3-manifolds Modulo Surgery Triangles
Lucas Culler

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 , we consider the abelian group generated by bordered 3-manifolds with boundary , modulo the relation that the three manifolds involved in any surgery triangle sum to zero. We show that 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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Finite Group Theory Research · Algebraic Geometry and Number Theory
