Configuration Space Models for Spaces of Maps from a Riemann Surface to Complex Projective Space

Abstract

Let ${\mathrm{Map}}_d^{\ast }({\mathrm{M}}_g,C{\mathrm{P}}^{n-1})$ denote the space consisting of all basepoint preserving continuous maps of degree d from a compact Riemann surface ${\mathrm{M}}_g$ of genus $g$ into an $(n-1)$-dimensional complex projective space $C{\mathrm{P}}^{n-1}$. In this paper, we construct a finite dimensional configuration space model ${\mathrm{SP}}_n^d({\mathrm{M}}_g^{\prime })$ for the infinite dimensional space ${\mathrm{Map}}_d^{\ast }({\mathrm{M}}_g,C{\mathrm{P}}^{n-1})$ and show that the Atiyah–Jones type theorem (cf. [1], [12]) holds for this model.