边界上群作用的欧拉示性数和Gysin序列
Euler characteristics and Gysin sequences for group actions on boundaries
摘要：
设G是一个局部紧群，X是一个泛真G-空间，Z是X的G-等变紧化，对于G的每个紧子群H，Z是H-等变可收缩的。设W为结果边界。假设系数为C和C（W）的G满足Baum-Connes猜想，我们构造了一个精确序列来计算G的约化群C*-代数嵌入到G与C（W）的交叉积中所诱导的K-理论上的映射.这个确切序列涉及X的等变欧拉示性数，我们使用双变K-理论中的庞加莱对偶的抽象概念来研究它。因此，如果G是无挠的并且轨道空间X/G的欧拉示性数非零，则边界交叉积的单位元是阶等于X/G的欧拉示性数的绝对值的扭元。进一步，我们得到了Lueck和Rosenberg关于等变K-同调中de Rham算子类的一个定理的新证明.
英文摘要：
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.
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