Abstract
We show that if $X$ is a cocompact $G\textrm{-}CW$-complex such that each isotropy subgroup $G_\sigma$ is $L^{(2)}$-good over an arbitrary commutative ring $k$, then $X$ satisfies some fixed-point formula which is an $L^{(2)}$-analogue of Brown’s formula in 1982. Using this result we present a fixed point formula for a cocompact proper $G\textrm{-}CW$-complex which relates the equivariant $L^{(2)}$-Euler characteristic of a fixed point $CW$-complex $X^s$ and the Euler characteristic of $X/G$. As corollaries, we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji–Mislin in 2009. We also show that if X is a free $G\textrm{-}CW$-complex such that $C_{*} (X)$ is chain homotopy equivalent to a chain complex of finitely generated projective $Z \pi_1 (X)$-modules of finite length and $X$ satisfies some fixed-point formula over $\mathbb{Q}$ or $\mathbb{C}$ which is an $L^{(2)}$-analogue of Brown’s formula, then $\chi (X/G) = \chi^{(2)} (X)$. As an application, we prove that the weak Bass conjecture holds for any finitely presented group $G$ satisfying the following condition: for any finitely dominated $CW$-complex $Y$ with $\pi_1 (Y)=G, \widetilde{Y}$ satisfies some fixed-point formula over $\mathbb{Q}$ or $\mathbb{C}$ which is an $L^{(2)}$-analogue of Brown’s formula.
Funding Statement
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03932318).
Citation
Jang Hyun Jo. "Equivariant $L^2$-Euler characteristics of $G\textrm{-}CW$-complexes." Ark. Mat. 55 (1) 155 - 164, September 2017. https://doi.org/10.4310/ARKIV.2017.v55.n1.a7
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