Free involutions on 𝐸_{4𝑚} lattices

Jastrzebowski, Wojtek

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

Free involutions on $E_ {4m}$ lattices
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Proc. Amer. Math. Soc. 123 (1995), 1941-1945 Request permission

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.

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