Some exotic actions of finite groups on smooth 4-manifolds (Q342980)

The problem of exotic smooth actions on smooth 4-manifolds has been widely studied since 1976, e.g. by \textit{S. E. Cappell} and \textit{J. L. Shaneson} [Ann. Math. (2) 104, 61--72 (1976; Zbl 0345.57003)], in a large number of papers. In a paper from 2009 the authors \textit{R. Fintushel} et al. [J. Topol. 2, No. 4, 769--778 (2009; Zbl 1221.57031)] constructed infinite families of exotic free or cyclic actions of finite cyclic groups on smooth closed 4-manifolds with nontrivial Seiberg-Witten invariant. In the present paper the author uses G-monopole invariants (for definition see [\textit{Y. Ruan}, in: Topics in symplectic \(4\)-manifolds. 1st International Press lectures presented in Irvine, CA, USA, March 28--30, 1996. Cambridge, MA: International Press. 101--116 (1998; Zbl 0939.57024)]) to give many non-free non cyclic exotic group actions on certain connected sums of 4-manifolds with vanishing Seiberg-Witten invariant.NEWLINENEWLINEThe following corollary of the main result of the paper provides concrete examples of the developed theory.NEWLINENEWLINECorollary 3.2. Let \(H\) be a finite group of order \(l\geq 2\) acting freely on \(S^3\). For any \(k\geq 2\), there exists an infinite family of topologically equivalent but smoothly distinct non-free actions of \(\mathbb{Z}_k\oplus H\) onNEWLINENEWLINENEWLINE\[NEWLINE (klm+l-1)(S^2\times S^2),NEWLINE\]NEWLINE NEWLINE\[NEWLINE (kl(n-1)+l-1)(S^2\times S^2)\#klnK3,NEWLINE\]NEWLINE NEWLINE\[NEWLINE (2n'-1)+l-1)\mathbb{C}P_2\#(kl(10 n'+m'-1)+l-1)\overline{\mathbb{C}P}_2NEWLINE\]NEWLINENEWLINENEWLINEfor infinitely many \(m\), and any \(m'\geq 1,n, n'\geq 2\).