2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/56517We 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.24pagesMathematical PhysicsAlgebraic GeometryAlgebraic Topologytext