Free involutions on 𝐸₄ₘ lattices

Jastrzebowski, Wojtek

doi:10.1090/S0002-9939-1995-1254844-5

Free involutions on $E_ {4m}$ lattices
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by Wojtek Jastrzebowski

Proc. Amer. Math. Soc. 123 (1995), 1941-1945

DOI: https://doi.org/10.1090/S0002-9939-1995-1254844-5

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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

Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279–315. MR 710056, DOI 10.4310/jdg/1214437665

The orientation of Yang-Mills moduli spaces and four manifold topology

17

Allan L. Edmonds, Involutions on odd four-manifolds, Topology Appl. 30 (1988), no. 1, 43–49. MR 964061, DOI 10.1016/0166-8641(88)90079-X
Allan L. Edmonds, Aspects of group actions on four-manifolds, Topology Appl. 31 (1989), no. 2, 109–124. MR 994404, DOI 10.1016/0166-8641(89)90075-8
Allan L. Edmonds and John H. Ewing, Realizing forms and fixed point data in dimension four, Amer. J. Math. 114 (1992), no. 5, 1103–1126. MR 1183533, DOI 10.2307/2374891
Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. MR 679066

The

programming language, 1987-1993

Sławomir Kwasik and Pierre Vogel, Asymmetric four-dimensional manifolds, Duke Math. J. 53 (1986), no. 3, 759–764. MR 860670, DOI 10.1215/S0012-7094-86-05341-X
John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372, DOI 10.1007/978-3-642-88330-9
V. A. Rohlin, New results in the theory of four-dimensional manifolds, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 221–224 (Russian). MR 0052101
Jean-Pierre Serre, Complex semisimple Lie algebras, Springer-Verlag, New York, 1987. Translated from the French by G. A. Jones. MR 914496, DOI 10.1007/978-1-4757-3910-7