now a fiber bundle M ÝÑ Bwith fiber F. If the fiber and the base are aspherical spaces, then so is M. That is, M,B and F are classifying spaces for their
4.1 Group cohomology and central extensions 69
fundamental groups. Then the long exact sequence of homotopy groups reduces to a short exact sequence of the form
0ÝÑπ1pFq ÝÑπ1pMq ÝÑπ1pBq ÝÑ0. Conversely, consider the following short exact sequence
0ÝÑK ÝÝÑi ΓÝÝÑp QÝÑ0. (4.1)
Then there exists a fiber bundle BΓ ÝÑ BQ with fiber BK inducing the above short exact sequence.
We can construct such a fiber bundle in the following way. For discrete groups one can consider the universal covering spaceEΓ ÝÑ BΓ. In fact, this is the universal bundle for principalΓ-bundles. Consider the spaceEΓˆΓEQ. SinceΓacts freely onEΓ, this is a classifying space BΓ. Moreover, EΓ{K is a classifying space BK. Therefore, by taking the quotient byK first we get
EΓˆΓEQ“BKˆQEQ.
One can regard this space as a fiber bundle π: BK ˆQEQ ÝÑ BQ associated to the principalQ-bundle EQ ÝÑ BQ. Furthermore, the spaceEΓˆΓ EQis the fiber bundle associated to the principalQ-bundleEQÝÑ BQbyp:ΓÝÑ Q. It is then clear that the bundle map πinduces the homomorphism pat the level of fundamental groups. Thus, the long exact sequence of homotopy groups of the bundleπ: BK ˆQ EQ ÝÑ BQis exactly (4.1).
We are particularly interested in central extensions Γ of a group Q by an abelian groupC. Namely, we are interested in short exact sequences of groups of the form
0ÝÑCÝÑÝi ΓÝÝÑp QÝÑ0 (4.2)
whereipCqlies in the center ofΓ.
Given a group extension as in (4.1), we derive theLyndon-Hochschild-Serre spec-tral sequence as a special case of Serre spectral sequence for the associated fibration described above. In particular, the second page of the Lyndon-Hochschild-Serre spectral sequence is given by
E2p,q “HppQ;HqpC;Rqq and it converges to the group cohomologyHp`qpΓ;Rq.
Remark 4.5. When C is torsion of order m and R is m divisible Example 4.4 shows that the cohomology groups HrpC;Rq vanish for all r ą 0. In this case the Lyndon-Hochschild-Serre spectral sequence with values in R degenerates at the second page and gives an isomorphismH˚pQ;Rq – H˚pΓ;Rq.
The caseC “ Zplays a special role for us. Therefore we describe this situation in further detail. WhenC “ Zthe fibration associated to the extension (4.2) is a principal
70 Group extensions and Kähler groups
S1-bundle. Conversely, every principalS1-bundle over a classifying spaceBQgives rise to a group extension by looking at the long exact sequence of homotopy groups. More-over, the total space of such anS1-bundle is aspherical, hence a classifying space for its fundamental group. Therefore central extensionsΓof a groupQbyZare classified by their Euler class of the associated principalS1-bundle inH2pQ;Zq. We will call this the Euler class of the central extensionand denote it byepΓq.
Remark4.6. This is not the usual definition of the characteristic class of a central ex-tension. In general central extensions
0ÝÑCÝÝÑi ΓÝÑÝp QÝÑ0
are classified by their characteristic class in H2pQ;Cq. Since we will not need this classification in full generality, we use our simplified definition.
Next we give a construction of the classifying space Bπ1pMqfor a manifold M. In fact, we will also construct a classifying mapM ÝÑBπ1pMq. LetΓbe the fundamental group of a manifold M. Then a classifying space BΓ for Γ can be constructed in the following way. We attach cells of dimension 3 toMalong generators ofπ2pMqin order to get a space M2 withπ2pM2q “ 0. Subsequently, we attach 4-cells in order to get a spaceM3such thatπ3pM3q “0 and so on. Since we only attached cells of dimension 3 or higher, the result is an aspherical spaceM8 which has the same fundamental group asM. HenceM8 “BΓ. Thus we have a natural inclusion
ι: MÝÑ BΓ. (4.3)
By definitionιinduces an isomorphism
ι˚: H1pBΓq ÝÑH1pMq (4.4)
and a injection
ι˚: H2pBΓq ÝÑH2pMq. (4.5) In particular it follows that b1pΓq “ b1pMqfor any manifold Mwithπ1pMq “Γ.
Now suppose X is an orbifold. Then we can replicate the above construction on the orbifold classifying spaceBX. We obtain a map ι: BX ÝÑ Bπorb1 pXqsuch that the homomorphisms (4.4) and (4.5) satisfy the same properties when replacing H˚pMqby Horb˚ pXq.