Bosonic Topological Insulators and Paramagnets: a view from cobordisms

We classify Bosonic Topological Insulators and Paramagnets in D ≤ 4 spatial dimensions using the cobordism approach. For D < 4 we confirm that the only such phase which does not fit into the group cohomology classification is the 3D Bosonic Topological Insulator protected by time-reversal symmetry whose surface admits an all-fermion topologically ordered state. For D = 4 there is a unique ”beyond group cohomology” phase. It is protected by gravitational anomalies of the boundary theory and is stable without any additional symmetry