Orientation-preserving homeomorphisms and the geometric definition of quasiconformality

• Christopher J. Bishop, Quasiconformal Mappings, Ch. 2 §2, printed pp. 51–53. Bishop defines a quadrilateral as a Jordan domain with two disjoint closed boundary arcs marked, assigns its modulus by a conformal rectangle, and defines geometric quasiconformality by the two-sided modulus bound for every quadrilateral.
• Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I, Ch. 2 §12.5, printed p. 188. QC2 states that moduli of quadrilaterals and annuli are $K$-quasi-invariant for an orientation-preserving homeomorphism.
• Lars Ahlfors and Arne Beurling, Conformal Invariants and Function-Theoretic Null-Sets, §§4–5, printed pp. 114–120, for the extremal-length convention and its conformal invariance.

Let $\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }}\subseteq \mathbb{C}$ be complex domains and $f:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ a homeomorphism (A complex domain is a nonempty connected open subset of $\mathbb{C}$, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

Orientation. For $a\in \mathrm{\Omega }$, $f$ induces an isomorphism $f_{\ast }:H_2(\mathrm{\Omega },\mathrm{\Omega }\setminus \{a\};\mathbb{Z})⟶H_2({\mathrm{\Omega }}^{\mathrm{\prime }},{\mathrm{\Omega }}^{\mathrm{\prime }}\setminus \{f(a)\};\mathbb{Z})\mathrm{.}$ Excision and the local-homology calculation identify both groups with $\mathbb{Z}$ (Excision for singular homology, Local homology detects manifold dimension, interior, and boundary). The generators are those determined by the standard orientation of $\mathbb{C}\cong {\mathbb{R}}^2$, with positively oriented basis $(1,i)$ (Orientation of a finite-dimensional real vector space, R-orientation of a topological manifold); the restriction isomorphisms from coordinate balls to points are supplied by Coordinate-ball classes identify local homology stalks. Write ${\epsilon }_f(a)\in \{+1,-1\}$ for the multiplier of $f_{\ast }$. Naturality of relative homology (Functoriality of relative homology) makes these maps compatible with coordinate-ball restrictions. In the local-system charts of R-orientation of a topological manifold, their multiplier is locally constant. Since $\mathrm{\Omega }$ is connected, ${\epsilon }_f$ is constant. The map $f$ is orientation-preserving when this sign is $+1$, and orientation-reversing when it is $-1$.

A quadrilateral. A quadrilateral in $\mathrm{\Omega }$ is a set $Q=\phi (\bar{\mathrm{\Pi }})$, where $\mathrm{\Pi }=(0,w)\times (0,h)$ with $w,h>0$ and $\phi :\bar{\mathrm{\Pi }}\to \mathbb{C}$ is continuous and injective. Its two marked sides are either $\phi (\{0\}\times [0,h])$ and $\phi (\{w\}\times [0,h])$, or the other pair of opposite sides. For one such choice, let $\mathrm{\Gamma }(Q)$ be the family of continuous paths in $Q$ whose endpoints lie on different marked sides and whose interior lies in $\mathrm{int}Q$. This is a curve family in $\mathrm{\Omega }$; its modulus $\mu (\mathrm{\Gamma }(Q))$ is defined in Extremal length and the curve-family modulus of a path family. The image $f(Q)$ is again a quadrilateral with marked sides carried by $f$, and $f\mathrm{\Gamma }(Q)=\mathrm{\Gamma }(f(Q))$.

Geometric definition. Let $K\ge 1$. The homeomorphism $f$ is $K$-geometrically quasiconformal when it is orientation-preserving and, for every quadrilateral $Q$ with $\bar{Q}\subseteq \mathrm{\Omega }$ and for each choice of its marked sides, $\frac{1}{K}\mu (\mathrm{\Gamma }(Q))\le \mu (\mathrm{\Gamma }(f(Q)))\le K\mu (\mathrm{\Gamma }(Q)),$ using the conventions of Extremal length and the curve-family modulus of a path family for zero and infinite values. It is geometrically quasiconformal if it is $K$-geometrically quasiconformal for some finite $K$; the least such $K$ is $K_f\ge 1$, its maximal dilatation.

The modulus condition uses the extremal-length construction and its Countable-Choice hypothesis; the orientation sign itself uses no choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). The inverse of a $K$-geometrically quasiconformal map is also $K$-geometrically quasiconformal: its local-homology map is the inverse isomorphism, and the two modulus inequalities rearrange to the same bounds for $f^{-1}$. Thus $K_{f^{-1}}=K_f$. The local orientation clause and the modulus inequality are distinct parts of this definition.