Tori detect invertibility of topological field theories
A topological field theory in the sense of [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053); \textit{M. Atiyah}, Turk. J. Math. 21, No. 1, 1--7 (1997; Zbl 0890.57019); \textit{G. B. Segal}, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060); \textit{G. Segal}, Lond. Math. Soc. Lect. Note Ser. 308, 421--577 (2004; Zbl 1372.81138)] is a symmetric monoidal functor \[ \mathcal{Z}:\mathrm{Cod}_{d}\rightarrow\mathrm{Vect} \] with \(\mathrm{Cod}_{d}\) being the symmetric monoidal category of closed compact \((d-1)\)-dimensional manifolds as objects and equivalence classes of \(d\)-dimensional bordisms as morphisms (the monoidal structure is simply the disjoint union) and \(\mathrm{Vect}\) being the category of vector spaces with its standard tensor products as its monoidal structure). The dimension of the vector space \(\mathcal{Z}(M)\) for each closed \((d-1)\)-dimensional manifold \(M\) is a very coarse measure of the complexity of the topological field theory, and invertible field theories assigning a one-dimensional vector space to each \((d-1)\)-dimensional manifold are by no means the simplest theories. Chao-Ming Jian [\url{http://mathoverflow.net/q/218982}] asked whether a \(d\)-dimensional theory assigning one-dimensional vector spaces \(\mathcal{Z}(S^{d-1})\) and \(\mathcal{Z}(T^{d-1})\) to the \((d-1)\)-dimensional sphere \(S^{d-1}\) and the \((d-1)\)-dimensional torus \(T^{d-1}\) must also assign one-dimensional vector space to all other \((d-1)\)-dimensional manifolds. This paper settles the question affirmatively in the sense that, as far as a topological field theory \(\mathcal{Z}\) is so extended that it assigns data to \(d\)-, \((d-1)\)- and \((d-2)\)-manifolds, \(\mathcal{Z}(T^{d-1})\) is one-dimensional if and only if the theory assigns one-dimensional vector space to all closed \((d-1)\)-dimensional manifolds. The principal result in this paper is a far-reaching generalization of this result: Theorem 11.1. Fix \(n\geq2\) and a tangential structure \((X,\xi)\) for \(d\)-manifolds. Assume either that \(d\geq3\) or that \((X,\xi)\) is spherophilic. Let \(\mathcal{Z}\) be an extended \(d\)-dimensional topological field theory \[ \mathcal{Z}:\mathrm{Bord}_{d}^{(X,\xi)}\rightarrow\mathcal{C} \] taking values in the symmetric monoidal \((\infty,n)\)-category \(\mathcal{C}\). Let \(T^{d-1}\) be the \((d-1)-\)torus. If for each \([x]\in\pi_{0}X\) we have that \(\mathcal{Z}(T^{d-1},x_{\ast}\theta_{+1\times\mathrm{Lie}})\) is invertible, then \(\mathcal{Z}\) is an invertible topological field theory. In the above, a theory is said to be \textit{invertible} if it assigns invertible values to all manifolds and \(\otimes\)-invertible objects to \((d-n)\)-manifolds. For each point \(x\in X\), \(x_{\ast}\) denotes the induced mapping from \(d\)-framing on \((X,\xi)\)-structures, and \(_{\ast}\theta_{+1\times\mathrm{Lie}}\) denotes the product \(d\)-framing on the \((d-1)\)-torus which is the \(+1\) bounding \(2\)-framing on the first \(S^{1}\)-factor and the Lie group \(1\)-framing on the remaining factors. For any tangential structure \(\xi:X\rightarrow BO(2)\) for \(2\)-manifolds, if \(X\) is connected, then \((X,\xi)\) is said to be \textit{spherophilic} provided that the \(2\)-sphere \(S^{2}\) admits an \((X,\xi)\)-structure. If \(X\) is disconnected, it is called so if each connected component is so. For each dimension \(d\) and each category number \(1\leq n\leq d\) (which is suppressed from the notation), \(\mathrm{Bord}_{d}^{(X,\xi)}\) denotes the symmetric monoidal \((\infty,n)\)-category of closed compact \((d-n)\)-manifolds as objects, \((d-n+1)\)-dimensional bordisms as \(1\)-morphisms, \((d-n+2)\)-dimensional bordisms between bordisms as \(2\)-morphisms, etc up until dimension \(d\). A key geometric ingredient in the above theorem, which partly explains why the above theorem holds when the category number \(n\geq2\), is that handle-decompositions for manifolds use handles with codimension-two corners, allowing us to implement certain geometric arguments in categorical terms within the higher category \(\mathrm{Bord}_{d}^{(X,\xi)}\). This includes handle decomposition and handle moves for \(d\)-manifolds, and surgery for \((d-1)\)-manifolds. Although the above theorem is a general result about extended field theories valued in symmetric monoidal \((\infty,n)\)-categories, it can be deduced from the case of \(n=2\), for which we can resort to the standard and long-established theory of weak \(2\)-categories (aka bicategories in the sense of [\textit{J. Bénabou}, Introduction to bicategories. Reports of the Midwest Category Seminar (1967; Zbl 1375.18001)]). The theorem can be interpreted in classical algebraic topology as the statement about maps between certain multisimplical spaces. One reason to single out the class of invertible topological field theories is that we can completely classify them with recourse to stable homotopy theory. A similar result is obtained in [\textit{D. S. Freed}, ``Short-range entanglement and invertible field theories'', \url{arXiv:1406.7278}].
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