RIGIDITY OF PERIODIC DIFFEOMORPHISMS OF HOMOTOPY K3 SURFACES

In this paper, we show that homotopy K3 surfaces do not admita periodic diffeomorphism of odd prime order 3 acting triviallyon cohomology. This gives a negative answer for period 3 toProblem 4.124 in Kirby's problem list. In addition, we givean obstruction in terms of the rationality and the sign of thespin numbers to the non-existence of a periodic diffeomorphismof odd prime order acting trivially on cohomology of homotopyK3 surfaces. The main strategy is to calculate the Seiberg–Witteninvariant for the trivial spin^c structure in the presence ofsuch a Z_p-symmetry in two ways: (1) the new interpretation ofthe Seiberg–Witten invariants of Furuta and Fang, and(2) the theorem of Morgan and Szabó on the Seiberg–Witteninvariant of homotopy K3 surfaces for the trivial Spin^c structure.As a consequence, we derive a contradiction for any periodicdiffeomorphism of prime order 3 acting trivially on cohomologyof homotopy K3 surfaces.