Important Question èr. Thomè: Is there a distinquished class or best class of Riemannian manifolds? Answer: Perhaps the answer is the class of: Einste

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1 Quaternionic Geometry and Einstein Manifolds We consider a Riemannian manifold èm;gè: On any manifold a linear connection r gives a way of comparing tangent vectors at diæerent points through the notion of parallel transport. The First Fundamental Theorem of Riemannian geometry says that there exists a unique torsionfree connection that leaves the metric tensor g invariant, i.e. rg = 0: The Riemann curvature tensor R measures the ëcurvature" of èm;gè: It is given by RèX; Y èz = èr X r Y, r Y r X, r ëx;y ë èz where X; Y; Z are vector æelds on M: Two important irreducible components of the Riemann curvature tensor R are: the Ricci curvature RicèX; Y è = Tr èz 7! RèX; ZèY è which is a symmetric tensor æeld and the scalar curvature which is a function on M: s g = Tr Ric 1

2 Important Question èr. Thomè: Is there a distinquished class or best class of Riemannian manifolds? Answer: Perhaps the answer is the class of: Einstein manifolds. Definition: A Riemannian manifold èm;gè is called an Einstein manifold if there exists a smooth function ç on M such that Ric = çg: The Bianchi identities imply that ç is actually a constant, called the Einstein constant. So there are 3 cases 1. ç é 0: 2. ç = 0: 3. ç é 0: There are two reasons why one might believe that the above answer is correct. 1. Einstein manifolds are the critical points of the ëhilbert Action" given by the total scalar curvature Sègè = Z s g ç g where the measure ç g is normalized to volume one, i.e. R çg = 1: 2. The class of Einstein manifolds is a natural generalization of the class of Riemannian manifolds of constant curvature. For dimension 2 the well known Uniformization Theorem says that every orientable surface admits a metric with curvature either 1; 0; or,1: These are automatically Einstein. In higher dimensions the condition of constant curvature is too strong, and the Einstein condition is a natural generalization. 2

3 Two natural questions arise: 1. Is there any relation between the geometry of an Einstein manifold and its topology? 2. How numerous are Einstein metrics and are they easy or diæcult to ænd? The answer to question 1 is generally yes in dimensions 3 and 4, but no in dimensions greater than 4. The following is well known: 1. In dimension 3, g is Einstein èè g has constant curvature. Not all 3-manifolds admit such a metric. 2. In dimension 4, the Hitchin-Thorpe inequality implies that if èm;gè is compact orientable Einstein manifold, then çèmè ç 3 2 jçèmèj where çèmè is the Euler characteristic and çèmè is the Hirzebruch signature. For example the connected sum è k C P 2 does not admit an Einstein metric if k é 4: In higher dimensions there are no known general obstructions to admitting an Einstein metric. Of course, there are obstructions to speciæc cases. For example, if èm;gè is Einstein with Einstein constant ç é 0 then Myers' Theorem implies that if èm;gè is complete it is compact with ænite fundamental group. As for question 2 very little is known as to how numerous Einstein metrics really are. There appear to be two basic methods for ænding Einstein metrics and both require the addition of more structure. 1. The analytic approach which proves existence of Einstein metrics by working directly with the PDE's. An 3

4 example of this is the celebrated proof by Yau and Aubin of Calabi's conjecture, and more recently the work of G. Tian on the case of positive ærst Chern class. 2 The constructive approach using the method of symmetry reduction. It is this approach that the remainder of this talk is dedicated to. A typical way of introducing more structure is through reduced holonomy. Recall that the Riemannian holonomy group Holègè of a metric g is the subgroup of the orthogonal group generated by the parallel transport of tangent vectors around closed loops. If we restrict ourselves to loops that are homotopic to the constant loop we obtain a subgroup Hol 0 ègè of Holègè called the restricted holonomy group. One can show that both Holègè and Hol 0 ègè only depend on the metric g: If èm;gè is reducible in the sense of de Rham, that is a product èm;gè = èm 1 æ M 2 ;g 1 æ g 2 è then the holonomy representation is reducible, that is, Holègè = Holèg 1 è æ Holèg 2 è: Conversely, if èm;gè is complete and simply connected, and its holonomy representation is reducible then èm;gè is a product. The main theorem on Riemannian holonomy is: 4

