In 5 or more dimensions [handlebody decomposition][1] and the associated handle moves. In dimension 4 [Kirby calculus][2]. In dimension 3 [Heegaard splittings][3]. (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*][4] while Kirby calculus is covered in Stipsicz and Gompf's [*4-Manifolds and Kirby Calculus*][5]. Both books touch upon Heegaard splittings.


  [1]: http://en.wikipedia.org/wiki/Handle_decomposition
  [2]: http://en.wikipedia.org/wiki/Kirby_calculus
  [3]: http://en.wikipedia.org/wiki/Heegaard_splitting
  [4]: http://www.amazon.com/Differential-Manifolds-Pure-Applied-Mathematics/dp/0124218504/ref=tmm_hrd_title_0
  [5]: http://www.amazon.com/4-Manifolds-Calculus-Graduate-Studies-Mathematics/dp/0821809946