# Density of compactly-supported homeomorphisms

**Disclaimer:**I posted the following question on MSE, but since there were no answers. I'm migrating it here.

Let $$Homeo_0(\mathbb{R}^n)$$ ($$Homeo_c(\mathbb{R}^n)$$) be the space of all (compactly-supported) orientation-preserving homeomorphisms on $$\mathbb{R}^n$$ to itself equipped with the topology of compact convergence. Is $$Homeo_c(\mathbb{R}^n)$$ dense in $$Homeo_0(\mathbb{R}^n)$$?

I tried to use Kirby's local contractibility result, but I couldn't figure if this is correct.

Note: To avoid confusion, as in the comments, recall that a homeomorphism is said to be compactly supported if it is equal to the identity map outside some compact subset of its domain.
See Definition 1.5 of these nice notes.

• I don't think you can approximate an orientation-reversing homeomorphism by compactly supported homeomorphisms Nov 9, 2020 at 17:54
• @DenisNardin Thank you for pointing that out, I polised up the post. Nov 9, 2020 at 17:55
• Possibly relevant may be papers by Morton Brown (and a coauthor), from about half a century ago. Nov 9, 2020 at 21:07

Lemma. Given an orientation-preserving homeomorphism $$h$$ of $$\mathbb{R}^n$$ there is a compactly-supported homeomorphsim $$h_1$$ which agrees with $$h$$ on the unit ball.
By conjugating by dilations one can find $$h_r$$'s that agree with $$h$$ on the ball of radius $$r$$, and this gives a sequence of compactly-supported homeomorphisms converging pointwise to $$h$$.
The lemma can be proved using the ideas in the proof of the Annulus Conjecture, but it is simpler here to deduce it from the Annulus Conjecture and the closely-related fact that $$Homeo^+(S^{n-1})$$ is path-connected.
Proof: Given $$h$$ consider the sphere $$h(S^{n-1}) \subset \mathbb{R}^n$$. This is compact so is contained inside a sphere $$r S^{n-1}$$ for some $$r \gg 0$$. Both these spheres are locally flat, so by the Annulus Theorem the region between them is homeomorphic to $$[0,1] \times S^{n-1}$$. Combined with $$h\vert_{D^n}$$ this gives a homeomorphism $$h': r D^n = D^n \cup [1,r] \times S^{n-1} \to r D^n = h(D^n) \cup \{annulus\}$$ which agrees with $$h$$ on the unit ball. On the boundary it induces an orientation-preserving homeomorphism of $$rS^{n-1}$$. Thisis topologically isotopic to the identity, and combined with $$h'$$ this gives a homeomorphism $$h'' : (r+1)D^n = rD^n \cup [0,1] \times S^{n-1} \to (r+1)D^n = rD^n \cup [0,1] \times S^{n-1}$$ which agrees with $$h$$ on the unit ball and which is the identity on the boundary: extending by the identity gives the required $$h_1$$.