On Sp(2) and Sp(2)· Sp(1) -structures in 8-dimensional vector bundles

Publicacions Matem`atiques, Vol 41 (1997), 383–401. ON Sp(2) AND Sp(2) ·Sp(1)-STRUCTURES IN 8-DIMENSIONAL VECTOR BUNDLES Martin ˇ Cadek and Jiˇ r´ iVan ˇ zura Abstract Let ξbe an oriented 8-dimensional vector bundle. We prove that the structure group SO(8) of ξcan be reduced to Sp(2) or Sp(2) ·Sp(1) if and only if the vector bundle associated to ξ via a certain outer automorphism of the group Spin(8) has 3 linearly independent sections or contains a 3-dimensional subbundle. Necessary and sufficient conditions for the existence of an Sp(2)- structure in ξover a closed connected spin manifold of dimension 8 are also given in terms of characteristic classes. 1. Introduction. In the last years great attention has been devoted to the study of hyper-K¨ahler and quaternion-K¨ahler manifolds of the dimension 4n. The structure group SO(4n) of the tangent bundles of these manifolds can be reduced to Sp(n) and Sp(n)·Sp(1), respectively. (See [Bes].) In the former case we will talk about an Sp(n)- structure, in the latter case about almost quaternionic structure, which is an Sp(n)·Sp(1)-structure in the tangent bundle. It is natural to ask about necessary and sufficient conditions for the existence of these structures in terms of characteristic classes. In dimension 4 the situation is easy since Sp(1) ∼ =SU(2) and Sp(1) ·Sp(1) ∼ =SO(4). We will show that in dimension 8 the existence of an Sp(2)-structure and an Sp(2) ·Sp(1)-structure can be reduced to the problems of existence of 3 linearly independent sections and a 3-dimensional subbundle in a certain other vector bundle, respectively. These problems have been solved at least partially. Research supported by the grant 201/93/2178 of the Grant Agency of the Czech Republic. Keywords. Cayley numbers, principle of triality, vector bundle, reduction of the structure group, classifying spaces, characteristic classes. 1991 Mathematics subject classifications: 57R22, 57R25, 55R25, 22E99. 384 M. ˇ Cadek, J. Vanˇ zura To prove the reduction theorems we explore the Cayley numbers, the principle of triality and the triality automorphism of Spin(8). We use the triality to describe the isomorhisms between Sp(2) and Spin(5), and between Sp(2) ·Sp(1) and Spin(5) ·Spin(3). All this is carried out in Section 2. The reduction theorems themselves are proved in Section 3. Section 4 has auxiliary character and contains necessary information on the cohomologies of classifying spaces and the triality automorphism in cohomology. In Section 5 the previous results together with the results of Crabb and Steer ([CS]) and Dupont ([Du]) are applied to obtain necessary and sufficient conditions for the existence of an Sp(2)-structure in oriented 8-dimensional vector bundles over closed connected spin manifolds of the same dimension. At the end we mention also the existence of Sp(1)-structures and some examples. The case of almost quaternionic structure needs some more effort and will be treated in the next paper ([CV2]). 