3-manifolds Modulo Surgery Triangles

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Surgery triangles are an important computational tool in Floer homology. Given a connected oriented surface Σ, we consider the abelian group K(Σ) generated by bordered 3-manifolds with boundary Σ, modulo the relation that the three manifolds involved in any surgery triangle sum to zero. We show that K(Σ) is a finitely generated free abelian group and compute its rank. We also construct an explicit basis and show that it generates all bordered 3-manifolds in a certain stronger sense. Our basis is strictly contained in another finite generating set which was constructed previously by Baldwin and Bloom. As a byproduct we confirm a conjecture of Blokhuis and Brouwer on spanning sets for the binary symplectic dual polar space.