1 + 1 dimensional cobordism categories and invertible TQFT for Klein surfaces - Juer, R (2012)
We discuss a method of classifying 2-dimensional invertible topological quantum field theories (TQFTs) whose domain surface categories allow non-orientable cobordisms. These are known as Klein TQFTs. To this end we study the 1+1 dimensional open-closed unoriented cobordism category K, whose objects are compact 1-manifolds and whose morphisms are compact (not necessarily orientable) cobordisms up to homeomorphism. We are able to compute the fundamental group of its classifying space BK and, by way of this result, derive an infinite loop splitting of BK, a classification of functors K → ℤ, and a classification of 2-dimensional open-closed invertible Klein TQFTs. Analogous results are obtained for the two subcategories of K whose objects are closed or have boundary respectively, including classifications of both closed and open invertible Klein TQFTs. The results obtained throughout the paper are generalisations of previous results by Tillmann [Til96] and Douglas [Dou00] regarding the 1+1 dimensional closed and open-closed oriented cobordism categories. Finally we consider how our results should be interpreted in terms of the known classification of 2-dimensional TQFTs in terms of Frobenius algebras.