Equivariant L2-Euler characteristics of G-CW-complexes

Jang Hyun Jo Sogang University

Complex Variables and Complex Analysis Rings and Algebras mathscidoc:1803.43013

Arkiv for Matematik, 55, 2017
We show that if X is a cocompact G-CW-complex such that each isotropy subgroup Gσ 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-CW-complex which relates the equivariant L(2)-Euler characteristic of a fixed point CW-complex Xs 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-CW-complex such that C∗(X) is chain homotopy equivalent to a chain complex of finitely generated projective Zπ1(X)-modules of finite length and X satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula, then χ(X/G)=χ(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 π1(Y )=G, Y satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula.
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@inproceedings{jang2017equivariant,
  title={Equivariant L2-Euler characteristics of G-CW-complexes},
  author={Jang Hyun Jo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180301170010570429948},
  booktitle={Arkiv for Matematik},
  volume={55},
  year={2017},
}
Jang Hyun Jo. Equivariant L2-Euler characteristics of G-CW-complexes. 2017. Vol. 55. In Arkiv for Matematik. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180301170010570429948.
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