Some applications of differential topology in general relativity

TL;DR

This paper explores how differential topology can reveal obstructions to certain spacetime structures in general relativity, clarifies relationships between cobordisms and causal properties, and provides an accessible overview of related results.

Contribution

It introduces new topological obstructions to spin-Lorentz and pin-Lorentz cobordisms and clarifies their equivalence and implications for spacetime causality.

Findings

Obstructions to spin-Lorentz and pin-Lorentz cobordisms identified

No direct link between homotopy type of Lorentz metric and causality in compact spacetimes

Spin-Lorentz and tetrad cobordism are shown to be equivalent

Abstract

Recently, there have been several applications of differential and algebraic topology to problems concerned with the global structure of spacetimes. In this paper, we derive obstructions to the existence of spin-Lorentz and pin-Lorentz cobordisms and we show that for compact spacetimes with non-empty boundary there is no relationship between the homotopy type of the Lorentz metric and the causal structure. We also point out that spin-Lorentz and tetrad cobordism are equivalent. Furthermore, because the original work [7] on metric homotopy and causality may not be known to a wide audience, we present an overview of the results here.