Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Free involutions on $ E\sb {4m}$ lattices

Author: Wojtek Jastrzebowski
Journal: Proc. Amer. Math. Soc. 123 (1995), 1941-1945
MSC: Primary 57N13; Secondary 11H06
MathSciNet review: 1254844
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We determine all the conjugacy classes of traceless involutions on the $ {E_{4m}}$ lattices. In particular, we show that for every $ m > 2$ there exist precisely two nonconjugate involutions which induce free $ {\mathbf{Z}}[{{\mathbf{Z}}_2}]$-module structures. By inspecting the parity of the $ {E_{4m}}$ form twisted by any such involution, we deduce that a closed, simply connected, topological 4-manifold with intersection form $ {E_{4m}}$ supports a locally linear involution if and only if m is odd and the Kirby-Siebenmann invariant of the manifold is trivial.

References [Enhancements On Off] (What's this?)

  • [1] C. W. Curtis and I. Reiner, Representation theory of finite groups and associate algebras, Wiley, New York, 1962. MR 0144979 (26:2519)
  • [2] S. K. Donaldson, An application of gauge theory to the topology of 4-manifolds, J. Differential Geom. 18 (1983), 269-316. MR 710056 (85c:57015)
  • [3] -, The orientation of Yang-Mills moduli spaces and four manifold topology, J. Differential Geom. 17 (1987), 397-428.
  • [4] A. L. Edmonds, Involutions on odd four-manifolds, Topology Appl. 30 (1988), 43-49. MR 964061 (89m:57041)
  • [5] -, Aspects of group actions on four-manifolds, Topology Appl. 31 (1989), 109-124. MR 994404 (90h:57050)
  • [6] A. L. Edmonds and J. H. Ewing, Realizing forms and fixed point data in dimension four, Amer. J. Math. 114 (1992), 1103-1126. MR 1183533 (93i:57047)
  • [7] M. H. Freedman, The topology of 4-manifolds, J. Differential Geom. 17 (1982), 357-453. MR 679066 (84b:57006)
  • [8] W. Jastrzebowski, The Slang programming language, 1987-1993.
  • [9] S. Kwasik and P. Vogel, Asymmetric four-dimensional manifold, Duke Math. J. 53 (1986), 759-763. MR 860670 (88c:57038)
  • [10] J. W. Milnor and D. Husemoller, Symmetric bilinear forms, Springer, New York, 1973. MR 0506372 (58:22129)
  • [11] V. A. Rohlin, New results in the theory of 4-dimensional manifolds, Dokl. Acad. Nauk SSSR 84 (1952), 221-224. MR 0052101 (14:573b)
  • [12] J.-P. Serre, Complex semisimple Lie algebras, Springer-Verlag, New York, 1987. MR 914496 (89b:17001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57N13, 11H06

Retrieve articles in all journals with MSC: 57N13, 11H06

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society