# zbMATH — the first resource for mathematics

Connective coverings of spaces of holomorphic maps. (English) Zbl 1031.58005
Let $$\text{Hol}_d (S^2,\mathbb{C} p^n)$$ and $$\text{Hol}^*_d (S^2,\mathbb{C} P^n)$$ be the spaces of holomorphic maps and based holomorphic maps $$S^2\to\mathbb{C} P^n$$ of degree $$d$$. Let $$h_n:S^{2n+1} \to\mathbb{C} P^n$$ be the Hopf fibering with fibre $$S^1$$ and $\widetilde{\text{Hol}_d}(S^2,\mathbb{C} P^n)= \bigl\{ (f,x)\in\text{Hol}_d (S^2,\mathbb{C} P^n) \times S^{2n+1}: ev(f)=h_n (x)\bigr\}.$ In [K. Yamaguchi, Kyushu J. Math. 56, 381-389 (2002; Zbl 1041.55005)] the author showed $$\widetilde {\text{Hol}_d}(S^2,S^2)$$ is the universal covering of $$\text{Hol}_d (S^2,S^2)$$ and homotopy equivalent to $$\widetilde {\text{Hol}^*_d} \times S^3$$, where $$\widetilde {\text{Hol}^*_d}$$ is the universal covering of $$\text{Hol}^* (S^2,S^2)$$. In this paper, assuming $$n\geq 2$$ and $$d\geq 1$$, $$\widetilde{\text{Hol}_d}(S^2,\mathbb{C} P^n)$$ is shown to be the 2-connected covering of $$\text{Hol}_d (S^2,\mathbb{C} P^n)$$. Existence of a fibration sequence $\text{Hol}^*_d (S^2,\mathbb{C} P^n) @>\widetilde j_d>> \widetilde {\text{Hol}}_d (S^2,\mathbb{C} P^n) @>\widetilde{ev}>> S^{2n+1},$ is also shown (Th. 1.3). This fibration have a section if and only if $$n\equiv `\pmod 2$$ or $$n\equiv d\equiv O \pmod 2$$, by the results on Whitehead product of complect projective spaces [G. W. Whitehead, Ann. Math. (2) 47, 460-475 (1946; Zbl 0060.41106)]. But there are isomorphisms of graded Abelian groups and graded rings $H_*\bigl( \widetilde {\text{Hol}_d} (S^2,\mathbb{C} P^n), A\bigr)\cong H_*\bigl( \text{Hol}^*_d (S^2,\mathbb{C} P^n), A\bigr) \otimes H_*(S{2n+1},A),$ $H^*\bigl( \widetilde {\text{Hol}_d} (S^2,\mathbb{C} P^n),A\bigr) \cong H^*\bigl(\text{Hol}^*_d(S^2,\mathbb{C} p^n), A\bigr)\otimes H^*(S^{2n+1},A),$ where $$A$$ is an Abelian group (Prop. 1.5), as a consequence of the computation of the homology of double loop space of $$S^{2n+ }$$ Lemma 3.1, cf. F. R. Cohen, T. I. Lada and J. P. May, [‘The homology of iterated loop spaces’, Lect. Notes Math. 533 (1976; Zbl 0334.55009)]. These are proved in Sect. 2 and 3. In Sect. 4, the last Section, homotopy types of $$\text{Hol}_1(S^2,\mathbb{C} P^n)$$ and $$\widetilde {\text{Hol}_1} (S^2,\mathbb{C} P^n)$$ are determined explicitely (Th. 1.6), analyzing $$U_{n+1}$$-action on $$\text{Hol}_1(\mathbb{C} p^k, \mathbb{C} P^n)$$ induced from the $$U_{n+1}$$-action on $$\mathbb{C} P^n$$. For example, the followings are shown $\widetilde {\text{Hol}}_1(S^2,\mathbb{C} P^2) \simeq SU_3,\quad \widetilde {\text{Hol}_1} (S^2,\mathbb{C} P^2) \simeq S^5\times S^7.$

##### MSC:
 58D15 Manifolds of mappings 55P35 Loop spaces 55R05 Fiber spaces in algebraic topology 32H99 Holomorphic mappings and correspondences