## A survey of twisted face-pairing 3-manifolds

### Abstract

The twisted face-pairing construction gives an efficient way to generate face-pairing descriptions for many interesting closed 3-manifolds. Our work in this paper is directed toward the goal of determining which closed, connected, orientable 3-manifolds can be generated from this construction. We succeed in proving that all lens spaces, the Heisenberg manifold (Nil geometry), $S^2 \times S^1$, and all connected sums of twisted face-pairing manifolds are twisted face-pairing manifolds. We show how to obtain most closed, connected, orientable, Seifert-fibered manifolds as twisted face-pairing manifolds. It still seems unlikely that all closed, connected, orientable 3-manifolds can be so obtained.

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