5 Berger's Classification Theorem: Let èm;gè be a Riemannian manifold of dimension n which is not locally symmetric and with Hol 0 ègè irreducible. Then Hol 0 ègè is one of the following: è1è SOènè: è2è Uèmè where n = 2m: è3è SUèmè where n = 2m: è4è Spèmè æ Spè1è where n = 4m and m ç 2: è5è Spèmè where n = 4m and m ç 2: è6è Spinè9è where n = 16: è7è Spinè7è where n = 8: è8è G 2 where n = 7: These geometries go by other names as follows: è1è Generic Riemannian geometry. è2è Kíahler geometry. è3è Special Kíahler geometry. è4è Quaternionic Kíahler geometry. è5è Hyperkíahler geometry. è6-8è Exotic holonomy. Remarks: 1. Geometries è3-8è are automatically Einstein. 2. Geometries è2è and è3è are related to the complex numbers. 3. Geometries è4è and è5è are related to the quaternions. 4. Geometries è6-8è are related to the octonions. 5

6 Our work concerns geometries of generic holonomy, i.e type è1è. Nevertheless, the Einstein metrics that we are interested in are quite special and, as the title of my talk suggests, related to the quaternions. The key is to consider the Riemannian cone of certain Riemannian manifolds. In recent work in progress with K. Galicki, we are studying Sasakian-Einstein manifolds using the cone technique. An important special case is what is called a 3-Sasakian structure. 3-Sasakian Manifolds Definition: Let ès;gè be a Riemannian manifold of real dimension m. We say that ès;gè is 3-Sasakian if the holonomy group of the Riemannian cone on S; ècèsè; çgè = èr + æ S; dr 2 + r 2 gè reduces to a subgroup of Spè m+1 è. In particular, m = 4n + 4 3;nç 1 and ècèsè; çgè is hyperkíahler. 3-Sasakian manifolds are manifolds with a certain metric quaternionic contact structure. They were ærst deæned by Kuo in 1970 using tensorial methods, and the deænition is quite technical. Shortly thereafter, Kashiwada proved that every 3-Sasakian manifold is Einstein with positive scalar curvature. Other important properties of 3-Sasakian manifolds were obtained early to mid 1970's by Ishihara, Konishi, Sasaki, Tachibana, Tanno, and Yu. Nevertheless, the only examples known until fairly recently were spheres and their quotients. Proposition: Let ès;gè be a Riemannian manifold Then S is 3-Sasakian if and only if it admits three characteristic vector æelds fç 1 ;ç 2 ;ç 3 g of 3 diæerent Sasakian structures such that gèç a ;ç b è = æ ab and ëç a ;ç b ë = 2æ abc ç c. 6

7 The triple fç 1 ;ç 2 ;ç 3 g deænes ç a èy è = gèç a ;Yè and æ a èy è = r Y ç a for each a = 1; 2; 3. We call fç a ;ç a ; æ a g the 3-Sasakian structure on ès;gè. The hyperkíahler geometry of the cone CèSè gives S a ëquaternionic contact structure" reæected by the composition laws of the è1,1è tensors æ a. æ a æ æ b, ç a æ ç b ç a èç b è = æ ab ; æ a ç b =,æ abc ç c ; =,æ abc æ c, æ ab id: éfrom our point of view, without refering to any of the tensorial properties, we notice that since a hyperkíahler manifold is Ricci-æat, the Deænition and Lemma immediately imply: Corollary: Every 3-Sasakian manifold ès;gè of dimension 4n + 3 is Einstein with Einstein constant ç = 2è2n + 1è. If S is complete it is compact with ænite fundamental group and diam ç ç: Moreover, every 3-Sasakian manifold admits two distinct homothety classes of positive Einstein metrics. CorollaryèB-Galickiè : Every complete 3-Sasakian manifold ès;gè is irreducible with holonomy SOè4n + 3è: Before discussing the fundamental structure theorems of a 3-Sasakian manifold we mention an important vanishing theorem of Galicki and Salamon: Theorem: Let ès;gè be a compact 3-Sasakian manifold of dimension 4n + 3. Then the odd Betti numbers b 2k+1 of S are all zero for 0 ç k ç n. 7

