The G 2 sphere of a 4-manifold

On the G 2 bundle of a Riemannian 4 - manifold

We expose the theory of the construction of a natural G 2 structure on the unit sphere tangent bundle SM of any given orientable Riemannian 4-manifold M , as laid in [4]. This time we work in the context of metric connections, or geometry with torsion. 1 Recalling the theory By a G 2 manifold it is understood a 7 dimensional Riemannian manifold with holonomy group contained in G 2 = Aut(O). The...

Associative Submanifolds of a G 2 Manifold

We study deformations of associative submanifolds of a G 2 manifold. We show that deformation spaces can be perturbed to be smooth and finite dimensional , and they can be made compact by constraining them with an additional equation and reducing it to Seiberg-Witten theory. This allows us to associate in-variants of certain associative submanifolds. More generally we apply this process to cert...

Associative Submanifolds of a G 2 Manifold

We study deformations of associative submanifolds Y 3 ⊂ M 7 of a G 2 manifold M 7. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative submanifolds of M. The local equations at each associative Y are restrictions of a global eq...

Spherical 2-categories and 4-manifold Invariants

A Hyperbolic 4-Manifold

There is a regular 4-dimensional polyhedron with 120 dodecahedra as 3-dimensional faces. (Coxeter calls it the "120-cell".) The group of symmetries of this polyhedron is the Coxeter group with diagram: For each pair of opposite 3-dimensional faces of this polyhedron there is a unique reflection in its symmetry group which interchanges them. The result of identifying opposite faces by these refl...