Exotic group actions on simply connected smooth 4-manifolds

Abstract

Abstract. We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.

Nonnegatively curved fixed point homogeneous manifolds in low dimensions

“…This classification holds in the smooth category, as a result of carrying out the constructions therein in this setting [9]. In particular, as an immediate consequence of Fintushel's results, work of Pao [30], and the validity of the Poincaré conjecture due to Perelman [23,26,32,33] one has the following theorem (cf.…”

Double point surgery and configurations of surfaces

“…As shown by Fintushel, Stern and Sunukjian [8], twisted rim surgery, applied to the branch set of a cyclic branched cover, can give rise to smoothly exotic group actions on simply connected 4-manifolds. The action (on the total space of the branched cover) is by definition semi-free.…”

“…In Section 7, we apply the double point surgery to the construction of group actions, in the fashion of a recent paper of Fintushel, Stern and Sunukjian [8].…”

“…As (m, n) = 1, the group Z m ⊕ Z n is isomorphic to Z mn ; writing the group as Z m ⊕ Z n is convenient for keeping track of the isotropy subgroups of the action. In particular, as the singular set consists of a configuration, rather than an embedded surface, these examples are different from those presented in [8]. In the terminology of group actions [2], the actions constructed in [8] are semi-free, whereas ours are not.…”

Conjugation spaces and 4-manifolds

“…13. Further examples of smooth conjugation 4-manifolds might also come from the rim surgery construction of Fintushel and Stern [14], [15], although it is not clear at present how to detect exotic smooth structures on the 2-fold branched coverings of Z 2 -homology 4-spheres.…”