topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, a branch of

mathematics 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 ...

, a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

''M'' may be decomposed or split by writing ''M'' as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form ''M''. Manifold decomposition works in two directions: one can start with the smaller pieces and build up a manifold, or start with a large manifold and decompose it. The latter has proven a very useful way to study manifolds: without tools like decomposition, it is sometimes very hard to understand a manifold. In particular, it has been useful in attempts to classify

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 lo ...

s and also in proving the higher-dimensional

Poincaré conjecture In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...

. The table below is a summary of the various manifold-decomposition techniques. The column labeled "''M''" indicates what kind of manifold can be decomposed; the column labeled "How it is decomposed" indicates how, starting with a manifold, one can decompose it into smaller pieces; the column labeled "The pieces" indicates what the pieces can be; and the column labeled "How they are combined" indicates how the smaller pieces are combined to make the large manifold. {, class="wikitable" , - ! Type of decomposition ! ''M'' ! How it is decomposed ! The pieces ! How they are combined , - !

Triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...

, Depends on dimension. In dimension 3, a theorem by Edwin E. Moise gives that every 3-manifold has a unique triangulation, unique up to common subdivision. In dimension 4, not all manifolds are triangulable. For higher dimensions, general existence of triangulations is unknown. , ,

Simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

, Glue together pairs of codimension-one faces , - ! Jaco-Shalen/Johannson torus decomposition ,

Irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...

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 is ...

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

s , Cut along embedded tori ,

Atoroidal In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non- boundary parallel, incompressible t ...

Seifert-fibered A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for a ...

3-manifolds , Union along their boundary, using the trivial homeomorphism , - !

Prime decomposition In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...

, Essentially

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 t ...

s and

s. The decomposition is unique when the manifold is orientable. , Cut along embedded

spheres The Synchronized Position Hold Engage and Reorient Experimental Satellite (SPHERES) are a series of miniaturized satellites developed by MIT's Space Systems Laboratory for NASA and US Military, to be used as a low-risk, extensible test bed for the ...

; then union by the trivial homeomorphism along the resultant boundaries with disjoint balls. ,

Prime 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 an ...

s ,

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 classifi ...

, - !

Heegaard splitting In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies. Definitions Let ''V'' and ''W'' be handlebodies of genus ''g'', an ...

Closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

s , , Two handlebodies of equal genus , Union along the boundary by some homeomorphism , - !

Handle decomposition In mathematics, a handle decomposition of an ''m''-manifold ''M'' is a union \emptyset = M_ \subset M_0 \subset M_1 \subset M_2 \subset \dots \subset M_ \subset M_m = M where each M_i is obtained from M_ by the attaching of i-handles. A handle dec ...

, Any compact (

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...

) n-manifold (and the decomposition is never unique) , Through

Morse function In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiabl ...

s a handle is associated to each critical point. , Balls (called

handles A handle is a part of, or attachment to, an object that allows it to be grasped and manipulated by hand. The design of each type of handle involves substantial ergonomic issues, even where these are dealt with intuitively or by following tra ...

) , Union along a subset of the boundaries. Note that the handles must generally be added in a specific order. , - !

Haken hierarchy In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in whi ...

, Any

Haken manifold In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in w ...

, Cut along a sequence of incompressible surfaces , 3-balls , , - ! Disk decomposition , Certain

s , Suture the manifold, then cut along special surfaces (condition on boundary curves and sutures...) , 3-balls , , - !

Open book decomposition In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3-manifold ''M'' into a union of surfaces (necessarily with boundary) and solid tori. Open books have relevance to contact geometry, with a ...

, Any

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

, , A link and a family of

2-manifold In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...

s that share a

boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...

with that link , , - ! Trigenus ,

Compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

3-manifolds ,

Surgeries Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pat ...

, Three orientable handlebodies , Unions along subsurfaces on boundaries of handlebodies

See also