F. Quinn proved that any open topological 4-manifold admits a smooth structure in Ends of maps III: dimensions 4 and 5.
He first proves the generalized annulus conjecture:
Suppose $h:D^j\times \mathbb{R}^{4-j}\to W$ is a homeomorphism of smooth manifolds, smooth on the boundary. If $j=0$ or $1$, then $h$ is isotopic rel boundary and a neighborhood of the end to a map which is smooth on a neighborhood of $D^j\times \{0\}$. When $j=2$, there is an isotopy to a map smooth on either a neighborhood of $D^2\times \{0\}$ after some finger moves to introduce self-intersections, or the image of a neighborhood of $D^2\times \{0\}$ under a topological isotopy (rel $\partial$ and $\infty$).
He then states that "immersion theory interprets the handle result as a calculation of the homotopy groups of classifying spaces" and concludes (Corollary 2.2.3)
$\mathrm{TOP}(4)/O(4)\to \mathrm{TOP}/O$ is 3-connected. Consequently, any 4-manifold has a smooths structure in the complement of a point.
Questions:
- Why does the generalized annulus conjecture imply 3-connectedness of $\mathrm{TOP}(4)/O(4)\to \mathrm{TOP}/O$?
- Why does 3-connectedness of $\mathrm{TOP}(4)/O(4)\to \mathrm{TOP}/O$ imply that open 4-manifolds are smoothable? I see that it implies that the tangent microbundle of an open 4-manifold admits an $O(4)$-structure, but can the $O(4)$-structure be integrated to a smooth structure on the base (see this post)? What is the argument Quinn is using?
