In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originat ...
, a Heegaard splitting () is a decomposition of a compact oriented
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds ...
that results from dividing it into two
handlebodies.
Definitions
Let ''V'' and ''W'' be
handlebodies of genus ''g'', and let ƒ be an orientation reversing
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
from the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
of ''V'' to the boundary of ''W''. By gluing ''V'' to ''W'' along ƒ we obtain the compact oriented
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds ...
:
Every 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 "counterclockwise". A space ...
three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to
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
isotopy.
The gluing map ƒ need only be specified up to taking a double
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
in the
mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mo ...
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 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.
A Heegaard splitting is reducible if there is an essential simple closed curve
on ''H'' which bounds a disk in both ''V'' and in ''W''. A splitting is irreducible if it is not reducible. It follows from
Haken's Lemma that in a
reducible manifold every splitting is reducible.
A Heegaard splitting is stabilized if there are essential simple closed curves
and
on ''H'' where
bounds a disk in ''V'',
bounds a disk in ''W'', and
and
intersect exactly once. It follows from
Waldhausen's Theorem that every reducible splitting of an
irreducible manifold In topology, a branch of mathematics, a prime manifold is an ''n''-manifold that cannot be expressed as a non-trivial connected sum of two ''n''-manifolds. Non-trivial means that neither of the two is an ''n''-sphere.
A similar notion is that of ...
is stabilized.
A Heegaard splitting is weakly reducible if there are disjoint essential simple closed curves
and
on ''H'' where
bounds a disk in ''V'' and
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
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
. 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 bodies and surfaces
such that
and
. The interiors of the compression bodies must be pairwise disjoint and their union must be all of
. The surface
forms a Heegaard surface for the submanifold
of
. (Note that here each ''V
i'' and ''W
i'' is allowed to have more than one component.)
A generalized Heegaard splitting is called strongly irreducible if each
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
; 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 ', where the index runs over the Heegaard surfaces in the generalized splitting. These multi-sets can be well-ordered by
lexicographical order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of ...
ing (monotonically decreasing). A generalized Heegaard splitting is thin if its complexity is minimal.
Examples
;
Three-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimens ...
: The three-sphere
is the set of vectors in
with length one. Intersecting this with the
hyperplane gives a
two-sphere. This is the standard genus zero splitting of
. Conversely, by
Alexander's Trick, all manifolds admitting a genus zero splitting are
homeomorphic to
. Under the usual identification of
with
we may view
as living in
. Then the set of points where each coordinate has norm
forms a
Clifford torus
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon ...
,
. This is the standard genus one splitting of
. (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
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classific ...
of the pair
with the pair
. 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
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.
In the 3-manifold case, a lens space can be visualiz ...
s: All have a standard splitting of genus one. This is the image of the Clifford torus in
under the quotient map used to define the lens space in question. It follows from the structure of the
mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mo ...
of the
two-torus that only lens spaces have splittings of genus one.
;
Three-torus: Recall that the three-torus
is the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ ...
of three copies of
(
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s). Let
be a point of
and consider the graph
. It is an easy exercise to show that ''V'', a
regular neighborhood of
, is a handlebody as is
. Thus the boundary of ''V'' in
is a Heegaard splitting and this is the standard splitting of
. 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) embedding of the two-sphere into the three-sphere. (In higher dimensions this is known as the
Schoenflies theorem. In dimension two this is the
Jordan curve theorem
In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exterio ...
.) This may be restated as follows: the genus zero splitting of
is unique.
; Waldhausen's theorem: Every splitting of
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
and
in ''M'' there is a third splitting
in ''M'' which is a stabilization of both.
; Haken's lemma: Suppose that
is an essential two-sphere in ''M'' and ''H'' is a Heegaard splitting. Then there is an essential two-sphere
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
are standard. The same holds for
lens space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.
In the 3-manifold case, a lens space can be visualiz ...
s (as proved by Francis Bonahon and Otal).
Splittings of
Seifert fiber spaces are more subtle. Here, all splittings may be isotoped to be vertical or horizontal (as proved by Yoav Moriah and
Jennifer Schultens).
classified splittings of
torus bundles (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
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
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
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
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
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
The following articles may also be useful; they either contain specialised vocabulary or provi ...
.
Meeks and
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
went on to use results of Waldhausen to prove results about the topological uniqueness of minimal surfaces of finite genus in
. The final topological classification of embedded minimal surfaces in
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
Heegaard Floer homology of
Peter Ozsvath
Peter may refer to:
People
* List of people named Peter, a list of people and fictional characters with the given name
* Peter (given name)
** Saint Peter (died 60s), apostle of Jesus, leader of the early Christian Church
* Peter (surname), a su ...
and
Zoltán Szabó. The theory uses the
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
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
s.
History
The idea of a Heegaard splitting was introduced by . While Heegaard splittings were studied extensively by mathematicians such as
Wolfgang Haken
Wolfgang Haken (June 21, 1928 – October 2, 2022) was a German American mathematician who specialized in topology, in particular 3-manifolds.
Biography
Haken was born in Berlin, Germany. His father was Werner Haken, a physicist who had Max ...
and
Friedhelm Waldhausen in the 1960s, it was not until a few decades later that the field was rejuvenated by , primarily through their concept of strong irreducibility.
See also
*
Manifold decomposition
*
Handle decompositions of 3-manifolds In mathematics, a handle decomposition of a 3-manifold allows simplification of the original 3-manifold into pieces which are easier to study.
Heegaard splittings
An important method used to decompose into handlebodies is the Heegaard splitting, w ...
*
Compression body
In the theory of 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 S be a compact, closed surface (not necessarily connected). Attac ...
References
*
*
*
*
*
*{{Citation , last1=Kobayashi , first1=Tsuyoshi , title=Heegaard splittings of exteriors of two bridge knots , url=https://projecteuclid.org/journals/geometry-and-topology/volume-5/issue-2/Heegaard-splittings-of-exteriors-of-two-bridge-knots/10.2140/gt.2001.5.609.full , year=2001 , journal=Geometry and Topology , volume=5 , issue=2 , pages=609–650, doi=10.2140/gt.2001.5.609 , s2cid=13991798
3-manifolds
Minimal surfaces
Geometric topology