$H{\Bbb C}P^3$的光滑自由对合及$S^3$嵌入$S^6$的Smith猜想
Smooth free involution of $H{\Bbb C}P^3$ and Smith conjecture for imbeddings of $S^3$ in $S^6$
摘要：
本文建立了$H{\Bbb C}P^3$上自由对合的存在性与$S^6$上具有嵌入$S^3$的不动点集的对合的存在性之间的等价性，然后利用$H{\Bbb C}P^3$上的已知结果给出了$S^6$中嵌入$S^3$的Smith猜想的一族反例.此外，本文还证明了每个光滑同伦复射影3-空间不存在保向光滑自由对合，从而回答了一个公开问题[Pe]。此外，还研究了$H{\Bbb C}P^3$上光滑方向保持对合的存在性问题。
英文摘要：
This paper establishes an equivalence between existence of free involutions on $H{\Bbb C}P^3$ and existence of involutions on $S^6$ with fixed point set an imbedded $S^3$, then a family of counterexamples of the Smith conjecture for imbeddings of $S^3$ in $S^6$ are given by known result on $H{\Bbb C}P^3$. In addition, this paper also shows that every smooth homotopy complex projective 3-space admits no orientation preserving smooth free involution, which answers an open problem [Pe]. Moreover, the study of existence problem for smooth orientation preserving involutions on $H{\Bbb C}P^3$ is completed.
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