Three-manifolds with boundary and the Andrews-Curtis transformations

Abstract

We investigate an extended version of the stable Andrews-Curtis transformations, referred to as EAC transformations, and compare it with a notion of equivalence in a family of $3$-manifolds with boundary, called the {\emph{simple balanced $3$-manifolds}}. A simple balanced $3$-manifold is a $3$-manifold with boundary, such that every connected component $N$ of it has unique positive and negative boundary components ${\partial }^{+}N$ and ${\partial }^{-}N$, such that ${\pi }_1(N)$ is the normalizer of the image of ${\pi }_1({\partial }^{\pm }N)$ in ${\pi }_1(N)$. Associated with every simple balanced $3$-manifold $N$ is the EAC equivalence class of a balanced presentation of the trivial group, denoted by $P_N$, which remains unchanged as long as $N$ remains in a fixed equivalence class of simple balanced $3$-manifolds. In particular, the isomorphism class of the corresponding group is unchanged. Motivated by the Andrews-Curtis conjecture, we study the equivalence class of a trivial balanced $3$-manifold (obtained as the product of a closed oriented surface with the unit interval). We show that every balanced $3$-manifold in the trivial equivalence class admits a {\emph{simplifier}} to a trivial balanced $3$-manifold.