Sphere eversions and realization of mappings

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P. M. Akhmetiev used a controlled version of the stable Hopf invariant to show that any (continuous) map N -> M between stably parallelizable compact n-manifolds, n≠ 1,2,3,7, is realizable in R²ⁿ, i.e. the composition of f with an embedding M⊂ R²ⁿ is C⁰-approximable by embeddings. It has been long believed that any degree 2 map S³ -> S³, obtained by capping off at infinity a time-symmetric (e.g. Shapiro's) sphere eversion S² x I -> R³, was non-realizable in R⁶. We show that there exists a self-map of the Poincaré homology 3-sphere, non-realizable in R⁶, but every self-map of Sⁿ is realizable in R²ⁿ for each n>2. The latter together with a ten-line proof for n=2, due essentially to M. Yamamoto, implies that every inverse limit of n-spheres embeds in R²ⁿ for n>1, which settles R. Daverman's 1990 problem. If M is a closed orientable 3-manifold, we show that there exists a map S³ -> M, non-realizable in R⁶, if and only if π₁(M) is finite and has even order. As a byproduct, an element of the stable stem Π₃ with non-trivial stable Hopf invariant is represented by a particularly simple immersion S³ -> R⁴, namely the composition of the universal 8-covering over Q³=S³/±1,± i,± j,± k and an explicit embedding Q³⊂ R⁴.