Let Y be a space and A a
differential graded algebra over the field Q of rationals corresponding in
Sullivan’s theory to the rational homotopy type of Y . Then to the rational
homotopy type of a fibration over Y equipped with a given cross-section
corresponds a differential graded Lie algebra L over A, free as an A-module. The
differential graded Lie algebra Q ⊗AL corresponds in Quillen’s theory to
the rational homotopy type of the fibre of the fibration. Furthermore, by
restriction of scalars, L can be considered as a differential graded Lie algebra
over Q. Then it contains a differential graded Lie sub-algebra over Q which
corresponds to the rational homotopy type of the space of cross-sections
which are homotopic to the given cross-section. Some examples illustrate this
result.
