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In the [[mathematics|mathematical]] field of [[geometric topology]], a '''Heegaard splitting''' {{IPAc-en|audio=En-heegaard.ogg|}} is a decomposition of a compact oriented [[3-manifold]] that results from dividing it into two [[handlebody|handlebodies]].

==Definitions==
Let ''V'' and ''W'' be [[handlebody|handlebodies]] of genus ''g'', and let ƒ be an orientation reversing [[homeomorphism]] from the [[Boundary (topology)|boundary]] of ''V'' to the boundary of ''W''.  By gluing ''V'' to ''W'' along ƒ we obtain the compact oriented [[3-manifold]]

:<math> M = V \cup_f W. </math>

Every closed, [[orientable]] three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to [[Edwin E. Moise|Moise]].  This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of [[Smale]] about handle decompositions from Morse theory.

The decomposition of ''M'' into two handlebodies is called a '''Heegaard splitting''', and their common boundary ''H'' is called the '''Heegaard surface''' of the splitting.  Splittings are considered up to [[Homotopy#Isotopy|isotopy]].

The gluing map ƒ need only be specified up to taking a double [[coset]] in the [[mapping class group]] of ''H''.  This connection with the mapping class group was first made by [[W. B. R. Lickorish]].

Heegaard splittings can also be defined for compact 3-manifolds with boundary by replacing handlebodies with [[compression body|compression bodies]].  The gluing map is between the positive boundaries of the compression bodies.

A closed curve is called '''essential''' if it is not homotopic to a point, a puncture, or a boundary component.<ref>{{cite book |last1= Farb |first1= B. |last2 = Margalit |first2 = D. |title=A Primer on Mapping Class Groups |publisher=Princeton University Press| page=22}}</ref>

A Heegaard splitting is '''reducible''' if there is an essential simple closed curve <math>\alpha</math> on ''H'' which bounds a disk in both ''V'' and in ''W''.  A splitting is '''irreducible''' if it is not reducible.  It follows from [[Heegaard splitting#Theorems|Haken's Lemma]] that in a [[irreducible manifold|reducible manifold]] every splitting is reducible.

A Heegaard splitting is '''stabilized''' if there are essential simple closed curves <math>\alpha</math> and <math>\beta</math> on ''H'' where <math>\alpha</math> bounds a disk in ''V'', <math>\beta</math> bounds a disk in ''W'', and <math>\alpha</math> and <math>\beta</math> intersect exactly once.  It follows from [[Heegaard splitting#Theorems|Waldhausen's Theorem]] that every reducible splitting of an [[irreducible manifold]] is stabilized.

A Heegaard splitting is '''weakly reducible''' if there are disjoint essential simple closed curves <math>\alpha</math> and <math>\beta</math> on ''H'' where <math>\alpha</math> bounds a disk in ''V'' and <math>\beta</math> bounds a disk in ''W''.  A splitting is '''strongly irreducible''' if it is not weakly reducible.

A Heegaard splitting is '''minimal''' or '''minimal genus''' if there is no other splitting of the ambient three-manifold of lower [[genus]]. The minimal value ''g'' of the splitting surface is the '''Heegaard genus''' of ''M''.

===Generalized Heegaard splittings===

A '''generalized Heegaard splitting''' of ''M'' is a decomposition into  [[compression body|compression bodies]] <math>V_i, W_i, i=1,\dots,n</math> and surfaces <math>H_i, i=1,\dots, n</math> such that <math>\partial_+ V_i = \partial_+ W_i = H_i</math> and <math>\partial_- W_i = \partial_- V_{i+1}</math>. The interiors of the compression bodies must be pairwise disjoint and their union must be all of <math>M</math>.  The surface <math>H_i</math> forms a Heegaard surface for the submanifold <math>V_i \cup W_i</math> of <math>M</math>.  (Note that here each ''V<sub>i</sub>'' and ''W<sub>i</sub>'' is allowed to have more than one component.)

A generalized Heegaard splitting is called '''strongly irreducible''' if each <math>V_i \cup W_i</math> is strongly irreducible.

There is an analogous notion of [[thin position]], defined for knots, for Heegaard splittings.  The complexity of a connected surface ''S'', ''c(S)'', is defined to be <math>\operatorname{max}\{0, 1 - \chi(S)\}</math>; the complexity of a disconnected surface is the sum of complexities of its components.  The complexity of a generalized Heegaard splitting is the multi-set ''{c(S_i)}'', where the index runs over the Heegaard surfaces in the generalized splitting.  These multi-sets can be well-ordered by [[lexicographical order]]ing (monotonically decreasing).  A generalized Heegaard splitting is '''thin''' if its complexity is minimal.

