## 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.## References

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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