Connective coverings of spaces of holomorphic maps
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 [\textit{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 [\textit{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. \textit{F. R. Cohen}, \textit{T. I. Lada} and \textit{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 \[ \widetilde {\text{Hol}}_1(S^2,\mathbb{C} P^2) \simeq SU_3,\quad \widetilde {\text{Hol}_1} (S^2,\mathbb{C} P^2) \simeq S^5\times S^7. \]
scientific article; zbMATH DE number 1997255