We propose that Symmetry Protected Topological Phases with a finite symmetry
group G are classified by cobordism groups of the classifying space of G. This
provides an explanation for the recent discovery of bosonic SPT phases which do
not fit into the group cohomology classification. We discuss the connection of
the cobordism classification of SPT phases to gauge and gravitational anomalies
in various dimensions.Comment: 17 pages, latex. v2: typos fixed, a footnote on terminology added.
v3: substantially reworked version which takes into account the possibility
of a nontrivial thermal Hall respons
