We use the space Map(
X,Y
) of
continuous maps
and its based version Map
_{
*
}
(
X,Y
) quite often in algebraic topology.
One of the reasons is Map(
X,

) and Map
_{
*
}
(
X,

) are right adjoint to functors (

)
×
X
and (

)
∧
X
. From the viewpoint of
homotopical algebra
, it is desirable that our category of spaces is
closed under various constructions. The mapping space
construction is one of the fundamental constructions.
When we study inﬁnite dimensional spaces such
as mapping spaces, we need to approximate them by
wellknown spaces. There are several such models for
mapping spaces. These are especially useful for
studying iterated loop spaces.
It is useful to regard the mapping space Map(
X,F
) as the space of cross sections on the trivial
bundle
. For example, it is used in the construction of
twisted cohomology theories
.
There are
spectral sequences
for computing the
homotopy groups
of mapping spaces and spaces of sections.

Federer spectral sequence
[
Fed56
]

Schultz’ spectral sequence
[
?
]
for computing the homotopy groups of the space of
sections
