We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings, including Matti Vuorinen injectivity problem. The construction of such mappings comes from our construction of nontrivial compact 4-dimensional cobordisms with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such bounded locally homeomorphic quasiregular mappings are defined in the unit 3-ball <$E3~symbol <172>~R sup 3> as mappings equivariant with the standard conformal action of uniform hyperbolic lattices <$EGAMMA~symbol <172>~Isom sup 3> in the unit 3-ball and with its discrete representation = <$Erho ( GAMMA )~symbol <172>~Isom sup 4>. Here is the fundamental group of our nontrivial hyperbolic 4-cobordism = <$E( nothing sup 4 union OMEGA (~))> and the kernel of the homomorphism <$Erho~:~GAMMA~symbol О~> is a free group 3 on three generators.

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