Sphere Eversions and Realization of Mappings

P. M. Akhmetiev used a controlled version of the stable Hopf invariant to show that any (continuous) map $N\to M$ between stably parallelizable compact $n$-manifolds, $n\ne 1,2,3,7$, is realizable in $\mathbb R^{2n}$, i.e., the composition of $f$ with an embedding $M\subset \mathbb R^{2n}$ is $C^0$-approximable by embeddings. It has been long believed that any degree-$2$ map $S^3\to S^3$ obtained by capping off at infinity a time-symmetric (e.g., Shapiro's) sphere eversion $S^2\times I\to \mathbb R^3$ is nonrealizable in $\mathbb R^6$. We show that there exists a self-map of the Poincaré homology 3-sphere that is nonrealizable in $\mathbb R^6$, but every self-map of $S^n$ is realizable in $\mathbb R^{2n}$ 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 $\mathbb R^{2n}$ for $n>1$, which settles R. Daverman's 1990 problem. If $M$ is a closed orientable 3-manifold, we show that a map $S^3\to M$ that is nonrealizable in $\mathbb R^6$ exists if and only if $\pi _1(M)$ is finite and has even order. As a byproduct, an element of the stable stem $\Pi _3$ with nontrivial stable Hopf invariant is represented by a particularly simple immersion $S^3\looparrowright \mathbb R^4$, namely, by the composition of the universal $8$-covering over $Q^3=S^3/\{\pm 1,\pm i,\pm j,\pm k\}$ and an explicit embedding $Q^3\hookrightarrow \mathbb R^4$.