Equivariant differential topology
Equivariant differential topology
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DOI：
10.1016/0040-9383(69)90005-6
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发表时间：
1969-04
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影响因子：
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通讯作者：
A. Wasserman
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文献类型：
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作者：
A. Wasserman
第2节讨论g向量束的分类。精确的表述是：M ‘ ’上的k维g向量束的等价类与M到Gk （V ‘）的映射的等变同伦类（V ’中k-平面的格拉斯曼量，如果t b> n+ k）的自然一一对应。通过一个横向论证证明了分类映射的存在性。从向量场解曲线的存在唯一性出发，证明了同伦映射诱导束的等价性。Atiyah[1]证明了紧拓扑空间的一个类似定理。第3节发展了g流形的协同理论。定义了等变同伦群，并证明了在k+ n= V的（2n+ 3）维的Gk （V2n+ 3$ R）泛束的Thorn空间上，如果G是阿贝的或有限的，从属于V的n维的G流形的无取向协伦群是同构于V2n+ 36r的球的映射的等变同伦类。对于g流形，即使建立一个弱的横截定理，也存在着严重的技术困难；因此，任意紧李群的同构性的存在性仍然是一个悬而未决的问题。第4节将R. Palais[14]关于Hilbert流形的Morse理论的结果推广到g流形的情况。证明了当M是有限维时，在M上不变实值函数集合中的“莫尔斯函数”是稠密的。并证明了传递莫尔斯函数的临界值对应于在轨道上或更一般地在非退化临界子流形上添加“柄束”。然后推导了临界子流形的摩尔斯不等式。本节的结果于2010年公布。本节中的一些结果是Meyer[6]独立获得的。我要感谢RS Palais教授的建议和鼓励，并向我提出这个问题。我也很感激与他进行了许多有益的讨论。本文的部分研究得到了DA31-124ARO (D) 128的支持。127
Section 2 concerns the classification of G-vector bundles. The precise statement is: The equivalence classes of k-dimensional G-vector bundles over M”“subordinate” to Pare in a natural one-to-one correspondence with the equivariant homotopy classes of maps of M into Gk (V’), the grassmannian of k-planes in V’, if t> n+ k. The existence of a classifying map is proved via a transversality argument. The equivalence of bundles induced by homotopic maps can be shown to follow from the existence and uniqueness of solution curves of vector fields. Atiyah [l] has proved a similar theorem for compact topological spaces. Section 3 develops a cobordism theory for G-manifolds. Equivariant homotopy groups are defined and it is shown that the unoriented cobordism group of G-manifolds of dimension n, subordinate to V are isomorphic to the equivariant homotopy classes of maps of the sphere in V2n+ 3 6 R into the Thorn space of the universal bundle over Gk (V2n+ 3$ R) where k+ n=(2n+ 3) dimension of V, if G is abelian or finite. There is a severe technical difficulty in establishing even a weak transversality theorem for Gmanifolds; hence, the existence of the isomorphism for arbitrary compact Lie groups is still an open question. Section 4 generalizes the results of R. Palais [14] on Morse Theory on Hilbert Manifolds to the case of G-manifolds. It is shown that “Morse functions” are dense in the set of invariant real valued functions on M if M is finite dimensional. Also it is shown that passing a critical value of a Morse function corresponds to adding on “handle-bundles” over orbits or more generally over non-degenerate critical submanifolds. Morse inequalities are then deduced for the case of critical submanifolds. The results in this section were announced in [15]. Some of the results in this section have been obtained independently by Meyer [6]. I wish to thank Professor RS Palais for his advice and encouragement and for suggesting this problem to me. I am also grateful for many helpful discussions with him. t Research for this paper was partially supported by DA31-124ARO (D) 128. 127