Rigidity of periodic diffeomorphisms of homotopy K3 surfaces

Abstract

In this paper, we show that homotopy K3 surfaces do not admit a periodic diffeomorphism of odd prime order 3 acting trivially on cohomology. This gives a negative answer for period 3 to Problem 4.124 in Kirby's problem list. In addition, we give an obstruction in terms of the rationality and the sign of the spin numbers to the non-existence of a periodic diffeomorphism of odd prime order acting trivially on cohomology of homotopy K3 surfaces. The main strategy is to calculate the Seiberg-Witten invariant for the trivial spin(c) structure in the presence of such a Z(p)-symmetry in two ways: (1) the new interpretation of the Seiberg-Witten invariants of Furuta and Fang, and (2) the theorem of Morgan and Szabo on the Seiberg-Witten invariant of homotopy K3 surfaces for the trivial Spin(c) structure. As a consequence, we derive a contradiction for any periodic diffeomorphism of prime order 3 acting trivially on cohomology of homotopy K3 surfaces.