Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems
Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems
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DOI：
10.1007/bf01390270
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发表时间：
1977
影响因子：
3.1
通讯作者：
E. Fadell;P. Rabinowitz
中科院分区：
文献类型：
--
作者：
E. Fadell;P. Rabinowitz
在另一项工作[1]中，作者采用了上同调指数（参见Yang [2，3]，Conner-Floyd [4]和Holm-Spanier [5]）代替通常的亏格概念[6，7]，这在对称群为7/2的对称情况下很有用，例如，考虑（奇数）映射f使得f（-x）=-f（x）。用这个上同调指数代替亏格是由于需要附加的性质--穿透性质（命题（3.9））。本文将上同调指数的概念推广到对称群为任意紧李群G的一般情形。证明了任何上同调类c~ H*（Bo），其中B e是G的泛分类空间，都产生一个整数指数X，其中X是任意仿紧自由G-空间，并且这个指数具有（w3），非常一般地，这种理论所要求的通常概念，包括刺穿性质。第四节讨论了三种重要的特殊情况，即当c~专门化为无穷射影空间F ~ F的上同调的生成元时，其中F ~ F是实数F ~ F、复数~ F或四元数IH，群G是F ~ F中的单位球面。对于这三种情况，我们使用符号index~ X，index e X，index~ X。第一个指标X在一个限制范畴中等价于杨E的上同调指标[2，3]。这是在[1]中使用的索引，并且在Conner-Floyd 1-4]中由co-indexz 2 X指定。在第5节中，我们在IF上的赋范线性空间~的背景下使用索引r X=索引r X+ 1的概念重新表述了理论。在感兴趣的应用程序中，其中基本的对称群是S1，由于存在任意阶的各向同性子群，所产生的作用可能不是自由的。相应地，在第6节中，我们使用为自由G-空间发展的指标理论，定义一般情况下的指标理论，即不假设自由作用的仿紧G-空间范畴。* 本研究部分由海军研究办公室根据合同号N 00014 -76-C-0300赞助，部分由美国陆军根据合同号DAAG 2 -75-C-0024赞助，部分由国家科学基金会根据批准号NSF MCS 76 -06373赞助。允许为美国政府的目的部分或全部复制
In another work [1] the authors employed a cohomological index (see also Yang [2, 3], Conner-Floyd [4] and Holm-Spanier [5]) in place of the usual notion of genus [6, 7] which is useful in symmetric situations with the group of symmetry being 7/2, eg, in the consideration of (odd) maps f such that f (-x)=-f (x). Replacing genus by this cohomological index was dictated by the need of additional property-the piercing property (Proposition (3.9)). In this paper we extend this idea of cohomological index to the general situation where the symmetry group is an arbitrary compact Lie group G. It turns out that any cohomology class c~ H*(Bo), where B e is the universal classifying space for G, gives rise to an integer, index, X, where X is an arbitrary paracompact free G-space, and this index enjoys (w 3), quite generally, the usual notions required of such a theory, including the piercing property. Section 4 is devoted to three important special cases, namely when c~ is specialized to the generator of the cohomology ofiFP, infinite projective space, where IF is either the reals F,,, the complex numbers~, or the quaternions IH, and the group G is the unit sphere in IF. We use the notation index~ X, index e X, index~ X, for these three cases. The first, index~ X, is equivalent in a restricted category, to the cohomological index of Yang E2, 3]. This is the index employed in [1] and it is designated in Conner-Floyd 1-4] by co-indexz2 X. In Section 5 we reformulate the theory in the setting of a normed linear space~ over IF using the notion Index r X= index r X+ 1. In applications of interest, where the underlying group of symmetry is S 1, the resulting action may not be free due to the presence of isotropy subgroups of arbitrary order. Accordingly, in Section 6, we employ the index theory developed for free G-spaces, to define index theories in the general situation, namely the category of paracompact G-spaces without the assumption of a free action. The* This research was sponsored in part by the Office of Naval Research under Contract No. N00014-76-C-0300, by the US Army under Contract No. DAAG2-75-C-0024, and in part by the National Science Foundation under Grant No. NSF MCS76-06373. Any reproduction in part or in full for the purposes of the US Government is permitted