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Semi-linear homology $G$-spheres and their equivariant inertia groups


Author: Zhi Lü
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
Published electronically: August 25, 2003
MathSciNet review: 2020024
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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

DOI: https://doi.org/10.1090/S0002-9947-03-03388-9
Keywords: Semi-linear homology $G$-sphere, equivariant inertia group, $G$-action, representation, surgery
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).
Article copyright: © Copyright 2003 American Mathematical Society

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