Osaka Journal of Mathematics

Some exotic actions of finite groups on smooth 4-manifolds

Chanyoung Sung

Full-text: Open access


Using $G$-monopole invariants, we produce infinitely many exotic non-free actions of $\mathbb{Z}_{k}\oplus H$ on some connected sums of finite number of $S^{2}\times S^{2}$, $\mathbb{C}P_{2}$, $\overline{\mathbb{C}P}_{2}$, and $K3$ surfaces, where $k\geq 2$, and $H$ is any nontrivial finite group acting freely on $S^{3}$.

Article information

Osaka J. Math., Volume 53, Number 4 (2016), 1055-1061.

First available in Project Euclid: 4 October 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX] 57M60: Group actions in low dimensions 57R50: Diffeomorphisms


Sung, Chanyoung. Some exotic actions of finite groups on smooth 4-manifolds. Osaka J. Math. 53 (2016), no. 4, 1055--1061.

Export citation


  • S. Akbulut: Variations on Fintushel–Stern knot surgery on 4-manifolds, Turkish J. Math. 26 (2002), 81–92.
  • S. Akbulut: Cappell–Shaneson homotopy spheres are standard, Ann. of Math. (2) 171 (2010), 2171–2175.
  • D. Auckly: Families of four-dimensional manifolds that become mutually diffeomorphic after one stabilization, Topology Appl. 127 (2003), 277–298.
  • S.E. Cappell and J.L. Shaneson: Some new four-manifolds, Ann. of Math. (2) 104 (1976), 61–72.
  • W. Chen: Pseudoholomorphic curves in four-orbifolds and some applications; in Geometry and Topology of Manifolds, Fields Inst. Commun. 47, Amer. Math. Soc., Providence, RI, 11–37, 2005.
  • W. Chen: Smooth $s$-cobordisms of elliptic 3-manifolds, J. Differential Geom. 73 (2006), 413–490.
  • R. Fintushel and R.J. Stern: An exotic free involution on $S^{4}$, Ann. of Math. (2) 113 (1981), no. 2, 357–365.
  • R. Fintushel, R.J. Stern and N. Sunukjian: Exotic group actions on simply connected smooth 4-manifolds, J. Topol. 2 (2009), 769–778.
  • R.E. Gompf: Killing the Akbulut–Kirby $4$-sphere, with relevance to the Andrews–Curtis and Schoenflies problems, Topology 30 (1991), 97–115.
  • R.E. Gompf: More Cappell–Shaneson spheres are standard, arXiv:0908.1914.
  • R.E. Gompf: Sums of elliptic surfaces, J. Differential Geom. 34 (1991), 93–114.
  • R.E. Gompf and A.I. Stipsicz: $4$-Manifolds and Kirby Calculus, Amer. Math. Soc., Providence, RI, 1999.
  • B. Hanke, D. Kotschick and J. Wehrheim: Dissolving four-manifolds and positive scalar curvature, Math. Z. 245 (2003), 545–555.
  • R. Mandelbaum: Decomposing analytic surfaces; in Geometric Topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York, 147–217, 1979.
  • J. Morgan and G. Tian: Ricci Flow and the Poincaré Conjecture, Amer. Math. Soc., Providence, RI, 2007.
  • J. Morgan and G. Tian: Completion of the proof of the geometrization conjecture, arXiv:0809.4040.
  • Y. Ruan: Virtual neighborhoods and the monopole equations; in Topics in Symplectic $4$-Manifolds (Irvine, CA, 1996), First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 101–116, 1998.
  • C. Sung: $G$-monopole classes, Ricci flow, and Yamabe invariants of 4-manifolds, Geom. Dedicata 169 (2014), 129–144.
  • C. Sung: Finite group actions and $G$-monopole classes on smooth 4-manifolds, arXiv: 1108.3875.
  • C. Sung: $G$-monopole invariants on some connected sums of 4-manifolds, Geom. Dedicata 178 (2015), 75–93.
  • C.H. Taubes: The Seiberg–Witten invariants and 4-manifolds with essential tori, Geom. Topol. 5 (2001), 441–519.
  • M. Ue: Exotic group actions in dimension four and Seiberg–Witten theory, Proc. Japan Acad. Ser. A Math. Sci. 74 (1998), 68–70. \endthebibliography*