mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory...

gluing the unit disk along the boundary. The genus of a 3-dimensional handlebody is an integer representing the maximum number of cuttings along embedded...

two types (i.e.: 0-handles and j-handles for some fixed j) is called a handlebody. When forming M union a j-handle H j {\displaystyle H^{j}} M ∪ f H j =...

3-manifolds, a compression body is a kind of generalized handlebody. A compression body is either a handlebody or the result of the following construction: Let...

oriented 3-manifold that results from dividing it into two handlebodies. Let V and W be handlebodies of genus g, and let ƒ be an orientation reversing homeomorphism...

orientable surface is null-cobordant, because it is the boundary of a handlebody. On the other hand, the 2n-dimensional real projective space P 2 n ( R...

fiber space Heegaard splitting Waldhausen conjecture Compression body Handlebody Incompressible surface Dehn's lemma Loop theorem (aka the Disk theorem)...

{\displaystyle S^{3}} is a handlebody. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality...

cut or boundary. This is closely related to, but not identical with, handlebody decompositions. More technically, the idea is to start with a well-understood...

subset of a genus g handlebody in S³ (whose complement is also a genus g handlebody). If a link is a subset of a genus two handlebody, it is a double torus...

boundary with that link Trigenus Compact, closed 3-manifolds Surgeries Three orientable handlebodies Unions along subsurfaces on boundaries of handlebodies...

geometric topology, most commonly studied are Morse functions, which yield handlebody decompositions, while in mathematical analysis, one often studies solution...

1960s by Stephen Smale, for differentiable manifolds. The development of handlebody theory allowed much the same proofs in the differentiable and PL categories...

manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable...

Andrews–Curtis conjecture. Andrews, J. J.; Curtis, M. L. (1965), "Free groups and handlebodies", Proceedings of the American Mathematical Society, 16 (2), American...

Similarly, such a cancelling pair can be created. Two different smooth handlebody decompositions of a smooth 4-manifold are related by a finite sequence...

cut or boundary. This is closely related to, but not identical with, handlebody decompositions. It is a major tool in the study and classification of...

manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable...

tunnels) that must be added to the knot so that the complement becomes a handlebody. The tunnel number can equally be defined for links. The boundary of a...

Berkeley as a student of Robion Kirby. In topology, he has worked on handlebody theory, low-dimensional manifolds, symplectic topology, G2 manifolds....

March 11, 2011. Andrews, J. J.; Curtis, M. L. (1965), "Free groups and handlebodies", Proceedings of the American Mathematical Society, 16 (2): 192–195,...

mirror orbifold with underlying space a ball. It is double covered by a handlebody; the corresponding index 2 subgroup is a Kleinian group called a Schottky...

defined by the conditions: K lies on a genus two Heegaard surface S in each handlebody bound by S, K meets some meridian disc exactly once. John Berge constructed...

International Journal of Mathematics vol. 1, no 1 (1990) 29–46. Robert Gompf, Handlebody construction of Stein surfaces, Annals of Mathematics 148, (1998) 619–693...

method used to decompose into handlebodies is the Heegaard splitting, which gives a decomposition in two handlebodies of equal genus. As an example:...

a compact oriented 3-manifold that results from dividing it into two handlebodies. Every closed, orientable three-manifold may be so obtained; this follows...

need not be complex, Annals of Mathematics, Volume 138, 1993, p. 61–111 Handlebody construction of Stein surfaces, Annals of Mathematics, Volume 148, 1998...

topologically equivalent (homeomorphic) to a sphere, torus, or more general handlebody, classified by its number of handles. A non-orientable surface must be...

is called an integral surgery. Such surgeries are closely related to handlebodies, cobordism and Morse functions. If all surgery coefficients are infinite...

geometric topology, most commonly studied are Morse functions, which yield handlebody decompositions, which generalize to Morse–Bott functions and can be used...