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 h : I → Map ⁡ ( X , Y ) {\displaystyle h:I\to \operatorname {Map} (X,Y)} in the mapping space is exactly a homotopy between the starting point and the end point.

From the category theory point of view, a mapping space provides the internal Hom (i.e., hom that is also an object) in the category of spaces.

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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

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.)

Here is another common one. We have:

Map ⁡ ( X , Y ) ↪ X × Y {\displaystyle \operatorname {Map} (X,Y)\hookrightarrow X\times Y}

given by f ↦ Γ f = {\displaystyle f\mapsto \Gamma _{f}=} the graph of f {\displaystyle f} . Then we can give Map ⁡ ( X , Y ) {\displaystyle \operatorname {Map} (X,Y)} the Whitney topology (also called the fine topology or the strong topology) where a basic open set consists of those g {\displaystyle g} such that Γ g ⊂ U {\displaystyle \Gamma _{g}\subset U} for some open subset U ⊂ X × Y {\displaystyle U\subset X\times Y} .^[1][2] The compact-open topology does not handle a behavior at infinity well and so sometimes the Whitney topology is used instead.

If X {\displaystyle X} is a paracompact^[3] and Y {\displaystyle Y} is a metric space, then the Whitney topology has a basic open set of the form

B ( f , ϵ ) := { g ∣ d ( g ( x ) , f ( x ) ) < ϵ ( x ) } {\displaystyle B(f,\epsilon ):=\{g\mid d(g(x),f(x))<\epsilon (x)\}}

for some f ∈ Map ⁡ ( X , Y ) {\displaystyle f\in \operatorname {Map} (X,Y)} and some continuous function ϵ : X → R > 0 {\displaystyle \epsilon :X\to \mathbb {R} _{>0}} . If, moreover, Y {\displaystyle Y} is complete, then we have the following important fact:

Let Q ⊂ Map ⁡ ( X , Y ) {\displaystyle Q\subset \operatorname {Map} (X,Y)} be a subset such that every uniform limit of a sequence in Q {\displaystyle Q} , if any, is in Q {\displaystyle Q} . Then Q {\displaystyle Q} is a Baire space.^[4]

This is proved by the same way Baire's category theorem is proved except we use the above family-version of a ball.

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For manifolds M , N {\displaystyle M,N} , there is the subset 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} .^[5]

See also: Grauert's approximation theorem

A basic result here is a theorem of Milnor which says that the mapping space Map ⁡ ( X , Y ) {\displaystyle \operatorname {Map} (X,Y)} has the homotopy type of a CW-complex if X {\displaystyle X} is a compact Hausdorff space and Y {\displaystyle Y} has the homotopy type of a CW-complex.^[6]

• Hirsch, Morris (1997). Differential Topology. Springer. ISBN 0-387-90148-5.
• Milnor, John (1959). "On spaces having the homotopy type of CW-complex". Transactions of the American Mathematical Society. 90 (2): 272–280. doi:10.2307/1993204. JSTOR 1993204.
• Wall, C. T. C. (4 July 2016). Differential Topology. Cambridge University Press. ISBN 9781107153523.
• M. Golubitsky and V. Guillemin, Stable mappings and their singularities, x+209 pp, Springer Graduate Texts 14, Springer-Verlag, 1973.

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