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In this paper we calculate an invariant in      W ( Z 2 ) W({{\mathbf {Z}}_2})      , the Witt ring of nonsingular, symmetric      Z 2 {{\mathbf {Z}}_2}      -inner product spaces, for orientation-preserving involutions on compact, closed, connected      4 n 4n      -dimensional manifolds      M M      . This invariant with the Atiyah-Singer index theorem uniquely determines the orthogonal representation of      Z 2 {{\mathbf {Z}}_2}       on      H 2 n ( M ; Z ) / TOR {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.

A 𝑊(𝑍₂) invariant for orientation preserving involutions