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Spin structures on $3$–manifolds via arbitrary triangulations

Riccardo Benedetti and Carlo Petronio

Algebraic & Geometric Topology 14 (2014) 1005–1054

Let M be an oriented compact 3–manifold and let T be a (loose) triangulation of M with ideal vertices at the components of M and possibly internal vertices. We show that any spin structure s on M can be encoded by extra combinatorial structures on T. We then analyze how to change these extra structures on T, and T itself, without changing s, thereby getting a combinatorial realization, in the usual “objects/moves” sense, of the set of all pairs (M,s). 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.

$3$–manifold, spin structure, triangulation, spine
Mathematical Subject Classification 2010
Primary: 57R15
Secondary: 57N10, 57M20
Received: 16 April 2013
Revised: 9 September 2013
Accepted: 15 September 2013
Published: 21 March 2014
Riccardo Benedetti
Dipartimento di Matematica
Università di Pisa
Largo Bruno Pontecorvo 5
I-56127 Pisa
Carlo Petronio
Dipartimento di Matematica
Università di Pisa
Largo Bruno Pontecorvo 5
I-56127 Pisa