8 The Fundamental Foliations The work of the author on 3-Sasakian manifolds has been done in a series of papers in collaboration with K. Galicki and B. Mann, including one with E. Rees. In what follows S will denote a complete hence compact 3-Sasakian manifold. The 3 characteristic vector æelds ç a for a = 1; 2; 3 generate a locally free action of the group Spè1è ' SUè2è acting as isometries on S and hence, deæne a 3-dimensional Riemannian foliation F 3 on S all leaves of which are compact. Theorem: Let ès;g;ç a è be a compact 3-Sasakian manifold of dimension 4n + 3. Then èiè The metric g is bundle-like with respect to the foliation F 3 : èiiè All leaves are totally geodesic. èiiiè Each leaf L is a 3-dimensional homogeneous spherical space form of constant sectional curvature one, and the generic leaves are either Spè1è = S 3 or SOè3è: èivè The space of leaves S=F 3 is a quaternionic Kíahler orbifold of dimension 4n with scalar curvature 16nèn+2è é 0. Recall that an orbifold or V-manifold is a generalization of a manifold where the local Euclidean condition is replaced by the condition that every point has a neighborhood that is homeomorphic to R n =, where, is a ænite group, called a local uniformizing group. So locally we have ænite branched covers. By a quaternionic Kíahler orbifold we mean an orbifold whose local uniformizing groups preserve the quaternionic Kíahler structure. 8

9 The regular version of this theorem, that is, the case when the space of leaves is a smooth manifold is essentially due to Ishihara and Konishi. In the regular case, Konishi also proved an important inversion theorem. More generally we have Theorem: Let èo;g O è be a quaternionic Kíahler orbifold of dimension 4n with positive scalar curvature 16nèn + 2è: Then there is a principal SOè3è V-bundle over O whose total space admits a 3-Sasakian structure with scalar curvature 2è2n + 1èè4n + 3è: Remark: The local geometry of a 3-Sasakian manifold S is determined completely by the local geometry of the quaternionic Kíahler orbifold O = S=F 3 : Moreover, the local uniformizing groups of O are just the leaf holonomy groups of F 3 ; and they are invariants of the 3-Sasakian structure. There is a 1-dimensional foliation F 1 on S obtained by choosing a 1-dimensional subalgebra of the Lie algebra spè1è: èactually there is a 2-sphere's worth of equivalent 1- dimensional foliations.è This gives the analogue of the Salamon twistor space of a quaternionic Kíahler orbifold, and its geometry is important in the study of 3-Sasakian geometry. We shall refer to the space of leaves Z = S=F as the twistor space of the 3-Sasakian manifold S: 9

10 Theorem: Let ès;gè be a compact 3-Sasakian manifold. Then èiè The leaves of F are all diæeomorphic to circles with cyclic leaf holonomy groups. èiiè The space of leaves Z = S=F is a compact Kíahler- Einstein orbifold èz;hè with scalar curvature 4nèn + 1è in such a way that ç : ès;gè,!èz;hè is an orbifold Riemannian submersion. èiiiè Z has a complex contact structure that is compatible with the Kíahler-Einstein structure. èivè Z is a simply connected normal projective algebraic variety of Kodaira dimension çèzè =,1: èvè Z has the structure of a Q -factorial Fano variety. Recall that a Fano variety means that the anti-canonical sheaf K,1 is ample, Q -factorial means that for every Weil divisor D on Z there is an integer m such that md is a Cartier divisor. Since an orbifold æbration is not a æbration in the usual sense, the usual techniques in topology for æbrations do not apply directly. However, Haeæiger ëhaeë has deæned orbifold homology, cohomology, and homotopy groups which do have an analogue in the standard theory. Let X be an orbifold of dimension n; and let P denote the bundle of orthonormal frames on X: It is a smooth manifold on which the orthogonal group Oènè acts locally freely with the quotient X: Let EOènè,!BOènè denote the universal Oènè bundle. Consider the diagonal action of Oènè on EOènè æ P and denote the quotient by BX: Now there is a natural projection p : BX,!X with generic æber the contractible space EOènè: 10

