Smooth free involution of $H\mathbb{C}{P}^3$ and Smith conjecture for imbeddings of $S^3$ in $S^6$

This paper establishes an equivalence between existence of free involutions on $H\mathbb{C}{P}^3$ 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 $H\mathbb{C}{P}^3$. 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 $H\mathbb{C}{P}^3$ is completed.