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To what extent have people classified $n$-dimensional TQFTs that assign a 1-dimensional vector space to every compact oriented $(n-1)$-manifold?

I have some vague reasons to suspect that the classification depends heavily on $n \bmod 8$, but I'm having trouble finding information on this.

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They are called "invertible field theories".

See https://www.ma.utexas.edu/users/dafr/M392C/, lectures 17-24 for the relation to Madsen-Tillmann spectra.

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See also: arxiv.org/abs/1406.7278 –  Qiaochu Yuan Aug 19 at 15:27

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