zbMATH — the first resource for mathematics

Spaces of holomorphic maps with bounded multiplicity. (English) Zbl 0993.32012
Let \(\text{Hol}^*_d(S^2,CP^{n-1})\) be the space consisting of all basepoint preserving holomorphic maps \(f:S^2{\rightarrow}CP^{n-1}\) of degree \(d\). The corresponding space of continuous maps is denoted by \({\Omega}^2_dCP^{n-1}\). The space \(\text{Hol}^*_d(S^2,CP^{n-1})\) is homeomorphic to the \(n\)-tuples \((p_1(z),...,p_n(z))\in C[\mathbf z]^{\mathbf n}\) of monic polynomials of degree \(d\) with no common root and G. B. Segal proved that it is a finite-dimensional model of \({\Omega}^2CP^{n-1}\).
In this paper the author considers a certain subspace of \(\text{Hol}^*_d(S^2,CP^{n-1})\) defined using the concept of multiplicity of roots, and proves that it is also a finite-dimensional model of a certain double loop space (Theorem 1.3).

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
55P10 Homotopy equivalences in algebraic topology
58D15 Manifolds of mappings
Full Text: DOI