Equivariant L<SUP>2</SUP>-Euler characteristics of G-CW-complexes

초록

We show that if X is a cocompact G-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-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-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 Q or 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, Y satisfies some fixed-point formula over Q or C which is an L-(2)-analogue of Brown's formula.

키워드