Smooth equivariant triangulations ofG-manifolds forG a finite group
Smooth equivariant triangulations ofG-manifolds forG a finite group
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DOI：
10.1007/bf01405351
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发表时间：
1978-10
影响因子：
1.4
通讯作者：
S. Illman
中科院分区：
文献类型：
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作者：
S. Illman
事实上，我们证明了上述结果的一种更一般的形式，即证明了（M， N）对的光滑等变三角的存在性和组合唯一性，其中M是一个光滑的无边界g流形，N是M的光滑g子流形，使得N在M内，见定理3.6-3.8。我们的结果的证明与怀特黑德对他的结果的原始证明完全一致。在Munkres[8]中对Whitehead的证明作了很好的阐述，并进行了修改。我们选择参考Munkres[8]，而不是Whitehead的原始论文[15]，建议读者熟悉Munkres[8]的第二部分。1977年11月7日添加的注释。根据审稿人的建议，我们在这里就本论文与其他一些有关的早期工作的关系发表一些评论。但让我们首先指出，我们对上述定理的主要兴趣在于唯一性部分。在[17]中，我们构造了有限G-变CW复合体的等变简单同伦理论，其中G可以是任意紧李群或任意离散群。等变Whitehead扭转不是一个拓扑不变量，即存在一个G-同胚f: X-~ Y不是一个等变单同伦等价，即使对于有限循环群G也是如此（参见Mitnor[18]）。但是等变Whitehead扭转是一个组合不变量，因此，
In fact we prove a more general form of the above result in that we prove existence and combinatorial uniqueness of smooth equivariant triangulations of pairs (M, N), where M is a smooth G-manifold without boundary and N is a smooth G-submanifold of M such that N is dosed in M, see Theorems 3.6-3.8. The proof of our result is completely along the lines of Whitehead's original proof of his results. A good exposition of Whitehead's proof, with modifications, is given in Munkres [8]. We have chosen to refer to Munkres [8], instead of to Whitehead's original paper [15], and it is advisable that the reader is familiar with Part II of Munkres [8].Note added November 7, i977. At the suggestion of the Referee we include here some remarks on the relation of the present paper with some other related earlier work. But let us first point out that our own main interest in the above theorem lies in the uniqueness part. In [17] we constructed an equivariant simple homotopy theory for finite G-cquivariant CW complexes, where in fact G can be an arbitrary compact Lie group or any discrete group. The equivariant Whitehead torsion is not a topological invariant, ie, there exists a G-homeomorphism f: X-~ Y which is not an equivariant simple homotopy equivalence, this even for G a finite cyclic group (cf. Mitnor [18]). But the equivariant Whitehead torsion is a combinatorial invariant and hence,