# Cyclic surgery theorem

In three-dimensional topology, a branch of mathematics, the **cyclic surgery theorem** states that, for a compact, connected, orientable, irreducible three-manifold *M* whose boundary is a torus *T*, if *M* is not a Seifert-fibered space and *r,s* are slopes on *T* such that their Dehn fillings have cyclic fundamental group, then the distance between *r* and *s* (the minimal number of times that two simple closed curves in *T* representing *r* and *s* must intersect) is at most 1. Consequently, there are at most three Dehn fillings of *M* with cyclic fundamental group. The theorem appeared in a 1987 paper written by Marc Culler, Cameron Gordon, John Luecke and Peter Shalen.[1]

## References

- M. Culler, C. Gordon, J. Luecke, P. Shalen (1987). Dehn surgery on knots. The Annals of Mathematics (
*Annals of Mathematics*) 125**(2)**: 237-300.

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