Inmathematics, ahandle decomposition of anm-manifoldM is a union$${\displaystyle \mathrm{\varnothing }=M_{-1}\subset M_0\subset M_1\subset M_2\subset \cdots \subset M_{m-1}\subset M_m=M}$$where each${\displaystyle M_i}$ is obtained from${\displaystyle M_{i-1}}$ by the attaching of${\displaystyle i}$-handles. A handle decomposition is to a manifold what aCW-decomposition is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world ofsmooth manifolds. Thus ani-handle is the smooth analogue of ani-cell. Handle decompositions of manifolds arise naturally viaMorse theory. The modification of handle structures is closely linked toCerf theory.

Consider the standardCW-decomposition of then-sphere, with one zero cell and a singlen-cell. From the point of view of smooth manifolds, this is a degenerate decomposition of the sphere, as there is no natural way to see the smooth structure of${\displaystyle S^n}$ from the eyes of this decomposition—in particular the smooth structure near the0-cell depends on the behavior of the characteristic map${\displaystyle \chi :D^n\to S^n}$ in a neighbourhood of${\displaystyle S^{n-1}\subset D^n}$.

The problem with CW-decompositions is that the attaching maps for cells do not live in the world of smooth maps between manifolds. The germinal insight to correct this defect is thetubular neighbourhood theorem. Given a pointp in an${\displaystyle m}$-manifoldM, its closed tubular neighbourhood${\displaystyle N_p}$ isdiffeomorphic to${\displaystyle D^m}$, thus we have decomposedM into the disjoint union of${\displaystyle N_p}$ and${\displaystyle M\setminus \mathrm{int}(N_p)}$ glued along their common boundary. The vital issue here is that the gluing map is a diffeomorphism. Similarly, take a smooth embedded arc in${\displaystyle M\setminus \mathrm{int}(N_p)}$, its tubular neighbourhood is diffeomorphic to${\displaystyle I\times D^{m-1}}$. This allows us to write${\displaystyle M}$ as the union of three manifolds, glued along parts of their boundaries:

(1)${\displaystyle D^m}$,

(2)${\displaystyle I\times D^{m-1}}$, and

(3) the complement of the open tubular neighbourhood of the arc in${\displaystyle M\setminus \mathrm{int}(N_p)}$.

Notice all the gluing maps are smooth maps—in particular when we glue${\displaystyle I\times D^{m-1}}$ to${\displaystyle D^m}$ the equivalence relation is generated by the embedding of${\displaystyle (\mathrm{\partial }I)\times D^{m-1}}$ in${\displaystyle \mathrm{\partial }{D}^m}$, which is smooth by thetubular neighbourhood theorem.

Handle decompositions are an invention ofStephen Smale.^[1] In his original formulation,the process of attaching aj-handle to anm-manifoldM assumes that one has a smooth embedding of${\displaystyle f:S^{j-1}\times D^{m-j}\to \mathrm{\partial }M}$. Let${\displaystyle H^j=D^j\times D^{m-j}}$. The manifold${\displaystyle M{\cup }_f{H}^j}$ (in words,M union aj-handle alongf) refers to the disjoint union of${\displaystyle M}$ and${\displaystyle H^j}$ with the identification of${\displaystyle S^{j-1}\times D^{m-j}}$ with its image in${\displaystyle \mathrm{\partial }M}$, i.e.,$${\displaystyle M{\cup }_f{H}^j=\left(M\bigsqcup (D^j\times D^{m-j})\right)/\sim }$$where theequivalence relation${\displaystyle \sim }$ is generated by${\displaystyle (p,x)\sim f(p,x)}$ for all${\displaystyle (p,x)\in S^{j-1}\times D^{m-j}\subset D^j\times D^{m-j}}$.

One says a manifoldN is obtained fromM by attachingj-handles if the union ofM with finitely manyj-handles is diffeomorphic toN. The definition of a handle decomposition is then as in the introduction. Thus, a manifold has a handle decomposition with only0-handles if it is diffeomorphic to a disjoint union of balls. A connected manifold containing handles of only two types (i.e.: 0-handles andj-handles for some fixedj) is called ahandlebody.

When formingM union aj-handle${\displaystyle H^j}$$${\displaystyle M{\cup }_f{H}^j=\left(M\bigsqcup (D^j\times D^{m-j})\right)/\sim }$$

${\displaystyle f(S^{j-1}\times \{0\})\subset M}$ is known as theattaching sphere.

${\displaystyle f}$ is sometimes called theframing of the attaching sphere, since it givestrivialization of itsnormal bundle.

${\displaystyle \{0{\}}^j\times S^{m-j-1}\subset D^j\times D^{m-j}=H^j}$ is thebelt sphere of the handle${\displaystyle H^j}$ in${\displaystyle M{\cup }_f{H}^j}$.

