Homotopy equivalence of maps with compact support and maps which vanish at infinity

Question

Let $X$ be a nice space, maybe a manifold, and let $Y$ be a based space.

What sort of conditions must we impose on $Y$ (and $X$ if need be) to get a homotopy equivalence $$ \mathcal C_c(X,Y) \simeq \mathcal C_0(X,Y)$$ between the space of continuous maps with compact support and that of maps which vanish (in the sense of mapping to the basepoint) at infinity?

Perhaps the fact that latter is same as the space of based maps $\mathcal C(\hat X, Y) $ for $\hat X$ the one-point compactification of $X$ is relevant.

Answer 1

Let's assume $X$ is locally compact Hausdorff, so that $\hat X$ is compact Hausdorff and you can indeed use $C(\hat X,Y)$.

If the inclusion $i:C_c(X,Y)\to C(\hat X, Y)$ is a homotopy equivalence for all $Y$, in particular for $\hat X$, then the identity map $\hat X\to \hat X$ is based homotopic to a map that takes a neighborhood of $\infty$ to $\infty$.

(This fails, for example, if $X$ is $\mathbb Z$, or $\mathbb Z\times \mathbb R$, or any ``surface of infinite genus'', because in these cases $X$ has arbitrarily small neighborhoods of infinity $N$ such that the homology of $(X,N)$ is finitely generated, while the homology of $\hat X$ is not. But if $X$ is the interior of a compact manifold with boundary, then using a collar you get what you want.)

Conversely, if there is a homotopy of the identity as above then by composing with it you get at least half of what you want: a right homotopy inverse of that inclusion $i$.