Given a sufficiently nice n-pivotal n-category, I'll describe an explicit state sum invariant for oriented n+1-dimensional manifolds. ("Sufficiently nice" refers to some finiteness and semisimplicity conditions; "n-pivotal" means pivotal for n=2 and something analogous for higher n.) The construction also works for unoriented, Spin and Pin manifolds. The universal state sum specializes to a long list of familiar state sums: Turaev-Viro, Crane-Yetter, Douglas-Reutter, Reshitikhin-Turaev Dehn surgery formula (thought of as a state sum), Brown-Arf (for Pin_- 2-manifolds), Dijkgraaf-Witten.
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A Universal State Sum