2. The action of Sp(2) and Sp(2) ·Sp(1) on the Cayley numbers. The letters Z,R,C,Hand Owill denote integers, real numbers, complex numbers, quaternions and the Cayley numbers, respectively. Sp(2) is the group of the quaternionic linear automorphisms acting from the left on a right quaternionic 2-dimensional vector space preserving a positive definite Hermitian form on it. If we identify Hwith the real 4-dimensional vector space R⊕iR⊕jR⊕kR, we get the inclusion β:Sp(2) →SO(8). Let π:Spin(8) →SO(8) be the standard double covering. Since Sp(2) is simply connected, there is a monomorphism γ:Sp(2) →Spin(8) such that the diagram Spin(8) u π Sp(2) NNN NP γ w βSO(8) commutes. Sp(2) ·Sp(1) is the group Sp(2) ×Sp(1)/{(1,1),(−1,−1)}. The following left action on a right quaternionic 2-dimensional space V (A, α)v=Av¯α where A∈Sp(2), α∈Sp(1), v∈Vand ¯αis a quaternionic conjugate to α, induces a homomorphism Sp(2) ×Sp(1) →SO(8) with kernel {(1,1)(−1,−1)}, whence the inclusion ν:Sp(2) ·Sp(1) →SO(8). The On Sp(2) and Sp(2) ·Sp(1)-structures 385 former homomorphism induces also a homomorhism µ:Sp(2)×Sp(1) → Spin(8). The kernel of this homomorphism is again {(1,1),(−1,−1)}. Taking the curve (expπit ⊕expπit,expπit)∈Sp(2) ×Sp(1) for 0 ≤t≤1, beginning at (1,1) and ending at (−1,−1), its image is the loop in SO(8) which is covered by a loop in Spin(8) beginning and ending at 1. That is why there is an inclusion ˆµ:Sp(2) ·Sp(1) →Spin(8) such that the diagram Sp(2) ×Sp(1) w µ u Spin(8) u π Sp(2) ·Sp(1) AAAA AC ˆµ w νSO(8) commutes. Now we convert the Cayley numbers into a right quaternionic vector space. Although His a subalgebra of O, the usual multiplication is not a right action. We define a new multiplication denoted by the dot ·:O×H→Oin the following way x·1=x, x ·i=xi, x ·j=xj, x ·k=(xi)j, where x∈Oand xy stands for the usual multiplication in O. This multiplication converts Ointo a right H-vector space with the basis 1 and e. (The basis of Oover Ris 1,i,j,k,e,f,g,h, the usual multiplication is given in the same way as in [Po].) Since old and new multiplication by iand jfrom the right are the same, it is easy to see that Sp(2) = {A∈SO(8) : A(xi)=A(x)i, A(xj)=A(x)jfor every x∈O}. For the corresponding Lie algebras it reads as (1) sp(2) = {a∈so(8) : a(xi)=a(x)i, a(xj)=a(x)jfor every x∈O}. According to [Fr] (see also [Bra]) there are outer automorphisms λ and κof so(8) such that the principle of triality holds. For every x, y ∈O and every a∈so(8) we have a(xy)=b(x)y+xc(y) where b=(λκ)(a),c=(κλ)(a). The automorphisms λand κare described in detail in [Fr] and [Bra]. For the moment we need only the following properties λ3=id,κ 2=id, κλκ =λ2,λ=id. Using them we get that (κλ)2= id. So, for a,b,cin the principle of triality we have a=(κλ)(c) and b=λ2(c). Let us note that κλ and λκ are standard spin representations + and −. 