2003/05/31 by Sergey A. Melikhov
Mathematics · #math.GT #msc:57R35 #msc:57R42 #msc:57Q35 #msc:57Q55
published as Proc. Steklov Inst. Math. 247 (2004), 1-21; Russian transl., Tr. Mat. Inst. Steklova 247 (2004), 159-181 · 24 pages, 1 figure
arxiv created 2004/12/23 · arxiv updated 2009/11/30
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.