Invariants of three-dimensional manifolds (Q1919137)

The authors consider the combinatorial approach to the construction of invariants of a three-dimensional manifold and construct an invariant starting from a monoidal category \(\mathbf C\) without any braiding conditions, i.e commutativity conditions of the form \(X\otimes Y\simeq Y\otimes X\). For any three-dimensional manifold \(M\) with boundary \(S\) and a triangulation \(D\) of \(M\), they construct a finite-dimensional linear space \(W(S,D')\) depending on the restriction \(D'\) of \(D\) to \(S\) (a triangulation of \(S\)) and a vector \(I_{\mathbf C}(M,S,D)\in W(S,D')\). The main result of the paper is that \(I_{\mathbf C}(M,S,D)\) depends only on \(D'\) and not on \(D\) itself. Taking the inductive limit of spaces \(W(S,D')\) over all triangulations \(D'\) of \(S\), the authors construct a space \(K(S)\) and a vector \(I_{\mathbf C}(M,S)\in K(S)\), which is an invariant of a three-dimensional manifold \(M\) with boundary \(S\). In particular, when \(M\) is a closed manifold, \(S=\emptyset\), they have \(K(\emptyset)=k\), and their construction gives an invariant \(I_{\mathbf C}(M)\in k\).