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The first author introduced a notion of equivalence on a family of$3$-manifolds with boundary, called (simple) balanced $3$-manifolds in anearlier paper and discussed the analogy between the Andrews-Curtis equivalencefor group presentations and the aforementioned notion of equivalence. Motivatedby the Andrews-Curtis conjecture, we use tools from Heegaard Floer theory toprove that there are simple balanced $3$-manifolds which are not in the trivialequivalence class (i.e. the equivalence class of $S^2\times [-1,1]$).

Simple balanced three-manifolds, Heegaard Floer homology and the Andrews-Curtis conjecture