Simple balanced three-manifolds, HeegaardFloer homology and the Andrews–Curtis conjecture

The first author introduced a notion of equivalence on a family of 3-manifolds with boundary, called (simple) balanced 3-manifolds in an earlier paper and discussed the analogy between the Andrews-Curtis equivalence for group presentations and the aforementioned notion of equivalence.Motivated by the Andrews-Curtis conjecture, we use tools from Heegaard Floer theory to prove that there are simple balanced 3-manifolds which are not in the trivial equivalence class (ie the equivalence class of S 2 OE 1; 1).57K18, 57R58 1. Introduction 4519 2. A Heegaard diagram for the mapping torus 4523 3. Simplifying computations using algorithmic calculations 4526 4. Nonpolygonal disks with holomorphic representatives 4528 5.The first nontrivial Heegaard-Floer group 4533 6.The second nontrivial Heegaard-Floer group 4536