arXiv:1710.02050v4 [math.GN] 31 Aug 2022 CHARACTERIZATIONS OF GENERALIZED JOHN DOMAINS VIA HOMOLOGICAL BOUNDED TURNING PAWEL GOLDSTEIN, ZOFIA GROCHULSKA, CHANG-YU GUO, PEKKA KOSKELA, AND DEBANJAN NANDI Abstract. In this paper, we extend the characterization of John disks obtained by Näkki and Väisälä [31] to generalized John domains in higher dimensions under mild assumptions. The main ingredient in this characterization is to use the higher dimensional analogues of the local linear connectivity (LLC) and homological bounded turning properties introduced by Väisälä in his study of metric duality theory [34]. Somewhat surprisingly, we constructed a uniform domain in R 3 , which is topologi- cally 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 three. 1. Introduction and main results One of the most important and ubiquitous notions in the theory of domains and its applications to modern analysis is that of a John domain. Recall that an open, connected and bounded set Ω ⊂ R n is a John domain if there is a point x 0 ∈ Ω and a constant C such that each x ∈ Ω can be connected with x 0 with a rectifiable John curve γ : [0, 1] → Ω, γ (0) = x, γ (1) = x 0 , satisfying for all t ∈ [0, 1] l(γ ([0,t])) ≤ Cd(γ (t),∂ Ω). (1.1) The term John domain was introduced by Martio and Sarvas [29] to honour F. John, who used this condition in his seminal work on elasticity [28]. The class of John domains includes all smooth domains, Lipschitz domains and certain fractal domains, such as the interior of the von Koch snowflake and other snowflake-type domains. In the planar case, Näkki and Väisälä [31] developed a rich theory of John disks, i.e., simply connected planar John domains and proved the following important character- ization of John disks (for definitions see Section 2). The exact formulation contains a few further equivalencies, which are however not in the scope of this paper. Proposition 1.1. [31, Theorem 4.5] Let Ω ⊂ ˙ R 2 be a simply connected domain such that Ω does not contain the point at infinity. Then the following conditions are quan- titatively equivalent: 1) Ω is a John disk, 2) Ω is LLC-2, 3) ˙ R 2 \ Ω is bounded turning. In [31, Theorem 4.5], the theorem is stated for conformal disks. By applying the Riemann mapping theorem, it thus holds for any simply connected domain Ω ⊂ ˙ R 2 such that Ω does not contain the point at infinity. 2000 Mathematics Subject Classification. 57N65,55M05. Key words and phrases. John domain, uniform domain, ball separation property, homological bounded turning, homotopical bounded turning. P. Goldstein was partially supported by FNP grant POMOST BIS/2012-6/3 and by NCN grant no 2012/05/E/ST1/03232. C.-Y. Guo is supported by the Young Scientist Program of the Ministry of Science and Technology of China (No. 2021YFA1002200), the National Natural Science Foundation of China (No. 12101362) and the Natural Science Foundation of Shandong Province (No. ZR2021QA003). P. Koskela was partially supported by the Academy of Finland grant 323960. 1