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Transactions of the American Mathematical Society

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$(Z_{2})^{k}$-actions whose fixed data has a section

Author: Pedro L. Q. Pergher
Journal: Trans. Amer. Math. Soc. 353 (2001), 175-189
MSC (2000): Primary 57R85; Secondary 57R75
Published electronically: June 21, 2000
MathSciNet review: 1783791
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Abstract: Given a collection of $2^{k}-1$ real vector bundles $\varepsilon _{a}$ over a closed manifold $F$, suppose that, for some $a_{0}, \varepsilon _{a_{0}}$ is of the form $\varepsilon _{a_{0}}^{\prime }\oplus R$, where $R\to F$ is the trivial one-dimensional bundle. In this paper we prove that if $\bigoplus _{a} \varepsilon _{a} \to F$ is the fixed data of a $(Z_{2})^{k}$-action, then the same is true for the Whitney sum obtained from $\bigoplus _{a} \varepsilon _{a}$ by replacing $\varepsilon _{a_{0}}$ by $\varepsilon _{a_{0}}^{\prime }$. This stability property is well-known for involutions. Together with techniques previously developed, this result is used to describe, up to bordism, all possible $(Z_{2})^{k}$-actions fixing the disjoint union of an even projective space and a point.

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Additional Information

Pedro L. Q. Pergher
Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, CEP 13.565-905, São Carlos, SP, Brazil

Keywords: $(Z_{2})^{k}$-action, fixed data, Stong's exact sequence, $((Z_{2})^{k},q)$-manifold-bundle, projective space bundle, bordism class, representation, Smith homomorphism
Received by editor(s): November 11, 1998
Published electronically: June 21, 2000
Additional Notes: The present work was partially supported by CNPq
Article copyright: © Copyright 2000 American Mathematical Society

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