A 𝑊(𝑍₂) invariant for orientation preserving involutions

Alexander, John P.

doi:10.1090/S0002-9939-1975-0377947-3

A $W(Z_{2})$ invariant for orientation preserving involutions
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Proc. Amer. Math. Soc. 51 (1975), 455-460 Request permission

Abstract:

In this paper we calculate an invariant in $W({{\mathbf {Z}}_2})$, the Witt ring of nonsingular, symmetric ${{\mathbf {Z}}_2}$-inner product spaces, for orientation-preserving involutions on compact, closed, connected $4n$-dimensional manifolds $M$. This invariant with the Atiyah-Singer index theorem uniquely determines the orthogonal representation of ${{\mathbf {Z}}_2}$ on ${H^{2n}}(M;{\mathbf {Z}})/\operatorname {TOR}$. We also give an example to show that this invariant detects actions that the Atiyah-Singer theorem cannot.

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