EquivariantL-theory II
EquivariantL-theory II
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等变L理论II
DOI：
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发表时间：
1990
期刊：
影响因子：
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通讯作者：
I. Madsen
中科院分区：
文献类型：
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作者：
W. Lück;I. Madsen
等变代数 K 理论分解为群环的普通代数 K 理论之和（参见 [4,5,7,12]）。对于 K t 也是如此，因为上三角矩阵的行列式是对角线上条目的乘积。对于等变 L 理论，情况更为复杂。我们确实获得了一组精确的轨道序列，类似于康纳和弗洛伊德在[2]中获得的邻近家族序列。但这些精确的序列并不总是以类似于等变 K 理论的分裂的方式分裂。 G=Z/2 有简单的反例。然而，如果变换群具有奇数阶，则等变 L 群实际上会以预期的方式分解，参见。下面的定理 2.11。我们在本文中同时研究光滑和局部线性 PL 类别。精确轨道序列存在的结果是，这两个流形类别的等变 L 群是相等的。本文基于 [9] 中给出的等变 L 理论的定义。我们建议读者参考该论文，了解其中有些繁琐的定义。参考文献 (I.?. ?) 始终指第一部分 [9]。我们希望当前等变 L 群的定义和这里介绍的计算技术将使进一步的计算成为可能。例如，在我们看来，评估某些标准 2 群的等变 L 群，以及确定等变 Rothenberg 序列是很有意义的。
Equivariant algebraic K-theory decomposes as a sum of ordinary algebraic Ktheory of group rings (see [4, 5, 7, 12]). This follows for K t because the determinant of a upper triangular matrix is the product of the entries on the diagonal. For equivariant L-theory the situation is more complicated. We do obtain a set of exact orbit sequences similar to the neighbouring family sequences obtained by Connor and Floyd in [2]. But these exact sequences do not always split in a way analogous to the splitting of equivariant K-theory. There are easy counterexamples for G=Z/2 . However, if the transformation group has odd order, then the equivariant L-groups do in fact decompose in the expected fashion, cf. Theorem 2.11 below. We work in this paper in the smooth and locally linear PL-category simultaneously. A consequence of the existence of the exact orbit sequence is that the equivariant L-groups are equal for these two manifold categories. The paper is founded upon the definition of equivariant L-theory given in [9]. We refer the reader to that paper for the somewhat cumbersome definitions. A reference (I.?. ?) always refers to the first part [9]. It is our hope that the present definitions of equivariant L-groups and the calculational techniques presented here will make further calculations possible. It seems to us to be of some interest to evaluate equivariant L-groups for some of the standard 2-groups, for example, and to determine the equivariant Rothenberg sequence.