A manifold obtained by attachinggk-handles to the disc${\displaystyle D^m}$ is an(m,k)-handlebody of genusg.

Ahandle presentation of a cobordism consists of a cobordismW where${\displaystyle \mathrm{\partial }W=M_0\cup M_1}$ and an ascending union$${\displaystyle W_{-1}\subset W_0\subset W_1\subset \cdots \subset W_{m+1}=W}$$whereM ism-dimensional,W ism+1-dimensional,${\displaystyle W_{-1}}$ is diffeomorphic to${\displaystyle M_0\times [0,1]}$ and${\displaystyle W_i}$ is obtained from${\displaystyle W_{i-1}}$ by the attachment ofi-handles. Whereas handle decompositions are the analogue for manifolds what cell decompositions are to topological spaces, handle presentations of cobordisms are to manifolds with boundary what relative cell decompositions are for pairs of spaces.

Given aMorse function${\displaystyle f:M\to \mathbb{R}}$ on a compact boundaryless manifoldM, such that thecritical points${\displaystyle \{p_1,\dots ,p_k\}\subset M}$ off satisfy, and providedthen for allj,${\displaystyle f^{-1}[t_{j-1},t_j]}$ is diffeomorphic to${\displaystyle (f^{-1}(t_{j-1})\times [0,1])\cup H^{I(j)}}$ whereI(j) is the index of the critical point${\displaystyle p_j}$. TheindexI(j) refers to the dimension of the maximal subspace of the tangent space${\displaystyle T_{p_j}M}$ where theHessian is negative definite.

Provided the indices satisfy${\displaystyle I(1)\le I(2)\le \cdots \le I(k)}$ this is a handle decomposition ofM, moreover, every manifold has such Morse functions, so they have handle decompositions. Similarly, given a cobordism${\displaystyle W}$ with${\displaystyle \mathrm{\partial }W=M_0\cup M_1}$ and a function${\displaystyle f:W\to \mathbb{R}}$ which is Morse on the interior and constant on the boundary and satisfying the increasing index property, there is an induced handle presentation of the cobordismW.

Whenf is a Morse function onM, -f is also a Morse function. The corresponding handle decomposition / presentation is called thedual decomposition.

• AHeegaard splitting of a closed, orientable 3-manifold is a decomposition of a3-manifold into the union of two(3,1)-handlebodies along their common boundary, called the Heegaard splitting surface. Heegaard splittings arise for3-manifolds in several natural ways: given a handle decomposition of a 3-manifold, the union of the0 and1-handles is a(3,1)-handlebody, and the union of the3 and2-handles is also a(3,1)-handlebody (from the point of view of the dual decomposition), thus a Heegaard splitting. If the3-manifold has atriangulationT, there is an induced Heegaard splitting where the first(3,1)-handlebody is a regular neighbourhood of the1-skeleton${\displaystyle T^1}$, and the other(3,1)-handlebody is a regular neighbourhood of thedual1-skeleton.
• When attaching two handles in succession${\displaystyle (M{\cup }_f{H}^i){\cup }_g{H}^j}$, it is possible to switch the order of attachment, provided${\displaystyle j\le i}$, i.e.: this manifold is diffeomorphic to a manifold of the form${\displaystyle (M\cup H^j)\cup H^i}$ for suitable attaching maps.
• The boundary of${\displaystyle M{\cup }_f{H}^j}$ is diffeomorphic to${\displaystyle \mathrm{\partial }M}$ surgered along the framed sphere${\displaystyle f}$. This is the primary link betweensurgery, handles and Morse functions.
• As a consequence, anm-manifoldM is the boundary of anm+1-manifoldW if and only ifM can be obtained from${\displaystyle S^m}$ by surgery on a collection of framed links in${\displaystyle S^m}$. For example, it's known that every3-manifold bounds a4-manifold (similarly oriented and spin3-manifolds bound oriented and spin4-manifolds respectively) due toRené Thom's work on cobordism. Thus every 3-manifold can be obtained via surgery on framed links in the3-sphere. In the oriented case, it's conventional to reduce this framed link to a framed embedding of a disjoint union of circles.
• TheH-cobordism theorem is proven by simplifying handle decompositions of smooth manifolds.

• Casson handle
• Cobordism theory
• CW complex
• Handlebody
• Kirby calculus
• Manifold decomposition

• A. Kosinski,Differential Manifolds Vol 138 Pure and Applied Mathematics, Academic Press (1992).
• Robert Gompf and Andras Stipsicz,4-Manifolds and Kirby Calculus, (1999) (Volume 20 inGraduate Studies in Mathematics),American Mathematical Society, Providence, RIISBN 0-8218-0994-6

• Geometric topology

• Articles with short description
• Short description matches Wikidata