Invertible topological field theories

A $d$-dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal $(\infty ,n)$-category of $d$-bordisms (embedded into $R^{\infty }$ and equipped with a tangential $(X,\xi )$-structure) that lands in the Picard subcategory of the target symmetric monoidal $(\infty ,n)$-category. We classify these field theories in terms of the cohomology of the $(n-d)$-connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the $(\infty ,n)$-category of bordisms with ${\Omega }^{\infty -n}MT\xi$ as an $E_{\infty }$-space. This generalizes the celebrated result of Galatius–Madsen–Tillmann–Weiss (Acta Math. 202 (2009), no. 2, 195–239) in the case $n=1$, 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 $n$-uple case. We also obtain results for the $(\infty ,n)$-category of $d$-bordisms embedding into a fixed ambient manifold $M$, generalizing results of Randal–Williams (Int. Math. Res. Not. IMRN 2011 (2011), no. 3, 572–608) in the case $n=1$. We give two applications: (1) we completely compute all extended and partially extended invertible TFTs with target a certain category of $n$-vector spaces (for $n⩽4$), 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).