==Examples==

'''[[Three-sphere]]''': The three-sphere <math>S^3</math> is the set of vectors in <math>\mathbb{R}^4</math> with length one.  Intersecting this with the <math>xyz</math> hyperplane gives a [[two-sphere]].  This is the '''standard''' genus zero splitting of <math>S^3</math>.  Conversely, by [[Alexander's Trick]], all manifolds admitting a genus zero splitting are [[homeomorphic]] to <math>S^3</math>.

Under the usual identification of <math>\mathbb{R}^4</math> with <math>\mathbb{C}^2</math> we may view <math>S^3</math> as living in <math>\mathbb{C}^2</math>.  Then the set of points where each coordinate has norm <math>1/\sqrt{2}</math> forms a [[Clifford torus]], <math>T^2</math>.  This is the standard genus one splitting of <math>S^3</math>.  (See also the discussion at [[Hopf bundle]].)

'''Stabilization''': Given a Heegaard splitting ''H'' in ''M'' the '''stabilization''' of ''H'' is formed by taking the [[connected sum]] of the pair <math>(M, H)</math> with the pair <math>(S^3, T^2)</math>.  It is easy to show that the stabilization procedure yields stabilized splittings.  Inductively, a splitting is '''standard''' if it is the stabilization of a standard splitting.

'''[[Lens space]]s''': All have a standard splitting of genus one.  This is the image of the Clifford torus in <math>S^3</math> under the quotient map used to define the lens space in question.  It follows from the structure of the [[mapping class group]] of the [[torus|two-torus]] that only lens spaces have splittings of genus one.

'''[[Three-torus]]''': Recall that the three-torus <math>T^3</math> is the [[Cartesian product]] of three copies of <math>S^1</math> ([[circle]]s).  Let <math>x_0</math> be a point of <math>S^1</math> and consider the graph
<math> \Gamma = 
S^1 \times \{x_0\} \times \{x_0\} \cup
\{x_0\} \times S^1 \times \{x_0\} \cup
\{x_0\} \times \{x_0\} \times S^1
</math>.  It is an easy exercise to show that ''V'', a [[regular neighborhood]] of <math>\Gamma</math>, is a handlebody as is <math>T^3 - V</math>.  Thus the boundary of ''V'' in <math>T^3</math> is a Heegaard splitting and this is the standard splitting of <math>T^3</math>. It was proved by Charles Frohman and [[Joel Hass]] that any other genus 3 Heegaard splitting of the three-torus is topologically
equivalent to this one. Michel Boileau and Jean-Pierre Otal proved that in general any Heegaard splitting of the three-torus is equivalent to the result of stabilizing this example.

==Theorems==
'''Alexander's Lemma''': Up to isotopy, there is a unique ([[piecewise linear homeomorphism|piecewise linear]]) embedding of the two-sphere into the three-sphere.  (In higher dimensions this is known as the [[Jordan–Schönflies theorem|Schoenflies theorem]].  In dimension two this is the [[Jordan curve theorem]].)  This may be restated as follows: the genus zero splitting of <math>S^3</math> is unique.

'''Waldhausen's Theorem''': Every splitting of <math>S^3</math> is obtained by stabilizing the unique splitting of genus zero.

Suppose now that ''M'' is a closed orientable three-manifold.

'''Reidemeister–Singer Theorem''': For any pair of splittings <math>H_1</math> and <math>H_2</math> in ''M'' there is a third splitting <math>H</math> in ''M'' which is a stabilization of both.

'''Haken's Lemma''':  Suppose that <math>S_1</math> is an essential two-sphere in ''M'' and ''H'' is a Heegaard splitting.  Then there is an essential  two-sphere <math>S_2</math> in ''M'' meeting ''H'' in a single curve.

==Classifications==

There are several classes of three-manifolds where the set of Heegaard splittings is completely known.  For example, Waldhausen's Theorem shows that all splittings of <math>S^3</math> are standard.  The same holds for [[lens space]]s (as proved by Francis Bonahon and Otal).

Splittings of [[Seifert fiber space]]s are more subtle.  Here, all splittings may be isotoped to be '''vertical''' or '''horizontal''' (as proved by Yoav Moriah and Jennifer Schultens).

