Lickorish–Wallace theorem
Encyclopedia
In mathematics
, the Lickorish–Wallace theorem in the theory of 3-manifold
s states that any closed
, orientable, connected 3-manifold may be obtained by performing Dehn surgery
on a framed link in the 3-sphere
with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted.
The theorem was proved in the early 1960s by W. B. R. Lickorish and Andrew H. Wallace, independently and by different methods. Lickorish's proof rested on the Lickorish twist theorem, which states that any orientable automorphism
of a closed orientable surface
is generated by Dehn twist
s along 3g − 1 specific simple closed curves in the surface, where g denotes the genus
of the surface. Wallace's proof was more general and involved adding handles to the boundary of a higher dimensional ball.
A corollary of the theorem is that every closed, orientable 3-manifold bounds a simply-connected compact 4-manifold
.
By using his work on automorphisms of non-orientable surfaces, Lickorish also showed that every closed, non-orientable, connected 3-manifold is obtained by Dehn surgery on a link in the non-orientable 2-sphere bundle over the circle. Similar to the orientable case, the surgery can be done in a special way which allows the conclusion that every closed, non-orientable 3-manifold bounds a compact 4-manifold.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Lickorish–Wallace theorem in the theory of 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s states that any closed
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
, orientable, connected 3-manifold may be obtained by performing Dehn surgery
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
on a framed link in the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted.
The theorem was proved in the early 1960s by W. B. R. Lickorish and Andrew H. Wallace, independently and by different methods. Lickorish's proof rested on the Lickorish twist theorem, which states that any orientable automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
of a closed orientable surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
is generated by Dehn twist
Dehn twist
In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface .-Definition:...
s along 3g − 1 specific simple closed curves in the surface, where g denotes the genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
of the surface. Wallace's proof was more general and involved adding handles to the boundary of a higher dimensional ball.
A corollary of the theorem is that every closed, orientable 3-manifold bounds a simply-connected compact 4-manifold
4-manifold
In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
.
By using his work on automorphisms of non-orientable surfaces, Lickorish also showed that every closed, non-orientable, connected 3-manifold is obtained by Dehn surgery on a link in the non-orientable 2-sphere bundle over the circle. Similar to the orientable case, the surgery can be done in a special way which allows the conclusion that every closed, non-orientable 3-manifold bounds a compact 4-manifold.