View source for Zorich theorem

In 1967, V.A. Zorich proved the following result for quasi-regular mappings in space (see also [[Quasi-regular mapping|Quasi-regular mapping]]): A locally homeomorphic quasi-regular mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301402.png" />, is, in fact, a [[Homeomorphism|homeomorphism]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301403.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301404.png" />. In 1971, O. Martio, S. Rickman and J. Väisälä proved a stronger quantitative result: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301406.png" /> there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301407.png" />, the radius of injectivity, such that every locally injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301408.png" />-quasi-regular mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z1301409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z13014010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z13014011.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z13014012.png" />, is injective in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130140/z13014013.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Rickman,   "Quasiregular mappings" , ''Ergeb. Math. Grenzgeb.'' , '''26''' , Springer  (1993)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Martio,   U. Sebro,   "Universal radius of injectivity for locally quasiconformal mappings"  ''Israel J. Math.'' , '''29'''  (1978)  pp. 17–23</TD></TR></table>