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The Quarterly Journal of Mathematics Advance Access originally published online on August 29, 2007
The Quarterly Journal of Mathematics 2008 59(2):237-256; doi:10.1093/qmath/ham033
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

RIGIDITY OF PERIODIC DIFFEOMORPHISMS OF HOMOTOPY K3 SURFACES

Jin Hong Kim{dagger}

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Kusong-dong, Yusong-gu, Daejon 305–701, Korea

{dagger} E-mail: jinkim{at}math.kaist.ac.kr

Received 10 October 2006; revised 25 May 2007
  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 spinc structure in the presence of such a Zp-symmetry in two ways: (1) the new interpretation of the Seiberg–Witten invariants of Furuta and Fang, and (2) the theorem of Morgan and Szabó on the Seiberg–Witten invariant of homotopy K3 surfaces for the trivial Spinc structure. As a consequence, we derive a contradiction for any periodic diffeomorphism of prime order 3 acting trivially on cohomology of homotopy K3 surfaces.


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