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

Abstract

n−1 ) denote the space consisting of all basepoint preserving continuous maps of degree d from a compact Riemann surface M g of genus g into a (n − 1)-dimensional complex projective space CP n−1 . In this paper, we construct a finite dimensional configuration space model SP (M g , X). In this paper, the author would like to study a finite dimensional model of

$$E_n$$ E n -cell attachments and a local-to-global principle for homological stability

“…Applying the local-to-global homological stability principle we get the following corollaries. This greatly improves on the range of * ≤ k d+1 − d + 3 established in [4] and agrees with the range recently proved in [5]. Homological stability for the spaces Div d δ(k) (Σ) was originally proven in [2].…”

“…This result greatly improves on the range of * ≤ k 2d established in [7] which was the only integral homological stability range known for bounded symmetric powers in high dimensions. We note that the arguments of [4] and Section 6.1.1 all use that M is twodimensional in a crucial way -either using that Sym(C) is a manifold or that Ω 2 S 3 × Z ≃ Ω 2 S 2 -and thus those techniques cannot be used to achieve this range in higher dimensions. A rational range with slope 1 was proven in [5] using the fact that Sym(M ) is an orbifold.…”

The topology of the space ofJ–holomorphic maps to ℂP2

“…We prove these properties of Σg−pt R 2 by proving that it is homotopy equivalent to the so called bounded symmetric product of Σ g − pt. The work of Kallel in [Kal01a] and Yamaguchi in [Yam03] show that these bounded symmetric products give models of the space of continuous maps from a surface to a complex projective space. These results imply that H * ( Σg R 2 ) approximates H * (Map * (Σ g , CP 2 )).…”

“…The above theorem was improved by Yamaguchi (Theorem 1.3 and Theorem 3.1 of [Yam03]). 2 ) induce isomorphisms on homology groups…”

“…The work of Kallel in [Kal01a] and Yamaguchi in [Yam03] show that these bounded symmetric products give models of the space of continuous maps from a surface to a complex projective space. These results imply that H * ( Σg R 2 ) approximates H * (Map * (Σ g , CP 2 )).…”