Mapping space

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

Viewing the set of all the maps as a space is useful because that allows for topological considerations. For example, a curve ${\displaystyle h:I\to \operatorname {Map} (X,Y)}$ 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

${\displaystyle \operatorname {Map} (X\times Y,Z)\simeq \operatorname {Map} (X,\operatorname {Map} (Y,Z))}$

and thus ${\displaystyle \operatorname {Map} }$ is an analog of the Hom functor. (For pathological spaces, this relation may fail.)

Smooth mappings

For manifolds ${\displaystyle M,N}$, there is the subspace ${\displaystyle {\mathcal {C}}^{r}(M,N)\subset \operatorname {Map} (M,N)}$ that consists of all the ${\displaystyle {\mathcal {C}}^{r}}$-smooth maps from ${\displaystyle M}$ to ${\displaystyle N}$. It can be equipped with the weak or strong topology.

A basic approximation theorem says that ${\displaystyle {\mathcal {C}}_{W}^{s}(M,N)}$ is dense in ${\displaystyle {\mathcal {C}}_{S}^{r}(M,N)}$ for ${\displaystyle 1\leq s\leq \infty ,0\leq r<s}$.^[1]

Homotopy type of a mapping space

A basic result here is a theorem of Milnor which says that the mapping space ${\displaystyle \operatorname {Map} (X,Y)}$ has the homotopy type of a CW-complex if ${\displaystyle X}$ is compact Hausdorff space and ${\displaystyle Y}$ has the homotopy type of a CW-complex.^[2]