# Reference request: Higson compactification

It seems that the idea of the Higson compactification first arose in the context of non-compact manifolds in a 1992 preprint of Higson called "The relative $K$-homology of Baum and Douglas".

It seems that this paper was either published under a different name or was not published; it's not on MathSciNet. Does anyone know where I could find a copy of it?

Many thanks.

Added question: in particular, I'm wondering if Roe's more abstract definition of the Higson compactification in terms of bounded continuous functions of vanishing variation at infinity is equivalent to Higson's original definition in terms of bounded smooth functions with vanishing derivative.

More precisely, consider a non-compact Riemannian manifold $M$ as a metric space in the usual way. For any bounded continuous function $f$ on $M$, define the variation of $f$ at scale $r$ to be the function $V_r(f):M\rightarrow\mathbb{R}^{\geq 0}$ given by

$$V_r(f)(x)=\sup\{|f(x)-f(y)|:y\in B_r(x)\}.$$

Then I'd like to know whether the set

$$\{f\in C_b(M):\text{ for all }r>0, V_r(f)(x)\rightarrow 0\text{ as } x\rightarrow\infty\}$$ is the same as the set

$$\overline{\{f\in C_b^\infty(M):|df(x)|\rightarrow 0\text{ as }x\rightarrow\infty\}}$$

where the closure is taken in $C_b(M)$ with the sup norm.

I think it is clear that the second set without the closure lies inside the first set, but other than this I'm not sure.

## 1 Answer

(Higson 1995, §0) explains that an account “has in the meantime appeared elsewhere (Roe 1993)”.

• Thanks. Roe's account abstracts the construction from manifolds to metric spaces using functions of bounded variation. However, I'd like to see the original, simpler construction in terms of functions of vanishing gradient at infinity. Commented Sep 16, 2017 at 13:12