3-manifolds Modulo Surgery Triangles

TL;DR

This paper studies the algebraic structure of bordered 3-manifolds modulo surgery triangles, proving it is finitely generated, explicitly constructing a basis, and confirming a related combinatorial conjecture.

Contribution

It establishes that the group of bordered 3-manifolds modulo surgery triangles is finitely generated, provides an explicit basis, and relates this to a conjecture in combinatorics.

Findings

$K( ext{Sigma})$ is finitely generated and free abelian.

Constructed an explicit basis for $K( ext{Sigma})$.

Confirmed a conjecture on spanning sets for a dual polar space.

Abstract

Surgery triangles are an important computational tool in Floer homology. Given a connected oriented surface $\Sigma$, we consider the abelian group $K(\Sigma)$ generated by bordered 3-manifolds with boundary $\Sigma$, modulo the relation that the three manifolds involved in any surgery triangle sum to zero. We show that $K(\Sigma)$ 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.