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1 FoliationsLet be an -manifold, possibly with boundary, and let be a decomposition of into connected, topologically immersed submanifolds of dimension . is said to be a codimension (smooth) foliation of if admits an (smooth) atlas of foliated charts, that is (diffeomorphisms) homeomorphisms
2 Defining differential form
If is a smooth, transversely orientable codimension foliation and its tangential plane field, then there is a nonsingular -form such that, for each ,
This implies that for some .
The leaves of are the immersed submanifolds . Each belongs to a unique leaf. The foliation determines its tangential plane field by if .
The space of leaves is with the quotient topology, where if and only if and belong to the same leaf of .
4 Holonomy CocycleThe holonomy cocycle of the atlas is given by
A smooth foliation is said to be transversely orientable if everywhere.