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We propose that Symmetry Protected Topological Phases with a finite symmetrygroup G are classified by cobordism groups of the classifying space of G. Thisprovides an explanation for the recent discovery of bosonic SPT phases which donot fit into the group cohomology classification. We discuss the connection ofthe cobordism classification of SPT phases to gauge and gravitational anomaliesin various dimensions.

Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology