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A transformation of the fibres (or of their homotopy invariants) of a fibre space corresponding to a path in the base. More precisely, let  $  p :  E \rightarrow B $
be a locally trivial [[Fibre space|fibre space]] and let  $  \gamma :  [ 0 , 1 ] \rightarrow B $
be a path in  $  B $
with initial point  $  a = \gamma ( 0) $
and end-point  $  b = \gamma ( 1) $. 
A trivialization of the fibration  $  \gamma  ^ {*} ( E) $
defines a homeomorphism  $  T _  \gamma  $
of the fibre  $  p  ^ {-} 1 ( a) $
onto the fibre  $  p  ^ {-} 1 ( b) $, 
$  T _  \gamma  :  p  ^ {-} 1 ( a) \rightarrow p  ^ {-} 1 ( b) $. 
If the trivialization of  $  \gamma  ^ {*} ( E) $
is modified, then  $  T _  \gamma  $
changes into a homotopically-equivalent homeomorphism; this also happens if  $  \gamma $
is changed to a homotopic path. The homotopy type of  $  T _  \gamma  $
is called the monodromy transformation corresponding to a path  $  \gamma $. 
When  $  a = b $, 
that is, when  $  \gamma $
is a loop, the monodromy transformation  $  T _  \gamma  $
is a homeomorphism of  $  F = p  ^ {-} 1 ( a) $
into itself (defined, yet again, up to a homotopy). This mapping, and also the homomorphisms induced by it on the homology and cohomology spaces of  $  F $, 
is also called a monodromy transformation. The correspondence of  $  \gamma $
with  $  T _  \gamma  $
gives a representation of the [[Fundamental group|fundamental group]]  $  \pi _ {1} ( B , a ) $
on  $  H  ^ {*} ( F  ) $.

The idea of a monodromy transformation arose in the study of multi-valued functions (see [[Monodromy theorem|Monodromy theorem]]). If  $  S \rightarrow P  ^ {1} ( \mathbf C ) $
is the [[Riemann surface|Riemann surface]] of such a function, then by eliminating the singular points of the function from the Riemann sphere  $  P  ^ {1} ( \mathbf C ) $, 
an unbranched covering is obtained. The monodromy transformation in this case is also called a covering or deck transformation.

The monodromy transformation arises most frequently in the following situation. Let  $  D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
be the unit disc in the complex plane, let  $  X $
be an [[Analytic space|analytic space]], let  $  f :  X \rightarrow D $
be a proper holomorphic mapping (cf. [[Proper morphism|Proper morphism]]), let  $  X _ {t} $
be the fibre  $  f ^ { - 1 } ( t) $, 
$  t \in D $, 
$  D  ^ {*} = D \setminus  \{ 0 \} $, 
and let  $  X  ^ {*} = f ^ { - 1 } ( D  ^ {*} ) $. 
Diminishing, if necessary, the radius of  $  D $, 
the fibre space  $  f :  X  ^ {*} \rightarrow D  ^ {*} $
can be made locally trivial. The monodromy transformation  $  T $
associated with a circuit around  $  0 $
in  $  D $
is called the monodromy of the family  $  f :  X \rightarrow D $
at  $  0 \in D $, 
it acts on the (co)homology spaces of the fibre  $  X _ {t} $, 
where  $  t \in D  ^ {*} $. 
The most studied case is when  $  X $
and the fibres  $  X _ {t} $, 
$  t \neq 0 $, 
are smooth. The action of  $  T $
on  $  H  ^ {*} ( X _ {t} , \mathbf Q ) $, 
in this case, is quasi-unipotent [[#References|[4]]], that is, there are positive integers  $  k $
and  $  N $
such that  $  ( T  ^ {k} - 1 )  ^ {N} = 0 $. 
The properties of the monodromy display many characteristic features of the degeneracy of the family  $  f :  X \rightarrow D $. 
The monodromy of the family  $  f :  X \rightarrow D $
is closely related to mixed Hodge structures (cf. [[Hodge structure|Hodge structure]]) on the cohomology spaces  $  H  ^ {*} ( X _ {0} ) $
and  $  H  ^ {*} ( X _ {t} ) $(
see [[#References|[5]]]–[[#References|[7]]]).

When the singularities of  $  f :  X \rightarrow D $
are isolated, the monodromy transformation can be localized. Let  $  x $
be a singular point of  $  f $(
or, equivalently, of  $  X _ {0} $) 
and let  $  B $
be a sphere of sufficiently small radius in  $  X $
with centre at  $  x $. 
Diminishing, if necessary, the radius of  $  D $, 
a local trivialization of the fibre space  $  B \cap f ^ { - 1 } ( D  ^ {*} ) $
can be defined. It is compatible with the trivialization of the fibre space  $  \partial  B \cap f ^ { - 1 } ( D) \rightarrow D $
on the boundary. This gives a diffeomorphism  $  T $
of the manifold of "vanishing cycles"  $  V _ {t} = B \cap X _ {t} $
into itself which is the identity on  $  \partial  V _ {t} $, 
and which is called the local monodromy of  $  f $
at  $  x $. 
The action of the monodromy transformation on the cohomology spaces  $  H  ^ {*} ( V _ {t} ) $
reflects the singularity of  $  f $
at  $  x $(
see [[#References|[1]]], [[#References|[2]]], [[#References|[7]]]). It is known that the manifold  $  V _ {t} $
is homotopically equivalent to a bouquet of  $  \mu $
$  n $-
dimensional spheres, where  $  n + 1 =  \mathop{\rm dim}  X $
and  $  \mu $
is the Milnor number of the germ of  $  f $
at  $  x $.

