A pseudo-riemannian manifold of signature (p,q) is a smooth manifold of dimension p+q endowed with a smooth metric g of signature (p,q). The case q=1, called Lorentzian geometry, has been studied extensively as it is used in the mathematical theory for general relativity. One of the conditions usually required for a manifold to be a reasonable model for a spacetime is global hyperbolicity. In particular, globally hyperbolic spacetimes are always homeomorphic to a product S x R where any inextendible non-positive curve is the graph of a map from R to S. Because R is the only connected simply connected one dimensional manifold, global hyperbolicity has a simplicity to its definitions which is difficult to reproduce in the case q>1, where the topology of the negative part plays a bigger role.

 

In this talk, we will discuss two ways to generalize global hyperbolicity in higher signature, one stronger than the other. To this end, we will introduce a new homological theory suited to the study of pseudo-riemannian spaces and we will use it to reformulate global hyperbolicity using intersection theory.