Zorich's theorem

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 edit

Every locally homeomorphic quasiregular mapping $f:R^{n}\rightarrow R^{n}$ 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.^[4]

^ Zorič, V. A. (1967). "Homeomorphism of quasiconformal space maps". Proceedings of the USSR Academy of Sciences. 176: 31–34. MR 0223568. As cited by Zorich (1992)
^ Lavrentieff, M. (1938). "Sur un critère différentiel des transformations homéomorphes des domaines à trois dimensions". Proceedings of the USSR Academy of Sciences. 20: 241–242. As cited by Zorich (1992)
^ Zorich, Vladimir A. (1992). "The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems". In Vuorinen, Matti (ed.). Quasiconformal Space Mappings: A collection of surveys 1960-1990. Germany: Springer-Verlag. pp. 132–148. doi:10.1007/BFB0094243. ISBN 3-540-55418-1. LCCN 92012192. OCLC 25675026. S2CID 116148715. Retrieved February 10, 2024.
^ Zorich (1992), p. 135.

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