Spin structures on $3$–manifolds via arbitrary triangulations

Abstract

Let $M$ be an oriented compact $3$–manifold and let $\mathcal{T}$ be a (loose) triangulation of $M$ with ideal vertices at the components of $\partial M$ and possibly internal vertices. We show that any spin structure $s$ on $M$ can be encoded by extra combinatorial structures on $\mathcal{T}$. We then analyze how to change these extra structures on $\mathcal{T}$, and $\mathcal{T}$ itself, without changing $s$, thereby getting a combinatorial realization, in the usual “objects/moves” sense, of the set of all pairs $\left(M,s\right)$. Our moves have a local nature, except one, that has a global flavour but is explicitly described anyway. We also provide an alternative approach where the global move is replaced by simultaneous local ones.