实向量空间与实向量丛的等变 K 理论
摘要
设 G 为作用在有限维实向量空间 V 上的有限群。记 P(V) 为 V 相伴的射影空间。本文利用 Atiyah 与作者先前的结果,以非常显式的方式计算了 V 与 P(V) 的等变复 K 理论的秩。该计算的意义在于 Baum-Connes-Slominska Chern 特征给出的显式公式,以及 V 的等变 K 理论自由这一基本事实。我们利用这些拓扑计算来证明代数结果,例如计算在中心扩张中分裂的 G 的共轭类个数。我们的主要例子是 V = R^n 且 G = 通过对坐标置换作用的 n 个字母的对称群的情形。该例子与著名的 Euler 五角恒等式以及(颇具讽刺意味地)V 的等变 K 理论的 Euler-Poincare 示性数相关。
引用
@article{arxiv.math/0509497,
title = {Equivariant K-theory of real vector spaces and real vector bundles},
author = {Max Karoubi},
journal= {arXiv preprint arXiv:math/0509497},
year = {2007}
}
备注
25 pages ; see also http://www.math.jussieu.fr/~karoubi/ One historical comment added in Jan. 2007 to this file : N. Kuhn has pointed out to me that the Baum-Connes-Slominska Chern character was described independently in his paper Character rings in algebraic topology, Advances in Homotopy Theory, Proceedings of Cortona 1988, LMS Lecture Note Series 139 (1989), 111--126. See in particular Theorem 6.4, which dates from 1986 and is closely related to the (then new) generalized character theory of Hopkins-Kuhn-Ravenel