title: Handle decompositions of 3-manifolds
text: # Handle decompositions of 3-manifolds
  
  In mathematics, a handle decomposition of a 3-manifold allows simplification of the original 3-manifold into pieces which are easier to study.
  
  ## Heegaard splittings
  
  An important method used to decompose into handlebodies is the Heegaard splitting, which gives us a decomposition in two handlebodies of equal genus.
  
  ## Examples
  
  As an example: lens spaces are orientable 3-spaces and allow decomposition into two solid tori, which are genus-one-handlebodies. The genus one non-orientable space is a space which is the union of two solid Klein bottles and corresponds to the twisted product of the 2-sphere and the 1-sphere: $\scriptstyle S^{2}{\tilde {\times }}S^{1}$.
  
  ## Orientability
  
  Each orientable 3-manifold is the union of exactly two orientable handlebodies; meanwhile, each non-orientable one needs three orientable handlebodies.
  
  ## Heegaard genus
  
  The minimal genus of the glueing boundary determines what is known as the Heegaard genus. For non-orientable spaces an interesting invariant is the tri-genus.
  
  