Equivariant $L^2$-Euler characteristics of $G\textrm{-}CW$-complexes

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.