The simplest case is that of a Morse singularity when, in a neighbourhood of  $  x $, 
$  f $
reduces to the form  $  f = z _ {0}  ^ {2} + \dots + z _ {n}  ^ {2} $(
cf. [[Morse lemma|Morse lemma]]). In this case  $  \mu = 1 $, 
and the interior  $  V _ {t}  ^ {0} $
of  $  V _ {t} $
is diffeomorphic to the tangent bundle of the  $  n $-
dimensional sphere  $  S  ^ {n} $. 
A vanishing cycle  $  \delta $
is a generator of the cohomology group with compact support  $  H _ {c}  ^ {n} ( V _ {t}  ^ {0} , \mathbf Z ) \cong \mathbf Z $, 
defined up to sign. In general, if  $  f :  X \rightarrow D $
is a proper holomorphic mapping (as above, having a unique Morse singularity at  $  x $), 
then a cycle  $  \delta _ {x} $
vanishing at  $  x $
is the image of a cycle  $  \delta \in H _ {c}  ^ {n} ( V _ {t}  ^ {0} ) $
under the natural mapping  $  H _ {c}  ^ {n} ( V _ {t}  ^ {0} ) \rightarrow H  ^ {n} ( X _ {t} ) $. 
In this case the specialization homomorphism  $  r _ {t}  ^ {*} :  H  ^ {i} ( X _ {0} ) \rightarrow H  ^ {i} ( X _ {t} ) $
is an isomorphism for  $  i \neq n , n + 1 $, 
and the sequence

$$ 
0  \rightarrow  H  ^ {n} ( X _ {0} )  \rightarrow  H  ^ {n} ( X _ {t} )
  \mathop \rightarrow \limits ^ { {(  , \delta _ {x} ) }}   \mathbf Z \rightarrow
$$

$$ 
\rightarrow \ 
H  ^ {n+} 1 ( X _ {0} )  \rightarrow  H  ^ {n+} 1 ( X _ {t} )  \rightarrow  0
$$

is exact. The monodromy transformation  $  T $
acts trivially on  $  H  ^ {i} ( X _ {t} ) $
for  $  i \neq n $
and its action on  $  H  ^ {n} ( X _ {t} ) $
is given by the Picard–Lefschetz formula: For  $  z \in H  ^ {n} ( X _ {t} ) $,

$$ 
T _ {z}  =  z \pm  ( z , \delta _ {x} ) \delta _ {x} .
$$

The sign in this formula and the values of  $  ( \delta _ {x} , \delta _ {x} ) $
are collected in the table.<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  n   \mathop{\rm mod}  4 $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  0 $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  1 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  2 $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  3 $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  \pm  $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  - $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  - $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  + $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  + $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  ( \delta _ {x} , \delta _ {x} ) $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  2 $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  0 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  - 2 $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  0 $
</td> </tr> </tbody> </table>

</td></tr> </table>

A monodromy transformation preserves the intersection form on  $  H  ^ {n} ( X _ {t} ) $.

