Mapping space explained

In mathematics, especially in algebraic topology, the mapping space between two spaces is the space of all the (continuous) maps between them.

h:I\to\operatorname{Map}(X,Y)

in the mapping space is exactly a homotopy.

Topologies

A mapping space can be equipped with several topologies. A common one is the compact-open topology or the k-ification of it. Typically, there is then the adjoint relation

\operatorname{Map}(X x Y,Z)\simeq\operatorname{Map}(X,\operatorname{Map}(Y,Z))

and thus

\operatorname{Map}

is an analog of the Hom functor. (For pathological spaces, this relation may fail.)

Smooth mappings

See main article: Whitney topologies. For manifolds

M,N

, there is the subspace

l{C}^r(M,N)\subset\operatorname{Map}(M,N)

that consists of all the

l{C}^r

-smooth maps from

. It can be equipped with the weak or strong topology.

A basic approximation theorem says that

	s(M,
l{C}
	W

is dense in

	r(M,
l{C}
	S

for

1\les\leinfty,0\ler<s

Homotopy type of a mapping space

A basic result here is a theorem of Milnor which says that the mapping space

\operatorname{Map}(X,Y)

has the homotopy type of a CW-complex if

is a compact Hausdorff space and

has the homotopy type of a CW-complex.

References

Book: Hirsch, Morris. Morris Hirsch. Differential Topology. Springer. 1997. 0-387-90148-5.
John . Milnor . John Milnor . On spaces having the homotopy type of CW-complex . . 90 . 2 . 1959 . 272–280 . 10.2307/1993204 . 1993204.
Book: Wall, C. T. C.. Differential Topology. 4 July 2016 . Cambridge University Press. 9781107153523.