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 h : I → Map ( X , Y ) {\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. Typically, there is then the adjoint relation
Map ( X × Y , Z ) ≃ Map ( X , Map ( Y , Z ) ) {\displaystyle \operatorname {Map} (X\times Y,Z)\simeq \operatorname {Map} (X,\operatorname {Map} (Y,Z))}and thus Map {\displaystyle \operatorname {Map} } is an analog of the Hom functor. (For pathological spaces, this relation may fail.)
Smooth mappings
For manifolds M , N {\displaystyle M,N} , there is the subspace C r ( M , N ) ⊂ Map ( M , N ) {\displaystyle {\mathcal {C}}^{r}(M,N)\subset \operatorname {Map} (M,N)} that consists of all the C r {\displaystyle {\mathcal {C}}^{r}} -smooth maps from M {\displaystyle M} to N {\displaystyle N} . It can be equipped with the weak or strong topology.
A basic approximation theorem says that C W s ( M , N ) {\displaystyle {\mathcal {C}}_{W}^{s}(M,N)} is dense in C S r ( M , N ) {\displaystyle {\mathcal {C}}_{S}^{r}(M,N)} for 1 ≤ s ≤ ∞ , 0 ≤ r < s {\displaystyle 1\leq s\leq \infty ,0\leq r<s} .1
- Hirsch, Morris (1997). Differential Topology. Springer. ISBN 0-387-90148-5.
- Wall, C. T. C. (4 July 2016). Differential Topology. Cambridge University Press. ISBN 9781107153523.
References
Hirsch 1997, Ch. 2., § 2., Theorem 2.6. - Hirsch, Morris (1997). Differential Topology. Springer. ISBN 0-387-90148-5. ↩