Spin structures on 3-manifolds via arbitrary triangulations

TL;DR

This paper develops a combinatorial method to encode and manipulate spin structures on 3-manifolds using triangulations, providing local moves to explore all such structures.

Contribution

It introduces a new combinatorial encoding of spin structures on 3-manifolds via triangulations and describes local moves to modify these structures without changing the underlying spin structure.

Findings

Encoding of spin structures through combinatorial data on triangulations

Local moves that alter triangulations while preserving spin structures

An alternative approach replacing a global move with local ones

Abstract

Let M be an oriented compact 3-manifold and let T be a (loose) triangulation of M, with ideal vertices at the components of the boundary 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.