Dimensions of fixed point sets of involutions
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- 作者：Pedro L. Q. Pergher and Fábio G. Figueira
- 关键词：Mathematics Subject Classification (2000). Primary 57R85 ; Secondary 57R75
- 刊名：Archiv der Mathematik
- 出版年：2006
- 出版时间：September, 2006
- 年：2006
- 卷：87
- 期：3
- 页码：280-288
- 全文大小：108 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics - 出版者：Birkh盲user Basel
- ISSN：1420-8938
文摘
Suppose the fixed point set F of a smooth involution T:M → M on a smooth, closed and connected manifold M decomposes into two components Fn and F2 of dimensions n and 2, respectively, with n > 2 odd. We show that the codimension k of Fn is small if the normal bundle of F2 does not bound; specifically, we show that k≦ 3 in this case. In the more general situation where F is not a boundary, n (not necessarily odd) is the dimension of a component of F of maximal dimension and k is the codimension of this component, and fixed components of all dimensions j, 0≦ j≦ n, may occur, a theorem of Boardman gives that k \leqq \frac32nk \leqq \frac{3}{2}n . In addition, we show that this bound can be improved to k≦ 1 (hence k = 1) for some specific values of n and some fixed stable cobordism classes of the normal bundle of F2 in M; further, we determine in these cases the equivariant cobordism class of (M, T).