In mathematics, a handle decomposition of a 3-manifold allows simplification of the original 3-manifold into pieces which are easier to study.
An important method used to decompose into handlebodies is the Heegaard splitting, which gives a decomposition in two handlebodies of equal genus.
[1] 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:
For non-orientable spaces an interesting invariant is the tri-genus.