球面翻转与映射的实现
Sphere eversions and realization of mappings
摘要：
P. M.阿赫梅季耶夫使用稳定的霍普夫不变量的受控版本来证明，在可稳定平行化的紧致n-流形n \ne 1，2，3，7之间的任何（连续）映射N -> M在R^{2n}中是可实现的，即f与嵌入M \subset R^{2n}的复合是C^0-可嵌入逼近的。长期以来，人们一直认为，任何2次映射S^3 -> S^3，通过在无穷远处对时间对称（例如夏皮罗）的球面外翻S^2 x I -> R^3进行封顶而得到，在R^6中是不可实现的。我们证明了存在一个Poincaré同调3-球面的自映射，它在R^6中是不可实现的，但对于每个n>2，S^n的每个自映射在R^{2n}中都是可实现的。后者连同n=2的十行证明，主要是由于M。Yamamoto等人的结论，暗示了当n>1时，n-球面的每个逆极限嵌入R^{2n}中，从而使R稳定。1990年的问题。若M是闭的可定向3-流形，我们证明了存在一个映射S^3 -> M，在R^6中不可实现，当且仅当π_1（M）是有限的且具有偶数阶.作为副产品，具有非平凡稳定霍普夫不变量的稳定茎Q^3的元素由特别简单的浸入S^3 -> R^4表示，即在Q^3=S^3/{ \pm 1，\pm i，\pm j，\pm k}$上的泛8-覆盖和显式嵌入Q^3 \subset R^4的合成。
英文摘要：
P. M. Akhmetiev used a controlled version of the stable Hopf invariant to show that any (continuous) map N -> M between stably parallelizable compact n-manifolds, n\ne 1,2,3,7, is realizable in R^{2n}, i.e. the composition of f with an embedding M\subset R^{2n} is C^0-approximable by embeddings. It has been long believed that any degree 2 map S^3 -> S^3, obtained by capping off at infinity a time-symmetric (e.g. Shapiro's) sphere eversion S^2 x I -> R^3, was non-realizable in R^6. We show that there exists a self-map of the Poincaré homology 3-sphere, non-realizable in R^6, but every self-map of S^n is realizable in R^{2n} for each n>2. The latter together with a ten-line proof for n=2, due essentially to M. Yamamoto, implies that every inverse limit of n-spheres embeds in R^{2n} for n>1, which settles R. Daverman's 1990 problem. If M is a closed orientable 3-manifold, we show that there exists a map S^3 -> M, non-realizable in R^6, if and only if π_1(M) is finite and has even order. As a byproduct, an element of the stable stem Π_3 with non-trivial stable Hopf invariant is represented by a particularly simple immersion S^3 -> R^4, namely the composition of the universal 8-covering over Q^3=S^3/{\pm1,\pm i,\pm j,\pm k}$ and an explicit embedding Q^3\subset R^4.
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