Algebraic topology of G2 manifolds

Abstract

In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G 2 . Some of these results are new. We give self-contained proofs here. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. For the sake of completeness we decided to collect them here in a self-contained way to be easily accessible for future usage in calibrated geometry. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of G 2 manifolds with special properties, which appear in the first named author's work with S. Salur about G 2 dualities. IntroductionRecall that G 2 ⊂ SO (7) is the 14-dimensional exceptional Lie group defined as the automorphisms of the imaginary octonions im(O) = R 7 preserving the cross product operation R 7 × R 7 → R 7 (e.g. [HL], [Br], [AS1], [AS2]). Octonions are the elements of the 8 dimensional division algebra O = H ⊕ lH = R 8 where H are the quaternions, O is generated by 1, i, j, k, l, li, lj, lk . The cross product operation × on im(O) is induced from the octonion multiplication on O by uWe say an oriented 7-manifold M 7 has a G 2 structure if its SO(7)-tangent frame bundle lifts to a G 2 -bundle by the canonical fibration:Alternatively G 2 can be defined by the special 3-frames in R 7 as follows:or as linear automorphisms of R 7 preserving a certain 3-form ϕ 0 ∈ Ω 7 (R 7 )By using this last definition, a G 2 structure on M 7 can be defined as a 3form ϕ ∈ Ω 3 (M 7 ) such that at each p ∈ M the pair (T p (M), ϕ(p)) is (pointwise)

Free immersions and panelled web 4-manifolds

“…In particular Ker h 4 ∈ C 7 . Besides that Ker h 4 is certainly contained in π 4 (G + 3 R 7 ) = Z ⊕ Z from [AK16] which is torsion free. C 7 is a class of torsion groups, and the only torsion subgroup of Z ⊕ Z is the trivial one, hence Ker h 4 = 0.…”

“…We have the following characteristic class relations for the bundles over the Grassmannian G + 3 R 7 , This is a combination of the results in Section 7 of the resource. Notice that, to be able to say that the generators stated in part (c) are the sole generators, one needs to know that there is no torsion as explained in [AK16]. Also note that we can realize the embeddings of CP 2 and CP 2 through the inclusion of G + 2 R 6 in G + 3 R 7 since the oriented Grassmannian is a double cover of the Grassmannian, we can include the projective space with both orientations.…”

“…Proof. Following [AK16], G + 3 R 7 is simply-connected so 1-connected, and possessing π 0123 = {0, 0, Z 2 , 0} as the first four homotopy groups none of which contains elements of order 7 k , k ∈ Z + hence of class C 7 .…”

Characteristic classes on Grassmannians

“…We have * ω 5 2 ω 5 3 ω 6 2 ω 6 3 ω 7 2 ω 7 3 ω 8 2 ω 8 3 | f (S 7 ) = ω 1 2 ω 1 3 ω 1 4 ω 5 4 ω 6 4 ω 7 4 ω 8 4 | f (S 7 ) = ω 1 2 ω 1 3 ω 1 4 ω 1 5 ω 1 6 ω 1 7 ω 1 8 = dV S 7 , * ω 1 2 ω 1 3 ω 6 2 ω 6 3 ω 7 2 ω 7 3 ω 8 2 ω 8 In the last section we have Cayley submanifold CAY = {π ∈ G(4, 8 Then the induced metric is…”

“…The computations on specific Grassmann manifolds like G (3,7) or G (4,8) have important implications on the theory of calibrated submanifolds like associative, coassociative, or Cayley submanifolds of Riemannian 7-8-manifolds of G 2 or Spin 7 holonomy. This work has many applications like [1,11] among potential others.…”