11 Haeæiger deænes the orbifold cohomology, homology, and homotopy groups by H i orbèx; Zè = H i èbx; Zè; Hi orb èx; Zè = H i èbx; Zè; çi orb èxè = ç i èbxè: This deænition of ç orb 1 is equivalent to Thurston's better known deænition in terms of orbifold deck transformations, and when X is a smooth manifold these orbifold groups coincide with the usual groups. The map p : BX,!X induces an isomorphism H i orb ès; Zè æ Q ' Hi ès; Zè æ Q : An important invariant of Z is the group Pic orb èzè of line V-bundles on Z; and this is related to the orbifold cohomology and homology groups of Haeæiger. Proposition: Let Z be the twistor space of the 3-Sasakian manifold S: Then PicèZè is free, H orb 1 èzè is torsion, and Pic orb èzè ' PicèZè æ H orb 1 èzè Hence, Pic orb èzè æ Q ' PicèZè æ Q : We have: Theorem: èb-galickiè If Pic orb èzè is torsionfree, then there is a 1-1 correspondence between a 3-Sasakian manifold and its twistor space, with the exception of the case that Z = C P 2n+1 in which case there are precisely two 3-Sasakian manifolds corresponding to this twistor space, namely S 4n+3 and RP 4n+3 : 11

12 The more general case when Pic orb èzè has torsion is not fully understood yet. In this case the 1-1 nature clearly fails in the orbifold category, that is when S is a 3-Sasakian orbifold. We suspect that the only case when S is smooth where there is not a 1-1, but a 2-1 correspondence is when S has constant curvature. The geometry of the fundamental foliations æts nicely into a diagram of orbifold æbrations: Z CèSè. &? è,,,,,,, y S: &. O All the spaces in this diagram are Einstein. S is 3- Sasakian, CèSè is hyperkíahler, Z is Kíahler-Einstein, and O is quaternionic Kíahler. The leaf holonomy groups for both foliations are important invariants of the 3-Sasakian structure on S: In fact, the leaf holonomy groups of the foliation F 3 give a rough classiæcation of 3-Sasakian manifolds. First, there is whether the generic leaves are an S 3 or an SOè3è. Then the leaf holonomy groups must be one of the ænite subgroup, of SUè2è or a Z 2 quotient of it, namely: è1è, = id; è2è, = Z m the cyclic group of order m; è3è, = D æ m a binary dihedral group with m an integer greater than 2, è4è, = T æ the binary tetrahedral group, è5è, = O æ the binary octahedral group, è6è, = I æ the binary ocosahedral group. 12

13 A 3-Sasakian manifold of type è1è is a regular 3-Sasakian manifold for which we have a conjecture equivalent to the LeBrun-Salamon Conjecture that any quaternionic Kíahler manifold of positive scalar curvature is symmetric. ConjectureèB-Galickiè: Let ès;gè be a compact regular 3-Sasakian manifold. Then S is homogeneous. 3-Sasakian Homogeneous Manifolds Definition: Let I 0 ès;gè ç IèS;gè be the subgroup of the isometry group which preserves the 3-Sasakian structure, that is if ç k 2 Diæ S corresponds to k 2 I 0 ès;gè then ç k æç a = ç a ; for all a = 1; 2; 3: Then I 0 ès;gè is called the group of 3- Sasakian isometries and when it acts transitively on ès;gè the space S is said to be a 3-Sasakian homogeneous space. The following describes 3-Sasakian homogeneous spaces, and is analogous to Boothby's classiæcation of all homogeneous complex contact manifolds and Wolf's classiæcation of homogeneous quaternionic Kíahler manifolds. TheoremèB-Galicki-Mannè: Any 3-Sasakian homogeneous space S = G=H is one of the following: Spèn + 1è ; Spènè SUèmè S, Uèm, 2èæUè1è æ ; Spèn + 1è SpènèæZ 2 ; SOèkè SOèk, 4èæSpè1è ; G 2 Spè1è ; F 4 Spè3è ; E 6 SUè6è ; E 7 Spinè12è ; E 8 E 7 : Here n ç 0, Spè0è denotes the trivial group, m ç 3, and k ç 7: 13

