| Project/Area Number |
15540087
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | The University of the Ryukyus |
|---|
Principal Investigator |
KAMIYAMA Yasuhiko The University of the Ryukyus, Department of Mathematics, Associate Professor, 理学部, 助教授 (10244287)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
SHIGA Hiroo The University of the Ryukyus, Department of Mathematics, Professor, 理学部, 教授 (40128484)
TEZUKA Michishige The University of the Ryukyus, Department of Mathematics, Professor, 理学部, 教授 (20197784)
|
|---|
| Project Period (FY) |
2003 – 2005
|
| Project Status |
Completed (Fiscal Year 2005)
|
| Budget Amount *help |
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
|
|---|
| Keywords | rational function / conjugation / polynomial / multiple root / configuration space / loop space / stable splitting / generating variety / 対合 / 充満 / スペクトル系列 / シュティーフェル多様体 / 一般の位置 / 正則写像 / 共通根 / ホモトピー同値 / ホモトピーファイバー / 偶数次元局面 |
|---|
| Research Abstract |
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.
|
