Zorich's theorem

In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967. The result was conjectured by M. A. Lavrentev in 1938.

Theorem

Every locally homeomorphic quasiregular mapping ƒ : Rⁿ → Rⁿ for n ≥ 3, is a homeomorphism of Rⁿ.

The fact that there is no such result for n = 2 is easily shown using the exponential function.

References

• V.A. Zorich, "The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems", M. Vuorinen (ed.), Quasiconformal Space Mappings, Lecture Notes in Mathematics, 1508 (1992) pp. 132–148