The homotopy fixed point set of Lie group actions on elliptic spaces
The homotopy fixed point set of Lie group actions on elliptic spaces
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DOI：
10.1112/plms/pdv015
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发表时间：
2014-07
影响因子：
1.8
通讯作者：
Urtzi Buijs;Yves F'elix;Sergio Huerta;A. Murillo
中科院分区：
文献类型：
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作者：
Urtzi Buijs;Yves F'elix;Sergio Huerta;A. Murillo
设G是一个紧连通李群，或者更一般地说是一个有限CW复形的同伦型的路连通拓扑群，并且设X是一个有理幂零G空间。本文分析了同伦不动点集XhG的同伦类型，以及自然内射k：XG <$XhG .我们证明了如果X是椭圆的，即它有有限维有理同伦和上同调，那么XhG的每个路分量也是椭圆的。我们还给出了包含k的一个显式代数模型，在此基础上我们可以证明，例如，对于G一个环面，π*（k）在有理同伦中是单射的，但通常远非一个有理同伦等价。
Let G be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW‐complex, and let X be a rational nilpotent G ‐space. In this paper, we analyze the homotopy type of the homotopy fixed point set XhG , and the natural injection k:XG↪XhG . We show that if X is elliptic, that is, it has finite‐dimensional rational homotopy and cohomology, then each path component of XhG is also elliptic. We also give an explicit algebraic model of the inclusion k based on which we can prove, for instance, that for G a torus, π*(k) is injective in rational homotopy but, often, far from being a rational homotopy equivalence.