In 5 or more dimensions handlebody decomposition and the associated handle moves. In dimension 4 Kirby calculus. In dimension 3 Heegaard splittings. (Dimensions less than 3 are left as an exercise for the interested reader.)

Handlebody decomposition and the associated handle moves are covered elegantly in Kosinski's Differential Manifolds while Kirby calculus is covered in Stipsicz and Gompf's 4-Manifolds and Kirby Calculus. Both books touch upon Heegaard splittings.