a holonomy group of translations of the fiber at m; if the structure group of B is G, it is a subgroup of G that measures the deviation of B from the product bundle M × G. Monodromy groupoid and foliations Analogous to the 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 X of a fibration . The result has the structure of a groupoid over the base space X. The advantage is that we can drop the condition of connectedness of X. Moreover the construction can also be generalized to foliations: Consider a (possibly singular) foliation of M. Then for every path in a leaf of 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 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. Definition via Galois theory Let F(x) denote the field of the rational functions in the variable x over the field F, which is the field of fractions of the polynomial ring F[x]. An element y = f(x) of F(x) determines a finite field extension [F(x) : F(y)]. This extension is generally not Galois but has Galois closure L(f). The associated Galois group of the extension [L(f) : F(y)] is called the monodromy group of f. In the case of F = C Riemann surface theory enters and allows for the geometric interpretation given above. In the case that the extension [C(x) : C(y)] is already Galois, the associated monodromy group is sometimes called a group of deck transformations. This has connections with the Galois theory of covering spaces leading to the Riemann existence theorem. See also Braid group Monodromy theorem Mapping class group (of a punctured disk) Notes References "Group-groupoids and monodromy groupoids", O. Mucuk, B. Kılıçarslan, T. ¸Sahan, N. Alemdar, Topology and its Applications 158 (2011) 2034–2042 doi:10.1016/j.topol.2011.06.048 R. Brown Topology and Groupoids (2006). P.J. Higgins, "Categories and groupoids", van Nostrand (1971) TAC Reprint H. Żołądek, "The Monodromy Group", Birkhäuser Basel 2006; doi: 10.1007/3-7643-7536-1 Mathematical analysis Complex analysis Differential geometry