Invertible topological field theories.

A d$d$‐dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal (∞,n)$(\infty,n)$‐category of d$d$‐bordisms (embedded into R∞$\mathbb {R}^\infty$ and equipped with a tangential (X,ξ)$(X,\xi)$‐structure) that lands in the Picard subcategory of the target symmetric monoidal (∞,n)$(\infty,n)$‐category. We classify these field theories in terms of the cohomology of the (n−d)$(n-d)$‐connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the (∞,n)$(\infty,n)$‐category of bordisms with Ω∞−nMTξ$\Omega ^{\infty -n}MT\xi$ as an E∞$E_\infty$‐space. This generalizes the celebrated result of Galatius–Madsen–Tillmann–Weiss (Acta Math. 202 (2009), no. 2, 195–239) in the case n=1$n=1$, and of Bökstedt–Madsen (An alpine expedition through algebraic topology, vol. 617, Contemp. Math., Amer. Math. Soc., Providence, RI, 2014, pp. 39–80) in the n$n$‐uple case. We also obtain results for the (∞,n)$(\infty,n)$‐category of d$d$‐bordisms embedding into a fixed ambient manifold M$M$, generalizing results of Randal–Williams (Int. Math. Res. Not. IMRN 2011 (2011), no. 3, 572–608) in the case n=1$n=1$. We give two applications: (1) we completely compute all extended and partially extended invertible TFTs with target a certain category of n$n$‐vector spaces (for n⩽4$n \leqslant 4$), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum (Forum Math. 25 (2013), no. 5, 1067–1106. arXiv:0912.4706). [ABSTRACT FROM AUTHOR]