Editing Handlebody

[[Image:Triple torus illustration.png|right|thumb|A genus three handlebody.]]
In the [[mathematics|mathematical]] field of [[geometric topology]], a '''handlebody''' is a decomposition of a [[manifold]] into standard pieces. Handlebodies play an important role in [[Morse theory]], [[cobordism theory]] and the [[surgery theory]] of high-dimensional manifolds. Handles are used to particularly study [[3-manifold]]s.

Handlebodies play a similar role in the study of manifolds as [[simplicial complex]]es and [[CW complex]]es play in [[homotopy theory]], allowing one to analyze a space in terms of individual pieces and their interactions.

==''n''-dimensional handlebodies==
If <math>(W,\partial W)</math> is an <math>n</math>-dimensional manifold with boundary, and 
:<math>S^{r-1} \times D^{n-r} \subset \partial W</math>

(where <math>S^{n}</math> represents an [[n-sphere]] and <math>D^n</math> is an [[Ball_(mathematics)|n-ball]]) is an embedding, the <math>n</math>-dimensional manifold with boundary
:<math>(W',\partial W') = ((W \cup( D^r \times D^{n-r})),(\partial W - S^{r-1} \times D^{n-r})\cup (D^r \times S^{n-r-1}))</math>

is said to be ''obtained from'' 
:<math>(W,\partial W)</math>

by attaching an ''<math>r</math>-handle''.
The boundary <math>\partial W'</math> is obtained from <math>\partial W</math> by [[Surgery theory|surgery]]. As trivial examples, note that attaching a 0-handle is just taking a disjoint union with a ball, and that attaching an n-handle 
to <math>(W,\partial W)</math> is gluing in a ball along any sphere component of  <math>\partial W</math>. [[Morse theory]] was used by [[René Thom|Thom]] and [[Milnor]] to prove that every manifold (with or without boundary) is a handlebody, meaning that it has an expression as a union of handles. The expression is non-unique: the manipulation of handlebody decompositions is an essential ingredient of the proof of the [[Smale]] [[h-cobordism]] theorem, and its generalization to the [[s-cobordism]] theorem. A manifold is called a "k-handlebody" if it is the union of r-handles, for r at most k. This is not the same as the dimension of the manifold. For instance, a 4-dimensional 2-handlebody is a union of 0-handles, 1-handles and 2-handles. Any manifold is an n-handlebody, that is, any manifold is the union of handles. It isn't too hard to see that a manifold is an (n-1)-handlebody if and only if it has non-empty boundary.
Any handlebody decomposition of a manifold defines a [[CW complex]] decomposition of the manifold, since attaching an r-handle is the same, up to homotopy equivalence, as attaching an r-cell. However, a handlebody decomposition gives more information than just the homotopy type of the manifold. For instance, a handlebody decomposition completely describes the manifold up to homeomorphism. In dimension four, they even describe the smooth structure, as long as the attaching maps are smooth. This is false in higher dimensions; any [[exotic sphere]] is the union of a 0-handle and an n-handle.

==3-dimensional handlebodies==
A handlebody can be defined as an [[orientable]] 3-manifold-with-boundary containing  pairwise disjoint, properly embedded 2-discs such that the manifold resulting from cutting along the discs is a 3-ball.  It's instructive to imagine how to reverse this process to get a handlebody.  (Sometimes  the orientability hypothesis is dropped from this last definition, and one gets a more general kind of handlebody with a non-orientable handle.)  

The ''genus'' of a handlebody is the [[genus (topology)|genus]] of its  [[boundary (topology)|boundary]] [[Surface (topology)|surface]].   [[Up to]] [[homeomorphism]], there is exactly one handlebody of any non-negative integer genus.

The importance of handlebodies in [[3-manifold]] theory comes from their connection with [[Heegaard splitting]]s.  The importance of handlebodies in [[geometric group theory]] comes from the fact that their [[fundamental group]] is free.

A 3-dimensional handlebody is sometimes, particularly in older literature, referred to as a '''cube with handles'''.

==Examples==
Let ''G'' be a connected [[finite set|finite]] graph embedded in [[Euclidean space]] of dimension n.  Let ''V'' be a [[closed (topology)|closed]] [[regular neighborhood]] of ''G'' in the Euclidean space. Then ''V'' is an n-dimensional handlebody.  The graph ''G'' is called a ''spine'' of ''V''.

Any genus zero handlebody is [[homeomorphic]] to the three-[[ball (topology)|ball]] B<sup>3</sup>. A genus one handlebody is [[homeomorphic]] to B<sup>2</sup> &times; S<sup>1</sup> (where S<sup>1</sup> is the [[circle]]) and is called a ''solid [[torus]]''. All other handlebodies may be obtained by taking the boundary-[[connected sum]] of a collection of solid tori.

==See also==
* [[Handle decomposition]]

==References==
*{{Citation | last1=Matsumoto | first1=Yukio | title=An introduction to Morse theory | url=https://books.google.com/books?id=TtKyqozvgIwC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Translations of Mathematical Monographs | isbn=978-0-8218-1022-4 | mr=1873233 | year=2002 | volume=208}}

[[Category:Geometric topology]]
[[Category:Surgery theory]]