attaching j-handles if the union of M with finitely many j-handles is diffeomorphic to N. The definition of a handle decomposition is then as in the introduction. Thus, a manifold has a handle decomposition with only 0-handles if it is diffeomorphic to a disjoint union of balls. A connected manifold containing handles of only two types (i.e.: 0-handles and j-handles for some fixed j) is called a handlebody. Terminology When forming M union a j-handle is known as the attaching sphere. is sometimes called the framing of the attaching sphere, since it gives trivialization of its normal bundle. is the belt sphere of the handle in . A manifold obtained by attaching g k-handles to the disc is an (m,k)-handlebody of genus g . Cobordism presentations A handle presentation of a cobordism consists of a cobordism W where and an ascending union where is -dimensional, is m+1-dimensional, is diffeomorphic to and is obtained from by the attachment of i-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. Morse theoretic viewpoint Given a Morse function on a compact boundaryless manifold M, such that the critical points of f satisfy , and provided then for all j, is diffeomorphic to where I(j) is the index of the critical point . The index I(j) refers to the dimension of the maximal subspace of the tangent space where the Hessian is negative definite. Provided the indices satisfy this is a handle decomposition of M, moreover, every manifold has such Morse functions, so they have handle decompositions. Similarly, given a cobordism with and a function 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 cobordism W. When f is a Morse function on M, -f is also a Morse function. The corresponding handle decomposition / presentation is called the dual decomposition. 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