#### Volume 14, issue 2 (2014)

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Spin structures on $3$–manifolds via arbitrary triangulations

### Riccardo Benedetti and Carlo Petronio

Algebraic & Geometric Topology 14 (2014) 1005–1054
##### Abstract

Let $M$ be an oriented compact $3$–manifold and let $\mathsc{T}$ 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 $\mathsc{T}$. We then analyze how to change these extra structures on $\mathsc{T}$, and $\mathsc{T}$ 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.

##### Keywords
$3$–manifold, spin structure, triangulation, spine
##### Mathematical Subject Classification 2010
Primary: 57R15
Secondary: 57N10, 57M20