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Proceedings of the American Mathematical Society

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

 

 

A $ W(Z\sb{2})$ invariant for orientation preserving involutions


Author: John P. Alexander
Journal: Proc. Amer. Math. Soc. 51 (1975), 455-460
MSC: Primary 57E15
DOI: https://doi.org/10.1090/S0002-9939-1975-0377947-3
MathSciNet review: 0377947
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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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DOI: https://doi.org/10.1090/S0002-9939-1975-0377947-3
Keywords: Witt ring, peripheral invariant
Article copyright: © Copyright 1975 American Mathematical Society