Manifolds admitting a $tilde G_2$-structure

arXiv:0704.0503v2 [math.AT] 9 Aug 2009 Manifolds admitting a ˜G2-structure. Hˆong-Vˆan Lˆe Abstract We find a necessary and sufficient condition for a compact 7-manifold to admit a ˜G2-structure. As a result we find a sufficient condition for an open 7-manifold to admit a closed 3-form of ˜G2-type. MSC: 55S35, 53C10 1 Introduction Recently a new class of geometries related with stable forms has been discovered [Hitchin2000], [Hitchin2001], [Witt2005], [Le2006], [LPV2007]. In some cases we can define easily a nec- essary and sufficient condition for a manifold M to admit a stable form of type ω in terms of topological invariants of M, for example if ω is a 3-form of G2-type [Gray1969]. But in general there is no method to solve the question how to find a necessary and sufficient condition for a manifold to admit a stable form. In a previous note [Le2006] we have wrongly stated a sufficient condition for an open manifold to admit a closed stable 3-form of ˜G2-type. We recall that [Bryant1987] a 3-form on R7 is called of ˜G2-type, if it lies on the Gl(R7)-orbit of a 3-form ω3 = θ1 ∧θ2 ∧θ3 + α1 ∧θ1 + α2 ∧θ2 + α3 ∧θ3 Here αi are 2-forms on V 7 which can be written as α1 = y1 ∧y2 + y3 ∧y4, α2 = y1 ∧y3 −y2 ∧y4, α3 = y1 ∧y4 + y2 ∧y3 and (θ1, θ2, θ3, y1, y2, y3, y4) is an oriented basis of (V 7)∗. The group ˜G2 can be defined as the isotropy group of ω3 under the action of Gl(R7). Bryant proved that [Bryant1987] ˜G2 coincides with the automorphism group of the split octonians. In this note we prove the following 1 Main Theorem. Suppose that M7 is a compact 7-manifold. Then M7 admits a 3-form of ˜G2-type, if and only if M7 is orientable and spinnable. Equivalently the first and second Stiefel-Whitney classes of M7 vanish. Suppose that M7 is an open manifold which admits an embedding to a compact orientable and spinnable 7-manifold. Then M7 admits a closed 3-form of ˜G2-type. 2 Proof of Main Theorem Our proof is based on the following simple fact on ˜G2. 2.1. Lemma. We have π1( ˜G2) = Z2. Hence its maximal compact Lie group is SO(4). This Lemma is well-known, (Bryant mentioned it but he omitted a proof in [Bryant1987]), but I did not find an explicit proof of it in popular lectures on Lie groups, though it could be given as an exercise. For a hint to a solution of this exercise we refer to [HL1982], p.115, for an explicit embedding of SO(4) into G2. The reader can also check that the image of this group is also a subgroup of ˜G2 ⊂Gl(R7). We shall denote this image by SO(4)3,4. The Cartan theory on symmetric spaces implies that SO(4)3,4 is a maximal compact Lie subgroup of ˜G2. Now let us return to proof of our Main theorem. Clearly if M7 admits a ˜G2-structure, then it must be orientable and spinnable, since a maximal compact Lie subgroup SO(4)3,4 of G2 is also a compact subgroup of G2. 2.2. Lemma. Assume that M7 is compact, orientable and spinnable. Then M7 admits a ˜G2-structure. Proof. Since M7 is compact, orientable and spinable, M7 admits a SU(2)-structure [Friedrich1997]. Now it is easy to see that it admits a SO(4)3,4-structure, where SO(4)3,4 is a maximal com- pact Lie subgroup of G2. Hence M7 admits a ˜G2-structure. ✷ To prove the last statement of the Main Theorem we shall use the following theorem due to Eliashberg-Mishachev to deform the 3-form ω3 to a closed 3-form ¯ω3 of ˜G2-type on M7. For a subspace R ⊂ΛpM we denote by CloaR a subspace of the space Sec R which consists of closed p-forms ω : M →R in the cohomology class a ∈Hp(M). Eliashberg-Mishashev Theorem [E-M2002,10.2.1] Let M be an open manifold, a ∈ Hp(M) a fixed cohomology class and R an open DiffM-invariant subset. Then the inclu- 2 sion CloaR ֒→Sec R is a homotopy equivalence. In particular, - any p-form ω : M →R is homotopic in R to a closed form ¯ω. - any homotopy of p-form ωt : M :→R which connects two closed forms ω0, ω1 ∈a can be deformed in R into a homotopy of closed forms ¯ωt connecting ω0 and ω1 ∈a. Let R be the space of all 3-forms of ˜G2-type on M = M7. Clearly this space is an open DiffM7-invariant subset of Λ3M7. Now we apply the Eliashberg-Mishashev theorem to our 3-form ω3 of ˜G2-type whose existence has been proved above. Hence M7 admits a closed 3-form ¯ω3 of ˜G2-type. ✷ 2.3. Remark. It seems that we can drop the closedness condition in our Main Theorem and use the classical obstruction theory to prove the main Theorem. Acknowledgement. This note is partially supported by grant of ASCR Nr IAA100190701. References [Adams1996] J.F. Adams, Lectures on exceptional Lie groups, The Chicago University Press, 1996. [Bryant1987] R. Bryant, Metrics with exceptional holonomy, Ann. of Math. (2), 126 (1987), 525-576. [E-M2002] Y. Eliashberg and N. Mishachev, Introduction to the h-Principle, AMS 2002. [Gray1969] A. Gray, Vector cross products on manifolds, TAMS 141, (1969), 465-504, (Errata in TAMS 148 (1970), 625). [Friedrich1997] Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann, On nearly parallel G2-manifolds, Journal Geom. Phys. 23 (1997), 259-286. [HL

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