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We study necessary and sufficient conditions for the existence of Lorentzianand weak Lorentzian cobordisms between closed smooth manifolds of arbitrarydimension 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 andSorkin on the existence of Lorentzian cobordisms. We compute the $\Spin(1,n)_0$-Lorentzian cobordism group for several dimensions. Restrictions on thegravitational kink numbers of $\Spin(1, n)_0$-weak Lorentzian cobordisms areobtained.

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