{{harvtxt|Cooper|Scharlemann|1999}} classified splittings of [[torus bundle]]s (which includes all three-manifolds with [[Sol geometry]]).  It follows from their work that all torus bundles have a unique splitting of minimal genus.  All other splittings of the torus bundle are stabilizations of the minimal genus one.

As of 2008, the only [[Geometrization conjecture|hyperbolic]] three-manifolds whose Heegaard splittings are classified are two-bridge knot complements, in a paper of Tsuyoshi Kobayashi.

==Applications and connections==

===Minimal surfaces===

Heegaard splittings appeared in the theory of [[minimal surface]]s first in the work of [[Blaine Lawson]] who proved that embedded minimal surfaces in compact manifolds of positive sectional curvature are Heegaard splittings. This result was extended by William Meeks to flat manifolds, except he proves that an embedded minimal surface in a flat three-manifold is either a Heegaard surface or [[totally geodesic]].
  
Meeks and [[Shing-Tung Yau]] went on to use results of Waldhausen to prove results about the topological uniqueness of minimal surfaces of finite genus in <math>\R^3</math>. The final topological classification of embedded minimal surfaces in <math>\R^3</math> was given by Meeks and Frohman. The result relied heavily on techniques developed for studying the topology of Heegaard splittings.

===Heegaard Floer homology===
Heegaard diagrams, which are simple combinatorial descriptions of Heegaard splittings, have been used extensively to construct invariants of three-manifolds. The most recent example of this is the [[Floer homology#Heegaard Floer homology|Heegaard Floer homology]] of [[Peter Ozsvath]] and [[Zoltán Szabó (mathematician)|Zoltán Szabó]]. The theory uses the <math>g^{th}</math> symmetric product of a Heegaard surface as the ambient space, and tori built from the boundaries of meridian disks for the two handlebodies as the [[Lagrangian submanifold]]s.

==History==

The idea of a Heegaard splitting was introduced by {{harvs|txt|authorlink=Poul Heegaard|last=Heegaard|first=Poul|year=1898}}.  While Heegaard splittings were studied extensively by mathematicians such as [[Wolfgang Haken]] and [[Friedhelm Waldhausen]] in the 1960s, it was not until a few decades later that the field was rejuvenated by {{harvs|txt|last1=Casson | first1=Andrew | author1-link=Andrew Casson| last2=Gordon | first2=Cameron | author2-link=Cameron Gordon (mathematician)|year=1987}}, primarily through their concept of '''strong irreducibility'''.

==See also==
*[[Manifold decomposition]]
*[[Handle decompositions of 3-manifolds]]
*[[Compression body]]

==References==
{{reflist}}
*{{Citation|last1= Farb |first1= Benson |author1-link=Benson Farb|last2 = Margalit |first2 = Dan |title=A Primer on Mapping Class Groups |publisher=Princeton University Press}}
*{{Citation | last1=Casson | first1=Andrew J. | author1-link=Andrew Casson| last2=Gordon | first2=Cameron McA. | author2-link=Cameron Gordon (mathematician)| title=Reducing Heegaard splittings | doi=10.1016/0166-8641(87)90092-7 | mr=918537 | year=1987 | journal=[[Topology and Its Applications]] | issn=0166-8641 | volume=27 | issue=3 | pages=275–283}}
*{{Citation | last1=Cooper | first1=Daryl | last2=Scharlemann | first2=Martin | url=http://mistug.tubitak.gov.tr/bdyim/toc.php?dergi=mat&yilsayi=1999/1 | mr=1701636 | year=1999 | journal=Turkish Journal of Mathematics | issn=1300-0098 | volume=23 | issue=1 | title=The structure of a solvmanifold's Heegaard splittings | pages=1–18}}
*{{Citation | last1=Heegaard | first1=Poul | title=Forstudier til en topologisk Teori for de algebraiske Fladers Sammenhang | url=http://www.maths.ed.ac.uk/~aar/papers/heegaardthesis.pdf | language=Danish | series=Thesis | jfm=29.0417.02 | year=1898}}
*{{Citation | last1=Hempel | first1=John | title=3-manifolds | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | isbn=978-0-8218-3695-8 | mr=0415619 | year=1976 | volume=86}}

[[Category:3-manifolds]]
[[Category:Minimal surfaces]]
[[Category:Geometric topology]]