In mathematics, a handle decomposition of a 3-manifold allows simplification of the original 3-manifold into pieces which are easier to study.
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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:
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.