Topology of spaces of conjugation-equivariant holomorphic maps

设RAG_k(Cp^n)是从^2到Cp^n的保基点全纯映射的空间，这是Ω^2_kCp^n的一个子空间.我和Cohen-Cohen-Mann-Milgram的一个定理告诉我们，用Ω^2S^&lt；2n+1&gt；的稳定和刻画了RAAT_k(Cp^n)的稳定同伦型.本研究的目的是从两个方向推广这一定理：(I)设Rrat_k(Cp^n)是通过复共轭与对合交换的映射的Ratat_k(Cp^n)的子空间。Brockett和Segal确定了Rrat_k(Cp^1)的同伦类型。但情况n[大于或等于]2是未知的。本研究的第一个成果是完全确定了Rrat_k(Cp^n)的稳定同伦类型。在这种情况下，对应的连续映射空间是ΩS^n×Ω^2S^&lt；2n+1&gt；(Ii)设P_&lt；k，n&gt；是多项式空间，使得n重根的个数至多为L。1970年，Arnold曾试图确定P^L_&lt；k，n&gt；的同调群，但大部分都是未知的。研究的第二个成果是完全确定了P^L；k，n&gt；的稳定同伦型，并由此确定了P^L_&lt；k，n&gt；的同调群。结果，我彻底解决了阿诺德的问题。粗略地说，主要结果是证明了单多项式空间和n元多项式空间之间的关系。对(II)中的成果给予了高度评价。例如，我在2005年7月在东京大学举行的欧洲共同体国际会议上作了一次全体演讲。

Let Rat_k(CP^n) be the space of basepoint-preserving holomorphic maps fromS^2 to CP^n. This is a subspace of Ω^2_kCP^n. A theorem by me and Cohen-Cohen-Mann-Milgram tells that the stable homotopy type of Rat_k(CP^n) is described in terms of stable summands of Ω^2S^<2n+1>. The purpose of this research is to generalize the theorem in two directions.(i)Let RRat_k(CP^n) be the subspace of Rat_k(CP^n) of maps which commute with an involution by complex conjugation. Brockett and Segal determined the homotopy type of RRat_k(CP^1). But the case n【greater than or equal】2 was unknown. The first achievement of this research is to determine the stable homotopy type of RRat_k(CP^n) completely. In this case, the corresponding continuous mapping space is ΩS^n×Ω^2S^<2n+1>.(ii)Let P_<k,n> be the space of polynomials such that the number of n-fold roots is at most l. In 1970, Arnold tried to determine the homology group of P^l_<k,n>, but most part was left unknown. The second achievement of research is to determine the stable homotopy type of P^l_<k,n> completely, and to show that the homology groups of P^l_<k,n> are determined from this. As a result, I solved Arnold's problem completely. Roughly, the main result is to prove a relationship between a space of single polynomials and a space of n-tuples of polynomials.The achievement in (ii) was highly evaluated. For example, I gave a plenary talk at the COE International Conference held at the University of Tokyo in July 2005.