386 M. ˇ Cadek, J. Vanˇ zura Lemma 2.1. The homomorphism κλ restricted to the Lie algebra (2) so(5) = {c∈so(8) : c(1) = c(i)=c(j)=0} is an isomorphism between so(5) and sp(2). Proof: Let c∈so(5), a=(κλ)(c), b=λ2(c). Using the principle of triality and the characterization of so(5) we get a(x)=a(x1) = b(x)1 + xc(1) = b(x). Next a(xi)=b(x)i+xc(i)=a(x)i a(xj)=b(x)j+xc(j)=a(x)j. So we have proved that a=(κλ)(c)∈sp(2) for every c∈so(5). Since κλ is a monomorphism and dim sp(2) = dim so(5), we get that κλ is an isomorhism between so(5) and sp(2). We will denote corresponding homomorphisms of Lie groups and Lie algebras by the same letters. Lemma 2.2. Let υ:Spin(5) →Spin(8) be the canonical inclusion. Then the diagram Spin(5) w υSpin(8) Sp(2) w γ ØØØØØ β u κλ Spin(8) u π u κλ SO(8) is commutative. Proof: The homomorphism γon the level of Lie algebras is the inclusion given by (1). Hence the upper square commutes on the level of Lie algebras according to the previous lemma since κλ is involution. So, it commutes on the level of the corresponding simply connected Lie groups as well. Finally, γwas chosen for the lower triangle to commute. Consider so(3) as the following subalgebra of so(8) = so(O) (3) so(3) = {c∈so(8) : c(k)=c(e)=c(f)=c(g)=c(h)=0}. On Sp(2) and Sp(2) ·Sp(1)-structures 387 The intersection of this algebra with the algebra so(5) from Lemma 2.1 is zero. The direct sum of these algebras can be characterized in the following way (4) so(5) ⊕so(3) = {c∈so(8) : c(1),c(i),c(j)∈R1,i,j} where R1,i,jis the real vector subspace of Ogenerated by 1,i,j. The Lie algebra sp(1) is the space of purely imaginary quaternions with the bracket [α1,α 2]=α1α2−α2α1. Its left action on the Cayley numbers equipped with a right multiplication ·by quaternions defined above is (α, x)−→ x·¯α for α∈sp(1), x∈O. So, we can consider sp(1) as a subalgebra of so(8) in the following way (5) sp(1) = {a∈so(8) : there is α∈H,¯α=−α, a(x)=x·¯α}. Lemma 2.3. The algebras sp(2) and sp(1) considered as subalgebras of so(8) by (1) and (5) have trivial intersection. Proof: Let a∈sp(2), α∈H,¯α=−αand for all x∈O a(x)=x·¯α. Then also x·(i¯α)=(x·i)·¯α=a(x·i)=a(x)·i=(x·¯α)·i=x·(¯αi). Hence i¯α=¯αi and similarly j¯α=¯αj and k¯α=¯αk, which implies α=0. In what follows we consider so(5) ⊕so(3), sp(2) and sp(1) only as Lie subalgebras of so(8) determined by (4), (1) and (5). Lemma 2.4. The image of sp(2) ⊕sp(1) under the isomorphism κλ is so(5) ⊕so(3). Proof: In Lemma 2.1 we have already proved that κλ(sp(2)) = so(5). Since κλ is an isomorphism and dim(so(5) ⊕so(3)) = dim(sp(2) ⊕sp(1)), it is sufficient to show that (κλ)(sp(1)) ⊂so(5) ⊕so(3). sp(1) as a Lie algebra is generated by elements a1,a 2, where a1(x)=x·¯ i=−xi, a2(x)=x·¯ j=−xj 388 M. ˇ Cadek, J. Vanˇ zura for every x∈O. It suffices to prove that (κλ)(a1) and (κλ)(a2) are in so(5) ⊕so(3). Let Rα,Lαbe usual right and left multiplication by the Cayley number α. Brada in [Bra] derived that Rα(xy)=(−Rαx)y+x((Rα+Lα)y). Comparing it with the principle of triality we get that c1=(κλ)(a1)=−(Ri+Li) c2=(κλ)(a1)=−(Rj+Lj). Hence c1(1) = −2i, c1(i)=2,c 1(j)=0, c2(1) = −2j, c1(i)=0,c 1(j)=2, which yields c1,c 2∈so(5) ⊕so(3) and completes the proof. Let θ:so(5) ⊕so(3) →so(8) be the canonical inclusion given by (4). Then the diagram Spin(5) ×Spin(3) w θSpin(8) Sp(2) ×Sp(1) w µ u κλ Spin(8) u κλ commutes since it commutes already on the level of Lie algebras according to the previous lemma. Moreover, ker θ={(1,1),(−1,−1)}and (κλ)(ker θ)=kerµ. Hence we can factor the homomorphisms θ,µand κλ to ϑ,ˆµand κλ, respectively, and get Lemma 2.5. The diagram Spin(5) ·Spin(3) w ϑSpin(8) Sp(2) ·Sp(1) w ˆµ ØØØØØ ν u κλ Spin(8) u π u κλ SO(8) commutes. From Lemma 2.2 and 2.5 we obtain immediately the following consequences. On Sp(2) and Sp(2) ·Sp(1)-structures 389 Lemma 2.6. The homogeneous space Spin(8)/Sp(2) determined by the inclusion γis diffeomorphic to the Stiefel manifold V8,3. The homogeneous space Spin(8)/Sp(2) ·Sp(1) determined by the inclusion ˆµis diffeomorphic to the Grassmann manifold G8,3. 