Characterizations of generalized John domains via homological bounded turning

We extend the characterization of John disks obtained by Näkki and Väisälä (1991) to generalized John domains in higher dimensions under mild assumptions. The main ingredient in this characterization is to use the higher-dimensional analogues of local linear connectivity (LLC) and homological bounded turning properties introduced by Väisälä in his 1997 study of metric duality theory.

Somewhat surprisingly, we construct a uniform domain in $\mathbb R^3$, which is topologically simple, such that the complementary domain fails to be homotopically $1$-bounded turning. In particular, this shows that a similar characterization of generalized John domains in terms of higher-dimensional homotopic bounded turning does not hold in dimension 3.