Geometry & Topology

Permutations, isotropy and smooth cyclic group actions on definite 4–manifolds

Ian Hambleton and Mihail Tanase

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Abstract

We use the equivariant Yang–Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4–manifolds.

Article information

Source
Geom. Topol., Volume 8, Number 1 (2004), 475-509.

Dates
Received: 29 July 2003
Revised: 17 January 2004
Accepted: 9 February 2004
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883372

Digital Object Identifier
doi:10.2140/gt.2004.8.475

Mathematical Reviews number (MathSciNet)
MR2033485

Zentralblatt MATH identifier
1072.58003

Subjects
Primary: 58D19: Group actions and symmetry properties 57S17: Finite transformation groups
Secondary: 70S15: Yang-Mills and other gauge theories

Keywords
gauge theory $4$–manifolds group actions Yang–Mills moduli space

Citation

Hambleton, Ian; Tanase, Mihail. Permutations, isotropy and smooth cyclic group actions on definite 4–manifolds. Geom. Topol. 8 (2004), no. 1, 475--509. doi:10.2140/gt.2004.8.475. https://projecteuclid.org/euclid.gt/1513883372


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References

  • Edward Bierstone, General position of equivariant maps, Trans. Amer. Math. Soc. 234 (1977) 447–466
  • Glen E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York (1972)
  • S K Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983) 279–315
  • S K Donaldson, The orientation of Yang–Mills moduli spaces and $4$–manifold topology, J. Differential Geom. 26 (1987) 397–428
  • S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, Oxford Science Publications, Clarendon Press and Oxford University Press, New York (1990)
  • Allan L Edmonds, Aspects of group actions on four-manifolds, Topology Appl. 31 (1989) 109–124
  • Allan L Edmonds, Periodic maps of composite order on positive definite 4–manifolds, preprint (2000) 20 pages
  • Ronald Fintushel, Terry Lawson, Compactness of moduli spaces for orbifold instantons, Topology Appl. 23 (1986) 305–312
  • Ronald Fintushel Ronald J Stern, Pseudofree orbifolds, Ann. of Math. 122 (1985) 335–364
  • Daniel S Freed, Karen K Uhlenbeck, Instantons and four-manifolds, second ed., Mathematical Sciences Research Institute Publications, vol. 1, Springer–Verlag, New York (1991
  • Ian Hambleton, Ronnie Lee, Perturbation of equivariant moduli spaces, Math. Ann. 293 (1992) 17–37
  • Ian Hambleton, Ronnie Lee, Smooth group actions on definite $4$–manifolds and moduli spaces, Duke Math. J. 78 (1995) 715–732
  • S ören Illman, Every proper smooth action of a Lie group is equivalent to a real analytic action: a contribution to Hilbert's fifth problem, from: “Prospects in topology (Princeton, NJ, 1994)”, Ann. of Math. Stud. vol. 138, Princeton Univ. Press, Princeton, NJ (1995) 189–220
  • John N Mather, Stratifications and mappings, from: “Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971)”, Academic Press, New York (1973) 195–232
  • J P Serre, Trees, Graduate Texts in Mathematics, Springer–Verlag, New York (1980)
  • Mihail Tanase, Smooth finite cyclic group actions on positive definite $4$–manifolds, PhD thesis, McMaster University (2003) viii+112 pages