3. Equivalent conditions for the existence. It is well known that for every topological group Gthere is a universal principal G-bundle EG →BG. Using Milnor’s construction of the functor B(see [Mi]) we can convert the commutative diagrams from Lemma 2.2 and 2.5 into commutative diagrams of classifying spaces. The mappings between classifying spaces corresponding to homomorphisms of groups will be denoted again by the same letters. Let Xbe a CW-complex. Applying the functor [X,−] we have [X, BSpin(5)] w υ∗[X, BSpin(8)] [X,BSp(2)] w γ∗ ØØØØØ β∗ u (κλ)∗ [X, BSpin(8)] u π∗ u (κλ)∗ [X,BSO(8)] and [X,B(Spin(5) ·Spin(3))] w ϑ∗[X, BSpin(8)] [X,B(Sp(2) ·Sp(1))] w ˆµ∗ ØØØØØ ν∗ u (κλ)∗ [X, BSpin(8)] u π∗ u (κλ)∗ [X,BSO(8)] where (κλ)∗and (κλ)∗are bijections. Classes of oriented 8-dimensional vector bundles over Xare in oneto-one correspondence with elements of [X, BSO(8)]. A vector bundle ξ∈[X,BSO(8)] has an Sp(2)-structure iff it is in the image of β∗and it has an Sp(2) ·Sp(1)-structure iff it is in the image of ν∗. A necessary condition for the existence of any of these structures is the existence of spin structure, i.e. ¯ ξ∈[X, BSpin(8)], π∗¯ ξ=ξ. So, let ξhave a spinor structure ¯ ξ. Then ξis in the image of β∗if and only if (κλ)∗(¯ ξ) is in the image of υ∗which is equivalent to the fact that the vector bundle π∗(κλ)∗¯ ξhas 3 linearly independent sections. 390 M. ˇ Cadek, J. Vanˇ zura Similarly, ξis in the image of ν∗if and only if (κλ)∗(¯ ξ) is in the image of ϑ∗, which is equivalent to the fact that the vector bundle π∗(κλ)∗(¯ ξ) has an oriented 3-dimensional subbundle. Hence we have proved Theorem 3.1. Let Xbe a CW-complex and let ξbe an oriented 8-dimensional vector bundle over X. Then ξhas an Sp(2)-structure if and only if it has a spinor structure ¯ ξand the vector bundle π∗(κλ)∗(¯ ξ) has 3linearly independent sections. Theorem 3.2. Let Xbe a CW-complex and let ξbe an oriented 8-dimensional vector bundle over X. Then ξhas an Sp(2) ·Sp(1)- structure if and only if it has a spinor structure ¯ ξand the vector bundle π∗(κλ)∗(¯ ξ)has an oriented 3-dimensional subbundle. 4. Triality automorphism in cohomology. In this section we summarize the facts on singular cohomology of BSpin(8) and κλ needed for the computation of necessary and sufficient conditions for the existence of Sp(2)-structure in terms of characteristic classes. We will use wm(ξ) for the m-th Stiefel-Whitney class of the vector bundle ξ,pm(ξ) for the m-th Pontrjagin class, and e(ξ) for the Euler class. For a complex vector bundle ξthe symbol cm(ξ) denotes the m-th Chern class. The letters wm,pm,eand cmwill stand for the characteristic classes of the universal bundles over the classifying spaces BSO(8), BSpin(8) and BU(4), respectively. The mapping ρm:H∗(X,Z)→ H∗(X,Zm) is induced from the reduction