Editing Monodromy (section)

===Monodromy groupoid and foliations===
[[File:Monodromy action.svg|thumb|upright=0.3|A path in the base has paths in the total space lifting it. Pushing along these paths gives the monodromy action from the fundamental groupoid.]]
Analogous to the [[Fundamental group#Fundamental groupoid|fundamental groupoid]] it is possible to get rid of the choice of a base point and to define a monodromy groupoid.  Here we consider (homotopy classes of) lifts of paths in the base space <math>X</math> of a fibration <math>p:\tilde X\to X</math>.  The result has the structure of a [[groupoid]] over the base space <math>X</math>.  The advantage is that we can drop the condition of connectedness of <math>X</math>.

Moreover the construction can also be generalized to [[foliation]]s:  Consider <math>(M,\mathcal{F})</math> a (possibly singular) foliation of <math>M</math>.  Then for every path in a leaf of <math>\mathcal{F}</math> we can consider its induced diffeomorphism on local transversal sections through the endpoints.  Within a simply connected chart this diffeomorphism becomes unique and especially canonical between different transversal sections if we go over to the [[Germ (mathematics)|germ]] of the diffeomorphism around the endpoints.  In this way it also becomes independent of the path (between fixed endpoints) within a simply connected chart and is therefore invariant under homotopy.