Editing Handle decomposition (section)

==Some major theorems and observations==

* A [[Heegaard splitting]] of a closed, orientable 3-manifold is a decomposition of a ''3''-manifold into the union of two ''(3,1)''-handlebodies along their common boundary, called the Heegaard splitting surface.  Heegaard splittings arise for ''3''-manifolds in several natural ways: given a handle decomposition of a 3-manifold, the union of the ''0'' and ''1''-handles is a ''(3,1)''-handlebody, and the union of the ''3'' and ''2''-handles is also a ''(3,1)''-handlebody (from the point of view of the dual decomposition), thus a Heegaard splitting.  If the ''3''-manifold has a [[triangulation (topology)|triangulation]] ''T'', there is an induced Heegaard splitting where the first ''(3,1)''-handlebody is a regular neighbourhood of the ''1''-skeleton <math>T^1</math>, and the other ''(3,1)''-handlebody is a regular neighbourhood of the [[Poincaré duality|dual ''1''-skeleton]].
* When attaching two handles in succession <math>(M \cup_f H^i) \cup_g H^j</math>, it is possible to switch the order of attachment, provided <math>j \leq i</math>, i.e.: this manifold is diffeomorphic to a manifold of the form <math>(M \cup H^j) \cup H^i</math> for suitable attaching maps.
* The boundary of <math>M \cup_f H^j</math> is diffeomorphic to <math>\partial M</math> surgered along the framed sphere <math>f</math>.  This is the primary link between [[Surgery theory|surgery]], handles and Morse functions.
* As a consequence, an ''m''-manifold ''M'' is the boundary of an ''m+1''-manifold ''W'' if and only if ''M'' can be obtained from <math>S^m</math> by surgery on a collection of framed links in <math>S^m</math>.  For example, it's known that every ''3''-manifold bounds a ''4''-manifold (similarly oriented and spin ''3''-manifolds bound oriented and spin ''4''-manifolds respectively) due to [[cobordism|René Thom's work on cobordism]].  Thus every 3-manifold can be obtained via surgery on framed links in the ''3''-sphere.  In the oriented case, it's conventional to reduce this framed link to a framed embedding of a disjoint union of circles.
* The [[h-cobordism|H-cobordism theorem]] is proven by simplifying handle decompositions of smooth manifolds.