Zorich's theorem explained

In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967.^[1] The result was conjectured by M. A. Lavrentev in 1938.^[2]

Theorem

Every locally homeomorphic quasiregular mapping

f : R ^{n} → R ^{n} for n \geq 3 , is a homeomorphism of R ^{n} .^[3]

The fact that there is no such result for

n = 2 is easily shown using the exponential function.

Notes and References

1. Zorič . V. A. . . 223568 . 31–34 . Homeomorphism of quasiconformal space maps . 176 . 1967. As cited by
2. Lavrentieff . M. . . 241–242 . Sur un critère différentiel des transformations homéomorphes des domaines à trois dimensions. . 20 . 1938. As cited by
3. Book: Zorich, Vladimir A. . Vladimir A. Zorich . Vuorinen . Matti . Matti Vuorinen . 1992 . The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems . 132–148 . Quasiconformal Space Mappings: A collection of surveys 1960-1990 . . Germany . 10.1007/BFB0094243 . 3-540-55418-1 . 92012192 . 25675026 . 116148715 . . February 10, 2024.