On the categorical and topological structure of timelike and causal homotopy classes of paths in smooth spacetimes

TL;DR

This paper explores the topological and categorical structures of timelike and causal homotopy classes in smooth spacetimes, revealing how these structures encode the spacetime's topology and conformal geometry without strong causality assumptions.

Contribution

It introduces a topology on spacetime based on timelike homotopy classes that refines the Alexandrov topology and shows how algebraic structures of homotopy classes encode the spacetime's topology and conformal structure.

Findings

The topology on spacetime from homotopy classes always matches the manifold topology.

The space of homotopy classes forms a semicategory or category, capturing essential geometric information.

The topology on the space of homotopy classes is locally Euclidean but generally not Hausdorff.

Abstract

For a smooth spacetime $X$, based on the timelike homotopy classes of its timelike paths, we define a topology on $X$ that refines the Alexandrov topology and always coincides with the manifold topology. The space of timelike or causal homotopy classes forms a semicategory or a category, respectively. We show that either of these algebraic structures encodes enough information to reconstruct the topology and conformal structure of $X$. Furthermore, the space of timelike homotopy classes carries a natural topology that we prove to be locally euclidean but, in general, not Hausdorff. The presented results do not require any causality conditions on $X$ and do also hold under weaker regularity assumptions.