  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 (i.e. the equivalence class of $S^2\times [-1,1]$).
