Constructions of exotic actions on product manifolds with an asymmetric factor (Q2105768)

Let \( M^m \) be an \( m \)-dimensional asymmetric (admitting no smooth, effective nontrivial finite group action) compact smooth manifold (without boundary). The authors construct effective exotic (not equivalent to the diagonal action) actions on \( M\times S^n\ (n=1,2) \). The main results in the article are: \textbf{Theorem 1}: There are exotic \( \mathbb{Z}_p \) (\(p\) prime) and \( S^1 \) (circle group) actions on \( M\times S^2 \) for all \( m\geq3 \). \textbf{Theorem 2}: There are exotic \( \mathbb{Z}_2 \) actions on \( M\times S^1 \) for all \( m\geq4 \). The proofs of these theorems use the existence of smooth homology spheres of every dimension (greater or equal to 4) bounding a contractible smooth manifold (proven in [\textit{M. A. Kervaire}, Trans. Am. Math. Soc. 144, 67--72 (1969; Zbl 0187.20401)]) and the \(h\)-cobordism theorem of S. Smale. \textbf{Theorem 5}: (If \( M \) belongs to a certain class of almost asymmetric manifolds) A free orientation-preserving \( \mathbb{Z}_p \)-action on \( M\times S^1 \) is equivalent to the diagonal action if and only if its orbit space is diffeomorphic to \( M\times S^1 \). Also, there are some results concerning the detection of the exotic and product actions in special cases.