Surgery on products with finite fundamental group
Surgery on products with finite fundamental group
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对有限基本群乘积进行手术
DOI：
10.1016/0040-9383(77)90054-4
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发表时间：
1977
期刊：
影响因子：
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通讯作者：
E. Stein
中科院分区：
文献类型：
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作者：
E. Stein
这篇论文关注的是同态，cp：&。(G)+ &（G），从G的有向边界到G的手术梗阻组。（定义和符号在Q1中给出。一般来说，这种同态并没有得到很好的理解。例如，已知诺维科夫关于高阶签名的猜想可以简化为计算cp的局部化版本，并且大多数理解cp的尝试都集中在这个问题上，其中同态的图像往往相当大（参见[3]在这个方向上的部分结果）。在这里，我们感兴趣的是一种完全不同的情况。也就是说，我们希望知道在什么情况下映射9是“几乎平凡的”，在这个意义上9的像是L。&e）C L&G）。这个观点至少有两个原因。首先，它涉及到证明某些手术障碍为零，增加了计算将导致手术理论之外的应用的可能性。第二，通过观察9是平凡的情况，避免了必须识别在大多数情况下未知的组中的元素的困难。因此，我们的研究将限于有限基本群，其中没有已知的例子9具有不等于L&e）的像。我们能够计算出一个大类的有限群的9。
THIS PAPER is concerned with the homomorphism, cp: &.(G)+ & (G), from the oriented bordism of the finitely presented group, G, to the surgery obstruction groups of G.(Definitions and notation are given in Ql.) In general, this homomorphism is not well understood. For example, it is known that Novikov’s conjecture on higher signatures can be reduced to a computation of a localized version of cp, and most of the attempts to understand cp have focused on this problem, where the image of the homomorphism tends to be quite large (see [3] for partial results in this direction). Here, we are interested in an entirely different situation. Namely, we wish to know under what circumstances the map 9 is “almost trivial”, in the sense that the image of 9 is L. &e) C L&G). This point of view is useful for at least two reasons. First, it involves proving that certain surgery obstructions are zero, increasing the likelihood that the calculations will lead to applications outside of surgery theory. Second, by looking at cases where 9 is trivial, one avoids the difficulty of having to identify elements in a group which is, in most cases, unknown. Therefore, our investigation will be confined to finite fundamental groups, where there is no known example of 9 having an image unequal to L&e). We are able to calculate 9 for a large class of finite groups.