Topology change and selection rules for high-dimensional $\Spin(1, n)_0$-Lorentzian cobordisms

TL;DR

This paper establishes conditions for the existence of Lorentzian cobordisms with $ ext{Spin}(1,n)_0$ structure between manifolds, extending previous results and computing related cobordism groups across dimensions.

Contribution

It generalizes previous work on Lorentzian cobordisms to higher dimensions with $ ext{Spin}(1,n)_0$ structure and computes the associated cobordism groups.

Findings

Necessary and sufficient conditions for $ ext{Spin}(1,n)_0$-Lorentzian cobordisms.

Computation of the cobordism group in several dimensions.

Restrictions on gravitational kink numbers for weak Lorentzian cobordisms.

Abstract

We study necessary and sufficient conditions for the existence of Lorentzian and weak Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is $\Spin(1, n)_0$. This extends a result of Gibbons-Hawking on $\Sl(2, \C)$-Lorentzian cobordisms between 3-manifolds and results of Reinhart and Sorkin on the existence of Lorentzian cobordisms. We compute the $\Spin(1, n)_0$-Lorentzian cobordism group for several dimensions. Restrictions on the gravitational kink numbers of $\Spin(1, n)_0$-weak Lorentzian cobordisms are obtained.