Zorich theorem

In 1967, V.A. Zorich proved the following result for quasi-regular mappings in space (see also Quasi-regular mapping): A locally homeomorphic quasi-regular mapping , , is, in fact, a homeomorphism of .

This result had been conjectured by M.A. Lavrent'ev in 1938. Note that the exponential function shows that there is no such result for . In 1971, O. Martio, S. Rickman and J. Väisälä proved a stronger quantitative result: For and there exists a number , the radius of injectivity, such that every locally injective -quasi-regular mapping , where and , for , is injective in .

References

[a1]	S. Rickman, "Quasiregular mappings" , Ergeb. Math. Grenzgeb. , 26 , Springer (1993)
[a2]	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
[a3]	O. Martio, U. Sebro, "Universal radius of injectivity for locally quasiconformal mappings" Israel J. Math. , 29 (1978) pp. 17–23