Zorich theorem

In 1967, V.A. Zorich proved the following result for quasi-regular mappings in space: A locally homeomorphic quasi-regular mapping $f : \mathbf{R}^n \rightarrow \mathbf{R}^n$, $n \ge 3$, is, in fact, a homeomorphism of $\mathbf{R}^n$.

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 $n=2$. In 1971, O. Martio, S. Rickman and J. Väisälä proved a stronger quantitative result: For $n \ge 3$ and $K > 1$ there exists a number $\psi(n,K) \in (0,1)$ , the radius of injectivity, such that every locally injective $K$-quasi-regular mapping $f : B^n \rightarrow \mathbf{R}^n$, where $B^b = B^n(1)$ and $B^n(r) = \{ x \in \mathbf{R}^n : |x| \le r \}$, for $r > 0$, is injective in $B^n(\psi(n,K))$.

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