Topology of completions of the space of rational functions

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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.