Topology of completions of the space of rational functions

对于瞬子模空间，我们得到了Uhlenbeck完备化，这在规范理论领域是有用的。本文的目的是定义从S^2到复流形V的有理函数空间的类似完备性。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次基全纯映射的空间，i_k：Rat_k(Cp^n)→Ω^2_kcp^n[相似或相等]Ω^2S^&lt；2n+1&gt；西格尔证明了i_k是k(2n-1)维的同伦等价。后来，我和独立的Cohen-Cohen-Mann-Milgram用Ω^2S^&lt；2n+1&gt；的稳定和刻画了Rat_k(Cp^n)的稳定同伦型。注意，Rat_k(Cp^n)由没有公根的一次k次复多项式的(n+1)元组组成。作为推广，我们定义了一个空间X^L_k(Cp^n)，它是由至多有L根的一次k次复多项式的(n+1)元组构成的。我们有X^0_k(Cp^n)=Rat_k(Cp^n)和X^k_k(Cp^n)=C^&lt；k(n+1)&gt；在本研究中，我证明了后者是前者的乌伦贝克完成。这意味着X^L_k(Cp^n)是一个从RAT_k(Cp^n)到它的完备化的空间。此外，我还成功地确定了X^L_k(Cp^n)的稳定同伦型.下一步，我将Cp^n转化为循环群ΩG.在这种情况下，从S^2到ΩG的有理函数空间恰好是瞬子模空间.我研究了它的完成性。在研究过程中，我证明了如下定理：设C是SU(2)在G中的中心化子，J：G/C→Ω^3_0是J(GC)(X)=gxg^&lt；-1&gt；-1&gt；-1&gt；定义的映射。则J_*：H_*(G/C；Z/2)→H_*(Ω^3_0G；Z/2)是内射的。请注意，这个结果是关于生成ΩG的映射的Bott定理的推广，并且本身非常有趣。

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.