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

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.