nLab compactly supported mapping space

Idea

A compactly supported mapping space is a subspace of a mapping space on those maps that have compact support.

Definition

Let $X$ be a topological space. For $C \subset X$ a compact subset, write $cof(X\setminus C \to X)$ for the cofiber of the inclusion of its complement, hence for the result of collapsing the complement to a base point.

For a pointed topological space $A$, the compactly supported mapping space from $X$ to $A$ is the following colimit of pointed mapping spaces:

(1)$$Map^c(X,A) \,\coloneqq\, \underset{\underset{C}{\longrightarrow}}{\lim} \, Map^\ast\big( cof(X \setminus C \to X) , A \big) \mathrlap{\,.}$$

Properties

Relation to one-point compactification

Now let $X$ be a locally compact topological space with one-point compactification $X_{cpt}$. Then we have a canonical homeomorphism

$$cof\big( X_{cpt} \setminus C \to X_{cpt} \big) \simeq cof\big( X \setminus C \to X \big)$$

and hence a system of comparison maps, natural in $C$, of the form:

(2)$$X_{cpt} \longrightarrow cof(X \setminus C \to X) \mathrlap{\,.}$$

which induces a comparison map from the compactly supported mapping space (1) to the pointed mapping space out of the one-point compactification:

(3)$$Map^c(X,A) \longrightarrow Map^\ast\big( X_{cpt} , A \big)$$

Proposition

A sufficient condition for the comparison map (3) to be a weak homotopy equivalence is that

• $X$ is a manifold which is finitary in that it is the complement of some faces (codimension=1 strata) of a compact manifold with corners.

(Mazel-Gee 2016 §2.5, Ayala & Francis 2025 p 6)

Related concepts

• compactly supported cohomology

References

• Aaron Mazel-Gee; §2.5 in: Locally Constant Factorization Algebras, lecture notes (2016) [pdf, pdf]

• David Ayala, John Francis; p. 6 of: A parametrized Pontryagin-Thom theorem [arXiv:2512.10274 math.AT]

In the generality of G-spaces:

• Jeremy Hahn, Asaf Horev, Inbar Klang, Dylan Wilson, Foling Zou; Prop. 4.4.2 in: Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra [arXiv:2006.13348]