Pulling back fixed points
Pulling back fixed points
复制标题
拉回固定点
DOI：
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发表时间：
1987
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通讯作者：
W. Browder
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文献类型：
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作者：
W. Browder
如果X，Y是G-空间（紧李群G作用于其上的空间），且f：X ~ Y是G-映射，则X中不动点的像在I1中是固定的。当一个不动点的逆像包含一个不动点时，特别是如果X，Y是G-流形，并且f的度与]GI（= G的阶）互素。在（未发表的工作的作者1），这是显示为初等阿贝尔p-群，与X，Y更一般的空间（“n-近流形”）。在本文中，我们证明了这是真实的阿贝尔p-群G光滑地作用在X上，与p奇数或与p=2与一个额外的复杂的假设。给出了非阿贝尔G的一个反例。该方法是证明一个定理的推广（Bredon，1973年）的7唇行动，这说，度素数到p的映射诱导注入模p上同调的固定集。我们将这个定理推广到交换p群，其中X上的作用是光滑的，并且p4 = 2（或其他假设）。即使在秩> 1的初等阿贝尔p-群的情况下，这似乎也是新的。我们实际上证明了一个更强的定理推广的程度，这可能是非零的映射之间的流形不同的维度。证明涉及复线性等变K理论在一个基本的方式，这需要光滑性和p + 2，或一个复杂性假设的X，以产生适当的复线性丛。在推论中，我们证明了阿贝尔p-群（p奇数）不能光滑地作用在具有恰好一个孤立不动点的闭流形上，推广了（Atiyah和Bott，1964）对G=Z/p的旧结果。我感谢伊卜·马德森的评论，特别是关于等变K理论的讨论。也要感谢迈克尔·戴维斯仔细阅读了手稿，并指出了几个难点。我也要感谢裁判的评论。
If X, Y are G-spaces (spaces on which a compact Lie group G acts), and f: X ~ Y is a G-map, then the image of a fixed point in X is fixed in I1. When does the inverse image of a fixed point contain a fixed point, in particular if X, Y are G-manifolds and f has degree prime to ]GI (=order of G). In (unpublished work of the author1), this is shown for elementary abelian p-groups, with X, Y more general spaces ("n-near manifolds"). In this paper we show that this is true for abelian p-groups G acting smoothly on X, with p odd or with p=2 with an additional complex hypothesis. We give a counterexample for nonabelian G. The method is to prove a generalization of a theorem of (Bredon, 1973) for 7lip actions, which says that maps of degree prime to p induce injections on the mod p cohomology of the fixed set. We generalize this theorem to abelian p groups, where the action on X is smooth, and p 4= 2 (or other hypothesis). This seems to be new even in the case of elementary abelian p-groups of rank > 1. We actually prove a stronger theorem for a generalization of the notion of degree, which may be non-zero for maps between manifolds of different dimension. The proof involves complex linear equivariant K-theory in an essential way, which requires smoothness and p + 2, or a complexity hypothesis on X in order to produce the appropriate complex linear bundles. Among the corollaries, we show that an abelian p-group (p odd) cannot act smoothly on a closed manifold with exactly one isolated fixed point, generalizing an old result of (Atiyah and Bott, 1964) for G=Z/p. This was found independently by (Ewing and Stong, 1986) by a different argument. I am indebted to Ib Madsen for his comments, in particular for discussions on equivariant K-theory. Thanks are due also to Michael Davis for carefully reading the manuscript, and pointing out several difficulties. I also wish to thank the referee for his comments.