Mathematical Research Letters

Volume 7 (2000)

Number 4

Introduction to twisted face-pairings

Pages: 477 – 491



J. W. Cannon (Brigham Young University)

W. J. Floyd (Virginia Tech)

W. R. Parry (Eastern Michigan University)


We give a mechanical recipe for creating simple face-pairing descriptions of closed 3-manifolds. We call the technique \emph{twisted face-pairing}. Among the simpler twisted face-pairings we have studied, we have discovered manifolds (usually infinite classes of manifolds) which admit geometries based on $S^3$, $S^2 \times \mathbf{R}$, $\mathbf{H}^3$, Solv, Nil, and the universal cover of $PSL(2,\mathbf{R})$. Our work suggests, but does not resolve, the \textbf{Conjecture:} \emph{It is impossible because of the twisting involved in the construction to obtain, by twisted face-pairing, manifolds based on the product geometries $\mathbf{E}^3$ and $\mathbf{H}^2 \times \mathbf{R}$}. The twisted face-pairing technique is easily simple enough for use by students, and if used in conjunction with some software package like SnapPea, serves as a great exploratory playground in 3-manifold theory.

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