Vanishing cycles and monodromy transformations are used in the Picard–Lefschetz theory, associating the cohomology space of a projective complex manifold and its hyperplane sections. Let  $  X \subset  P  ^ {N} $
be a smooth manifold of dimension  $  n + 1 $, 
and let  $  \{ X _ {t} \} $, 
$  t \in P  ^ {1} $, 
be a pencil of hyperplane sections of  $  X $
with basic set (axis of the pencil)  $  Y \subset  X $; 
let the following conditions be satisfied: a)  $  Y $
is a smooth submanifold in  $  X $; 
b) there is a finite set  $  S \subset  P  ^ {1} $
such that  $  X _ {t} $
is smooth for  $  t \in P  ^ {1} \setminus  S $; 
and c) for  $  s \in S $
the manifold  $  X _ {s} $
has a unique non-degenerate quadratic singular point  $  x _ {s} $, 
where  $  x _ {s} \in Y $. 
Pencils with these properties (Lefschetz pencils) always exist. Let  $  \sigma :  \overline{X}\; \rightarrow X $
be a monoidal transformation with centre on the axis  $  Y $
of the pencil, and let  $  f :  \overline{X}\; \rightarrow P  ^ {1} $
be the morphism defined by the pencil  $  \{ X _ {t} \} $; 
here  $  f ^ { - 1 } ( t) \cong X _ {t} $
for all  $  t \in P  ^ {1} $. 
Let a point  $  0 \in P  ^ {1} \setminus  S $
be fixed; then the monodromy transformation gives an action of  $  \pi _ {1} ( P  ^ {1} \setminus  S , 0 ) $
on  $  H  ^ {i} ( X _ {0} ) $(
non-trivial only for  $  i = n $). 
To describe the action of the monodromy on  $  H  ^ {n} ( X _ {0} ) $
one chooses points  $  s  ^  \prime  $, 
situated near  $  s \in S $, 
and paths  $  \gamma _ {s}  ^  \prime  $
leading from  $  0 $
to  $  s  ^  \prime  $. 
Let  $  \gamma _ {s} \in \pi _ {1} ( p  ^ {1} \setminus  S , 0 ) $
be the loop constructed as follows: first go along  $  \gamma _ {s}  ^  \prime  $, 
then once round  $  s $
and, finally, return along  $  \gamma _ {s}  ^  \prime  $
to  $  0 $. 
In addition, let  $  \delta _ {s} $
be a cycle vanishing at  $  x _ {s} $(
more precisely, take a vanishing cycle in  $  H  ^ {n} ( X _ {s  ^  \prime   } ) $
and transfer it to  $  H  ^ {n} ( X _ {0} ) $
by means of the monodromy transformation corresponding to the path  $  \gamma _ {s}  ^  \prime  $). 
Finally, let  $  E \subset  H  ^ {n} ( X _ {0} , \mathbf Q ) $
be the subspace generated by the vanishing cycles  $  \delta _ {s} $, 
$  s \in S $(
the vanishing cohomology space). Then the following hold.

1)  $  \pi _ {1} ( P  ^ {1} \setminus  S , 0 ) $
is generated by the elements  $  \gamma _ {s} $, 
$  s \in S $;

2) the action of  $  \gamma _ {s} $
is given by the formula

$$ 
\gamma _ {s} ( z )  =  z \pm  ( z , \delta _ {s} ) \delta _ {s} ;
$$

3) the space  $  E \subset  H  ^ {n} ( X _ {0} ) $
is invariant under the action of the monodromy group  $  \pi _ {1} ( P  ^ {1} \setminus  S , 0 ) $;

4) the space of elements in  $  H  ^ {n} ( X _ {0} ) $
that are invariant relative to monodromy coincides with the orthogonal complement of  $  E $
relative to the intersection form on  $  H  ^ {n} ( X _ {0} ) $, 
and also with the images of the natural homomorphisms  $  H _ {n} ( \overline{X}\; ) \rightarrow H _ {n} ( X _ {0} ) $
and  $  H  ^ {n} ( X ) \rightarrow H  ^ {n} ( X _ {0} ) $;

5) the vanishing cycles  $  \pm  \delta _ {s} $
are conjugate (up to sign) under the action of  $  \pi _ {1} ( P  ^ {1} \setminus  S , 0 ) $;

6) the action of  $  \pi _ {1} ( P  ^ {1} \setminus  S , 0 ) $
on  $  E $
is absolutely irreducible.

The formalism of vanishing cycles, monodromy transformations and the Picard–Lefschetz theory has also been constructed for  $  l $-
adic cohomology spaces of algebraic varieties over any field (see [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.I. Arnol'd, "Normal forms of functions in neighbourhoods of degenerate critical points" ''Russian Math. Surveys'' , '''29''' : 2 (1974) pp. 10–50 ''Uspekhi Mat. Nauk'' , '''29''' : 2 (1974) pp. 11–49 {{MR|}} {{ZBL|0304.57018}} {{ZBL|0298.57022}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968) {{MR|0239612}} {{ZBL|0184.48405}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Deligne (ed.) N.M. Katz (ed.) , ''Groupes de monodromie en géométrie algébrique. SGA 7.II'' , ''Lect. notes in math.'' , '''340''' , Springer (1973) {{MR|0354657}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> C.H. Clemens, "Picard–Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities" ''Trans. Amer. Math. Soc.'' , '''136''' (1969) pp. 93–108 {{MR|0233814}} {{ZBL|0185.51302}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" ''Invent. Math.'' , '''22''' (1973) pp. 211–319 {{MR|0382272}} {{ZBL|0278.14003}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J. Steenbrink, "Limits of Hodge structures" ''Invent. Math.'' , '''31''' (1976) pp. 229–257 {{MR|0429885}} {{ZBL|0303.14002}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> J.H.M. Steenbrink, "Mixed Hodge structure on the vanishing cohomology" P. Holm (ed.) , ''Real and Complex Singularities (Oslo, 1976). Proc. Nordic Summer School'' , Sijthoff &amp; Noordhoff (1977) pp. 524–563 {{MR|0485870}} {{ZBL|0373.14007}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Lefschetz, "L'analysis situs et la géométrie algébrique" , Gauthier-Villars (1924) {{MR|0033557}} {{MR|1520618}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S. Lefschetz, "A page of mathematical autobiography" ''Bull. Amer. Math. Soc.'' , '''74''' : 5 (1968) pp. 854–879 {{MR|0240803}} {{ZBL|0187.18601}} </TD></TR></table>