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Some results on orientation preserving involutions


Author: David E. Gibbs
Journal: Trans. Amer. Math. Soc. 218 (1976), 321-332
MSC: Primary 57D85
DOI: https://doi.org/10.1090/S0002-9947-1976-0410770-5
MathSciNet review: 0410770
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Abstract: The bordism of orientation preserving differentiable involutions is studied by use of the signature-like invariant $ {\text{ab}}: {\mathcal{O}_\ast}({Z_2}) \to {W_0}({Z_2};Z)$. The equivariant Witt ring $ {W_0}({Z_2};Z)$ is calculated and is shown to be isomorphic under ab to the effective part of $ {\mathcal{O}_4}({Z_2})$. Modulo 2 relations are established between the representation of the involution on $ {H^{2k}}({M^{4k}};Z)/{\operatorname{torsion}}$ and $ {\chi _0}(F)$ and $ {\chi _2}(F)$, where $ {\chi _i}(F)$ is the Euler characteristic of those components of the fixed point set with dimensions congruent to i modulo 4. For manifolds of dimension $ 4k + 2$, it is shown that $ {\chi _0}(F) \equiv {\chi _2}(F) \equiv 0\;(\bmod 2)$. Finally the ideal $ {E_0}({Z_2};Z)$ consisting of those elements of $ {W_0}({Z_2};Z)$ admitting a representative of type II is determined.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0410770-5
Keywords: Equivariant bordism, Euler characteristic, orientation preserving involution, representation, Witt ring
Article copyright: © Copyright 1976 American Mathematical Society

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