Configuration Spaces and the Topology of Curves in Projective Space

TL;DR

This paper surveys and extends the understanding of the topology of spaces of maps from positive genus curves into complex projective spaces, including fundamental group computations and special cases.

Contribution

It expands on prior work by computing fundamental groups of these mapping spaces and fully characterizing the case when the target is the complex projective line.

Findings

Fundamental groups of mapping spaces are computed.

The case for $n=1$ is fully determined with a non-trivial extension.

Both holomorphic and continuous map categories are analyzed.

Abstract

We survey and expand on the work of Segal, Milgram and the author on the topology of spaces of maps of positive genus curves into $n$-th complex projective space, $n\geq 1$ (in both the holomorphic and continuous categories). Both based and unbased maps are studied and in particular we compute the fundamental groups of the spaces in question. The relevant case when $n=1$ is given by a non-trivial extension which we fully determine.