Periodic maps of composite order on positive definite 4–manifolds

Edmonds, Allan L¹

¹ Department of Mathematics, Indiana University, Bloomington, Indiana 47405, USA

Geometry & topology, Tome 9 (2005) no. 1, pp. 315-339.

Résumé

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.

Détail

DOI : 10.2140/gt.2005.9.315

Keywords: periodic map, 4–manifold, positive definite, permutation representation, pseudofree

Bibliographie

[1] C Allday, V Puppe, Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics 32, Cambridge University Press (1993)

[2] G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press (1972)

[3] A I Borevich, I R Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press (1966)

[4] A L Edmonds, Construction of group actions on four-manifolds, Trans. Amer. Math. Soc. 299 (1987) 155

[5] A L Edmonds, Aspects of group actions on four-manifolds, Topology Appl. 31 (1989) 109

[6] A L Edmonds, Automorphisms of the $E_8$ four-manifold, from: "Geometric topology (Athens, GA, 1993)", AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 282

[7] A L Edmonds, J H Ewing, Locally linear group actions on the complex projective plane, Topology 28 (1989) 211

[8] A L Edmonds, J H Ewing, Realizing forms and fixed point data in dimension four, Amer. J. Math. 114 (1992) 1103

[9] N D Elkies, A characterization of the $\mathbb{Z}^n$ lattice, Math. Res. Lett. 2 (1995) 321

[10] W Franz, Über die Torsion einer Überdeckung, J. Reine Angew. Math. 173 (1935) 245

[11] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357

[12] C M Gordon, On the $G$–signature theorem in dimension four, from: "À la recherche de la topologie perdue", Progr. Math. 62, Birkhäuser (1986) 159

[13] I Hambleton, R Lee, Smooth group actions on definite 4–manifolds and moduli spaces, Duke Math. J. 78 (1995) 715

[14] I Hambleton, M Tanase, Permutations, isotropy and smooth cyclic group actions on definite 4–manifolds, Geom. Topol. 8 (2004) 475

[15] S Kwasik, T Lawson, Nonsmoothable $Z_p$ actions on contractible 4–manifolds, J. Reine Angew. Math. 437 (1993) 29

[16] M Tanase, Smooth finite cyclic group actions on positive definite 4–manifolds, PhD thesis, McMaster University (2003)