Stable homology as an indicator of manifoldlikeness in causal set theory

We present a computational tool for calculating the “spatial ” homology groups of a causal set. Localisation in the causal set is provided by an inextendible antichain which is the analog of a spacelike hypersurface, and a one parameter family of nerve simplicial complexes is constructed by “thickening” this antichain to obtain a covering. Earlier analytical work showed that for an inextendible antichain in a causal set which can be approximated by a globally hyperbolic spacetime region, there is a one parameter sub-family of these simplicial complexes which are homological to the continuum, provided the antichain satisfies certain conditions. Using causal sets that are approximated by a set of 2d spacetimes our numerical analysis suggests that these conditions are generically satisfied by inextendible antichains. In both 2d and 3d simulations, as the thickening parameter is increased, the continuum homology groups tend to appear as the first region in which the homology is “stable”. Our stability analysis suggests a necessary though not sufficient criterion to test for manifoldlikeness of a causal set. 1