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 T, there is an induced Heegaard splitting where the first (3,1)-handlebody is a regular neighbourhood of the 1-skeleton , and the other (3,1)-handlebody is a regular neighbourhood of the dual 1-skeleton. When attaching two handles in succession , it is possible to switch the order of attachment, provided , i.e.: this manifold is diffeomorphic to a manifold of the form for suitable attaching maps. The boundary of is diffeomorphic to surgered along the framed sphere . This is the primary link between 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 by surgery on a collection of framed links in . 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 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 theorem is proven by simplifying handle decompositions of smooth manifolds. See also Casson handle Cobordism theory CW complex Handlebody Kirby calculus Manifold decomposition References Notes General references 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 in Graduate Studies in Mathematics), American Mathematical Society, Providence, RI Geometric topology