Handle Decomposition - Some Major Theorems and Observations

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 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.