Lickorish–Wallace theorem
- Lickorish–Wallace theorem
In mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds 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 twists along 3"g" − 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.
References
*W. B. R. Lickorish, "A representation of orientable combinatorial 3-manifolds." Ann. of Math. (2) 76 (1962), 531–540.
*W. B. R. Lickorish, "Homeomorphisms of non-orientable two-manifolds." Proc. Cambridge Philos. Soc. 59 (1963), 307–317.
*A. H. Wallace, "Modifications and cobounding manifolds." Canad. J. Math. 12 (1960), 503–528.
Wikimedia Foundation.
2010.
Look at other dictionaries:
Lickorish — W. B. R. Lickorish in Berkeley William Bernard Raymond Lickorish (* 19. Februar 1935), meist zitiert als W. B. R. Lickorish oder W. B. Raymond Lickorish, ist ein britischer Mathematiker, der sich mit geometrischer Topologie, speziell… … Deutsch Wikipedia
W. B. Raymond Lickorish — W. B. R. Lickorish in Berkeley William Bernard Raymond Lickorish (* 19. Februar 1935), meist zitiert als W. B. R. Lickorish oder W. B. Raymond Lickorish, ist ein britischer Mathematiker, der sich mit geometrischer Topologie, speziell… … Deutsch Wikipedia
William Bernard Raymond Lickorish — W. B. R. Lickorish in Berkeley William Bernard Raymond Lickorish (* 19. Februar 1935), meist zitiert als W. B. R. Lickorish oder W. B. Raymond Lickorish, ist ein britischer Mathematiker, der sich mit geometrischer Topologie, speziell… … Deutsch Wikipedia
W. B. R. Lickorish — in Berkeley William Bernard Raymond Lickorish (* 19. Februar 1935), meist zitiert als W. B. R. Lickorish oder W. B. Raymond Lickorish, ist ein britischer Mathematiker, der sich mit geometrischer Topologie, speziell Knotentheorie und 3… … Deutsch Wikipedia
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. Dehn surgery can be thought of as a two stage process … Wikipedia
List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… … Wikipedia
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 usually made in whether we are dealing with say, topological 3… … Wikipedia
Y-homeomorphism — In mathematics, the y homeomorphism, or crosscap slide, is a special type of auto homeomorphism in non orientable surfaces. It can be constructed by sliding a Möbius band included on the surface around an essential 1 sided closed curve until the… … Wikipedia
Kirby calculus — In mathematics, the Kirby calculus in geometric topology is a method for modifying framed links in the 3 sphere using a finite set of moves, the Kirby moves. It is named for Robion Kirby. Using four dimensional Cerf theory, he proved that if M… … Wikipedia
Property P conjecture — In mathematics, the Property P conjecture is a statement about 3 manifolds obtained by Dehn surgery on a knot in the 3 sphere. A knot in the 3 sphere is said to have Property P if every 3 manifold obtained by performing (non trivial) Dehn surgery … Wikipedia
