Invertible topological field theories

Abstract

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