14 The Quotient Construction Let ès;gè be a 3-Sasakian manifold with a nontrivial group I 0 ès;gè of automorphisms of the 3-Sasakian structure. Now I 0 ès;gè extends to a group of hyperkíahler automorphisms I 0 ès; çgè on the cone CèSè by deæning each element to act trivially on R + : Recall that the hyperkíahler quotient of Hitchin, Karlhede, Linstríom, and Roçcek shows that any subgroup G ç I 0 èm; çgè gives rise to a hyperkíahler moment map ç : M,!g æ æ R 3 ; where g denotes the Lie algebra of G and g æ is its dual. Thus, we can deæne a 3-Sasakian moment map ç S : S,,! g æ æ R 3 simply by restriction ç S = çjs: Such a moment map takes the simple form é ç a S;ç é = 1 2 ça èx ç è: Then we have Theorem èb-g-mè: Let ès;gè be a 3-Sasakian manifold with a connected compact Lie group G acting on S smoothly and properly by 3-Sasakian isometries. Let ç S be the corresponding 3-Sasakian moment map and assume both that 0 is a regular value of ç S and that G acts freely on the submanifold ç,1 S è0è. Furthermore, let ç : ç,1 S è0è,,!s and ç : ç,1 S è0è,,!ç,1 S è0è=g denote the corresponding embedding and submersion. Then ès =G = ç,1 S è0è=g; çgè is a smooth 3- Sasakian manifold of dimension 4èn,dim gè+3 with metric çg and Sasakian vector æelds ç a determined uniquely by the two conditions ç æ g = ç æ çg and ç æ èç a jç,1 è0èè = ç S a : 14

15 3-Sasakian Toric Manifolds We can use the 3-Sasakian reduction theorem to obtain new 3-Sasakian manifolds by toral reductions from the 3-Sasakian sphere S 4n+3 : On S 4n+3 the group I 0 ès;gè of 3-Sasakian automorphisms is Spèn + 1è: We choose a k- dimensional subtorus T k in the maximal torus T n : Every quaternionic representation of a k-torus T k on H n+1 can be described via a homomorphism f æ : T k,!t n+1 f æ èç 1 ;:::;ç k è = 0 ky i=1. ç ai 1 i ::: 0 0 :::.... ky i=1 ç ai n+1 i where èç 1 ; ::; ç k è 2 S 1 æææææs 1 = T k are the complex coordinates on T k ; and a i j 2 Z are the coeæcients of a kæèn+1è integral weight matrix æ = èa l æè l=1;:::;k æ=1;:::;n+1 2 M k;n+1 èzè: Let fe l g k l=1 denote the standard basis for tæ k ' R k : Then the 3-Sasakian moment map ç æ : S 4n+3,!t æ k æ R 3 of the k- torus action deæned by ' èç1 ;:::;ç k èèuè = f æ èç 1 ;:::;ç k èu; is given by ç æ = P l çl æ e l with 1 C A ; ç l æèuè = X æ çu æ ia l æu æ : Let us denote further denote the triple èt k ;f æ ;' èç1 ;:::;ç k èè by T k èæè. We want to study the reduced space Sèæè = S 4n+3 =T k èæè = ç,1 è0è=t k æ èæè 15

