A remark on the exotic free actions in dimension 4

摘要

Donaldson's polynomial invariants are powerful tools for studying smooth 4-manifolds. For example the diffeomorphism types of elliptic surfaces with positive geometric genus were completely classified by them. The examples of smooth closed 1-connected noncomplex 4-manifolds with infinitely many smooth structures were first given by [9] and then were constructed by various methods. In this paper we will give some examples . of infinitely many exotic 4-manifolds whose universal coverings are mutually diffeomorphic. In fact we will show that the fundamental group of any spherical 3-manifold other than the 3-sphere acts freely on certain 4-manifolds in infinitely many different ways so that their orbit spaces are exotic (Main Theorem). Throughout this paper we denote the K3 surface by K, and for any finite group G we denote by |G| the order of G. For any closed oriented 4-manifold X, we denote by b_2~+=b_2~+(X) the rank of the maximal positive subspace H~+=H~+(X) of the intersection form q_x of X, and nX denotes the connected sum of n copies of X.