Citation: Galatius, Soren, et al. "The homotopy type of the cobordism category." (2009): 195-239.
Web: https://arxiv.org/abs/math/0605249
Tags: Mathematical, TQFT, Invertible-phases
In this fantastic paper, Galatius-Madsen-Tillmann-Weiss compute the homotopy type of the cobordism category. This means the following. Associated to the cobordism category, one can assign a Picard infinity-groupoid by formally inverting all of the objects and morphisms. This Picard infinity-groupoid has a corresponding homotopy type. What is it? Here, the authors give the answer. They compute the spectrum explicity. They call this new spectrum the Madsen-Tillmann spectrum.
Something to observe is that this is a hard and deep theorem. Even in the case that we work with the d=2 dimensional cobordism category, this implies a new proof of the generalized Mumford conjecture:
> Madsen, Ib, and Michael S. Weiss. "The stable moduli space of Riemann surfaces: Mumford's conjecture." arXiv preprint math/0212321 (2002).
This makes it easy to imagine that proving a lattice version of this theorem could be very hard, because it might imply Mumford's conjecture, which is a hard theorem. It was proposed by Mumford in 1983 and remained open until 2002 -
> Mumford, David. "Towards an enumerative geometry of the moduli space of curves." Arithmetic and Geometry: Papers Dedicated to IR Shafarevich on the Occasion of His Sixtieth Birthday. Volume II: Geometry (1983): 271-328.
The main theorem from this paper is quite important (especially in the context of invertible phases), and has been re-proved several times. One proof is in
> Bökstedt, Marcel, and Ib Madsen. "The cobordism category and Waldhausen's K-theory." arXiv preprint arXiv:1102.4155 (2011).
and Lurie + Ayala-Francis both state it as a corollary of the cobordism hypothesis:
> Lurie, Jacob. "On the classification of topological field theories." Current developments in mathematics 2008.1 (2008): 129-280.
> Ayala, David, and John Francis. "The cobordism hypothesis." arXiv preprint arXiv:1705.02240 (2017).