16 ., n Consider the kæ minor determinants æ æ1 :::æ k = det 0 a B 1 æ 1 ::: a 1 æ 1 C A a k æ 1 ::: a k æ k obtained by deleting n + 1, k columns of æ: Definition: Let æ 2 M k;n+1 èzè be the weight matrix. èiè If æ æ1 æææææææ k 6= 0; 8 1 ç æ 1 é æææ é æ k ç n + 1, then we say that æ is non-degenerate. Suppose æ is non-degenerate and let g be the kth determinantal divisor, i.e., the gcd of all the k by k minor determinants æ æ1 æææææææ k : Then æ is said to be admissible if in addition we have èiiè gcdèæ æ2 ææææ k+1 ; :::; æ æ1 æææ ^æ s ææææ k+1 ; :::; æ æ1 ææææ k è = g for all sequences of length èk+1è such that 1 ç æ 1 é æææ é æ s é æææé æ k+1 ç n + 1: Theorem èb-g-mè: Let Sèæè be a 3-Sasakian toral quotient space. Then èiè if æ is non-degenerate, Sèæè is an orbifold. èiiè If æ is degenerate, then either Sèæè is a singular stratiæed manifold which is not an orbifold or it is an orbifold obtained by reduction of a lower dimensional sphere S 4n,4r,1 by a torus T k,r èæ 0 è or a ænite quotient of such, where 1 ç r ç k and æ 0 is non-degenerate. èwhen r = k the quotient is the sphere S 4n,4k,1 è: èiiiè Assuming that æ is non-degenerate Sèæè is a smooth manifold if and only if æ is admissible. 16

17 When can we ænd admissible æ? Generally this question is non-trivial. Admissible æ clearly exist for k = 1 for any n; i.e. any dimension by simply choosing the elements of æ to be pairwise relatively prime. They also exist in dimension 7 for any k as the following æ shows: æ = ::: 0 a 1 b ::: 0 a 2 b æææ 1 a k b k 1 C A : We simply choose èa i ;b i è to be relatively prime, and if for some pair i; j a i = æa j or b i = æb j then we must have b i 6= æb j or a i 6= æa j ; respectively. However, there are mod 2 obstructions to smoothness which show somewhat surprisingly: Theorem èb-g-mè: Let æ be admissible and k é 1: Then dim Sèæè ç 15: Furthermore, if k é 4; then dim Sèæè = 7: This says that there are smooth toral quotients Sèæè only in the cases indicated, and we shall see shortly that k is precisely the second Betti number b 2 èsèæèè: Moreover, a theorem of R. Bielawski says that any 3-Sasakian manifold ès;gè whose automorphism group I 0 ès;gè contains a torus T n+1,k is homeomorphic to a Sèæè: 17

18 Furthermore, we have an existence theorem: Theorem èb-g-m-rees,b-g-mè: There exist smooth toric quotients Sèæè in dimension 7 with any second Betti number b 2 èsèæèè = k; and in dimensions 11 and 15 with second Betti numbers b 2 èsèæèè = 2; 3; and 4: The Topology of Sèæè The full isometry group of Sèæè is Tn+ 1, k æ Spè1è and in the case dim Sèæè = 7 one can obtain a stratiæcation of S by analysing the quotient Qèæè = Sèæè=èT 2 æ Spè1è: For general dimension Bielawski has obtained the rational homology of Sèæè by viewing Sèæè as a boundary at inænity of a hyperkíahler orbifold that is the limit of certain toric hyperkíahler orbifolds studied by Bielawski and Dancer. and: The orbit space Qèæè can be stratiæed as follows: Qèæè = Q 0 èæè t Q 1 èæè t Q 2 èæè Lemma èb-g-m-rè: Let dim Sèæè = 7: Then èiè The orbit space Qèæè is homeomorphic to the closed disc ç D 2 ; and the subset of singular orbits Q 1 èæètq 0 èæè is homeomorphic to the ç D 2 ' S 1 : èiiè Q 2 èæè is homeomorphic to the open disc D 2 : èiiiè Q 1 èæè is homeomorphic to the disjoint union of k + 2 copies of the open unit interval. èivè Q 0 èæè is a set of k + 2 points. 18

