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Topology of completions of the space of rational functions

有理函数空间的补全拓扑

基本信息

批准号：
13640085
负责人：
KAMIYAMA Yasuhiko
金额：
$ 1.28万
依托单位：
University of the Ryukyus
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640085/
关键词：
rational function completion loop space homology stable homotopy homotopy fiber instanton Lie group 実多項式ループ空間インスタントモジュライ空間

项目摘要

For instanton moduli spaces we have the Uhlenbeck completion, which is useful in the field of gauge theory. The purpose of this study is to define a similar completion for spaces of rational functions from S^2 to a complex manifold V.First we study the typical case V = CP^n. Let Rat_k(CP^n) be the space of based holomorphic maps of degree k from S^2 to CP^n. Let i_k : Rat_k(CP^n) → Ω^2_kCP^n 【similar or equal】 Ω^2S^<2n+1> be the inclusion. Segal showed that i_k is a homotopy equivalence up to dimension k(2n - 1). Later I and independently Cohen-Cohen-Mann-Milgram described the stable homotopy type of Rat_k(CP^n) in terms of stable summands of Ω^2S^<2n+1>.Note that Rat_k(CP^n) consists of (n + 1)-tuples of monic degree k complex polynomials without common roots. Generalizing this, we define a space X^l_k(CP^n) by the set of (n + 1)-tuples of monic degree k complex polynomials with at most l roots in common. We have X^0_k(CP^n) = Rat_k(CP^n) and X^k_k(CP^n) = C^<k(n+1)>. In this study I proved that the latter is the Uhlenbeck completion of the former. This implies that X^l_k(CP^n) is a space which appears when we shift from Rat_k(CP^n) to its completion. Moreover, I succeeded in determining the stable homotopy type of X^l_k(CP^n).Next I change CP^n to a loop group ΩG. In this case the space of rational functions from S^2 to ΩG is exactly the instanton moduli space. I studied its completion. In the process of the study, I was able to prove the following theorem: Let C be the centralizer of SU(2) in G and let J : G/C → Ω^3_0 be the map defined by J(gC)(x) = gxg^<-1>x^<-1>. Then J_* : H_*(G/C; Z/2) → H_*(Ω^3_0G; Z/2) is injective. Note that this result is a generalization of the Bott's theorem about generating maps of ΩG, and very interesting in itself.

对于瞬子模空间，我们有Uhlenbeck完备化，这在规范理论领域很有用。本文的目的是定义一个从S^2到复流形V的有理函数空间的类似完备化。首先我们研究了典型的情形V = CP^n。设Rat_k（CP^n）是从S^2到CP^n的k次基全纯映射空间。设i_k：Rat_k（CP^n）→ Ω ^2_kCP ^n [相似或相等] Ω^2S^<2n+1>为包含。Segal证明了i_k是k（2n - 1）维同伦等价。后来我和Cohen-Cohen-Mann-Milgram独立地用Ω^2S^<2n+1>的稳定和项描述了Rat_k（CP^n）的稳定同伦类型，注意Rat_k（CP^n）由没有公共根的一元k次复多项式的（n + 1）-元组组成。推广这一点，我们定义一个空间X^l_k（CP^n）由一次k阶复多项式的（n + 1）元组的集合，最多有l个共同的根。我们有X^0_k（CP^n）= Rat_k（CP^n）和X^k_k（CP^n）= C^<k（n+1）>。本文证明了后者是前者的Uhlenbeck完备化。这意味着X^l_k（CP^n）是当我们从Rat_k（CP^n）移到它的完备时出现的空间。此外，还成功地确定了X^l_k（CP^n）的稳定同伦类型，并将CP^n化为环群ΩG。在这种情况下，从S^2到ΩG的有理函数空间就是瞬子模空间。我研究了它的完成。在研究的过程中，我证明了如下定理：设C是SU（2）在G中的中心化子，设J：G/C → Ω^3_0是由J（gC）（x）= gxg^ x^定义的映射<-1><-1>。则J_*：H_*（G/C; Z/2）→ H_*（Ω^3_0G; Z/2）是内射的.注意，这个结果是关于生成ΩG的映射的博特定理的推广，并且本身非常有趣。