mod m. We say that x∈H∗(X;Z) is an element of order m(m=2,3,4,...) if and only if x= 0 and mis the least positive integer such that mx =0 (if it exists). The description of cohomologies of BSpin(8) comes from [Qu] and [CV1]. Lemma 4.1. The cohomology rings of BSpin(8) are H∗(BSpin(8); Z2)∼ =Z2[w4,w 6,w 7,w 8,ε] and H∗(BSpin(8); Z)∼ =Z[q1,q 2,e,δw 6]/2δw6 where q1,q2and εare defined by the relations p1=2q1,p 2=q2 1+2e+4q2,ρ 2q2=ε. On Sp(2) and Sp(2) ·Sp(1)-structures 391 Moreover, ρ2q1=w4,ρ 2e=w8. Let ξbe an oriented 8-dimensional vector bundle over a CW-complex Xgiven by the homotopy class of some mapping ξ:X→BSO(8). ξ has a spinor structure iff w2(ξ) = 0. If some lifting ¯ ξ:X→BSpin(8) is fixed we can define spin characteristic classes q1(ξ)=¯ ξ∗q1,q 2(ξ)=¯ ξ∗q2. The first spin characteristic class is always independent of the choice of ¯ ξ. Moreover, if H4(X;Z) has no element of order 4, then it is uniquely determined by the relations 2q1(ξ)=p1(ξ),ρ 2q1(ξ)=w4(ξ). The second spin characteristic class is independent of the spinor structure ¯ ξif Xis simply connected or H8(X;Z)∼ =Z. In the case of an 8-dimensional manifold q2(ξ) is uniquely determined by the relation 16q2(ξ)=4p2(ξ)−p2 1(ξ)−8e(ξ). See [CV1]. Lemma 4.2. For κ:BSpin(8) →BSpin(8) and λ:BSpin(8) → BSpin(8) we have κ∗(q1)=q1 κ∗(q2)=q2+e κ∗(e)=−e λ∗(q1)=q1 λ∗(q2)=−e−q2 λ∗(e)=q2. Proof: Lemma is an analoque of Theorem 2.1 from [GG]. Unfortunately, there is a mistake there caused by a bad sign in the formula for the Euler class in the proof. (However, this mistake does not influence the other results in [GG].) So, we outline the proof once more. First, we need more information on λand κ. Put e0=1,e1=i, e2=j,e3=k,e4=e,e5=f,e6=g,e7=hin O, and define gstev=δtves−δsvet. Consider the Cartan subalgebra of so(8) generated by g10,g23,g45 and g76. This basis obviously satisfies the orientation requirements of [BH] 398 M. ˇ Cadek, J. Vanˇ zura Hence the existence of Sp(2)-structure on this manifold implies e(M)[M]≡0mod12 according to Corollary 5.5 (iii). Remark 5.8. Existence of Sp(1)-structure. It is well known that Sp(1) ∼ =SU(2). Using the Postnikov tower for the fibration BSU(2) → BSO(4) it can be easily proved that the structure group of an oriented 4-dimensional vector bundle ξover a CW-complex Xof the same dimension can be reduced to Sp(1) ∼ =SU(2) if and only if w2(ξ)=0 and p1(ξ)+2e(ξ)=0. Consider a simply connected closed smooth 4-manifold M. According to the remark after Rochlin’s Theorem in [FU], the condition w2(M)= 0 is equivalent to the fact that the intersection form ωof Mis even. Rochlin’s Theorem ([FU, Theorem 1.2]) asserts that its signature σ(ω) is divisible by 16 and Donaldson’s Theorem ([FU, Theorem 1.3]) says that ωis indefinite. Using the classification of indefinite forms over Zwe get (∗)ω=−2nE8⊕m01 10  where m∈N,n∈Z,E8being described in [FU], rank E8=8,σ(E8)= 8. Then the signature of Mis σ(M)=−16nand the Euler characteristic is 16n+2m+ 2. Moreover, the Signature Theorem yields p1(M)[M]=3σ(M). Hence the tangent bundle of Mhas Sp(1) ∼ =SU(2)-structure if and only if it has the intersection form (∗) and {p1(M)+2e(M)}[M]=4m−16n+4=0, which means m=4n−1,n≥1. Example 