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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Semi-linear homology $G$-spheres and their equivariant inertia groups
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by Zhi Lü PDF
Trans. Amer. Math. Soc. 356 (2004), 61-71 Request permission

Abstract:

This paper introduces an abelian group $H\Theta _V^G$ for all semi-linear homology $G$-spheres, which corresponds to a known abelian group $\Theta _V^G$ for all semi-linear homotopy $G$-spheres, where $G$ is a compact Lie group and $V$ is a $G$-representation with $\dim V^G>0$. Then using equivariant surgery techniques, we study the relation between both $H\Theta _V^G$ and $\Theta _V^G$ when $G$ is finite. The main result is that under the conditions that $G$-action is semi-free and $\dim V-\dim V^G\geq 3$ with $\dim V^G >0$, the homomorphism $T: \Theta _V^G\longrightarrow H\Theta _V^G$ defined by $T([\Sigma ]_G)=\langle \Sigma \rangle _G$ is an isomorphism if $\dim V^G\not =3,4$, and a monomorphism if $\dim V^G=4$. This is an equivariant analog of a well-known result in differential topology. Such a result is also applied to the equivariant inertia groups of semi-linear homology $G$-spheres.
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Additional Information
  • Zhi Lü
  • Affiliation: Institute of Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China
  • Address at time of publication: Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
  • Email: zlu@fudan.edu.cn
  • Received by editor(s): July 3, 2000
  • Published electronically: August 25, 2003
  • Additional Notes: This work was supported by the Japanese Government Scholarship, and partially supported by the research fund of the Ministry of Education in China and the JSPS Postdoctoral Fellowship (No. P02299).
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 61-71
  • MSC (2000): Primary 57S15, 57S17, 57R91, 57R55, 57R67
  • DOI: https://doi.org/10.1090/S0002-9947-03-03388-9
  • MathSciNet review: 2020024