19 Hence, the quotient Qèæè can be depicted by the polygon: : v k æ e k æ v k+2.. ç 2 e k+2 v 2 æ æ v k+1 e 1 æ v 1 e k+1 The orbit space Qèæè The topology of Sèæè can then be analysed by using a Leray spectral sequence on the induced stratiæcation of Sèæè: This gives Theorem èb-g-m-r,b-g-mè: Let Sèæè be a smooth 3- Sasakian toric quotient of dimension 7. Then Sèæè is simply connected and H 2 èsèæè; Zè ' Z k where k = b 2 èsèæèè: Furthermore, the rational homology is given by b i èsèæèè = è 1 if i = 0; 7; 0 if i = 1; 3; 4; 6; k if i = 2; 5. 19

20 There is certainly torsion in H 3 èsèæè; Zè; but we have only been able to determine this in the case b 2 èsèæèè = k = 1: In this case we have determined the complete cohomology ring in all dimensions. Here, we have æ = p = èp 1 ; æææ;p n è with the p i 's relatively prime. Then Theorem èb-g-mè: Let Sèpè be a 3-Sasakian manifold with b 2 = k = 1 obtained by a circle reduction from S 4n+3 : Then, as rings, H æ, Sèpè; Z æ ç = è Zëe 2 ë ëe n+1 2 = 0ë æ Eëf 2n+1ë! =Rèpè: Here e 2 and f 2n+1 are generators with the subscripts indicating the cohomological dimension of each generator. Furthermore, the relations Rèpè are generated by ç n èpèe n 2 = 0 and f 2n+1 e n 2 = 0, where ç n èpè = P n+1 j=1 p 1 æææ ^p j æææp n+1 is the ènè th elementary symmetric polynomial in the entries of p: Notice this theorem shows that H 2n èsèpè; Zè = Z çn èpè: In more general dimension there is: Theorem èbielawskiè: Let Sèæè is a toric 3-Sasakian orbifold of dimension 4èn, kè + 3. Then we have b 2i èsèæèè = ç k + i, 1 k ç ; i é n + 1, k: 20

21 Consequences We give a list of some of the consequences that follow from our work together with other known results. Corollary 1: The quotients èsèpè;gèpèè give inænitely many non-homotopy equivalent simply-connected compact inhomogeneous 3-Sasakian manifolds, hence positive Einstein manifolds, in dimension 4n, 1 for every n ç 2: A remarkable theorem of Gromov says that if there is a negative number ç and a Riemannian metric with bounded diameter such that all sectional curvatures are ç ç; then the total Betti number of the manifold must be bounded. This together with our results give Corollary 2: For any negative real number ç there are inænitely many 3-Sasakian 7-manifolds, hence positive Einstein 7-manifolds, which do not admit metrics whose sectional curvatures are all greater than or equal to ç: Corollary 3: There are inænitely many 3-Sasakian 7- manifolds with arbitrarily small injectivity radius. Corollary 4: In dimension seven there exist 3-Sasakian manifolds with every rational homology type not excluded by the Galicki-Salamon Vanishing Theorem. Corollary 5: There exist 7-manifolds with arbitrary second Betti number having two distinct positive Einstein metrics of weak G 2 holonomy. Corollary 6: There exist Q -factorial contact Fano 3-folds X with b 2 èxè = l for any positive integer l: A theorem Mori and Mukai says that for smooth Fano 3-folds, b 2 ç 10: 21

22 Corollary 7: If the second Betti number b 2 èsèæèè = k é 3; the 3-Sasakian manifolds Sèæè are not homotopy equivalent to any homogeneous space. Corollary 8: There exist compact, self-dual, T 2 symmetric Einstein orbifolds of positive scalar curvature with arbitrary second Betti number. So the rigidity that is known to hold in the smooth quaternionic Kíahler category fails drastically in the orbifold category. In the case of a regular 3-Sasakian manifold S Galicki and Salamon used an index calculation to derive the following remarkable relation among the Betti numbers of S : nx k=1 kèn + 1, kèèn + 1, 2kèb 2k = 0: Bielawski's formula for the Betti numbers of toric quotients shows that in the non-regular case this formula holds if and only if k = 1: There are no relations in dimension 7 and our construction gives explicit examples where these relations fail in dimensions 11 and