5.9. Let n≥1 and M(n) be a simply connected closed smooth 4-manifold with the intersection form (∗) where m=4n−1. Then the tangent bundles of the manifolds M(n1)×M(n2) have an Sp(1) ×Sp(1)-structure and that is why also an Sp(2)-structure. On Sp(2) and Sp(2) ·Sp(1)-structures 399 Example 5.10. The manifolds S2×S6and S8do not carry an Sp(2)-structure since 4p2(M)−p2 1(M)−8e(M)=−8e(M)=0. The tangent bundle to the manifold S3×S5admits an Sp(2)-structure since all the characteristic classes are zero. (In fact, it is trivial.) Example 5.11. Quaternionic projective space HP2does not carry an Sp(2)-structure. (It has not even almost complex structure, which was proved in [Hi]). Borel and Hirzebruch ([BH]) computed p1(HP2)=2u, p2(HP2)=7u2,e(HP2)=3u2 where u∈H4(HP2;Z) and H∗(HP2;Z)=Z[u]/u3. So (iii) of Corollary 5.5 is not satisfied. Example 5.12. Complex Grassmann manifold G4,2(C) does not admit an Sp(2)-structure. From [BH] we know that H∗(G4,2(C); Z)=Z[u, v]/u3−2uv, v2−u2v where u∈H2(G4,2(C); Z) and v∈H4(G4,2(C); Z) and c1(G4,2(C)) = −4uc 2(G4,2(C)) = 7u2 c3(G4,2(C); Z)=−12uv c4(G4,2(C)) = 6u2v. But the condition (ii) of Corollary 5.6 is not satisfied. Example 5.13. In [Bea] and [Bes] there are two examples of closed simply connected hyper-K¨ahler manifolds of dimension 8. They are obtained by special constructions applied to the Kummer surface K3 and the complex torus. Using Remark 5.7 we can conclude that their Euler characteristics are divisible by 12. Example 5.14. Complex 4-dimensional projective surfaces Vd={(z0,z 1,... ,z 5)∈CP5;zd 0+zd 1+···+zd 5=0} considered as closed oriented smooth manifold of real dimension 8 do not carry an Sp(2)-structure. The equation {4p2(Vd)−p2 1(Vd)−8e(Vd)}[Vd]=d(d−2)(d−6)(−5d2+8d−8) = 0 400 M. ˇ Cadek, J. Vanˇ zura has in positive integers the only solutions d= 2 and d= 6. But for both these d e(Vd)[Vd]≡2mod4. Acknowledgement. The authors are grateful to the referee for the helpful comments which have improved this work. References [Bea] A. Beauville, Vari´et´es K¨ahleriennes dont la 1`ere classe de Chern est nulle, J. Differential Geom. 18 (1983), 755–782. [Bes] A. L. Besse,“Einstein manifolds,” Chapter 14, Springer-Verlag, Berlin, Heidelberg, 1987. [Br] C. Brada, Elements de la g´eometrie des octaves de Cayley, Thesis, Publications du D´epartement de Math´ematiques de l’Universit´e de Lyon I (1986). [BH] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces I, Amer. J. 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Milnor, Construction of universal bundles I, Ann. of Math. 63 (1956), 272–284. On Sp(2) and Sp(2) ·Sp(1)-structures 401 [Po] M. M. Postnikov,“Lie Groups and Lie Algebras, Lectures in Geometry,,” Semester V, translation from Russian, Mir, Moscow, 1986. [Qu] D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197–212. [Th] E. Thomas, Postnikov invariants and higher order cohomology operations, Ann. of Math. 85 (1967), 184–217. Academy of Sciences of the Czech Republic Institute of Mathematics ˇ Ziˇzkova 22 616 62 Brno CZECH REPUBLIC e-mail: [email protected] e-mail: v[email protected] Primera versi´o rebuda el 12 de Gener de 1996, darrera versi´o rebuda el 8 de Novembre de 1996