topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

, a branch of 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 ...

''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 ...

s and also in proving the higher-dimensional

Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured b ...

. 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 ...

, 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,

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 ...

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 or Seifert-fibered 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 su ...

, 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 th ...

; 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 a ...

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 classific ...

, - !

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'', and ...

, Closed,

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 de ...

, Any compact ( smooth) 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 differentiab ...

s a handle is associated to each critical point. , Balls (called handles) , Union along a subset of the boundaries. Note that the handles must generally be added in a specific order. , - ! Haken hierarchy , 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 wh ...

, 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 , Any closed

, , A link and a family of 2-manifolds that share a

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 ...

with that link , , - !

Trigenus In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple (g_1,g_2,g_3). It is obtained by minimizing the genera of three '' orientable'' handle bodies — with no intersection between th ...

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 ...

, closed 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