Geometry & Topology

Periodic maps of composite order on positive definite 4–manifolds

Allan L Edmonds

Full-text: Open access

Abstract

The possibilities for new or unusual kinds of topological, locally linear periodic maps of non-prime order on closed, simply connected 4–manifolds with positive definite intersection pairings are explored. On the one hand, certain permutation representations on homology are ruled out under appropriate hypotheses. On the other hand, an interesting homologically nontrivial, pseudofree, action of the cyclic group of order 25 on a connected sum of ten copies of the complex projective plane is constructed.

Article information

Source
Geom. Topol., Volume 9, Number 1 (2005), 315-339.

Dates
Received: 8 July 2004
Revised: 23 January 2005
Accepted: 21 February 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799568

Digital Object Identifier
doi:10.2140/gt.2005.9.315

Mathematical Reviews number (MathSciNet)
MR2140984

Zentralblatt MATH identifier
1087.57024

Subjects
Primary: 57S17: Finite transformation groups
Secondary: 57S25: Groups acting on specific manifolds 57M60: Group actions in low dimensions 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx]

Keywords
periodic map 4–manifold positive definite permutation representation pseudofree

Citation

Edmonds, Allan L. Periodic maps of composite order on positive definite 4–manifolds. Geom. Topol. 9 (2005), no. 1, 315--339. doi:10.2140/gt.2005.9.315. https://projecteuclid.org/euclid.gt/1513799568


Export citation

References

  • \itemsep 2pt plus 1pt
  • C Allday, V Puppe, Cohomological Methods in Transformations Groups, Cambridge studies in Advanced Mathematics 32, Cambridge Univ. Press (1993)
  • G E Bredon, Introduction to Compact Transformation Groups, Pure and Applied Mathematics 46, Academic Press (1972)
  • Z I Borevich, I R Shafarevich, Number Theory, Academic Press (1966)
  • A L Edmonds, Construction of group actions on four-manifolds, Trans. Amer. Math. Soc. 299 (1987) 155–177
  • A L Edmonds, Aspects of group actions on four-manifolds, Topology Appl. 31 (1989) no. 2, 109–124.
  • A L Edmonds, Automorphisms of the $E\sb 8$ four-manifold, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. 2.1, Amer. Math. Soc. Providence, RI (1997) 282–299
  • A L Edmonds, J H Ewing, Locally linear group actions on the complex projective plane, Topology 28 (1989) 211–223
  • A L Edmonds, J H Ewing, Realizing forms and fixed point data in dimension four, Amer. J. Math. 114 (1992) 1103–1126
  • N D Elkies, A characterization of the $Z\sp n$ lattice, Math. Res. Lett. 2 (1995) 321–326
  • W Franz, Über die Torsion einer Überdeckung, J. Reine Angew. Math. 173 (1935) 245–254
  • M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357–453
  • C Gordon, On the $G$–signature theorem in dimension four, from: “À la Recherche de la Topologie Perdue”, (L Guillou and A Marin, editors) Progr. Math. 62, Birkhäuser (1986) 159–180
  • I Hambleton, R Lee, Smooth group actions on definite $4$–manifolds and moduli spaces, Duke Math. J. 78 (1995) 715-732
  • I Hambleton, M Tanase, Permutations, isotropy and smooth cyclic group actions on definite 4–manifolds, \gtref8200411475509
  • S Kwasik, T Lawson, Nonsmoothable $Z\sb p$ actions on contractible $4$–manifolds, J. Reine Angew. Math. 437 (1993) 29–54
  • M Tanase, Smooth finite cyclic group actions on positive definite 4–manifolds, PhD Thesis, McMaster University (2003)