mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Lickorish–Wallace theorem in the theory of

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

s states that any closed,

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 "anticlockwise". A space is o ...

, connected 3-manifold may be obtained by performing

Dehn surgery In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then '' ...

on a

framed link In mathematics, a knot is an embedding of the circle () into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation o ...

in the

3-sphere In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ...

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 Andrew Hugh Wallace (1926 – 18 January 2008) was a Scottish-American mathematician. Biography Andrew Hugh Wallace was born and raised in Edinburgh, Scotland. He received in 1946 an MA in mathematics from Edinburgh University and in 1949 a PhD ...

, independently and by different methods. Lickorish's proof rested on the Lickorish twist theorem, which states that any orientable

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 automorphism ...

of a closed orientable

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

is generated by

Dehn twist In geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area dis ...

s along 3''g'' − 1 specific simple closed curves in the surface, where ''g'' denotes the

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoint ...

compact

4-manifold In mathematics, a 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. T ...

. 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.

References

* * * {{DEFAULTSORT:Lickorish-Wallace theorem 3-manifolds Theorems in topology Theorems in geometry