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Euler characteristics and Gysin sequences for group actions on boundaries
Authors:Heath Emerson  Ralf Meyer
Institution:1. Mathematics and Statistics, University of Victoria, PO BOX 3045 STN CSC, Victoria, B.C, Canada, V8W 3P4
2. Mathematisches Istitut, Georg-August-Universit?t G?ttingen, Bunsenstr. 3-5, 37073, G?ttingen
Abstract:Let G be a locally compact group, let X be a universal proper G-space, and let ></img>                              </span> be a <em>G</em>-equivariant compactification of <em>X</em> that is <em>H</em>-equivariantly contractible for each compact subgroup <span class= ></img>                              </span>. Let <span class= ></img>                              </span>. Assuming the Baum-Connes conjecture for <em>G</em> with coefficients <span class= ></img>                              </span> and <em>C</em>(?<em>X</em>), we construct an exact sequence that computes the map on <strong class=K-theory induced by the embedding ></img>                              </span>. This exact sequence involves the equivariant Euler characteristic of <em>X</em>, which we study using an abstract notion of Poincaré duality in bivariant <strong class=K-theory. As a consequence, if G is torsion-free and the Euler characteristic ></img>                              </span> is non-zero, then the unit element of <span class= ></img>                              </span> is a torsion element of order <span class= ></img>                              </span>. Furthermore, we get a new proof of a theorem of Lück and Rosenberg concerning the class of the de Rham operator in equivariant <strong class=K-homology.
Keywords:19K35  46L80
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