Sphere eversions and realization of mappings

Abstract

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\ne 1,2,3,7, is realizable in R^{2n}, i.e. the composition of f with an embedding M\subset R^{2n} is C^0-approximable by embeddings. It has been long believed that any degree 2 map S^3 -> S^3, obtained by capping off at infinity a time-symmetric (e.g. Shapiro's) sphere eversion S^2 x I -> R^3, was non-realizable in R^6. We show that there exists a self-map of the Poincar\'e homology 3-sphere, non-realizable in R^6, but every self-map of S^n is realizable in 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 R^{2n} 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 S^3 -> M, non-realizable in R^6, 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 non-trivial stable Hopf invariant is represented by a particularly simple immersion S^3 -> R^4, namely the composition of the universal 8-covering over Q^3=S^3/{\pm1,\pm i,\pm j,\pm k}$ and an explicit embedding Q^3\subset R^4.