Local surgery and applications to the theory of quadratic forms
Local surgery and applications to the theory of quadratic forms
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局部手术及其在二次形式理论中的应用
DOI：
10.1090/s0002-9904-1976-13992-4
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发表时间：
1976
期刊：
影响因子：
--
通讯作者：
W. Pardon
中科院分区：
文献类型：
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作者：
W. Pardon
设A是一个具有对合的酉环，L^（A）表示n维的外科阻塞群，由C定义. T. C. Wall在[13]的第5章和第6章中，对于同伦等价的外科手术。在这篇文章中，一个新的局部手术理论[8]，[9]被用来产生沃尔L -群的局部化序列（参见。实施例2遵循定理1）。该序列与从KUt的Sharpe酉周期性导出的MayerVietoris序列一起，0 < / < 2，可用于进行计算，该计算包括并扩展了巴克[1]，Bass [2]，Karoubi [6]和Wall [14]的许多结果。我们概述的方法来确定手术障碍应该更容易拓扑学家比其他最近的治疗，因为分析只涉及两个不平凡的，但基本上是几何工具：本地化序列和迈耶-Vietoris序列。第一个可以实现几何的情况下，群环;第二个涉及到一个正式的建设，在代数/^-理论，连同夏普的研究酉斯坦伯格群，这也可以实现几何[在。定位序列。设A是一个具有对合的环，A是关于乘法子集2 <$A的经典同分环，Karoubi [6]独立地得到了五项（从L%k+1（A）开始）。
Let A be a unitary ring with involution and let L^(A) denote the surgery obstruction group in dimension n, defined by C. T. C. Wall in Chapters 5 and 6 of [13], for surgery to a homotopy equivalence. In this note a new local surgery theory [8], [9] is used to produce a localization sequence for Wall's L -groups (cf. Example 2 following Theorem 1). This sequence, together with a MayerVietoris sequence derived from Sharpe's unitary periodicity for KUt, 0 < / < 2, can be used to make computations which include and extend many of the results of Bak [1], Bass [2], Karoubi [6], and Wall [14]. The approach we outline to the determination of surgery obstructions should be more accessible to topologists than other recent treatments, because the analysis involves only two nontrivial but essentially geometric tools: the localization sequence and the Mayer-Vietoris sequence. The first can be realized geometrically in the case of group rings; the second involves a formal construction in algebraic /^-theory, together with Sharpe's study of the unitary Steinberg group, which can also be realized geometrically [in. The localization sequence. Let A be a ring with involution and A a classical ring of quotients with respect to a multiplicative subset 2 Ç A , Five terms (starting with L%k+l(A)) °^ ^ following localization sequence have been obtained by Karoubi [6] independently.