In
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
, a branch of mathematics, the cyclic surgery theorem states that, for a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
,
connected,
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space ...
,
irreducible three-manifold ''M'' whose boundary is a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not ...
''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.
[M. Culler, C. Gordon, J. Luecke, P. Shalen (1987). Dehn surgery on knots. The Annals of Mathematics (''Annals of Mathematics'') 125 (2): 237-300.]
References
Geometric topology
3-manifolds
Knot theory
Theorems in topology
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