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Euler characteristics and Gysin sequences for group actions on boundaries | | Heath Emerson
; Ralf Meyer
; | | Date: |
3 May 2005 | | Subject: | K-Theory and Homology MSC-class: 19K35, 46L80 | math.KT | | Abstract: | Let G be a locally compact group, let X be a universal proper G-space, and let Z be a G-equivariant compactification of X that is H-equivariantly contractible for each compact subgroup H of G. Let W be the resulting boundary. Assuming the Baum-Connes conjecture for G with coefficients C and C(W), we construct an exact sequence that computes the map on K-theory induced by the embedding of the reduced group C*-algebra of G into the crossed product of G by C(W). This exact sequence involves the equivariant Euler characteristic of X, which we study using an abstract notion of Poincare duality in bivariant K-theory. As a consequence, if G is torsion-free and the Euler characteristic of the orbit space X/G is non-zero, then the unit element of the boundary crossed product is a torsion element whose order is equal to the absolute value of the Euler characteristic of X/G. Furthermore, we get a new proof of a theorem of Lueck and Rosenberg concerning the class of the de Rham operator in equivariant K-homology. | | Source: | arXiv, math.KT/0505050 | | Services: | Forum | Review | PDF | Favorites | |
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