Zorich's theorem

In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967.{{cite journal

| last = Zorič | first = V. A.

| journal = Proceedings of the USSR Academy of Sciences

| mr = 223568

| pages = 31–34

| title = Homeomorphism of quasiconformal space maps

| volume = 176

| year = 1967}} As cited by {{harvtxt|Zorich|1992}} The result was conjectured by M. A. Lavrentev in 1938.{{cite journal

| last = Lavrentieff | first = M.

| journal = Proceedings of the USSR Academy of Sciences

| pages = 241–242

| title = Sur un critère différentiel des transformations homéomorphes des domaines à trois dimensions.

| volume = 20

| year = 1938}} As cited by {{harvtxt|Zorich|1992}}

Theorem

Every locally homeomorphic quasiregular mapping $f : R^{n} \rightarrow R^{n}$ for $n \geq 3$ , is a homeomorphism of $R^{n}$ .{{cite book |last=Zorich |first=Vladimir A. |author-link=Vladimir A. Zorich |editor-last=Vuorinen |editor-first=Matti |editor-link=Matti Vuorinen |year=1992 |chapter=The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems |pages=132–148 |title=Quasiconformal Space Mappings: A collection of surveys 1960-1990 |series=Lecture Notes in Mathematics |volume=1508 |publisher=Springer-Verlag |publication-place=Germany |doi=10.1007/BFB0094243 |isbn=3-540-55418-1 |lccn=92012192 |oclc=25675026 |s2cid=116148715 |chapter-url={{GBurl|fo58CwAAQBAJ|p=132}} |access-date=February 10, 2024}}

The fact that there is no such result for $n = 2$ is easily shown using the exponential function.{{sfnp|Zorich|1992|p=135}}

References

Category:General topology

Category:Theorems in topology

Category:Conjectures that have been proved